Stochastic Processes: Harmonizable Theory (Multivariate Analysis) 9811213658, 9789811213656

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SERIES  ON  MULTIVARIATE  ANALYSIS Editor: M M Rao ISSN: 1793-1169 Published Vol. 1: Martingales and Stochastic Analysis J. Yeh Vol. 2: Multidimensional Second Order Stochastic Processes Y. Kakihara Vol. 3: Mathematical Methods in Sample Surveys H. G. Tucker Vol. 4: Abstract Methods in Information Theory Y. Kakihara Vol. 5:

Topics in Circular Statistics S. R. Jammalamadaka and A. SenGupta

Vol. 6:

Linear Models: An Integrated Approach D. Sengupta and S. R. Jammalamadaka

Vol. 7:

Structural Aspects in the Theory of Probability: A Primer in Probabilities on Algebraic-Topological Structures H. Heyer

Vol. 8:

Structural Aspects in the Theory of Probability (Second Edition) H. Heyer

Vol. 9:

Random and Vector Measures M. M. Rao

Vol. 10: Abstract Methods in Information Theory (Second Edition) Y. Kakihara Vol. 11: Linear Models and Regression with R: An Integrated Approach D. Sengupta and S. R. Jammalamadaka Vol. 12: Stochastic Processes: Harmonizable Theory M. M. Rao

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Rao, M. M. (Malempati Madhusudana), 1929– author. Title: Stochastic processes : harmonizable theory / M.M. Rao, University of California, Riverside. Description: 1st edition. | Hackensack, NJ : World Scientific Publishing Co. Pte. Ltd., [2020] | Series: Series on multivariate analysis, 1793-1169 ; vol. 12 | Includes bibliographical references and index. Identifiers: LCCN 2020012114 | ISBN 9789811213656 (hardcover) | ISBN 9789811213663 (ebook) | ISBN 9789811213670 (ebook other) Subjects: LCSH: Stochastic processes. Classification: LCC QA274 .R3725 2020 | DDC 519.2/3--dc23 LC record available at https://lccn.loc.gov/2020012114

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To the Memories of Friends P. R. Krishnaiah for changing my attempts to join the Indian Air Force Academy in DehraDun to Minnesota’s Stochastic Analysis study and Herbert Heyer for the advocacy and example to use abstract methods in Stochastic Analysis

b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Preface

The following work constitutes the third and final part of the original planned threevolume treatment of ‘Stochastic Analysis’. The first one appeared in 1995, as the second edition of an earlier publication (the 1979 version), and the second part, termed ‘Inference Theory’ (of stochastic processes) appeared in 2010 (as a revised, and enlarged work of the original 2000 year publication). These two volumes are published originally by Netherlands’ publishers. That publication house was acquired by Springer and then reissued under their own name (2nd edition of 2nd volume). The current volume completes the originally planned coverage of the trilogy. Many new developments are covered in these volumes. I shall now indicate the contents of this volume for a brief view of the included topics presented in six chapters. Each chapter also has its contents outlined in an early paragraph. Chapter 1 details a characterization of the general concept of (weak) harmonizability extending the classical weak stationarity of processes, initiated by Khintchine, extended by Lo´eve and the general form obtained by Bochner (after some initial special cases treated separately by L. Schwartz and Yu. A. Rozanov). They showed the central part of Fourier or harmonic analysis is essential in stochastic theory as well, and the first two chapters of this volume, using this framework, are devoted to analyzing the structural aspects based on the first two moments. Most of the traditional treatments concentrate on the second order (i.e., covariance) properties, assuming the processes to be centered. But the mean values are not generally constants. First, they are characterized for harmonizable classes and then the primary focus shifts to the positive definiteness of covariances which are characterized. As a key consequence, the existence of L´evy’s Brownian Motion and its properties are established. Thereafter the general harmonizability aspect becomes the key component of our study. From this point on, the property of V -boundedness of process, becomes primary and Bochner’s characterization of it, and consequences take on a central place in the ensuing analysis. Chapters 2 and 3 consider extensions of weakly harmonizable processes, as well as a detailed study in much of the (new) classes by Karhunen and Cram´er, and their dilations to larger (or super) Hilbert spaces. They have several consequences. Also integral representations of (Gel’fand’s) local functionals given for vii

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the first time, as well as a new and notable probabilistic proof of the long open and interesting Riemann hypothesis, with a detailed discussion, are included here. In these chapters some applications of the work to the harmonizable processes and other extensions including the Cram´er, Karhunen classes using generalized Fourier analysis methods as well as some summability analysis of Kamp´e de Feriet and Frankiel (and independently of Parzen’s) on nonstationary processes are studied. They are of interest in applications involving works on signal extraction problems. When the indexing set is higher (2 or more) dimensional, then one has to study random fields so that new methods and analyses are needed. This is considered from Chapter 4 on. Here the concept of isotropy arises and we need to use properties of Bessel and related special functions. In these cases, a weakening of Lebesgue’s integral is needed and fortunately it is found in a work of M. Morse and W. Transue. It is refined by D. K. Chang and the author (1986) to obtain a suitable form to use here. This is employed and with it we consider the isotropic random fields analysis which is detailed in Chapter 4. They include the corresponding (integral) representations of random fields for both harmonizable, Karhunen and Cram´er classes that extend the former. Chapter 5 is devoted to some extensions of the preceding analysis if the indexing is an LCA group as well as an extension when it is a hypergroup. The work indicates many possibilities of the study when the indexing is considered on these structures that have both practical and theoretical consequences. There is also some analysis given in this chapter if the optimality criterion is not the usual squared loss but a general nonnegative convex function that may be specialized for many applications. The last chapter contains an analysis of bistochastic operators, their characterizations as well as their convergent properties. Also some accounts of isotropic analysis of random fields are included, and the area is active. It is also useful for readers to learn more about random measures with applications here. They are detailed in my earlier work in the volume entitled Random and Vector Measures (2012), especially in its last three chapters. This gives a good motivation as well as some new ideas on extending the present work in other directions, motivated by the work in (the long) Chapter 7 of the above book. An extended presentation of Cram´er and Karhunen classes as a far-reaching generalizations of (weak) harmonizability (and weak stationarity in many respects) is also detailed in last two chapters that exhibit the extent of growth of the subjects with possible further applications and exploration for second order processes. In the preparation of this monograph, I have been awarded a UCR Senate’s Edward A. Dickson Professorship which has really assisted me in completing this volume. The material is composed using the LATEX version with the great help of Ms. Ambika Vanchinathan for a period of over a year, from Chennai, India, and I am very grateful to her for this assistance. I am also much obliged to my friends Drs. Y. Kakihara and J. H. Park, for their great help in proofreading and effective corrections of the TeX version. Also from our Department office, I had been assisted by James Marberry, as well as Crissy Reising and Joyce Sphabmisai in some preparations

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of this work. I hope the material in this volume, along with its predecessors, will encourage new as well as seasoned researchers in the subject, to significantly advance this work into the future.

Riverside, CA August 18, 2020

M. M. Rao

b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1

2

3

Harmonizability and Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Second Order Processes and Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Admissible Means for Stationary Processes and Extensions . . . . . . . . . 1.3 Positive Definiteness as a Basis of Stochastic Analysis . . . . . . . . . . . . . 1.4 Important Remarks on Abstract and Concrete Versions of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 17

Harmonic Approaches for Integrable Processes . . . . . . . . . . . . . . . . . . . . 2.1 Morse-Transue Integration Method and Stochastic Analysis . . . . . . . . . 2.2 V -Boundedness, Weak and Strong Harmonizabilities . . . . . . . . . . . . . . 2.3 Harmonizability and Stationary Dilations for Applications . . . . . . . . . . 2.4 Domination of Vector Measures and Application to Cram´er and Karhunen Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Multiple Generalized Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Local Functionals in Probability; Their Integral Representations and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 A Probabilistic Proof of Riemann’s Hypothesis . . . . . . . . . . . . . . . . . . . . 2.8 Admissible Means of Second Order Processes . . . . . . . . . . . . . . . . . . . . 2.9 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 59 72

32 36 39

77 87 102 109 115 122 127

Applications and Extensions of Harmonizable Processes . . . . . . . . . . . . . 131 3.1 Special Classes of Weak Harmonizability . . . . . . . . . . . . . . . . . . . . . . . . 131 3.2 Linear Models for Weakly Harmonizable Classes . . . . . . . . . . . . . . . . . . 141 3.3 Application to Signal Extraction from Noise, and Sampling . . . . . . . . . 146 3.4 Class (KF) and Nonstationary Processes Applications . . . . . . . . . . . . . . 157 3.5 Further Classifications and Representations of Second Order Processes 164 xi

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3.6 3.7

Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4

Isotropic Harmonizable Fields and Applications . . . . . . . . . . . . . . . . . . . . 4.1 Harmonizability for Multiple Indexed Random Classes . . . . . . . . . . . . . 4.2 A Classification of Isotropic Covariances . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Representations of Multiple Generalized Random Fields . . . . . . . . . . . . 4.4 Remarks on Harmonizability and Isotropy for Generalized Fields . . . . 4.5 Summability Methods for Second Order Random Processes . . . . . . . . . 4.6 Prediction Problems for Stochastic Flows . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 185 195 201 203 206 223 227

5

Harmonizable Fields on Groups and Hypergroups . . . . . . . . . . . . . . . . . . 5.1 Bimeasures and Morse-Transue (or MT-) Integrals . . . . . . . . . . . . . . . . . 5.2 Harmonizability on LCA Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Harmonizability on Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Remarks on Strict Harmonizability and V -Boundedness . . . . . . . . . . . . 5.5 Vector-Valued Harmonizable Random Fields . . . . . . . . . . . . . . . . . . . . . 5.6 Cram´er and Karhunen Extensions of Harmonizability Compared . . . . . 5.7 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 229 234 245 247 249 250 251 256

6

Some Extensions of Harmonizable Random Fields . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Harmonizability, Isotropy and Their Analyses . . . . . . . . . . . . . . . . . . . . . 6.3 Some Moving Averages and Sampling of Harmonizable Classes . . . . . 6.4 Multivariate Harmonizable Random Fields . . . . . . . . . . . . . . . . . . . . . . . 6.5 Optimum Harmonizable Filtering with Squared Loss . . . . . . . . . . . . . . . 6.6 Applications and Extensions of Harmonizable Fields . . . . . . . . . . . . . . . 6.7 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 259 260 269 282 289 294 297 301

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

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In this chapter the well-known and powerful harmonizability technique will be introduced as a motivation for studying several intrinsic aspects of stochastic or random processes, and random fields. It is not a mere descriptive expression to say that Fourier analytic tools are among the most important techniques in such a detailed analysis of random functions and fields. First, it is natural to consider second order processes which indicate generalizations to many classes influenced by applications and develop an extended theory as well as the consequent analysis. 1.1 Second Order Processes and Stationarity If an experiment is conducted and its outcome is observed denoting it by {Xt , t ∈ T }, during the time period T one may consider its moment structure, based on an underlying probability model, denoted by (Ω, Σ, P ), whose first two moments, defined as E(Xt ) and E(Xt2 ), are assumed to exist and analyzed as a basic step in its analysis. Thus if
mt = E(Xt ) and r(s, t) = E (Xs − ms )(Xt − mt ) , where the Xt may be complex-valued, are called respectively the means and covariances of the process defined for a subset T of the reals R, one may study the structural (or moment) properties of the process based on the mean {mt , t ∈ T } and the covariance function {r(s, t); s, t ∈ T }. To simplify writing, let Yt = Xt − mt , the centered process, [E(Yt ) = 0] with the same covariances r(s, t)(= E(Ys Y¯t )) which may be analyzed in detail without the interference of the means. But the covariance function r given by r(s, t) = E(Ys Yt )[= E(Ys Y¯t )] inP the complex case], has the key property of positive definiteness in that ni,j=1 ai a ¯j r(ti , tj ) ≥ 0 1

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and prompts its analysis. The important property of positive definiteness led by A. Ya. Khintchine (1934) to introduce and study the fundamental concept of covariance stationarity of the centered process ˜ t = Xt −mt , t ∈ T } with two moments, implying that the covariance {X is invariant under translations, so that 

 ¯ t ) = r˜(s − t). r(s, t) = E (Xs − E(Xs ) X t − E(X (1) The continuity condition of r (and hence r˜) is natural for many applications. This immediately leads to a recognition of the following representation of r˜ (hence r) due to the fundamental Fourier integral characterization by S. Bochner (1933). Z r˜(s − t) = ei(s−t)λ dF (λ), s, t ∈ R, (2) R

where F : R → R+ is a nondecreasing bounded function. This result is also valid if r˜(·) is only a measurable function, noted by F. Riesz, after studying Bochner’s theorem. In applications, the nondecreasing bounded function F ≥ 0 is usually termed the spectral distribution of the process Xt . Prior to the representation (2) by Bochner, there existed a special result if R is replaced by the integers Z, in which case (2) has a simpler form, discovered by G. Herglotz, so that the process {Xn , n ∈ Z} has the corresponding representation for r(·, ·) as, Z π r(m, n) = r˜(m − n) = ei(m−n)λ dF (λ), m, n ∈ Z, (3) −π

where Z denotes the integers and the function F (·) is, as before, nondecreasing and bounded. The continuity of the covariance is equivalent to the mean square continuity of the process, i.e., E(|X(t) − X(s)|2 ) → 0 as s → t in both cases. Now the intervention of the Fourier analysis as integral in (2) and (3) above suggests a powerful motivation to weaken the hypothesis of mean continuity (i.e., the continuity of r(·, ·)) as well as weakening of the hypothesis of stationarity of the process, so that r(s, t) may not be r˜(s−t) but merely retains the positive definiteness property. These ideas will be explored and extended in what follows. The preceding discussion shows that the second moment, or more precisely its covariance function is the key parameter for an analysis of

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second order processes. But then one may ask if the first moment function t 7→ f (t) = E(Xt ), t ∈ R, can be an arbitrary constant or may be adjusted by taking it to be zero after a subtraction for covariance? This point is not as simple as one may wish. Its nontriviality was shown by A. V. Balakrishnan (1959) and is instructive to consider as it will be of interest and a motivation for studying extensions of stationarity and this will be discussed in order to include some useful general classes of the basic stationary family that we started with. Such extensions are also useful for many applications. We then analyze the corresponding problem for families of nonstationarity processes below which are termed Karhunen and Cram´er classes introduced by the researchers with these names. This discussion implies and indicates that one may consider the pro¯ t ), cess t 7→ Xt in L2 (P ) which satisfies the condition r˜(s, t) = E(Xs X ˜ (E(Xt = 0)) to be stationary if r˜(s, t) = r˜(s − t) and apply the Khintchine–Bochner method so that r˜˜ satisfies (2) as a Fourier trans˜ form of a bounded measurable function F˜ and then study X(t) = X(t)−m(t), the mean-corrected process so that X(t) is the sum of a random and nonrandom (measurable) elements. Then the method used for ˜ X(t) and X(t) involves the (mean) function that also admits a Fourier transform. Its explicit form is determined first by Balakrishnan which is seen to be a nontrivial property of the Fourier representation of the first moment indicating that some extensions for higher moments are possible. First, we treat the representations of the mean and covariances, and then generalize stationarity, called “harmonizability”, of the process and later discuss some other extensions and of higher moment studies. 1.2 Admissible Means for Stationary Processes and Extensions Let {Xt , t ∈ R} be a covariance stationary process with mean m : t 7→ m(t) = E(Xt ), and covariance r(s, t) given by Z ¯ r(s, t) = E(Ys Yt ) = ei(s−t)λ dF (λ), [= r˜(s − t)] (4) R

where Yt = Xt −m(t), m(t) = E(Xt ), and F (·) as its “spectral distribution” to mean that it is a nonnegative and nondecreasing bounded function (left continuous by a convenient normalization), thus giving r(·), a Fourier integral representation. The precise form of the mean function

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m(·) is later determined by Balakrishnan (1959), without demanding that it be a constant which was done in earlier times, as follows: Theorem 1.2.1 Let X = {Xt , t ∈ R} be a process with mean m(·), and covariance r˜(·) which is L2 (P )-stationary and continuous so that
r˜(s − t) = E [Xs − m(s)][Xt − m(t)] where m(t) = E(Xt ). Then for r(·) represented by (4), m(·) must also have a Fourier representation   Z Z itλ itλ dµ (λ) dF (λ), (5) m(t) = e dµ(λ) = e dF R R in order that r(s, t) = r˜(s − t) − m(s)m(t) ¯ is a covariance in the sense dµ lies in the unit ball of L2 (F ), where F is the measure that µ  F and dF representing r(·, ·) by (4). This result serves as a motivation for us to see that the mean function of the process is also important to consider along with the main parameter, namely the covariance function. [Note that both r(·, ·) and r˜(·) cannot be covariances of the same process.] The problem will be analyzed for a general class of processes in which harmonic (or Fourier) analysis plays a key role and subsumes the above result. The class we study here is the harmonizable family containing the stationary class that was raised in the earlier works which motivated these questions. To present a comprehensive analysis that includes random fields such as {Xt , t ∈ Rn } ⊂ L2 (P ), 1 ≤ n < ∞, let us replace Rn by G, a locally compact abelian group, denoted as (G, +) and {Xt , t ∈ G} be the new family, usually termed a random field (and process if G = R). The next extension from stationarity considered by Khintchine, is to study classes of nonstationary processes whose covariance functions r(s, t) = E([Xs − ms ][Xt − mt ]) admit integral representations relative to ‘bimeasures’ β : B(G) × B(G) → C so that r(·, ·) can be represented by a double integral relative to a “positive definite bimeasure” which, when concentrated on the diagonal of G × G, gives back the (stationary) representation (4). Such an extension was considered (perhaps) independently by M. Lo´eve and Yu. A. Rozanov in early 1950’s who then obtained a Fourier integral representation of the covariance, termed a harmonizable class when G = Rn . We now describe this class in detail since in Rn , n > 1, the integrals (4) are different accordingly as the new F (·, ·) has a finite Vitali or Fr´echet variation based on the Lebesgue or other integrals.

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These two concepts play key roles in our analysis, but there are several other “variations” which were also analyzed and the details are found in Clarkson and Adams (1933). We only employ here the result that the relation: 0 ≤ Fr´echet(F ) ≤ Vitali(F ) ≤ ∞, with strict inequality between the first two variations when there is equality between the last two. An illustration will be given below. As is known from the standard measure theory, a complex measure on a σ-algebra is bounded, the (bi-) measure β : B(G) × B(G) → C is bounded as a complex measure and is nonnegative definite if n X

β(Ai , Aj )ai a ¯j ≥ 0,

n ≥ 1, Ai ∈ B(G).

i,j=1

Also if r : G × G → C is given by Z Z r(s, t) = hs, xiht, yiβ(dx, dy) ˆ G

(6)

ˆ G

ˆ is the dual of G and hs, ·i is a function on G, ˆ then r(·, ·) : G × where G G → C is a bounded continuous positive definite (hence a covariance) function. This β of (6) plays a key role in extending the stationary case of (4) and hence a characterization of it is useful, but not simple, prompting Lo´eve to ask for its characterization. It will be sketched in the complements section later. We also discuss the problem of their means here. The simple example that r(s, t) = f (s)f¯(t) where f (·) is a continuous mapping that is not the Fourier transform of an integrable function (such functions are known to exist) shows that an r(·, ·) being positive definite (hence a covariance) does not admit a representation as (6) above. So it is nontrivial. Any characterization of r(·, ·) of (6) depends on the integral used there since the bimeasure β(·, ·) need not define an absolute integral to invoke Lebesgue’s integration properties. Along with this, we need to characterize the mean function as an extension of theorems above. For this one finds that bimeasure β(·, ·) of (6) has several variations, and the Vitali and Fr´echet are the most relevant ones to use here. Following Clarkson and Adams two of these variations are given as follows and are stated for a set function: Definition 1.2.2 Let G be an LCA (= locally compact abelian) group and B(G) denote its Borel field. Let β : B(G) × B(G) → C be a bimeasure on the indicated product σ-algebra with values in the complex

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field C so that β(·, A), β(A, ·) are measures for each A ∈ B(G). Then β(·, ·) is said to have finite Fr´echet variation on the product σ-algebra B(G) ⊗ B(G) if kβk(G, G) is finite where, with ai ∈ C, n o n X ai a ¯j β(Ai , Aj ) : Ai ∈ B(G), |ai | ≤ 1 (7) kβk(G, G) = sup i,j=1

for all disjoint Ai , i = 1, . . . , n n ≥ 1. The β(·, ·) here is of finite Vitali variation, if the ai = 1 for all i in (7) and |β(Ai , Aj )| replaces β(Ai , Aj ), thus moving the absolute value signs inside. It is clear that if β(·, ·) has finite Vitali variation, then it automatically has finite Fr´echet variation but not true conversely. Further, if β(·, ·) has finite Vitali variation, denoted |β|(G, G) < ∞, then (3) is a familiar Lebesgue-Stieltjes integral. But in the more general case that kβk(G, G) < ∞ (and |β|(G, G) = ∞), the integral in (3) cannot be defined in the Lebesgue sense. The necessary weakening of Lebesgue’s work is given by M. Morse and W. Transue (1956) (to be called the MTintegral here) and that will be recalled now. It reduces to the Lebesgue case if |β|(G, G) < ∞ but is weaker and hence more general. Let CC (Ω) be the standard complex continuous function space on a locally compact Hausdorff space Ω with f ∈ CC (Ω) having compact support. Then each positive linear functional I : CC (Ω) → R is R uniquely representable as I(f ) =R Ω f dµ for a Baire measure µ on Ω, which is also denoted as I(f ) = Ω f dI. If I(·) is complex valued it can be expressed as I = I1 − I2 + i(I3 − I4 ), Ij ≥ 0 and I is then termed a “C-measure” by Morse and Transue who also considered bilinear forms. If Ωi , i = 1, 2 are locally compact spaces as above and ∧ : CC (Ω1 )× CC (Ω1 ) → C is a bilinear form and if ∧(f, ·), ∧(·, g) are C-measures then (f, g) is Morse-Transue (or MT-)integrable iff f is ∧(·,R g) and g is ∧(f, ·) integrable for each f and g, and both the numbers Ω1 f (ω1 ) ∧ R (dω1 , g), Ω2 g(ω2 ) ∧ (f, dω2 ) exist and are equal. The common value is denoted by the double integral: Z Z Z (f (ω1 ), g(ω2 )) ∧ (dω1 , dω2 ) = f (ω1 ) ∧ (dω1 , g) Ω1 Ω2 Ω1 Z = g(ω2 ) ∧ (f, dω2 ). (8) Ω2

The MT-integral is defined using the (equivalent) Daniell procedure than that of Lebesgue’s and applies to [and admits] larger classes of

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7

functions. Further discussion is omitted here (but given in Chang and Rao (1986)). With this background, it is possible to present some useful second order processes more general than stationarity but utilize fully the benefits of Fourier analysis and its sharp forms. These are naturally termed harmonizable classes, and are defined as follows: Definition 1.2.3 (a)Let G be an LCA (locally compact abelian) group and B(G) be its Borel σ-algebra. Then β : B(G) × B(G) → C is called a bimeasure if β(·, B) and β(A, ·) are complex measures on B(G), for each A, B ∈ B(G). [This does not imply that β(·, ·) is a C-measure on the product σ-algebra B(G) ⊗ B(G).] (b)The bimeasure β : B(G) × B(G) → C is said to have finite Fr´echet (or Vitali) variation as in Definition 1.2.2 above. (c) A random family X : G → L2 (X) with mean functional m(g) = E(Xg ), and covariance r(g1 , g2 ) = E[(Xg1 − E(Xg1 ))(Xg2 − E(Xg2 ))− ] is weakly harmonizable if it admits a representation, for its covariance r(·, ·) with MT-integral, as: Z Z hs, λiht, λ0 iβ(dλ, dλ0 ) r(s, t) = (9) ˆ G

ˆ G

where the bimeasure β(·, ·) has finite Fr´echet variation, and if β(·, ·) has finite Vitali variation, then it is called strongly harmonizable. (d)A random family X : G → L2 (P ) is termed weakly or strongly harmonizable accordingly as its covariance r(·, ·) of (9), is weakly or strongly harmonizable respectively. It is desirable to have a usable or simpler criterion to study the structure of the harmonizable field {Xg , g ∈ G} ⊂ L2 (P ), and also a characterization of its mean functional as given in the above theorem. This will be obtained for the fields now. After characterizing the Khinchine stationarity, with S. Bochner’s theorem on the Fourier integral representation of a positive definite function, a corresponding result by the same author is again available. It uses a new concept called variation (or V -) boundedness in two forms to be used for the weak and strong harmonizabilities. Definition 1.2.4 (a)On a locally compact abelian group G, a continuous positive definite function r : G × G → C is weakly V -bounded if krk < ∞ where

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r(s, t)f (s)¯ g (t)dµ(s)dµ(t) : kfˆk∞ ≤ 1, G×G  1 kˆ g k∞ ≤ 1, f, g ∈ L (G, µ) . (10)

 Z krk = sup

Here µ is a Haar measure on G and fˆ and gˆ is the Fourier transform of f and g. ˜ = G × G be the di(b)With the same notation as in (a) above let G rect product (with the product topology) which is an LCA group with its own Haar measure µ ˜ (see, for instance, Hewitt and Ross (1963), Chapter 6), then r is strongly V -bounded if  Z  ∼ ˜ ˜ krk = sup r(˜ s)f (˜ s) d˜ µ(s) : kf k∞ ≤ 1 < ∞, (11) ˜ G×G=G

˜ µ where (G, ˜) is the product (Haar) measure space, given in the above ˜ µ reference. Here f˜ ∈ L1 (G, ˜) and is not merely the product type. With these concepts we now can obtain characterizations of weak and strong harmonizable covariances and then of the means of the respective classes, extending Theorem 1.2.1 above given in the particular case of G = R and for L2 (P )-valued stationary processes. To help the reading of our work, it will be useful to briefly restate the Morse-Transue (or MT-) integral as applied to the present needs. If the Ωi , i = 1, 2, are locally compact and CC (Ωi ) are complex continuous compactly based function spaces, a bilinear form ∧ : CC (Ω1 ) × CC (Ω2 ) → C defined through the Daniell–Bourbaki procedure, the desired MT-integral is obtained if f ∈ CC (Ω1 ) is ∧(·, v) and g ∈ CC (Ω2 ) is ∧(u, ·)-integrableR for each u ∈ CC (Ω1 ), v R∈ CC (Ω2 ) and the (Lebesgue) integrals Ω1 f (ω1 ) ∧ (dω1 , g(ω2 )) and Ω2 g(ω2 ) ∧ (f (ω1 ),Rdω2R) exist as well as agree or equal; the common value is denoted: Ω1 Ω2 (f, g)(ω1 , ω2 ) ∧ (dω1 , dω2 ). If β : B(Ω1 ) × B(Ω2 ) → C is a bimeasure, then Ra pair (f, g) is β-integrable (or R MT-integrable for β), where µ1 : A 7→ Ω2 g(y)β(A, dy), µ2 : B 7→ Ω1 f (x)β(dx, B) exist as measures for A ∈ B(Ω1 ), B ∈ B(Ω2 ) and are equal, so that the common value is denoted as: Z Z Z Z (f (x), g(y))β(dx, dy) = f (x)µ1 (dx) = g(y)µ2 (dy). Ω1

Ω2

Ω1

Ω2

(12)

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9

Since the integral is not absolute, a further condition is required for this work, somewhat resembling the needed restrictions in the Riemann (nonabsolute) integrals. We now include the desired versions, and work with them. These will be referred to respectively as the MT- and the strict MT-integrals. Thus a bimeasure µ : B × B → C, and Borel functions f, g : B → C define an MT-integrable pair as above and is strictly MT-integrable, if for each pair A, B ∈ B, one has Z Z ∗ Z Z B (f, g)(x, y)µ(dx, dy) = f (x)ν1 (dx) = g(y)ν2A (dy) (13) A

B

A

B

where ν2A = ν2f |B(A); ν1B = ν1g |B(B), for all A, B ∈ B, B(A), B(B) being the restricted (or trace) σ-algebra of B to A (and B). The strengthening ensures the validity of a dominated convergence theorem for such a bimeasure µ : B×B → C. This distinction is needed since the Fr´echet variation of µ being finite, its Vitali variation can be infinite! (Cf. Exercise 1.) It is also discussed in the companion volume (2014), on Inference. With this preparation let us present a characterization of weakly as well as strongly harmonizable covariances on an LCA group G and then describe the corresponding mean functions. Theorem 1.2.5 (a)A covariance function r : G × G → C is weakly harmonizable if and only if it is continuous and weakly V-bounded. When these conditions hold r is representable as a strict MT-integral relative to a unique bimeasure β as: Z Z ∗ hs, xiht, yiβ(dx, dy) r(s, t) = (14) ˆ G

ˆ G

ˆ is the dual group of G. where G (b)On the other hand, r : G × G → C is a strongly harmonizable covariance function if and only if it is strongly V-bounded and has the representation as a standard MT-integral Z Z r(s, t) = hs, xiht, yiF (dx, dy) (15) ˆ G

ˆ G

for a positive definite F of finite Vitali variation which thus admits an extension to be a bounded scalar measure on the Borel σ-algebra ˆ ⊗ B(G). ˆ B(G)

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Remark 1. If r(·, ·) is a stationary covariance, then the function F of (15) concentrates on the diagonal x = y of the product space so that the representation reduces to Bochner’s classical result of a continuous positive definite function as desired. After establishing this result we characterize the means, extending the result of Theorem 1.2.1, and present some complements. Proof. The sufficiency is easy. Indeed if r(·, ·) is weakly harmonizable then for any f, g ∈ L1 (G, µ), µ a Haar measure on G, we have Z Z r(s, t)f (s)g(t) dµ(s)dµ(t) G G Z Z Z Z ∗  = f (s)g(t) hs, xiht, yi dβ(x, y) dµ(s)dµ(t) ˆ G ˆ G ZG ZG∗ ˆ ¯ = f (x)gˆ(y) dβ(x, y) , by Fubini’s Theorem, ˆ G

ˆ G

ˆ G), ˆ using a property of the MT-integration. ≤ kfˆk∞ kˆ g k∞ |β|(G, ˆ G) ˆ < ∞, this implies that r is weakly V Since by hypothesis |β|(G, bounded, and the assertion follows in this direction. For converse part, let r be weakly V -bounded. Consider the functional defined by the (Haar) integral: Z Z r(s, t)f (s)g(t) dµ(s) dµ(t) (16) l(f, g) = G

G

and set T (f, g) = l(F −1 (fˆ), F −1 (ˆ g )) : C0 (G) × C0 (G) → C. Then T is well-defined since F : f 7→ fˆ, f ∈ L1 (G), being the Fourier transform, ˆ Now T = l ◦ (F −1 , F −1 ) is bilinear is one-to-one on L1 (G) → C0 (G). and the V -boundedness condition implies the boundedness of T : sup{|T (fˆ, gˆ)| : kfˆk∞ ≤ 1, kˆ g k∞ ≤ 1} = krk < ∞.

(17)

ˆ × L1 (G) ˆ and has a Thus T is a bounded bilinear functional on L1 (G) ˆ ˆ bound preserving extension to C0 (G)×C 0 (G). Then by the multidimenˆ obtained sional extension of Riesz’s representation theorem on C0 (G), by Dobrakov (1989), there exists a unique bounded bimeasure ν which is positive definite and is of finite Fr´echet variation so that the following strict MT-integration holds:

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T (fˆ, gˆ) =

Z Z ˆ G

11

∗

fˆ(x)ˆ g (y)ν(dx, dy).

(18)

ˆ G

[For more details of bimeasure integrals, see also Chang and Rao (1986), p. 21 on these properties.] From (16) and (18) one has l(f, g) = T (fˆ, gˆ) which implies Z Z r(s, t)f (s)g(t) dµ(s) dµ(t) G

G

= l(f, g)  Z Z ∗ Z Z hx, sif (s)hy, tig(t)µ(ds)µ(dt) ν(dx dy). = ˆ G

ˆ G

G

(19)

G

It follows from (19) and Fubini’s theorem that  Z Z  Z Z ∗ hx, sihy, tiν(dx, dy) f (s)g(t)dµ(s)dµ(t) = 0. r(s, t) − G

G

ˆ G

ˆ G

Since f, g ∈ L1 (G) are arbitrary, we get [ ] = 0 a.e. [µ ⊗ µ] and by continuity of r, everywhere. This implies (14). ˜ = G×G For the strongly harmonizable case, the direct product G will be an LCA group with its Haar measure µ ˜ = µ ⊗ µ which is the usual product measure (cf. Hewitt and Ross (1963), e.g. (23.33)). Then ˜ and r : G ˜ → C is seen to be a strongly if x = (s, t) ∈ G × G = G, V -bounded mapping, so r(·) strongly V -bounded, gives: Z r(x)f (x) d˜ ≤ Kkf k∞ µ (x) ˜ G

for all µ-integrable f with f˜(x) = f (x)g(t), x = (s, t). Then by the classical Bochner representation theorem again one has Z r(x) = hx, uidF (u) ˜ G

˜ = G × G and µ for an F of bounded variation. With G ˜ = µ ⊗ µ it can be expressed (by the uniqueness in that representation) as: Z Z Z r(x) = r(s, t) = hx, yidF (y) = hs, λiht, λ0 idF (λ, λ0 ), ˜ G

ˆ G

ˆ G

ˆ × G. ˆ But since r(·, ·) where now F is of bounded (Vitali) variation on G is positive definite, it follows from this representation that F must be

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positive definite. This gives (15), and hence the proof of the theorem is complete. The essential distinction between weak and strong harmonizabilities is clarified.  As a consequence of the above result, it is possible to present an integral representation of a (weakly or strongly) harmonizable process {Xt , t ∈ G} on an LCA group G, generalizing the stationary case of the Bochner-Khintchine (classical) theorem as follows: Theorem 1.2.6 Let {Xt , t ∈ G} ⊂ L20 (P) be a (mean) continuous strongly or weakly V -bounded process where G is an LCA group. Then it admits a unique (stochastic) integral representation as: Z (20) Xt = ht, ui dZ(u), t ∈ G, ˆ G

ˆ → L2 (P ) whose spectral bimearelative to a vector measure Z : B(G) 0 sure β : (A, B) → E(Z(A)Z(B)) is weakly or strongly V -bounded. Proof. Let r(s, t) = E(Xs X t ) be the covariance of the centered process X which is V -bounded with continuous paths on G. Then for f, g : G → C bounded Borel functions with compact supports, consider Z Z r(s, t)f (s)g(t) dµ(s) dµ(t) G G  Z Z Xs f (s) dµ(s) Xt g(t) dµ(t) =E G

G

where µ is the Haar measure on G. Now using the V -boundedness hypothesis on X and taking f = g, the right side becomes: " Z 2 # Z Z r(s, t)f (s)f (t) dµ(s) dµ(t) E Xs f (s) dµ(s) = G

G

G

≤ Kkfˆk2∞ using (10), for some 0 < K < ∞. Hence

Z



Xs f (s) dµ(s) G

2,P

≤

√

Kkfˆk∞ .

SinceR a bounded part in L2 (P ) is relatively weakly compact, one has the set { G Xs f (s) dµ(s), f ∈ L1 (G), kfˆk∞ ≤ 1} ⊂ L2 (P ) to be bounded.

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ˆ → L2 (G) given by Consider the mapping T : C0 (G) Z ˆ Xs f (s) dµ(s) T (f ) = l(f ) =

13

(21)

G

which is bounded and linear, is weakly compact. So by the Riesz ˆ → L2 (P ), such that representation theorem, there exists a Z : B(G) 0 E(Z(A)) = 0 and represents T as Z Z Z ˆ ˆ T (f ) = f (u) dZ(u) = f (t)ht, uidµ(t) dZ(u). (22) ˆ G

ˆ G

G

This may be written in detail as: Z Z Z ˆ Xt f (t) dµ(t) = T (f ) = f (t)ht, uidµ(t) dZ(u) G

G

ˆ G

where the left is the Bochner and the right the Dunford-Schwartz (or Vector) integral. Hence by Fubini’s theorem (in vector form)  Z  Z Xt − ht, ui dZ(u) f (t) dµ(t) = 0. (23) G

ˆ G

It follows that, from the arbitrariness of f (·), that [ ] = 0 holds a.e. and by the continuity of Xt , the µ-null set must be empty. The same argument applies (with slight simplifications) to the strongly harmonizable case. The last statement on the spectral bimeasure of Z(·) is immediate.  So far the mean function is assumed to be zero, i.e. the harmonizable processes are centered. We now consider the possible mean function of such processes and establish a generalized form (i.e., subsuming the result of Theorem 1.2.1) of the means for such Xt . Theorem 1.2.7 Let X = {Xt , t ∈ G} be a weakly harmonizable process with a continuous covariance function r having F as its spectral bimeasure. Then a mapping m : G → C is the mean value of the random field X on the space (Ω, Σ, P ) with r as its covariance if and only if thereR is a unique g ∈ L2 (F ) such that (i) (g, g)F ≤ 1 and (ii) m(t) = Gˆ ht, ui dµ(u) where µ is a scalar measure given by Z ∗ µ : A 7→ g(v)F (A, dv), A ∈ B(G). (24) G

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Here r(·, ·) is the covariance of the X, and F (·, ·) is its (positive definite) spectral bimeasure and the integral with a star (∗) is the strict MTintegral. Proof. First, observe that L2 (F ) is a semi-inner product space since F is a positive definite bimeasure. It is also true that the space is complete, and separable when G is separable. This fact is not entirely simple and can be tried as a problem. Now using a few properties of the (MT)-integral, one can verify that the space L2 (F ) is complete for the inner product defined earlier using the (strict) MT-integral. The point here is that the MT-integral (and its strict version) is essential in this (generalized) analysis. Thus if r(s, t) = E(Xs X t ), which is now harmonizable, it admits an integral (vector or Dunford-Schwartz type) representation: Z Xt = ht, ui dZ(u), t ∈ G, (25) ˆ G

by (20) and Z(A) 6= 0 since E(Xt ) 6= 0 by assumption, A ∈ B(G) and mt = E(Xt ). Hence by a classical theorem of Hille (as in DunfordSchwartz (1958), IV.10.8) one has from (25): Z Z (26) mt = E(Xt ) = ht, uiE(dZ(u)) = ht, uiµ(du), ˆ G

ˆ G

ˆ and where µ(du) = E(Z(du)), so µ is a scalar measure on B(G) Z ft (u)µ(du), ft ∈ L2 (F ), (260 ) mt = l(ft ) = G

is a well-defined linear functional for some g of l(ft ) = (ft , g), and (g, g)F = klk ≤ 1. R R We now turn to the converse. Suppose m = Gˆ ft dµ where µ(A) = ¯(v)F (A, dv), the F (·, ·) being a bimeasure determined by Z(·). It ˆg G is to be verified that mt is the mean of a harmonizable field Xt so that if Yt = Xt − mt , then Yt is centered and r(s, t) = E[(Xs − ms )(Xt − mt )] = E(Xt Xt ) − ms mt . For this, it is enough to verify that r˜ is positive definite and continuous. Continuity being clear, it suffices to verify the positive Pn definiteness. Let f (u) = i=1 ai hsi , ui, ai ∈ C and consider the strict MTintegrals:

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1.2 Admissible Means for Stationary Processes and Extensions n X i,j=1

Z Z ai a ¯j r(si , sj ) =

ˆ G

∗

n X

ˆ G i,j=1

15

ai a ¯j hsi , uihsj , viF (du, dv) n Z X 2 − ai hsi , ui dµ(u) ˆ G i=1

Z 2 f (u)f¯(u )F (du, du ) − f (u) dµ(u) = ˆ ˆ ˆ G G G 2 Z Z ∗ 0 0 g¯(u )F (du, du ) = (f, f )F − f (u) ˆ ˆ G G Z Z ∗ 2 0 0 = (f, f )F − f (u)¯ g (u )F (du, du ) ≥ 0, Z Z

∗

0

ˆ G

0

ˆ G

since the last integral is dominated by (f, f )F (g, g)F ≤ (f, f )F by the CBS inequality and the fact that kgkF ≤ 1. So mt is the mean of Xt .  In the strongly harmonizable case, the MT-integral is already strict and so the ‘star’ on the integral can be omitted. This may be stated for a convenient reference as: Corollary 1.2.8 Let {Xt , t ∈ G} be a strongly harmonizable process with covariance r(·, ·). Then m : G → C is its mean function, so that mt = E(Xt ), if and only if there is aR function g ∈ L2 (F ) such that R(i) (g, g)F ≤ 1 and (ii) m(t) = G (t, u) dµg (u), where µg : A → G g¯(v)F (A, dv), A ∈ B(G), the integral being the (ordinary not strict) MT-integral. Remark 2. 1. We now record some consequences. If the process is second order stationary so that r(s, t) = r(s − t), then the ‘spectral function’ F concentrates on theRdiagonal so that µg (·) of the above corollary satisfies µg (A) = A g¯(u) dH(u) where H(·) is g = g ∈ L2 (H) and spectral measure of r. So dµ dH Rthe (bounded) 2 ˆ |g| dH ≤ 1. G In the stationary case, one has r(s, t) = r˜(s − t) = E[Xs Xt ] − ms m ¯t so that (with s = t), |mt |2 = E[|Xt |2 ] − r˜(0) is a constant. Thus one has |mt |2 = E[|Xt |2 ] − r˜(0) and mt = a(t, m0 ) for some m0 ∈ G, and t ∈ R.

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2. If r and r˜ are a pair of general covariances (not necessarily even harmonizable), the fact that (s, t) 7→ ms m ¯ t is always positive definite implies that r˜ − r is positive definite, and hence by the well-known reproducing kernel Hilbert space theory due to Aronszajn, the resulting Hilbert spaces Hr , Hr˜ also often called Aronszajn spaces (see below) satisfy the inclusion Hr˜ ⊂ Hr corresponding to the positive definiteness property of r˜ − r, and hence k · kr˜ ≥ k · kr , from which one obtains kmkr ≤ kmkr˜ ≤ 1, if mt is the mean of a harmonizable process with covariance r. This fact holds also for some non harmonizable process, e.g. for a pair of Brownian motion processes which are not necessarily harmonizable. That was observed by Ylvisaker (1961), where G = [0, 1] with addition (mod 1) as group operation and he then noted that m : G → R is a possible mean value of a R dmt 2 process with r as covariance if and only if G dt ≤ 1. This does not yet give the result of Theorem 1.2.8, since the corresponding characterization of Hr for the processes is not at hand. 3. A function g : G → R can be positive definite and nonmeasurable. For instance let f : R → R be a nonmeasurable function such that f (x + y) = f (x) + f (y) and g(x) = e−tf (t) for t > 0, so that f is conditionally positive definite. Then ‘g’ is positive definite and even nonmeasurable. Existence of such an f is well-known. (See Exercise 3 for more detail.) 4. In place of Fourier transform in the above work, one may consider other one-one mappings such as Mellin and a certain “L-transform” discussed by Kawata (1965), and the work can be extended for such classes also. Some analogs for unimodular groups will be considered later in this book. If r : S × S → C is a positive definite function (or a covariance) then the Aronszajn space, denoted Hr , defined on S of complex functions, is a Hilbert space satisfying the conditions: 1. r(·, t) ∈ Hr , t ∈ S, and 2. f ∈ Hr ⇒ (f, r(·, t)) = f (t). An easy consequence of the above definition is the fact that m : S → C, r(s, t) = m(s)m(t), ¯ is a covariance and this Hr is the (vector) space of all multiples of m, with norm kmkHr = 1. The basic property established by Aronszajn (1950), useful in the analysis of this space, is given by:

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17

Theorem 1.2.9 If ri : S × S → C, i = 1, 2 are covariances such that r1 dominates r2 (i.e., r1 −r2 is positive definite), we have then the inclusion Hr1 ⊃ Hr2 in that kf kHr2 ≤ kf kHr1 , f ∈ Hr2 arbitrary. This result based on Aronszajn’s theory (cf. Aronszajn (1950)) implies the following extension of corollary 1.2.8, due to Ylvisaker (1959), and is of interest here. Theorem 1.2.10 Let {Xt , t ∈ T ⊂ L2 (P )} be a process with the mixed ¯t ), and product moment K : T × T → C, defined by K(s, t) = E(Xs X m : T → C be a function. Then (s, t) → K(s, t) − m(s)m(t) ¯ is the covariance function of the Xt -process, if and only if m ∈ HK of unit norm, i.e., belongs to the unit ball of HK . These two results indicate adjuncts to the theory considered above, and will be of interest in extension of the earlier analysis. We omit the details here, since this aspect is not related to harmonizable analysis of the process. 1.3 Positive Definiteness as a Basis of Stochastic Analysis In a substantial part of stochastic analysis, the concept and properties of positive definiteness play fundamental roles. This has been basic both for stationary as well as harmonizable analyses. Here we shall sketch some of the key roles of it for the concept of Brownian motion. An English botanist named Robert Brown, observed in 1826 that a particle suspended in fluid makes irregular spontaneous movements caused by the molecular impacts on it by those of the medium. There appeared to be negligible bonds between the particles and the surrounding medium. The impacts on the particle ω are irregular and the displacement X ω (t + s) − X ω (t) is the sum of a large number of (centered) impacted displacements. [A similar phenomenon was noticed later around 1900 by a French researcher named L. Bachelier termed ‘speculation theory’, about stock market movements.] Thus the X(t)-process has independent (centered) increments, and the already available central limit theorem suggests that the random element X(t + s) − X(t), s > 0, is (approximately) Gaussian (or Laplacian as termed in France). This conclusion is also due, in 1905, to A. Einstein in Germany (and to M. von Smoluchawski in Poland). The result was proved mathematically rigorously by N. Wiener in 1923 in the U.S., and so it is also termed a Wiener

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process. Thus we have a process {Xt , t ≥ 0} which is termed Brownian motion, or the Wiener process, such that (Xtk − Xtk−1 ) is Gaussian with mean zero and variance c(tk − tk−1 ) > 0 for any 0 < t1 < · · · < tn and has stationary independent increments. This process formed a powerful motivation for an extension to second order processes whose covariances lead to weakly stationary classes as a first generalization and then to (weakly) harmonizable classes as its natural extension. The crucial fact in this extension is the positive definiteness property ¯ t ), where s, t ∈ T (= G) which has a of the covariance r(s, t) = E(Xs X semi-group structure under addition. This process motivates a study of the more general harmonizable classes where the index G is allowed to be a locally compact group. Recall that r : T × T → C is positive definite if for ti ∈ T, ai ∈ C n X

r(ti , tj )ai a ¯j ≥ 0,

n ≥ 1.

i,j=1

However, there is no direct or simple method of verifying this property. An integral representation of r(·, ·) is obtained by various mathematicians including S. Bochner, H. Cram´er, P. L´evy, F. Riesz and others. This is really a characterization of functions that are Fourier transforms, or those that admit just vector integral representations using V -boundedness again. This is restated, more generally, as follows and used to characterize vector Fourier transforms of interest here. Definition 1.3.1 If G is an LCA group and X a Banach space, a mapping f : G → X is V-bounded if (i) f (G) ⊂ X is bounded, (ii) f is strongly measurable relative to the Borel σ-algebras of G and X, (iii) the ¯ in X is weakly compact, where following set W R g k∞ ≤ 1, g ∈ L1 (G, µ) ⊂ X. (27) W = G f (t)g(t) dt : kˆ [Here ‘dt’ denotes a Haar measure on G.] Theorem 1.3.2 Let G be an LCA group and X a reflexive Banach space. A mapping X : G → X is a Fourier transform of a vector measure ˆ → X if and only if X is V-bounded and weakly continuous. ν : B(G) Thus Z Xg = hg, g 0 i dZ(g 0 ), g∈G (28) ˆ G

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ˆ B G) ˆ so {Xg , g ∈ where Z(·) is the unique vector measure on (G, G} is a weakly harmonizable random field, valued in X(= L2 (P ) is possible). Proof. Let X : G → X be V -bounded and weakly continuous. If L1 (G) denotes the (Lebesgue) space of integrable functions relative to the Haar 1 ˆ measure, denoted R dt, and f ∈ L (G), let f (·) be its Fourier 1transform, then TX : fˆ → G Xt f (t) dt is well-defined and linear on L (G) to X, satisfying, for some constant K > 0: kTX (fˆ)kX ≤ Kkfˆk∞ ,

by (27).

(29)

Also as a consequence of the Riemann–Lebesgue lemma, TX (fˆ) vanishes at infinity, i.e., TX (fˆ) ∈ C0 (G, X) and the V -boundedness hypothesis allows us to invoke the (vector) Riesz representation theorem ˆ → X we have so that for a unique vector measure ν : B(G) Z Tf (ˆ g) = gˆ(s) dν(s), g ∈ L1 (G). (30) G

This may be rewritten g ) is also an integral of R on using the fact that Tf (ˆ f, g, i.e., Tf (ˆ g ) = G f (t)g(t) dt, and one can invoke the classical Fubini theorem for the scalar (signed) measures x∗ ◦ ν, x∗ ∈ X∗ , to have: Z Z ∗ ∗ x (f (t))g(t) dt = x (Tf (ˆ g )) = gˆ(s) d(x∗ ◦ ν)(s). (31) ˆ G

G

The left integral can be simplified as: Z  Z   Z ∗ ∗ g(t) dˆ v (s) dt . f (t)g(t) dt = x x G

G

(32)

ˆ G

This implies that fˆ = νˆ, a.e., so that the V -bounded f is the Fourier ˆ → X. transform of ν : B(G) Conversely, if f is the Fourier transform of ν, then the weak compactness of the set (27) is to be established, since the function is clearly bounded, (kf (t)k ≤ kνk(G) < ∞). The weak compactness of W of (27) is also clear since it is bounded by kˆ g k∞ ≤ 1 and ν has finite variation, and that in a reflexive space bounded sets have weakly compact closures. This completes the argument. 

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Remark 3. The boundedness hypothesis in the above result is used in concluding the compactness of the closure of W in (27). However, the result holds for nonreflexive X also, with the rest of the hypothesis on ˘ using some abstract analysis which invokes the Eberlein–Smulian Theorem. The detail is sketched in the author’s book (Rao (2004), Measure Theory and Integration, 2nd Ed. Exercise 1, on p. 558), and some related extensions were also discussed there for interested readers. We shall return later to consider some extensions and applications of the general analysis touched on above. It is essential to discuss the positive definiteness property of the second order random fields, starting with the Brownian motion when the indexing is Rn , n > 1, as it depends on some new ideas. From the earlier discussion of this section, a Brownian motion, BM, {Xt , t ≥ 0} is a centered Gaussian process with stationary independent increments so that from the elementary identity (Xs − Xt )2 = Xs2 + Xt2 − 2Xs Xt , the cov (Xs , Xt ) = E(Xs Xt ) implies ¯ t ) = 1 [E(|Xs |2 ) + E(|Xt |2 ) − E(|Xs − Xt |2 )] CX (s, t) = E(Xs X 2 and hence for the BM the covariance must satisfy:
1 min(s, t) = CX (s, t) = [ksk+ktk−ks−tk], s, t ∈ R+ . (33) 2 Here k · k stands for the norm of R. Since we consider the processes {Xt , t ∈ T } with T ⊂ Rn , n ≥ 1, and even T = G an LCA group in our study, it becomes essential to know if CX (·, ·) of (33) defines a covariance, i.e., a positive definite function on T × T . This is not easy and P. L´evy was able to construct a Gaussian process {Xt , t ∈ IR2 } and show that its covariance is given by (33) so that it is positive definite, but an independent proof of the latter property could not be found by him! We present a general solution. Definition 1.3.3 Let (T, k · k) be a normed vector space and let X = {Xt , t ∈ T } be a real Gaussian process on (Ω, Σ, P ), a probability space. The process is called a L´evy–Brownian motion with T as its parameter set if (i) X0
= 0, a.e., (ii) E(Xt ) = 0, t ∈ T , and for t, t0 ∈ T, E (Xt − Xt0 )2 = kt − t0 k. Clearly if T = R this is the BM considered above, and the case that T is of higher dimension is not an easy extension and new ideas are

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needed. We present two (different) methods, if T = Rn , and later if T is more general. Thus let T = Rn , n > 1. Consider a family of complex functions on Rn , as φs : Rn → C, n > 1 defined by −(n+1)/2

φs (x) = Cn |x|

i(s,x)

(e

n

− 1), s ∈ R , (s, x) =

n X

si xi . (34)

i=1 2

n

Observe that the complex function φs ∈ L (R , dx), the classical Lebesgue space. The class {φs , s ∈ Rn } forms a dense set in the Lebesgue space L2 (Rn , dx), and φ0 (x) = 0. Note that for suitable Cn Z Z i(s,x) 2 dx 2 2 e |φs − φt | (x)dx = Cn − ei(t,x) = |s − t|. (35) |x|(n+1) Rn Rn A computation by evaluating this integral shows that setting Xt = φt , t ∈ Rn , is a process with X0 = 0, and for s, t ∈ Rn , one has E[(Xs − Xt )2 ] = E[|φs − φt |2 ] Z |φs (x) − φt (x)|2 dx = |s − t|. = Rd

But the left side is the real process which on expansion gives |s| + |t| − 2E(Xs Xt ) so that the covariance is: 1 cov (Xs , Xt ) = [|s| + |t| − |s − t|]. 2 This gives (33) which thus depends on several tricks and it is adopted from Neveu (1968). We state this result as: Proposition 1.3.4 For each integer n ≥ 1 there exists a real Brownian motion {Xt , t ∈ Rn } such that 0 = 0 a.e., and   Xp Pn 2 2 E[(Xs − Xt ) ] = |s − t| = i=1 (si − ti ) , and 1 E(Xs Xt ) = [|s| + |t| − |s − t|]. (36) 2 A different argument is needed if Rn is replaced by an infinitedimensional Hilbert space. This will now be given as it is also important and useful. For some concrete work below, it is useful to know that an abstract Hilbert space can isometrically be realized as a subspace of the familiar L2 (P ) on a suitable probability space (Ω, Σ, P ). The desired statement is as follows:

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Theorem 1.3.5 If H is a nontrivial Hilbert space, then it is always realizable isometrically and isomorphically as a closed subspace of L2 (P ) on a probability space (Ω, Σ, P ), even with P as Gaussian. Proof. We sketch the (well-known) argument here for completeness. Let {ei , i ∈ I} ⊂ H be a complete orthonormal set which exists. Note that such a set is obtained and is maximal (i.e., cannot be enlarged) by using Zorn’s lemma. Thus if Ωi = R, Σi = B (Borel σ-algebra of R) and R −t2 Pi : A → √12π A e 2 dt, A ∈ Σi , let (Ω, Σ, P ) = ⊗i∈I (Ωi , Σi , Pi ). If Πi : Ω → Ωi is the coordinate projection, then P ◦ Πi−1 = Pi and (Ω, Σ, P ) = ⊗i∈I (Ωi , Σi , Pi ), gives the desired probability space, and P ◦ Πi−1 = Pi and if {φi , i ∈ I} is a complete orthonormal set in H, let {Xi , i ∈ I} ⊂ L2 (Ω, Σ, P ) = L2 (P ) be an orthonormal set so that τ (φi ) = Xi , i ∈ I. Then (τ φi , τ φj ) = (Xi , Xj )L2 (P ) = (φi , φj )I = δij and an isomorphism between the spaces H and L2 (P ) is obtained. Because of the maximality (and Zorn’s lemma), the space H and L2 (P ) are isomorphic (or H is identifiable with a closed subspace of L2 (P )), so that the abstract space H is realizable as an L2 (P ).  A comparison of this result with the preceding representation in which the indexing of the process is restricted and thereby demanding the corresponding class of the (Gaussian) process—here a Brownian motion class—is found to be more intricate than expected and we need to consider the processes (or random fields) with special (multiple) indexing again as emphasized by P. L´evy. It is also crucial that the covariance given by (33) is real-valued in addition to it is being a multidimensional domain. The detail depends on the basic works of L. Schwartz and I. J. Schoenberg on real positive definite forms which will be presented here, following the simplification by Cartier (1971). Recall that for a standard Brownian Motion {Xt , t ≥ 0}, it is seen that E(Xt ) = 0, E[(Xt −Xt0 )2 ] = E(Xt2 )+E(Xt20 )−2E(Xt Xt0 ), which thus can be written as: 1 E(Xt Xt0 ) = [E(Xt2 ) + E(Xt20 ) − E(Xt − Xt0 )2 ] 2  1 = |t| + |t0 | − |t − t0 | (= cov X (t, t0 )), 2

(37)

for the classical Brownian motion with the time T = R+ , and the product moment gives a positive definite function. If T = Rn+ , the positive

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orthant, the fact that (37) is positive definite is not so easy. P. L´evy has shown this to be true for n = 2 and the problem for n > 2 was nontrivial. It was solved independently by L. Schwartz and I. J. Schoenberg and later more abstractly simplified by P. Cartier (1971) whose proof we now present. It is valid for T as a Hilbert space and is definitely sharper than the preceding result. Theorem 1.3.6 For each real Hilbert space T , there exists a L´evyBrownian motion X = {Xt , t ∈ T }, indexed by T , on a probability space (Ω, Σ, P ), E(Xt ) = 0 and the mapping (t, t0 ) 7→ E(Xt Xt0 ) satisfies (37), as its covariance expression. Proof. By the preceding theorem an abstract Hilbert space can be realized isomorphically as L2 (Ω, Σ, P ) on a probability space where P is a Gaussian (probability) measure. Thus for the present proof, it is enough to show the existence of a function f : T → H, and a Hilbert space H, such that f (0) = 0, kf (t) − f (t0 )k2H = kt − t0 kT since T and H are g 0 0 different spaces. But such an f exists iff (t, Pnt ) 7→ kt − t kT is negative definite in that for any reals (a1 , . . . , an ), i=1 ai = 0, one has g(ti , ti ) = 0,

n X

g(ti , tj )ai aj ≤ 0, g(−ti , −tj ) = g(ti , tj ).

i,j= 1

This ensures that C(t, t0 ) = E(Xt Xt0 ) of (37) is positive definite. Indeed, n X

n 1X C(ti , tj )ai aj = − g(ti , tj )ai aj ≥ 0, (38) 2 i,j=1 i,j=2 P where t1 = 0 and ni=1 ai = 0. Thus an inner product on T can be defined with C(·, ·), and let H2 be the completion of T using C(·, ·). Define a mapping h : T → H2 with h1 (0) = 0, (h(t), h(t0 )) = C(t, t0 ), so that g(t, t0 ) = kh(t) − h(t0 )k2H2 = kt − t0 kT . Then g(·, ·) is (termed) conditionally negative definite because: n X i,j=1

g(ti , tj )ai aj =

n X

(h(ti ) − h(tj ), h(ti ) − h(tj ))ai aj

i,j=1 n X

=2

i,j=1

2

h(ti ) ai aj − 2

n X i,j=1

h(ti )h(tj )ai aj ≤ 0

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Pn

since i=1 ai = 0 making the first term on the right vanish, and h is positive definite. We now exhibit such a g on any (finite) subspace. Let T0 = sp {ti , 1 ≤ i ≤ d} ⊂ T , a d-dimensional subspace which may be identified as Rd with a Haar measure, corresponding to the Lebesgue d 2 measure on R R , denoted by µ and let H1 = L0 (T0 , µ) with inner product hg, hi = T0 Re (g(t)h(t)) dµ(t), so that H1 is a Hilbert space. For α, t ∈ P T0 − {0}, let hα, ti = di=1 αi ti and hα (t) = ktk−(d+1)/2 (1 − eihα,ti ), t ∈ T0 − {0},

α ∈ T0 .

(39)

To see that hα ∈ H1 , note that 2

2

|hα (t)| ≤ kαk ktk

1−d



sin(hα, ti/2) hα, ti/2)

2 .

(40)

Here we used the result that |1 − eihα,ti |2 = 2(1 − cos(hα, ti)). Thus from (40) weR note that |hα (t)|2 is integrable so that hα ∈ H1 as desired. If V (α) = T0 |hα (t)|2 dµ(t), then V (0) = 0, and V (aα) = |a|V (α). Also for any automorphism U : T0 → T0 , V (U α) = V (α). Then it is known from analysis that such a mapping V must be of the form V (α) = Cα kαk for some Cα > 0. Hence integrating: khα − hα0 k2H1 = khα−α0 k2H1 = V (α − α0 ) = Cα−α0 kα − α0 kT . (41) −1

Letting fα = Cα 2 hα : Tα → H1 , it is seen that f : α → fα verifies all the requirements, and hence gives the desired assertion.  If T is a subset of Rd , d > 1, then a specialization of the above result in such a case is of interest for some applications and will be a companion of Proposition 1.3.4 above, established somewhat differently, by deducing it from the above, with the Dunford-Schwartz method. Thus we let Z : B0 (T ) → L2 (T ) be a vector measure, T ⊂ Rd andRuse the D-S integration as indicated above, for f ∈ L2 (T, µ). So f 7→ T f (t)dZ(t) is defined. We thus will state the following: Proposition 1.3.7 If T = Rk , 1 ≤ k < ∞, then the L´evy-Brownian motion {Xt , t ∈ T } admits the following vector integral representation: Z Xt = ft (u) dZ(u), t ∈ T (= Rk ), T

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where {Z(A), A ∈ B0 } is the noise process noted, and B0 is the δ-ring −1

of bounded Borel sets of T , ft (·) = Ct 2 ht (·) where Ck and ht are given respectively in (39) and (41) above. Another way of stating L´evy’s Brownian Motion (BM), useful for some applications, is as follows: A gaussian random field {Xt : t ∈ RN } is termed L´evy’s BM, if the following conditions hold: (i) E(Xt ) = 0, (ii) X0 = 0, a.e., and (iii) E[(Xs − Xt )2 ] = r(s, t), and its covariance Γ (s, t) = E(Xs Xt ) = 21 [r(0, s) + r(0, t) − r(s, t)]. Note that t 7→ Xt is continuous with probability one. The following is a class considered for some analysis by P. L´evy himself. Let SN (t) be a sphere in RN centered at the origin and radius t > 0. If σt (·) is the uniform probability distribution on SN (t), define a new (Gaussian) process MN = {MN (t), t ≥ 0} by the path integral: Z X(u)σt (du), t > 0, (42) MN (t) = SN (t)

called L´evy’s MN (t)-process. Note that E(MN (t)) = 0, and 1 ΓN (s, t) = E(MN (s)MN (t)) = (s + t − ρN (s, t)), 2 R R where the function ρN : (s, t) 7→ SN (s) SN (t) r(u, v)σs (du)σt (dv). An explicit evaluation of ρN (·, ·) is difficult, but for some special cases, these functions have been evaluated and their properties discussed in the monograph by Hida and Hitsuda (1993) where some other interesting properties of Gaussian random fields have been considered to which we refer the interested readers. We consider certain other aspects of L´evy’s B.M. of several parameters as discussed by R. Gangolli (1967) indicating new directions. The preceding result motivates the ensuing analysis. Also, the classical Schur observation that the pointwise product of a pair of Hermitian nonnegative matrices is a nonnegative Hermitian matrix is of particular interest in our analysis on the L´evy-Schoenberg kernels which uses as essential elements of the L´evy-B.M. theory. A generalized form is thus: Proposition 1.3.8 Let S be a topological space and r : S × S → R be a kernel such that r(a, b) = r(b, a), r(0, 0) = 0 for a distinguished point 0 ∈ S, and let f (a, b) = 21 [r(a, 0) + r(b, 0) − r(a, b)], a, b ∈ S. Then f is positive definite if and only if the kernel θ : S × S → R defined by θt (a, b) = exp{−t · r(a, b)} is positive definite, where t > 0.

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Proof. Now f (a, b) = 21 [r(a, 0) + r(b, 0) − r(a, b)] being real valued, its positive definiteness follows if for any signed measure µ(·), compactly supported, one has Z Z f (x, y) dµ(x) dµ(y) ≥ 0. (43) S

S

Since f (a, 0) = f (b, 0) = 0, it can be assumed that µ(S) = 0 to prove (43). Hence this becomes Z Z f (a, b) dµ(a) dµ(b) S S Z Z 1 = (r(a, 0) + r(b, 0) − r(a, b)) dµ(a) dµ(b) 2 S S Z Z 1 r(a, b) dµ(a) dµ(b). (44) =− 2 S S Then (43) is equivalent to verifying Z Z r(a, b) dµ(a) dµ(b) ≤ 0. S

Now in the forward direction θt (·, ·) is positive definite, so Z Z 0≤ exp{−t · r(a, b)} dµ(b) dµ(a) ZS ZS (1 − t · r(a, b) + o(t2 )) dµ(a)µ(b) = S ZS Z = −t r(a, b) dµ(a) dµ(b) + o(t2 ), t ↓ 0, S

(45)

S

(46)

S

where o(t2 ) is uniformly small on compact subset supports of µ, as t > 0. This gives (45) and the direct part of the result follows. For the converse, suppose the given f (·, ·) is positive definite. Then et·f is positive definite for t ≥ 0. Hence for each n ≥ 1, an ∈ C n X

ai a ¯j exp{−t · r(ai , aj )}

i,j=1

=

n X i,j=1

ai a ¯j exp{−t[r(ai , 0) + r(aj , 0) − 2f (ai , aj )]}

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by hypothesis. Set βi = αi e−t·r(ai ,0) in the above so that n X

βi β¯j e2t·f (ai ,aj ) ≥ 0,

(47)

i,j=1

since e2t·f , t ≥ 0, is positive definite so that θt (a, b) is also, as asserted.  Remark 4. This result has a complex as well as an operator valued version with essentially the same argument, as observed by P. Masani (1973). It will not be detailed here. The weak stationarity of a square integrable process has been formulated in an extended and very useful form by S. Bochner (1955), initially termed L2,2 -bounded processes, whose structural analysis is found to be very useful both for applications and an advancement of the theory including the stationary and harmonizable classes. The L2,2 -class is first discussed for a motivation as well as initial applications. Definition 1.3.9 Let {Xt , t ∈ I ⊂ R} ⊂ L2 (P ) be a process. It 2,2 is Pncalled L -bounded if forR each simple function f : I → R, f = ai χ(ti ,ti+1 ] , the integral I f (t) dXt , defined as the usual sum τ f = Pi=1 n i=0 ai (Xti+1 − Xti ), satisfies for some C > 0,
R R (48) E | I f (t) dXt |2 ≤ C I |f (t)|2 dt. A standard example of an L2,2 -bounded process is the Brownian Motion with variance parameter σ 2 (= C, here) and independent centered increments so that (48) becomes Z
2 2 E |τ f | = σ |f (t)|2 dt. (49) I

Since a stochastic measure on a ring into a Banach space is a vector measure, it has several types of variations, as detailed by Clarkson and Adams (1933), we recall here two of the most useful ones called the Fr´echet and Vitali concepts that are of immediate use. Thus if B(I) is the Borel σ algebra of an interval I ⊂ R, X is a Banach space with norm k·k, and the mapping Z : B(I) → X is σ-additive (or a vector measure), then consider the variations: n  X kZk(A)= sup k ai Z(Ai )k : |ai | ≤ 1, Ai ∈ B(I), Ai ⊂ A, disjoint i=1

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called the semi-variation (or Fr´echet variation) and another: |Z|(A) = sup

n nX

o kZ(Ai )k : Ai ∈ B, disjoint, Ai ⊂ A

i=1

called the Vitali (or the usual) variation of Z. It can be seen from the above pair of authors work that the |Z|(·) is always σ-additive but often nonfinite while kZk(·) is always finite but only σ-subadditive. If fn : I R→ R is a simple B(I)-measurable function, fRn → f pointwise, R and { I fn dZ, n ≥ 1}R ⊂ X is Cauchy, then lim f dZ = f dZ n→∞ n I I R is well-defined, f 7→ I f dZ is linear, and k I f dZkX ≤ kf k∞ kZk(I) also holds. The vector integrals here are taken in the Dunford-Schwartz sense. However, the properties of Lebesgue’s integration are not always valid. We present the following result to clarify matters, and the last observation, noting the distinction, is due to P. A. Meyer (1985). Proposition 1.3.10 Let X = {Xt , t ∈ I} be an L2,2 -bounded process. Then X induces a vector measure Z : B(I) → L2 (P R) such that the Fr´echet and Vitali variations of Z agree, and τ : f 7→ I f dX can be R expressed as τ (f ) = I f (t)X(t) dt, (a Bochner integral) so that the induced measure Z(·) can be treated as a Lebesgue-Stieltjes set function only if X(·) has finite variation on I in the classical sense. R Proof. The mapping τ : L2 (I, dt) → L2 (P ) defined as τ (f ) = I f dX, is a continuous linear operator (cf. (49)). Since L2 (P ) is reflexive, τ is weakly compact and by the Riesz-Dunford theorem R there is a2 unique 2 vector measure Z : B(I) → L (P ), with τ (f ) = I f dZ, f ∈ L (I, du) and Z is the induced measure of X. Now treat I = (a, b) as an LCA group under addition mod (b − a), and let Iˆ be its dual. So by the Plancherel theorem, we have Z Z
2 2 E |τ f | ≤ C |f | dt = C |fˆ| dµ I

Iˆ

where µ is the dual measure on Iˆ and fˆ = F(f ) is the Fourier transform of f , the µ is also (a constant multiple of) the Lebesgue measure. Hence if T = τ ◦ F−1 (so T (fˆ) = τ (f )), then T is a continuous linear mapping ˆ dt) and there is a vector measure Z˜ : B(I) → L2 (P ), such that on L2 (I, Z ˆ ˜ ˆ dµ). T (f ) = fˆ(x)Z(dx), f ∈ L2 (I, I

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Now using Fubini’s theorem, we get  Z Z ixt ˆ ˜ τ (f ) = T (f ) = e f (t) dt Z(dx) Iˆ I Z  Z ixt ˜ = f (t) e Z(dx) dt I Iˆ  Z  Z = f (t)Y (t)dt = f (t) dXt . I

29

(50)

I

˜ Here Y (t) is the Fourier transform of the vector measure Z(·), and the integral is in Bochner’s sense, so Y (·) is weakly harmonizable. Note that in (50) we cannot write dXt = Y (t) dt if X is possibly a Brownian motion which has no (Lebesgue) derivative anywhere, so that the stochastic integral is not of Lebesgue-Stieltjes type. For the last part, let Z(·) be the induced measure of X, and the integral can be taken in the Stieltjes sense. We now show that Z(·) must then have finite variation. Indeed consider πn : a = t0 < t1 < . . . < tn = b on I = [a, b] and for f : I → R, a bounded Borel mapping Sn is defined by Z n−1 X Sn (f ) = fn dZ = f (ti )[Z(ti+1 ) − Z(ti )] ∈ L2 (P ) ⊂ L1 (P ), I

i=0

Pn−1

where fn = i=0 f (ti )χ[ti ,ti+1 ] (t). Then Sn (f ) → τ (f ) as the partitions πn are refined and supn kSn (f )kX < ∞ for each f ∈ B(I), space of bounded function on I. By the uniform boundedness principle, it follows that supn kSn k = α0 < ∞, and the set {Sn , n ≥ 1} is uniformly bounded. If we put hωn

=

n−1 X


sgn Z(ti+1 ) − Z(ti ) (ω)χ[ti ,ti+1 ] ,

ω ∈ Ω,

i=0

then hωn ∈ B(I, BI ), (bounded Borel), khωn k∞ ≤ 1, and Sn (hωn )

=

n X

|Z(ti+1 ) − Z(ti )|(ω) ≤ kSn k · khωn k∞ ≤ α0 < ∞.

i=0

Integrating this on Ω, one sees that Z : B(I) → L1 (P ) must have finite variation so that Z(·) can be treated as a Lebesgue-Stieltjes measure. 

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Remark 5. It should be noted that L2,2 -boundedness and weak (or K-) stationarity are not the same, although there are some important processes that belong to both. It can be seen that Brownian motion process belongs to both classes. For the following work, it will be convenient to extend L2,2 -bounded to Lρ,p -bounded, where ρ > 0, p > 0 for an additive X : S → Lp (P ) where S is a semi-ring of half-open intervals of Rn . If there is a constant C(= Cρ,p > 0) such that for real S-simple φ on Rn one has, relative to the Lebesgue measure µ: Z

Z

p

φ dX ≤ Cρ,p |φ|ρ dµ. (51)

p

Rn

Rn

If p = ρ R= 2 we have the L2,2 -boundedness already considered. Now τ : φ 7→ Rn φ dX is a bounded linear mapping on simple Rfunctions φ and has an extension to Lp (P ); denoted as τ (φ) or F (φ) = Rn φdX. It will be useful for our analysis to see if the above stochastic integral can be given a vector integral form of the Dunford-Schwartz type so that we can use much of their (by now classical) vector integration arguments. Proposition 1.3.11 Let S be a semi-ring of Rn of half-open intervals and X : S → L2 (P ) be an L2,2 -bounded additive function. Then there 2 is R a vector (or stochastic) measure Z : B → L (P ) such that the F (φ) = φ(t) dX(t) can be represented by another vector integral (as given Rn in Dunford-Schwartz) Z ˆ F (φ) = φ(λ)Z(dλ), (52) Rn

where φˆ is the Fourier transform of φ ∈ L2 (µ), and B above is the Borel σ-algebra of Rn and the integral in (52) is the classical vector integral of Dunford-Schwartz type. Proof. By the L2,2 -boundedness hypothesis and the classical fact that the Fourier transform on L2 (Rn , µ) is an onto isometry, we have Z Z

Z

2

2 2 ˆ φ dX ≤ C1 |φ(t)| dµ(t) = C1 |φ(λ)| dµ(λ),

Rn

2

Rn

Rn

where φˆ = F(φ), the Fourier transform of φ ∈ L2 (µ) and τ (φ) = R ˆ = τ ◦ F−1 (φ) ˆ = τ (φ), is φ dX. Hence T = τ ◦ F−1 with T (φ) Rn

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31

a continuous linear mapping on L2 (µ) → L2 (P ), and it is well-known that τ (·) is representable as Z ˆ = ˆ τ (φ) = T (φ) φ(λ)Z(dλ), φ ∈ L2 (µ) Rn

for a unique vector measure Z : B → L2 (P ), (cf. Dinculearu’s ‘Vector Measures’ (1967), p. 259). This gives (52) as desired.  More detailed and general concepts coming from L2,2 -bounded concept and the basic Khintchine stationarity, to be detailed in the following chapters, we present another useful but fundamental extension, noted by K. Karhunen (1947), and develop its fundamental properties and applications later. Thus if r(s, t) is a (continuous) stationary covariance so that r(s, t) = r˜(s − t) has the Fourier representation with a spectral density f (λ) = F 0 (λ) ≥ 0. Z Z p p i(s−t)λ e dF (λ) = (eisλ f (λ))(eitλ f (λ)) dλ r(s, t) = r˜(s−t) = R

R

which pmay be expressed by Plancherel’s theorem if we express g(a, ·) = eia(·) f (·) ∈ L2 (R, dλ), and hence as: Z g(s, λ)¯ g (t, λ)dλ, s, t ∈ R. (53) r(s, t) = r˜(s − t) = R

Taking this as a motivation we can consider a general class of processes valued in L20 (P ), called of Karhunen type if {Xt , t ∈ T }, Xt ∈ L20 (P ), and if there is a family {g(t, ·), t ∈ T } ⊂ L2 (S, S, ν), with Z ¯ r(s, t) = E(Xs Xt ) = g(s, u)¯ g (t, u) dν(u), s, t ∈ T. (54) S

The class {g(s, ·), s ∈ S} ⊂ L2 (S, S, ν) and the class {Xt , t ∈ T } ⊂ L20 (P ) can be quite general and subsumes the Klintchine (stationary) process. Since by Theorem 1.3.5 above every (abstract) Hilbert space can be realized isometrically as a subspace of L2 (P ) on a probability space (Ω, Σ, P ), the general analysis of Karhunen processes will be useful in our study and its detailed treatment will be included later.

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1.4 Important Remarks on Abstract and Concrete Versions of Hilbert Spaces It is natural that many practical problems are first formulated concretely on a space of square summable sequences and their properties investigated for a basic understanding, and then abstraction. This presented the numerical sequence l2 -spaces, generalized to L2 (µ)-classes of Lebesgue’s, and on noting the key concept and its properties of positive definiteness led J. von Neumann to formulate an abstract space axiomatically, calling it an abstract Hilbert space, which helped an enormous growth of the subject, both in theory and applications. On the other hand, Theorem 1.3.5 above shows that an abstract Hilbert space can always be realizable concretely as a (sub-) space of L2 (P ) on some probability triple (Ω, Σ, P ). Thus this dichotomy is what makes our subject and the treatment so essential and useful for a study. An abstract version presents an overview of the subject suggesting newer and far-reaching aspects of the problems to be studied, while the concrete version leads us to detail the intricate aspects of the underlying structures, and thus both points are essential for us. The concrete and abstract idea mix was originated by Kolmogorov (1941) himself to indicate how useful and natural it is to develop the subject. Thus if Xt ∈ L20 (P ), with means zero and ¯ t ) = r(s + h, t + h), s, h ∈ R, stationary covariance r(s, t) = E(Xs X then Ut : Xs 7→ Xs+t , defines a linear operator on the closed subspace L = sp ¯ {Xt , t ∈ R} ⊂ L20 (P ), the Hilbert space of centered square integrable random variables Xt , so that the shift mapping Ut : Xs 7→ Xs+t is well-defined on the linear space of {Xt , t ∈ R} ⊂ L20 (P ), satisfying kUt Xs k22 = kXs+t k22 = r(s + t, s + t) = r(s, s) = kXs k22 , so that kUt Xs k2 = kXs k2 . A similar computation shows: (Ut Xs , Xh ) = (Xs+t , Xh ) = r(s + t, h) = r(s, h − t) = (Xs , Xh−t ) = (Xt , U−t Xh ), h, s ∈ R,

(55)

so that the linear operator Ut satisfies Ut∗ = U−t , where Ut∗ is the adjoint of Ut . Then (55) implies that {Ut , t ∈ R}, defined on L20 (P ), is a weakly continuous unitary group of operators and Xt = Ut X0 , so that using the spectral representation of the Ut -group one has Z  Z itλ Xt = Ut X0 = e dEλ X0 = eitλ dZ(λ), (56) R

R

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33

where Z : A 7→ EA X0 , A ∈ B(R), the Borel σ-algebra, and the integral is well-defined. Here Z(·) is orthogonally valued when the Xt -process is stationary in which case (Z(A), Z(B)) = α(A ∩ B) for a finite positive measure α : B(R) → R+ , called the spectral distribution of the process. In general β(A, B) = (Z(A), Z(B)) defines just a bimeasure and β on B(R) ⊗ B(R) need not have finite variation and one has to turn to bimeasure integration theory of M. Morse and W. Transue (briefly seen above but) for a detailed analysis of the subject; that study will start in the next chapters under the more general class called weakly harmonizable family. The bimeasure β : (A, B) 7→ (Z(A), Z(B)) which is thus σ-additive in A and B separately for disjoint families An and Bn , and β(·, ·) can have finite Vitali or only finite Fr´echet variation, as given in Definition 1.2.2 above. The importance of these concepts is that we can enlarge the study of second order classes from the Khintchine stationary family to much larger sets of classes of interest in filtering, sampling and other applications. The following is a useful and motivational illustration. Theorem 1.4.1 Let {Xt , t ∈ T } ⊂ L20 (P ) be a random field, centered, and whose covariance r(s, t) is representable as: Z ¯t) = r(s, t) = E(Xs X g(s, λ)g(t, λ) dσ(λ), s, t ∈ T, (57) S

for some {g(s, ·), s ∈ T } ⊂ L2 (S, S, σ(·)). Then there exists {Bt , t ∈ T } a class of bounded commuting family of operators on L20 (P ) such that one has Z (58) Xt = Ut Xt0 = g(t, λ) dZXt0 (λ), t ∈ T, S

for a unique orthogonally valued random measure ZXt0 : S → L20 (P ), such that E(ZXt0 (A)Z¯Xt0 (B)) = α(A ∩ B), A, B ∈ S, the vector integral in (58) being in the usual Dunford-Schwartz sense. In this result, one can have if T = R+ , S = Rn and Xt0 = X0 then the family {Ut , t ≥ 0} can be a closed densely defined set of linear operators, as noted by Getoor (1956). The function g of (58) is called the kernel representing the process. If it belongs to some larger classes, the problem leads to some other generalized classes of interest which may be indicated here. An extension of stationarity is a shift VsP : Xt 7→ Xt+s and it to be Pn linear, one must have j=1 aj Xtj = 0 ⇒ nj=1 aj Xt+s = 0 for n ≥

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P  Pn n 1 and kVs a X j=1 j tj k2 ≤ Ck j=1 aj Xtj k2 for n ≥ 2 for some C > 0. As a shift one should also have Vs1 Vs2 = Vs1 +s2 , s1 , s2 ≥ 0, with V0 = identity. If Vs = τ s for a bounded linear τ : L → L and Yi ∈ L ⊂ L2 (P ), linear subspaces, one must have for Y0 , Y1 , . . . , Yt in L, either p X

n

m

(τ Ym , τ Yn ) =

m,n=0

p X

(Yn+m , Ym+n ) ≥ 0,

(59)

m,n=0

or better p X

p X
(τ n Ym , τ m Ym ) . τ n+1 Yn , τ m+1 Yn ≤ C0

(60)

m,n=0

m,n=0

Such τ is called a subnormal operator and C0 = kτ 2 k. It was shown by J. Bram (1955) that this operator τ can be extended to an inclu˜ ⊃ L on which τ and its adjoint τ ∗ commute sive Hilbert space L (i.e. τ τ ∗ = τ ∗ τ ) and one can take the probability space (Ω, Σ, P ) rich ˜ ⊂ L2 (P ). Then under (60), τ s = Vs , s > 0, the constants enough so L 0 Cs replace C0 in (60) and {Vs , s ≥ 0} will be a semi-group of normal op˜ ⊂ L2 (P ), giving a nonstationary process generalization of erators on L 0 the unitary group of the stationary class considered earlier, to include the Karhunen class when the semi-group is weakly continuous and without further restrictions. Then this process {Xt = Vt X0 , t ≥ 0} ⊂ L20 (P ), though nonstationary, can be detailed for an integral representation, that includes the stationary processes, since {Vt , t ≥ 0} forms a weakly continuous normal semi-group. We can now present the general representation of the process as: Theorem 1.4.2 Let {Xt , t ≥ 0} ⊂ L20 (P ) be a (covariance) continuous process admitting a right translation operator τs : Xt 7→ Xt+s which is a bounded linear subnormal mapping, on the span of the process and hence has a normal extension to L20 (P ), using an enlargement of the probability space if necessary after which (60) holds. If the class {Vs , s ≥ 0} forms a strongly continuous semi-group then {Xt = Vt X0 , t ≥ 0} is a Karhunen process which admits a unitary spectral representation whose covariance has the integral formulation, on a Borel set ∆ ⊂ R2 , as ZZ ¯ r(s, t) = esλ+tλ dβ(λ) (61) ∆

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35

where β(A) = E(|ZX0 (A)|2 ), β : B(∆) → R+ . For a normal semigroup one has the decomposition τt = Rt Ut where {Rt , t ≥ 0} is a positive self-adjoint semi-group commuting with the unitary group {Ut , t ∈ R} given by Xt = Vt X0 = Rt Ut X0 , B(∆) = B|∆, restriction. Proof. Since by (60), the process {Xt = Vt X0 , t ≥ 0} in L2 (P ) and the family {Vt , t ≥ 0} forms a weakly continuous semi-group of bounded normal operators on L20 (P ) one can invoke the spectral theorem for such operators (cf. Hille and Phillips (1957), Theorem 22.4.2), to obtain the representation: Z Z Z Z tλ Xt = Vt X0 = e E(dλ)X0 = etλ dZX0 (λ), (62) ∆

∆

where ZX0 : B(∆) → L20 (P ) is an orthogonally valued measure. This represents a Karhunen field with covariance given by Z Z 0 esλ−tλ dβ(λ) r(s, t) = (63) ∆ 2

where β(A) = E(|ZX0 (A)| ), so that (63) is similar to the weakly (or K-) stationary case. This can be refined using the fact that Vt = Rt Ut = Ut Rt with {Ut , t ∈ R} being a unitary group where U−t = Ut∗ , t > 0, and {Rt , t > 0} a positive self-adjoint semi-group of operators and Rt , Ut commute. The semi-group property of the Vt yields the following: Vt+s = Rt+s Ut+s = Vt Vs = Rt Rs Ut Us = Rt+s Ut+s .

(64)

It follows that {Rt , t ≥ 0} and {Ut , t ≥ 0} are both strongly continuous semi-groups of operators on L20 (P ), and the Rt -family is positive in addition. This implies, along with Ut , Rt commuting, the following: r(s, t) = (Xs , Xt ) = (Vs X0 , Vt X0 ) = (Rs Us X0 , Rt Ut X0 ) = (Ut X0 , Rs Rt Ut X0 ) = (Ut∗ Us X0 , Rs+t X0 ) = (Us−t X0 , Rs+t X0 ) = (Rs+t X0 , Us+t X0 ), s + t > 0, s − t ∈ R ¯ ) = r˜(s − t, s + t), (say). ¯ s−t , X = E(X (65) s+t Thus r˜ is positive definite and defined on the cone {(s, t) : |t| ≤ s} and   s+t s−t r˜(s, t) = r , , 0 < t < s. (66) 2 2

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This shows that r˜ is positive definite on convex sets and its integral representation is not a direct consequence of the Bochner theorem. Such functions were discussed by Devinatz (1954) which in our case leads to the following, with (63), for (s, t) ∈ C = {(s, t) ∈ R+ × R : |t| ≤ 2s }: n X

ai a ¯j r˜(si + sj , ti − tj ) =

i,j=1

=

n X

ai a ¯j Rsi +sj X0 , Uti −tj X0




i,j=1 n X


ai a ¯j Rsi Uti X0 , Rsj Utj X0 ,

ij=1

since Vt = Rt Ut etc. =

n X

(ai Rsi Usi X0 , ai Rsi Usi X0 ) ≥ 0.

i=1

(67) The key converse implication is a consequence of Devinatz’s theorem. With the commutativity of Rs and Us and their spectral representations yield for (62) the following: Z Z 0 t Xt = Vt X0 = Rt Ut X0 = λ dE1λ eiλ dEλ0 X0 (Rt = R1t ) R+ R Z 0 et(log λ+λ ) dE˜λλ0 X0 (E˜λλ; = E1λ Eλ0 ), (68) = [Re λ>0]

where the commutativity of E1λ and Eλ0 is used. This alternative method of representation (62) for normal operators is valid for the closed densely defined case as well.  Remark 6. This result adapted from the author (cf. Rao (2008)) is extendable also for semi-groups {τt , t ≥ 0} defining Xt = τt X0 , t ≥ 0, which are closed and densely defined (see Getoor (1957)). 1.5 Complements and Exercises 1. This problem shows that a positive definite bimeasure has finite Fr´echet but not Vitali variation. It is a modification of one given in Clarkson and Adams (1933) showing the difference from Lebesgue’s theory which cannot be used here in the harmonizable treatments in

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37

general. Consider an n × 2n−1 matrix An = (anij , 1 ≤ i ≤ n, 1 ≤ j ≤ 2n−1 , aij = ±1) for n = 2m + 1, m ≥ 0 with all combinations of ±1, except that annj = 1, j = 1, . . . , 2n−1 . Define kAn k = P P n−1 P2n−1 n sup{| ni=1 2i=1 j=1 aij bi cj | : bi ∈ C, |bi | ≤ 1, |cj | ≤ 1}. Set Pn P2n−1 n |An | = i=1 j=1 |aij | = n2n−1 . Verify next (after detailed computation) that kAn k/|An | = rn → 0 as n → ∞ with x = P2m + 1. Now consider the bimeasure β, defined by β(A, B) = { βt (P ) : P ∈ A × B}, where βk (P ) = (−1)k aki /tk 2tk −1 if P = Pijtk , and 0 otherwise. Then βk (·, ·) is well-defined on B(R). |βk |(R, R) = 1, and β(·, ·) is a bimeasure and has finite Fr´echet variation. However Pk |β|(R, R) ≥ j=1 |βj |(R, R) = k → ∞, so that its Vitali variations infinite. [The details are not simple, and are in Chang and Rao (1986). This is a somewhat modified version of the original Clarkson-Adams work.] 2. Let G be an LCA group, ξ be a random variable on (Ω, Σ, P ) with mean zero and E(ξ 2 ) = 1, and Xt = ξf (t), t ∈ G. Verify that {Xt , t ∈ G} need not be harmonizable even if f : G → C is continuous and f (∞) = 0. If g(x) = e−t·f (x) , t > 0, f (x+y) = f (x)+f (y), verify that g : G → R is positive definite and nonmeasurable. 3. This problem shows the role of V -boundedness in our analysis. Let Xt = f (t)ξ where f : G → C is a mapping and ξ is a random variable with mean zero and unit variance. Then the process {Xt , t ∈ G} is not necessarily harmonizable even if f is continuous and vanishes at infinity. It is V -bounded only if f is the Fourier transform of an integrable function (i.e., f = gˆ). 4. This problem presents an estimation of a parameter, extending a result of Balakrishnan’s (1959) from weekly stationary errors to the (weakly) harmonizable case. So let Yt = αmt + Xt , t ∈ R, and α ∈ R as signal and Xt the noise having mean zero and a harmonizable covariance r with a spectral bimeasure F . The previous study was when the noise was taken stationary. The problem is to estimate α, as a weighted linear unbiased estimator on observing the Yt -process on an interval, [−B, B], using a weight funcRB tion p(·) of bounded variation, thus by α ˆ B = −B Yt dp(t), so that E(ˆ αB ) = α, the unbiasedness condition. Typically one should get the “best” lower-bound for the variance of this ‘unbiased estimator’ α ˆ B since the actual distribution function of α ˆ B is quite difficult, to

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study its behavior. The lower bound analysis is the next best thing and is thus of interest. [See Rao (1961) on the problems and its philosophy of use in many such studies.] 5. This result presents how classes of L´evy-Schoenberg kernels can be generated, supplementing Proposition 1.3.8. As before let our G be a separable topological group and K ⊂ G a closed subgroup. If f : G/K → R is a L´evy-Schoenberg kernel as defined in Proposition 1.3.8, then the kernel defined by: r(a, b) = f (a, a) + f (b, b) − 2f (a, b), a, b ∈ G/K, satisfies r(a, b) = r(b, a) : r(a, a) = 0, and r(xa, xb) = r(a, b), for a, b ∈ G/K and x ∈ G, and the kernel θt : (a, b) → exp{−t·r(a, b)} is positive definite for t ≥ 0. On the other hand if r(a, b) ∈ R, where a, b ∈ G/K and satisfying the preceding conditions then for any 0 ∈ G/K the function f : (a, b) 7→ 21 [r(a, 0) + r(b, 0) − r(a, b)], defines a L´evy-Schoenberg kernel. Thus there are plenty of L´evySchoenberg kernels on infinite dimensional (very) general topological spaces, and hence the L´evy-BM classes are large. 6. This problem presents a condition for a process, or random field {Xt , t ∈ G} ⊂ Lp (P ), 1 ≤ p ≤ 2, to have a Fourier integral representation, to be termed a strictly harmonizable random field. Thus {Xt , t ∈ G} is strictly harmonizable if it satisfies the V -boundedness condition (G is a compact abelian group)

n

( n )

X

X

˜ ai ∈ C ai Xi ≤ C sup ai χi : χi ∈ G,

i=1

i=1

T

˜ for some 0 < C < ∞, and χi are characters Rof G i.e., χi ∈ G int (dual of G). Then the representation that Xn = G e dZ(t), holds ˆ = (π, π]). (See Hosoya (1982).) if G = Z, the integer group (so G 7. The concept of ‘weak harmonizability’ can be extended for the Lp (P ), p ≥ 1, spaces (and even Orlicz spaces). Thus a sequence {Xn , n ∈ Z} is Lp -harmonizable if it admits a representation: Z Xn = einu dµ(u), n ∈ Z, I ⊂ R, I

where µ : B(I) → Lp (P ) is a vector measure and B(I) is a Borel σ-algebra.

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1.6 Bibliographical Notes

39

Here µ is often termed a spectral stochastic measure. The series {Xn , n = 0, ±1, ±2, . . .} is called an Lp -harmonizable process. I. Kluv´anek (1967) has shown that the following precise statement holds: [Let X be a Banach space and G an LCA group, X : G → X is a mapping. Then X is a generalized FourierRtransform of a regular ˆ → X, so X(g) = ˆ hg, siv(ds), g ∈ G, vector measure v : B(G) G ˆ is the dual group of the LCA group G, iff X(·) is weakly where G R continuous and V -bounded, in that the set { G f (t)X(t)dt : kfˆk∞ ≤ 1, f ∈ L1 (R)} as a subset of X is relatively weakly compact in X.] (For a detailed background of the work, see the another’s somewhat long paper (Rao (1982)), and for a good background and application, see Dehay (1991) discussing products of pairs of harmonizable processes.) 1.6 Bibliographical Notes The work of this volume starts with analyses of the second order processes as the ideas really give a familiar feeling for many others, motivate a study and explore new areas in this research. The basis for a large part of our analysis is the positive definiteness property appearing in different forms. Its characterization under minimal conditions, seen as a Fourier transform of a suitable measure obtained by S. Bochner, has aided enormous growth of stochastic processes, starting with A. I. Khintchine’s analysis and his characterization of (weakly) stationary classes, detailing the covariance structure. Many developments were based on the centered second order processes. The mean functions, if not taken to be zero (for ‘convenience’) is a problem that depends on some serious (Fourier) analysis was shown by A. V. Balakrishnan (1959), and it prompted further analysis that is included in this chapter. The original positive definiteness property and its characterization leading to harmonizable classes, first indicated briefly by Lo´eve and Rozanov, has been extended much further again with the key concept of V -boundedness introduced by S. Bochner himself which gave the weak and strong harmonizabilities of the covariances of second order processes, and the consequent work on means of the processes that were characterized. All this is included here.

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Some of the deeper analysis of harmonizable classes, in contrast with the stationary processes demands some advanced but interesting techniques, to be used and extended in the following chapters. Here a use of M. Morse and W. Transue analysis of (multiple) integrals which are weaker than Lebesgue’s are needed and their use will continue in these chapters. The ‘positive definite analysis’ leads to L´evy’s Brownian motions opening up new aspects of stochastic analysis, which takes to characterizations of abstract Hilbert spaces isometrically as certain Lebesgue spaces L2 (P ) on a Gaussian measure space, connecting and extending aspects of abstract Hilbert space analysis with concrete L2 (P )-type theory. This combination is benefitting in both areas. In the following chapters some specializations of the analysis as well as extensions, bringing in several ideas and applications of abstract harmonic analysis will be made. Since the Fourier analysis plays such a key role here, it may be of interest to record the following somewhat unusual experience for others to read and reflect. This is about knowing and retelling the actual evolution of our subject. For this I have noticed an AMS publication on the ‘History of Mathematics’ by a Greek distinguished mathematician, Nicolas K. Artemiadis, who obtained his Doctor d’Etat (like our Ph.D) in Paris and taught mathematics as a Professor at the University of Wisconsin and Southern Illinois University, and later returned to his native Greece and wrote this scholarly book in Greek language which later was translated and published in English by the AMS in 2004. I bought the volume and found among others an interesting account on Fourier and his problems on publishing it in the French Academy for many years (nearly 20 years) delayed before that classic work was finally published. [It seems that the merit of the ideas, was judged and appreciated by the Stalwarts, Lagrange, Laplace, and Legendre, but the work was not accepted for not being mathematically rigorous! His persistence and final publication in early 1820s saved us the Fourier analysis.] This historical account, published in Greek language originally quoted a passage from M. Klein’s book on Math History, published by a British publisher in England, was paid for and given in the Greek version, but the original passage was given in the AMS version. Somebody in the U.S. has thought it was “plagiarization” and made the AMS to withdraw the book, although the author produced letters of permission and purchase of a passage for the Greek edition! When I saw

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the note on withdrawing the English version by the AMS, to avoid court litigation, I have written a letter in 2005, to the AMS notices in which the book’s withdrawal was noted asking them to have the book made available, and mentioning that any historical account reported by various writers naturally will be the same and could not be “plagiarized”, and after all the Klein account could not be the original one about an ancient instance! (Cf. AMS Notices, 52 (2005), p. 1174.) After seeing my letter in the notices, Prof. Artemiadis wrote me a letter, dated October 20, 2005, stating “your letter to the NOTICES was for me an unexpected ray of life (literally). I cannot find strong enough words to express my deepest and most profound gratitude.” It was nice to see a scholar, who was much disturbed for withdrawing his cherished work’s circulation, to feel somewhat relieved. In a later issue of the Notices (a couple of years later), I have read the sad news that Prof. N. K. Artemiadis has passed away. It was some relief to think that the improper understanding of his work is slightly abated. The important role that the Fourier analysis, in its modern forms, playing in the following chapters will be self-evident. The random processes {Xt , t ∈ T } where T ⊂ Rn , n ≥ 1, subgroup, and T an LCA group will be seen as central to much of what follows.

b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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As seen in Chapter 1, stochastic analysis is deepened as well as enriched, both in content and form for theory as well as applications, by a use of the methods of harmonic analysis and related integration methods. We start with an analysis of second order processes that consider together with the first and second moments, as shown in Chapter 1, implying that the usual assumption of centering at their means is somewhat restricting the general structural analysis of the (second order) processes. Also included here are the integral determination of all local functionals, and also a probability solution of the classical Riemann hypothesis. Thus the chapter contains some key aspects of the subject. 2.1 Morse-Transue Integration Method and Stochastic Analysis We start with necessary integrability properties of second order processes, their basic structures and a general analysis for classes motivated by (and centering around) harmonizable classes, as seen in Chapter 1. It depends on explaining, detailing and employing seriously, the MorseTransue (or MT-) integration method which is weaker than the classical Lebesgue’s version, but is important for our work. This was briefly considered in Chapter 1. A few related results and some applications will be discussed here for quick reuse. Thus let Si be locally compact, K(Si ), the space of continuous compactly based complex functions, i = 1, 2 and B : K(S1 ) × K(S2 ) → C be a bilinear mapping, called a complex (or C) bimeasure on S1 ×S2 if B(·, v) : K(S1 ) → C and B(u, ·) : K(S2 ) → C are relatively bounded linear functionals for each u ∈ K(S1 ) and v ∈ K(S2 ). Here B(·, v) is said to be relatively 43

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bounded if for each compact K ⊂ S1 and K(K) ⊂ K(S1 ) is considered, then B(·, v) : K(K) → C is bounded. Similarly Λ(u, ·), u ∈ K(S1 ), has an analogous meaning for B(u, ·). This is designed to use the MT-work. Similarly, we associate a set-theoretical function β : S1 ×S2 → C, where Si is σ-Borel algebra of Si , and call it a bimeasure if β(·, F ), β(E, ·) are σ-additive for each E ∈ S1 and F ∈ S2 . Their Fr´echet and Vitali variations are defined and denoted by kβk(·, ·) and |β|(·, ·), where (Vitali) ( n ) X |β|(S1 , S2 ) = sup |β(Ai , Bj )| : Ai ∈ Si , Bj ∈ Sj , (1) i,j=1

and (for the Fr´echet), kβk(S , S2 ) = o n 1P n ¯j β(Ai , Bj ) : Ai ∈ Si , Bj ∈ Sj , |ai | ≤ 1, |aj | ≤ 1 . sup i,j=1 ai a (2) Then β has finite Vitali variation if |β|(S1 , S2 ) < ∞, and has finite Fr´echet variation if kβk(S1 , S2 ) < ∞, so that kβk(S1 , S2 ) ≤ |β|(S1 , S2 ). There will actually be strict inequality if the Vitali variation is infinite. For the following analysis, it will be useful to include a few more details of the Morse-Transue integration as it extends the Lebesgue integral by weakening its absolute integration condition. The concept of MT-integration is set down as follows. Definition 2.1.1 Let fi : Si → C, i = 1, 2 be Baire functions, and Λ be a C bimeasure on S1 × S2 . Then (f1 , f2 ) is Morse-Transue (or MT)integrable if (i) fi is Λ(·, g2 ) integrable and f2 is Λ(g1 , ·) integrable for gi ∈ K(Si ), i = 1, 2, the linear functionals f1 → Λ(f1 , ·)(g2 ) and f2 → Λ(·, f2 )(g1 ) are Radon measures, and that the integrals satisfy Λ(f1 , ·)(g2 ) = Λ(·, f2 )(g1 ) so that the common value is denoted by Λ(f1 , g2 ) = Λ(g1 , f2 ): Z Z Λ(f, g) = (M T ) (f, g)dΛ = Λ(f, ·)(g) = Λ(·, g)(f ). (3) S1

S2

This formulation is a slightly extended version of the original MTdefinition, applicable to measurable functions (not only the continuous ones), as shown (extended) by Erik Thomas (1970), and we use this extended form in the following applications as needed.

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The desired result for us is given as follows: Theorem 2.1.2 Let Si be a locally compact space with Si as its Borel σ-algebra and Λ : S1 × S2 → C be a bimeasure. Let fi ∈ K(Si ) be such that (f1 , f2 ) is Λ-integrable R R as in (3) so that the pair (f1 , f2 ) is Λintegrable and (f1 , f2 ) 7→ S1 S2 (f1 , f2 )dΛ defines a C-bimeasure on K(S1 ) × K(S2 ). Also any Λ-integrable pair (f1 , f2 ) is MT-integrable, and both integrals agree. Proof. We sketch the argument for comprehension of the MT-method of the (weaker) integration in which one utilizes some results (detailing the MT-method by Ylinen (1978)) for understanding certain points, which help shortening the usual detailed analysis. Thus let Λ : S1 × S2 → C be a measure and let (f, g) be Λintegrable. Note that the concept of Λ-integrability implies that each pair of bounded Baire functions (u, v) is so integrable and |Λ(f, g)| ≤ kΛk|(S1 , S2 )|kf k∞ kgk∞ , the latter symbols on K(Si ) are the ‘sup’ norms, so that Λ(·, ·) is a bounded C-bimeasure and that Λ(·, g), Λ(f, ·) define bounded Radon measures on B by the usual measure theory analysis. Thus one has: Z Z g(y)Λf (S, dy), (4) f (x)Λg (dx, S) = Λ(f, ·)(g) = Λ(·, g)(f ) = S

S

using the known properties of the MT-integrals (Morse-Transue, 1956). From this and the standard arguments, (4) gives since (f, g) is Λintegrable: Z Z Z g(y)Λ(f, dy) (f, g)dΛ = S

S

S

= Λ(f, ·)(g).

(5)

Analogous argument shows that Λ(·, g)(f ) and Λ(f, ·)(g) exist and are equal, i.e., Λ(f, ·)(g) = Λ(·, g)(f ) so that (f, g) is MT-integrable. Regarding the last statement, let Λ be a bounded C-bimeasure on K(S) × K(S), then by the M T -theory each pair of bounded Baire functions (f, g) is MT-integrable, since Λ is a bounded C-bimeasure. By the theory of these authors, β : B(S) × B(S) → C defined by β(E, F ) = Λ(χE , χF ) will be seen R R to be a complex bimeasure so that 0 it is bounded. If Λ : (h, k) 7→ S S (h, k)dβ, is defined for a pair of bounded Baire (h, k), then by the earlier part above Λ0 is a C-bimeasure

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agreeing with Λ on (χE , χF ) and so Λ0 (u, v) = Λ(u, v) for u, v ∈ K(S). It then follows by the extension method, that Λ = Λ0 as asserted.  Remark 7. As Ylinen (1978) shows by example that the hypothesis of boundedness of Λ after extension to K(S1 ) × K(S2 ) is essential to employ the result for more (not always continuous) functions. Thus the MT-integral is not absolute, contrasting it with that of Lebesgues. We include an example to this effect in the Complements below (cf. Ex. 1). For our further analysis, a restriction of the MT-integral is desirable. That is called a strict bimeasure integral (or strict MT-integral). It is defined as follows. Definition 2.1.3 Let (Ωi , Σi ), i = 1, 2 be a measurable pair and fi : Ωi → C be measurable and β : Σ1 × Σ2 → C, a bimeasure. Then the pair (f1 , f2 ) is called strictly β-integrable if we have: (a)f1 is β(·, F ) and R f2 is β(E, ·) integrable (Lebesgue), R so that f (ω1 )β(dω1 , B), and βf2 (A, F ) = F f2 (ω2 ), f1 β(E, B) = E 1 β(A, dω2 ), for A ∈ Σ1 , B ∈ Σ2 are complex integrals for each E ∈ Σ1 , F ∈ Σ2 , and (b)f1 is βf2R(·, F ) and f2 is f1 β(E, ·)-integrable for each E ∈ Σ1 , F ∈ R Σ2 and E f1 (ω1 )βf2 (dω1 , F ) = F f2 (ω2 )f1 β(E, dω2 ) holds. When these conditions obtain, the double integral is given by: Z Z ∗ Z Z Z (f1 , f2 )dβ = (χE f1 , χF f2 ) dβ = f1 (ω1 )βf2 (dω1 , F ). E

F

S1

S2

E

(6)

It can be verified that each strictly integrable pair for β, is naturally βintegrable as defined before, both have the same values and that the βintegrability results in Ylinen (1978) are valid for our strict β-integrals. The following change of variables formula can be established with some slightly modified (usual) arguments. Proposition 2.1.4 Let (f, g) be strictly β-integrable on (Ωi , Σi ), i = R1, 2 Rfor a bimeasure β : Σ1 × Σ2 → C, and let µ : (A, B) → (χA f, χB f )dβ for A ∈ Σ1 , B ∈ Σ2 . If h : Ω1 → C is bounded Ω1 Ω2 measurable and k : Ω2 → C is similarly obtained, then writing µ = µf,g , we get

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Z Z

∗

Z Z

∗

A

B

A ∈ Σ1 , B ∈ Σ2 .

(h, k)dµ,

(f h, gk)dβ = A

47

(7)

B

Proof. Since µ is evidently a bimeasure on Σ1 × Σ2 , define the linear functional on Σ1 × Σ2 → ca(Ω2 , Σ2 ) as: Z 0 k(ω2 )λ(dω2 ), F ∈ Σ2 , λ ∈ ca(Ω2 , Σ2 ), kF (λ) = F

k(·) being that given in the statement. But then there exist step functions k n → k pointwise, |kn | ≤ |k|, and we denote a typical kn as kn = P n n n j=1 bj χBj . Then for each E ∈ Σ1 , one has Z βgk (E, F ) = (gk)(ω2 )β(E, dω2 ), by definition, F Z (gkn )(ω2 )β(E, dω2 ), by dominated convergence, = lim n→∞

= lim

n→∞

F n X

bnj βg (E, F ∩ Bjn )

j=1

Z

Z n→∞

k(ω2 )βg (E, dω2 )

kn (ω2 )βg (E, dω2 ) =

= lim

F

F

= kF0 (βg (E, ·)). Now the standard Dunford-Schwartz theory gives:  Z Z 0 (gk)(ω2 )f β(E, dω2 ) = kF g(ω2 )f β(·, dω2 ) F E Z  0 = kF f (ω1 )βg (dω1 , ·) E Z = f (ω1 )βgk (dω1 , F ). E

Thus (f, gk) is strictly β-integrable and moreover, Z Z ∗ Z 0 (g, k)dβ = kF (µ(E, ·)) = k(ω2 )µ(E, dω2 ) = µ ˜E (E, F ). E

F

F

In a similar way, one shows that (f h, gk) is strictly β-integrable and deduces

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Z Z

∗

E

F

∗

h(ω1 )˜ µE (dω1 , F ) =

(f h, gk)dβ = E

Z Z

Z

(h, k)dµ, E

F

which completes the proof.  The corresponding statement for C-bimeasure and MT-integrals is: Proposition 2.1.5 Let (Si , Bi ), i = 1, 2 be σ-compact Borel spaces, and f, g be B1 , B2 -measurable scalar functions. If Λ is a bounded Cbimeasure on B1 × B2 , then (f h, gk) is MT-integrable for Λ and for all bounded scalar functions h, k on S1 , S2 iff (χA f, χB g) is MT-integrable for all A ∈ B1 and B ∈ B2 . The proof in one direction is immediate, and the converse is established as in the preceding proposition, with an interplay of (Radon) measures and linear functionals given in Bourbaki, and the details are left to the reader. The following consequence is often useful for applications. Corollary 2.1.6 Let (Si , Bi ), i = 1, 2 be measurable spaces and β : B1 × B2 → C be a bimeasure. Then we obtain: (i) A pair (f, g) of complex functions is β-integrable strictly if and only if the pair (|f |, |g|) is so integrable. (ii) If fi : Si → C, i = 1, 2 are measurable and |f1 | ≤ |f |, |f2 | ≤ |g| and (f, g) is strictly β-integrable, then so is (f1 , f2 ). (iii) If (fm , gn ) is a sequence of measurable functions, fn → f˜, gn → g˜ pointwise and |fn | ≤ |f |, |gn | ≤ |g|, (f, g) as in (i) then (f˜, g˜) is also strictly β-integrable and one has Z Z ∗ Z Z ∗ ˜ (fn , gm )dβ, E ∈ B1 , F ∈ B2 . (f , g˜)dβ = m→∞ lim E

F

n→∞

E

F

The result can be established with the standard argument familiar in Real Analysis, and need not be repeated. If (χA f, χB g) is MT-integrable relative to a bimeasure, the result holds for them also. Hereafter when the result holds for strict as well as the general case of a bimeasure, the distinction will be omitted in the statements. The class of second order processes that is covered by our work includes the harmonizable families, but the methods are adapted to some general classes introduced first by Karhunen (1947) and independently by Lo´eve (1948). Soon after, H. Cram´er (1951) has given a formulation

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that includes both, amplifying the structure. We analyze these classes to clarify and enhance the application-potential of the classes, all of which are of second order. The existence of these classes is an immediate consequence of the fundamental Kolmogorov-Bochner projective limit theorem discussed and detailed in the initial part of our trilogy (cf. Rao (1995) Chapter I). Also it is noted that each positive definite continuous function r : G × G → C on an LCA group G is the covariance function of a (even a Gaussian) centered process (= random field) with this r as its covariance function. Then for such r we may also suppose the existence of a positive definite function F : G × G → C of locally finite Fr´echet variation kF k(K) < ∞, in that on each compact product set K ×K ⊂ G×G, for each set of (scalar) Baire functions {g(s, ·), s ∈ G} one has (G being an LCA group as always) Z Z g(s, λ)¯ g (s, λ0 )F (dλ, dλ0 ) < ∞, s ∈ G. (8) G

G

For such a family we can introduce the general Cram´er and Karhunen classes of processes that subsume the stationary and then the harmonizable classes enhancing the study of our stochastic analysis in many ways. The preceding account on bimeasure integration (of MT-type) is utilized crucially in obtaining integral representations of harmonizable as well as Karhunen and Cram´er classes of processes which form a substantial part of the second order (continuous parameter and other) processes of great interest in applications. We have already introduced the Karhunen class in second order processes in Chapter 1 (Section 3), as the second order class with means ¯ t ), Xt ∈ L2 (P ), given by zero, and covariance r(·, ·) : (s, t) 7→ E(Xs X 0 Z g(s, u)¯ g (t, u)dν(u), s, t ∈ T, (9) r(s, t) = S

where {g(s, ·), s ∈ S} ⊂ L2 (S, S, ν) for some measure ν on (S, S). Thus the class of second order processes {Xs , s ∈ S} ⊂ L20 (P ) whose covariance is representable as (9) is termed the Karhunen class which properly includes the (centered) Khintchine or weak stationary class. But we have also seen that the (weakly) stationary class is included properly in the (weakly) harmonizable class. Similarly, Cram´er (1951) has extended the Karhunen class in the following (natural) way.

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Definition 2.1.7 Let {Xt , t ∈ R} ⊂ L20 (P ) be a process with covari¯ t ) that is representable as, ance r : (s, t) 7→ E(Xs X Z Z g(s, u)¯ g (t, v)β(du, dv), s, t ∈ R, (10) r(s, t) = R

R

where β is a positive definite set function (bimeasure) of bounded Vitali variation on R2 , where the integral in (10) is in the standard LebesgueStieltjes sense in the complex plane. Note that if g(s, u) = eisu in (10), this reduces to the Lo`eve definition of a (in the present terminology) strongly hormonizable process. Clearly this goes over to the stationary case if g(s, u) = eisu and β(·, ·) concentrates on the line u = v in R2 . It also extends the Karhunen class, if g(·, ·) is more general but β(·, ·) concentrates on u = v. For a clear understanding of the extension involved, we first characterize the (all important) Karhunen fields. Theorem 2.1.8 Let {Xt = Nt X0 : t ∈ T }, X0 ∈ L20 (P ), be a process generated by a family of bounded commuting set of normal operators on L20 (P ), of a centered random variable X0 : Ω → R, N0 = id., and T to be a separable topological space. Then there exist a continuous function b : T × [0, 1] → R, and an orthogonally valued σ-additive Z : B(I) → L20 (P ), I = [0, 1], such that we have the (vector) integral representation: Z Xt = b(t, λ)dZ(λ), t ∈ T, (11) I

¯ t ) is given by and its covariance function r : (s, t) 7→ E(Xs X Z r(s, t) = b(s, λ)¯b(t, λ)dα(λ), s, t ∈ T,

(12)

I

where the measure α : B(I) → R+ satisfies α(A) = kZ(A)k22 for all Borel sets A ⊂ I; B(I) being the Borel σ-algebra of I. Proof. The original argument depended on a classical theorem due to J. von Neuman, but we give an elegant short modern proof due to B. R. Gelbaum (1964) of interest. Let a be the set of continuous linear operators, made into a closed (sub) algebra of (continuous) linear mappings from B(L22 (P )) where a is determined by the set {I, Ns∗ , Nt , s, t ∈ T } under the uniform

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(or operator) norm. Since T is a separable set, the family a is a separable Banach*-algebra with identity and is isomorphic to the space C(M ), the space of continuous complex functions on the compact metric space M which is the space of “maximal ideals” of a. This is a consequence of I. M. Gel’fand’s representation theorem for Banach*-algebras (cf. Loomis (1953), Sec. 26). It is the new or modern application of the abstract Harmonic Analysis, that gives a clearer structure of the problem although we need to use the abstract ideas. [If a is not separable, then M will not be metric, and still compact, but our argument applies to both cases.] The compact M is thus a complete separable metric (or Polish) space but M is uncountable. Now one can invoke a well-known theorem, due to Kuratowski of 1930, stating that there exists an f : I → M , which is one-to-one and onto so f, f −1 are Borel functions, called a Borel equivalence (cf. e.g. Royden (1988), p. 406). Since a continuous complex function is uniformly approximable by (complex) simple functions (by the Stone-Weierstrass theorem), if a < tk < 1, and Ik = [0, tk ) ↑ I, their correspondents denoted Etk in a, form a resolution of the identity, and the isomorphism between a and C(M Pn) gives for each B ∈ a,n a unique bi ∈ C(M ) (or Bc ↔ bi ), so that i=1 λi (χIi − χIi−1 ) = bi satisfies, for the uniform norm, kbt − bnt k∞ < , (13) and this gives (using operator norm on βi ) by Gel’fand’s isomorphism kBt − Btn k < 

(14)

for each  > 0. Using the Riemann-Stieltjes approximation one gets Z (15) Bt = b(t, λ)dEλ , t ∈ T. I

It now follows, on taking Bt = Mt , for an element Xt0 ∈ L20 (P ), Z Xt = Nt Xt0 = b(t, λ)dz(λ), t ∈ T. I

Thus Z(·) is a vector measure satisfying the conditions of the theorem and kZ(A)k22 = α(A) gives (α(·) is a finite measure): Z r(s, t) = b(s, λ)¯b(t, λ)dα(λ). (16) I

This is (12) and the result follows. 

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Remark 8. 1. The key function b(s, ·) above can be given a concrete construction, also following Gelbaum (1964) who refined Kuratowski’s method. It is noted that there is a Cantor set S ⊂ I = [0, 1] which is (isometrically) equivalent to M , and let f : S → M be an onto isomorphism. Now set Ms = f ([0, s) ∩ S) ⊂ M, s ∈ I and let χMs correspond, under isometric equivalence, to Es ∈ a. Then {Es , s ∈ I} is a resolution of the identity and the collection Bs ↔ b(s, ·) defined as: b(t, u) = Bt (f (a))χS (u) + αb(t, u1 ) + βb(t, u2 ), if u = αu1 + βu2 , where (u1 , u2 ) is a deleted interval for S in I, α ∈ I, β = 1 − α. It can be seen that this b(t, ·) satisfies (16). A more detailed computation is given, with discussion in the above noted interesting Gelbaum paper. 2. If T is not separable, then the Kuratowski theorem used above is not applicable. Then by a type of counting method Gelbaum shows that the above work needs a drastic change and (16) does not hold if card (T ) > 2card(M ) . The corresponding result will be sketched here which is still of Karhunen’s class. A generalized (less sharp) version can be given as follows: Theorem 2.1.9 On a Lebesgue space L20 (P ) of centered random variables on (Ω, Σ, P ), and an indexed set {Bt , t ∈ T } of a commuting normal collection of bounded linear operators on L20 (P ), if X0 ∈ L20 (P ) is a centered random variable, consider Xt = Bt X0 ∈ L20 (P ), t ∈ T . Then {Xt , t ∈ T } forms a Karhunen field, in that there exists a compact set M , a stochastic measure Z : B(M ) → L20 (P ) of orthogonal values, and a continuous family of functions f (t, ·) : M → C, relative to which the following stochastic integral representation holds: Z Xt = Bt Xt0 = f (t, λ)dZ(λ), t ∈ T, (17) M

Z(·) = ZX0 (·), and the covariance function r of the Xt -field is representable as: Z r(s, t) = (Xs , Xt ) = f (s, λ)f¯(t, λ)dα(λ), s, t ∈ T, (18) M

where α(A) =

kZ(A)k22 , A

∈ B(M ), Z(·) being orthogonally valued.

Proof. The argument uses an interesting theorem due to N. Dunford (cf. Dunford–Schwartz, Part III (1963), Theorem X.2.1) which is an extension of a result due to J. von Neumann and uses some abstract analysis.

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Thus let a be a B ∗ -algebra generated by {I, Bs∗ , Bt , s, t ∈ T }. Then by the Gel’fand–Neumark theorem a is isomorphic and *-isometric to C(M ), the continuous complex function space on the set M of maximal ideals of a, which is a compact Hausdorff-space in the ‘hull-kernel’ topology. Thus for each t ∈ T , there exists an ft = f (t, ·) ∈ C(M ), ˆt ). With this, we can invoke Dunford’s generso that f ↔ Bt (ft = B alized spectral theorem [detailed in Dunford-Schwartz (1963), Part II of this book, Theorem X.2.1] so that there is a unique spectral measure E(·) : B(M ) → a such that (the construction of E(·) being the new element): Z f (t, λ)dE(λ), t ∈ T, Bt = M

and then Z Xt = Bt Xt0 =

Z f (t, λ)dE(λ)Xt0 =

M

f (t, λ)dZX0 (λ), M

which gives (7), the projection E(λ) commuting with all elements of a, E(·) being strongly σ-additive and orthogonally valued. Then (17) and (18) are immediate consequences.  The point of the Karhunen field represented by the above theorem is to place the second order process or field in an abstract Hilbert space context, taking most second order processes into its fold. To make this idea clear, by showing how several of the random fields are brought under its fold, we present these points in a general setting, and show how H. Cram´er gave an essential “ultimate” formulation, making again the harmonizable class a key part here. Theorem 2.1.10 Let {Xt , t ∈ T } ⊂ L20 (P ) be a process or field with mean zero and covariance r(s, t) which has the integral representation: Z ¯ r(s, t) = E(Xs Xt ) = g(s, u)¯ g (t, u)dα(u) (19) S

for some measure space (S, S, α) and a square integrable set {g(s, ·), s ∈ T } ⊂ L2 (α) so that the process (or random field) is of Karhunen class. Then it has an operator representation, relative to a set of commuting operators {Br , Bs∗ , r, s ∈ T } ⊂ B(L20 (P )) so that Z Xt = Bt Xt0 = g(t, λ)dZXt0 (λ), t ∈ T, (20) S

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for a unique orthogonally valued measure ZXt0 : S → L20 (P ) such that α(A) = kZXt0 (A)k22 , A ∈ S. On the other hand, every abelian family {Bt , t ∈ T } ⊂ B(L20 (P )) of normal operators defines a Karhunen field {Xt = Bt Xt0 , t, t0 ∈ T }, by the preceding theorem. Remark 9. This result can be given a somewhat generalized form with the associated concept introduced by H. Cram´er, to be detailed below and this set of ideas connects nicely the above vector measures on L2 (P ). It may be noted in passing that the general class of our Karhunen fields is representable by an abelian set of bounded operators on L20 (P ) whose generalized spectrum (in the Gel’fand sense) is locally compact. If the process {Xt , t ∈ R} also admits a bounded shift operator and is weakly continuous, then there is a strongly continuous semi-group {Nt , t ∈ R} of normal operators such that {Xt = Nt X0 , t > 0} and a spectral set Λ ⊂ R. If the process is stationary, then Xt = Ut X0 , where {Ut , t ∈ R} is a unitary group on L20 (P ). Thus there is always an operator representation and the process admits a stochastic integral formulation with an orthogonally valued measure on B(Λ) into L20 (P ). This result specialized to Karhunen processes was obtained by Getoor (1956) using a different method. If T = R (or an LCA group) and Xt is weakly continuous, then one has Xt = At Ut Y0 where At commutes with Ut , and As is a closed densely defined mapping, {Ut , t ∈ R} being a unitary group. [See Mizel and Rao (2009) for some extensions.] As in the harmonizable case, the orthogonally valued condition on Z(·) can also be relaxed for the Karhunen class as follows: Proposition 2.1.11 Let the mean continuous Karhunen process {Xt , t ∈ T } ⊂ L20 (P ) with representation Xt = Bt X0 be as in (20) above. If the abelian class {Bt , t ∈ T } forms a group, T being an LCA group, then the Xt admits a representation Z Xt = ht, λidZ(λ), t ∈ T, (ht, ·i ∈ Tˆ), (21) T

where the measure is Z : B(R) → L20 (P ), so that the field is weakly harmonizable. (Tˆ is the dual of T .) Proof. The result is a slight refinement of the preceding consideration, as it depends on a modification due to Wermer (1954, and see Dunford-

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Schwartz, Part III, Lemma XV.6.1) on commuting normal operators, according to which there exists a bounded self-adjoint operator C with a bounded inverse so that Ut = CBt C −1 is unitary. Consequently, there is a spectral resolution of Ut giving the representation (by a classical result due to Hille, used before), Z −1 Bt = C Ut C = ht, λidC −1 Eλ C, t ∈ T, (22) Tˆ

and hence we have Z Xt = Bt Xt0 =

eitλ dZ(λ), t ∈ T,

(23)

TR

where Z : B(R) → L20 (P ) is σ-additive but not orthogonally valued. Hence {Xt , t ∈ T } is just weakly harmonizable.  An extension of the preceding results based on Karhunen’s work was given by Cram´er (1951), and it will now be discussed as it presents a completion of this set of ideas leading to a generalized harmonizability. Thus Cram´er considers a centered second order process Xt , t ∈ R, ¯ t ) is representable as: whose covariance r : (s, t) → E(Xs X Z Z r(s, t) = g(s, u)¯ g (t, v)β(du, dv), s, t ∈ R, (24) R

R

where β(·, ·) is a positive definite set function of bounded Vitali variation so that (24) is defined as a Lebesgue-Stieltjes integral. It is natural to follow the basic format of the (weakly) harmonizable process and its integral representation, extending the ideas and methods of the harmonizable classes and we can present the generalized representation for the (weak) Cram´er class itself. It is useful to restate the matter to avoid ambiguity. Thus if Z : B(R) → L20 (P ) is an orthogonally valued (vector) measure and V : L20 (P ) → L20 (P ) is a bounded linear operator then the function Z˜ = V Z : B(R) → L20 (P ) is σ-additive and ˜ ˜ β defined as β(C, D) = (Z(C), Z(D)) is a bimeasure, and the representation (21) becomes in this case: Z ˜ Yt = V Xt = g(t, λ)dZ(λ), t ∈ R. (25) R

Then the Yt -process is termed a weak Cram´er process with β(·, ·) as its bimeasure of finite-Fr´echet (not Vitali) variation. Its covariance is then given by (for s, t ∈ R)

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Z Z r(s, t) = (Ys , Yt ) = R

g(s, u)¯ g (t, u0 )β(du, du0 ).

(26)

R

Here the bimeasure β(·, ·) of (26) is not of (even locally) finite Vitali variation, and the integral now should be defined in strict MT-sense, to be useful in the next chapter. Here we include a dilation result connecting the general weak Cram´er and Karhunen processes somewhat extending the corresponding harmonizable cases to round out the present analysis. This will clarify the underlying structures based on Fourier or harmonic analysis that is at the base of this problem. Theorem 2.1.12 Let {Xt , t ∈ T } be a centered process of weak Cram´er class relative to a family {g(t, ·), t ∈ T } of strictly β-integrable functions defining its covariance r that is representable as: Z Z g(s, u)¯ g (t, u0 )β(du, du0 ), s, t ∈ T. (27) r(s, t) = (Xs , Xt ) = S

S

Then there exists a stochastic measure Z : B(S) → L20 (P ) such that Z Xt = g(t, λ)dZ(λ), t ∈ T, (28) S

¯ holds as a Dunford-Schwartz vector integral, where E(Z(C)Z(D)) = β(C, D), C, D ∈ S0 , the δ-ring of bounded Borel sets from S contained in compact sets. Conversely, every process defined by (28) is of weak Cram´er class with covariance (27) as a strict M T -integral. The result is detailed with necessary background material on (strict) Morse-Transue integrals in the article entitled ‘Bimeasures and Nonstationary Processes’ by Chang and Rao (1986), going over 100 pages and hence it will be referred to, but not reproduced here. That every Cram´er process may be ‘dilated’ to a Karhunen process on a superHilbert space will now be stated. This is similar to (but extends) the result that a (weakly) harmonizable process can be dilated to a stationary one on an inclusive Hilbert space which is analogous to, and using, the classical result of Na˘imark’s with its extension by Sz.-Nagy which in our case implies that a (weak) Cram´er family can be dilated to a Karhunen class on a larger Hilbert space. In the classical context, this says that a harmonizable class can be dilated to a stationary family on a super Hilbert space, just guides to further work.

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The desired dilation of Cram´er process to the Karhunen class on a larger L2 (P )-space can be given as follows. [There are however some differences between this and the harmonizable case.] Theorem 2.1.13 Let {Xt , t ∈ T } ⊂ L20 (P ) be a process of weak Cram´er class so that it is representable as: Z Xt = g(t, λ)dZ(λ), t ∈ T, (29) S

as in the preceding result. Then there exists a dilation to a large Hilbert space K containing the vector space L generated by the Xt -process, denoted as {Yt , t ∈ T }, which is a Karhunen process relative to a measure µ on (S, S), such that Xt = QYt where Q is the orthogonal projection of K onto L determined by the X-process, the space spanned by the weak Cram´er process, Z ˜ Yt = g(t, u)dZ(u), t ∈ T, (30) S

˜ ˜ ˜ where the Z(·) has orthogonal increments so that (Z(C), Z(D)) = µ(C ∩ D), C, D ∈ S. But a Yt -process given by (30) defined as Xt = V Yt , t ∈ T is always of a weak Cram´er process whose covariance can be given as an integral for s, t ∈ R: Z Z g(s, u)¯ g (t, v)β(du, dv), (31) r(s, t) = (Ys , Yt ) = R

R

only when {g(t, ·), t ∈ T } is strictly MT-integrable. We omit the details of proof as it uses some works that are needed for the preceding result, and are available in the same reference. The work shows that the Cram´er and Karhunen classes are the abstractions of the harmonizable and the stationary families that opened the probabilistic analysis and use extensions of Fourier methods. It will be seen below that several other areas are opened up by this abstraction which enriches and deepens the analysis further. A few related results will be included in the Complements at the end of this chapter. Several of these ideas admit interesting Hilbert space extended operator analysis with probabilistic insights, leading to studies of new problems of use with stochastic applications. The above discussion implies the following easy (abstract) characterization of harmonizable processes in the way stationarity was conceived by Khintchine in the earlier times:

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Theorem 2.1.14 Let {Xt , t ∈ T ⊂ R} ⊂ L20 (P ) be a centered second order process. Then it is weakly harmonizable iff (i) E(|Xt |2 ) ≤ M < ∞, t ∈ T , whose covariance r(·, ·) is continuous on T × TR and (ii) sup{E(| f (t)Xt dt|2 ) : kfˆk∞ ≤ 1} < ∞, where fˆ: t 7→ T R itλ e f (λ)dλ, is the Fourier transform of f . Here in (ii) the symbol R is the (vector or) Bochner integral. The following result illustrates the above theorem and gives a feeling for the subject. Thus let {S(t), t ≥ 0} be a contraction semi-group of linear operators on L20 (P ). This means kS(t)k ≤ 1, (operator norm), S(u + v) = S(u)S(v), S(0) = id, u, v ≥ 0, and kS(u)f − f k → 0 as u → 0+ , where kf k2 = E(|f |2 ). Now let Y (t) = S(t)X0 , if t ≥ 0, and = S ∗ (−t)X0 if t < 0, where S ∗ (t) is the adjoint operator of S(t). Hence E((S(u)f ), g¯) = E((f, S ∗ (u)¯ g )), f, g ∈ L20 (P ). Setting S(−u) = S ∗ (u), the process {Y (t), t ∈ R} becomes weakly harmonizable, and the family {S(u), u ∈ R} is positive definite: n X n X

E((S(ui − uj )fi , f¯j )) ≥ 0, fi ∈ L20 (P ),

i=1 j=1

for each set {u1 , . . . , un } ⊂ T . This is easy if T = Z and the case T = R can then be reduced to the former using some standard analysis. The relation between the covariance function and its measure for (weakly) harmonizable processes, an analog of the classical inversion theorem, can be given which is of interest in this analysis. The result was considered by Lo`eve for the strongly harmonizable case and its analog for the weak harmonizability is not obvious. We include the result as it motivates further analysis and insight in this theory. The following inversion formula, an analog of the classical L´evy case and of Lo`eve’s for the strongly harmonizable result, is obtainable with some standard computations and some modifications. Here is the formula: Proposition 2.1.15 Let r(·, ·) be a weakly harmonizable covariance function of {Xt , t ∈ R} ⊂ L20 (P ) with F (·, ·) as its spectral bimeasure. If A = (a1 , a2 ), B = (b1 , b2 ) are continuity intervals of F , then

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Z F (A, B) = a→∞ lim b→∞

a

−a

Z

b

−b

59

e−ia2 s − e−ia1 s e+ib2 t − e+ib1 t · r(s, t)dsdt. −is it

It will be of interest to find an estimator of F “optimally” using some extended statistical analogs of the stationary case. This has not been done, and may be a useful problem to consider. The preceding discussion and analysis on the importance of a weak (and strong) harmonizable process and its dependence on Fourier analysis, shows a need to present characterizations of the key weak harmonizability, extending the classical (weak) stationarity in the next section and then use it in the following analysis. 2.2 V -Boundedness, Weak and Strong Harmonizabilities Since V -boundedness plays a key role in our analysis, it may be useful to present some of its basic properties and results here and use them freely later on. Recall that a centered process Xt , t ∈ R, is (strongly) harmonizable if its covariance r admits the integral representation as a Fourier transform relative to a bimeasure F , so that Z Z 0 r(s, t) = eisλ−itλ dF (λ, λ0 ), s, t ∈ R, (32) R

R

where F is of bounded Vitali variation. However, this is not really broad enough for applications. In fact consider {Xn , n ∈ Z}, Xn ∈ L20 (P ), and be an orthonormal so that the set is stationary, and r(m, n) = R π family 1 i(m−n)λ r˜(m − n) = 2π e dλ, m, n ∈ Z, but the truncated series −π X˜m = Xn , n > 0, = 0 for n ≤ 0, has the property that their covari¯˜ ) = 1 if m = n, and = 0 otherwise, so that ˜mX ance r : (m, n) → E(X n r˜ cannot be expressed as (32), by a nontrivial extension of a theorem of F. and M. Riesz, due to S. Bochner, that r˜ must be absolutely continuous for all m = n, and r˜(m − n) → 0 as |m| + |n| → ∞ which does not hold here. Hence r˜ cannot be represented as (32), using the Lebesgue integral. In order to advance the subject, we need to go for a weaker integral, due to M. Morse and W. Transue (1955/56) which is found suitable here. We shall see that the (Morse–Transue or) MT-integral recalled in the last section, and a slight modification of it, will play an important role in our analysis as well as in many other applications.

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Let Ω1 , Ω2 be locally compact Hausdorff spaces and Λ : Cc (Ω1 ) × Cc (Ω2 ) → C be a bilinear form on the product compactly based continuous scalar function spaces such that Λ(f, ·) and Λ(·, g) are linear and are C-measures (or that each is a continuous complex linear functional). Thus by the usual Daniell formulation we have, I = I1 − I2 + i(I3 − I4 ) and each Ij corresponds to a “C-measure”. As a motivation for the main characterization of weakly harmonizable processes, we recall the following earlier result due to Niemi (1975) which gives a feeling for the subject. Theorem 2.2.1 Let X : R → L20 (P ) be a process which is weakly continuous and valued in a ball, so that kX(t)k2 ≤ M < ∞, t ∈ R. Then the process X is weakly harmonizable relative to a covariance bimeasure F of finite semi-variation, if and only if there exists a stochastic measure Z : B(R) → L20 (P ) with F (·, ·) as its bimeasure F : (A, B) 7→ (Z(A), Z(B)) and representation. Z X(t) = eitλ Z(dλ), t ∈ R, (33) R

as a Dunford-Schwartz integral, and kZk(R) < ∞. Particularly X(·) is strongly harmonizable iff the bimeasure F of Z in (33) is of bounded variation (in Vitali’s sense) in R2 . In any case the harmonizable process X : R → L20 (P ) is uniformly continuous, and is represented as a vector integral given by (33). This result contains an important spectral representation due to Rozanov (1959), and Niemi (1975) has given an independent demonstration. It will be presented here with a different demonstration using an important new concept naturally due to Bochner (1956), called V -boundedness of great interest. It was also considered by Phillips (1950) in a slightly different context, and the ideas are used here which illustrate the key role of the Fourier analysis in our subject. Definition 2.2.2 Let X = {X(t), t ∈ R} ⊂ X, where X is a Banach space. It is V -bounded (V for variation) if the set {X(t) : t ∈ R} ⊂ X is in a ball, X(·) is strongly measurable (i.e., X(R) is separable and X −1 (B) ∈ B, for each Borel set B ⊂ R), and the set C is relatively weakly compact in X where Z  ˆ C= f (t)X(t)dt : kf ku ≤ 1, f ∈ L(R) ⊂ X (34) R

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61

and fˆ: t 7→ R eitλ f (λ)dλ, the integral of the vector function f (·) is in Bochner’s sense (or the standard Bochner integral). R

A fundamental characterization of weak harmonizability due to Bochner (1956), with some earlier analysis by Phillips (1950), is given by the following key result when X = L20 (P ): Theorem 2.2.3 A stochastic process X : R → L20 (P ) is weakly harmonizable iff it is V -bounded, and weakly continuous. Proof. First, let X be weakly continuous and V -bounded, so

Z



f (t)X(t)dt ≤ ckfˆku , f ∈ L1 (R), c > 0,

(35)

2

R

by definition of V -boundedness. Also Y = {fˆ: f ∈ L1 (R)} ⊂ C0 (R) the continuous complex function space each fˆ vanishing at infinity, by the Riemann-Lebesgue lemma. Since Y is a real algebra in C0 (R) and separates points of R, it is uniformly dense by the clasR sical Stone-Weierstrass theorem. If F : f 7→ R f (λ)e−itλ dλ, t ∈ R, then F : L1 (R) → C0 (R) is one-to-one and contractive. The mapping R 2 ˆ T : Y → X = L0 (P ), as T (f ) = R f (t)X(t)dt ∈ X is well-defined. The following diagram is commutative where T1 (·) is given by: Z T1 (f ) = f (t)X(t)dt ∈ X. R

F

L1 (R)

-

Y

-



T

T1

.

X By hypothesis T is bounded and since Y ⊂ C0 (R) is dense, it has a norm-preserving extensions T˜ to C0 (R). Since X is reflexive, the mapping T˜ is weakly compact and by a classical theorem in representing such operators (cf., Dunford-Schwartz (1958), VI.73), it can be given an integral form as follows: However, details are unfortunately unavailable, in order to make the work (result) clearer. Details are sketched ˜ be the ‘one point’ compactification of L here. Thus let L = R and L

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˜ Now C0 (L) can be identified as a suband consider the space C(L). ˜ whose elements vanish at ‘∞’. Since T˜ : C0 (L) → X is space of C(L) continuous and C0 (L) is an “abstract M -space”, there is a continuous extension of T˜, agreeing with the original of course on C0 (L). [This uses the well-known theorem due to Kelley-Nachbin-Goodner, and then ˜ → X, we have T = T˜ = Q.] All this work implies that there is a Z˜ : L a vector measure, such that Z ˜ ˜ f (t)Z(dt), f ∈ C(L), (36) T (f ) = ˜ L

˜ using the D-S integral on the right. If we set ˜ L), and kT˜k = kZk( ˜ ∩ A), A ∈ B(L), then Z(·), a vector measure, satisfies Z : A → Z(L ˜ and if f0 = f |L , we have: kZk ≤ kZk, Z Z ˜ ˜ ˜ T (f ) = f0 (t)dZ(t) + f (t)Z(dt), f ∈ C(L) L

{∞}

= T¯(f0 ),

since f (∞) = 0.

Hence T˜(f ) = T¯(f ), f ∈ C0 (L), kT˜k ≤ kT¯Qk ≤ kT˜k, and Z T˜(f ) = f (t)Z(dt), f ∈ C0 (L).

(37)

L

Thus writing R for L, it follows now that

  Z

˜

kT k = sup f (t)Z(dt) : f ∈ C0 (R), kf k ≤ 1 R

˜ = kZk(R) = kZk(R),

(38)

and T ⇔ Z corresponds uniquely. Also kT k = kZk(R). Let l ∈ X∗ and applying it to (37) with fˆ in place of f , we get Z Z ˆ f (t)(l ◦ Z)(dt) = f (t)l ◦ X(t)dt, f ∈ L0 (R), (39) R

R

R since T˜|Y = T so that T (fˆ|R) = R f (t)X(t)dt for all f ∈ L1 (R). It follows from (39) with properties of Lebesgue integrals that Z Z Z f (t)dt et (λ)l ◦ Z(dλ) = f (t)l ◦ X(t)dt. R

R

R

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Hence on rearrangement, we have Z  Z f (t)l et (λ)Z(dλ) − X(t) dt = 0, f ∈ L(R). R

63

(40)

R

Since f is arbitrary, and l ∈ X∗ is also unrestricted, (40) implies Z Z et (λ)Z(dλ) = eitλ Z(dλ), t ∈ R. X(t) = R

(41)

R

This shows that X is weakly harmonizable by (32). In the opposite direction, let X : R → L20 (P ) be weakly harmonizable so that it can be expressed as (41), with kZk(R) < ∞. Hence kX(t)k2 ≤ M0 < ∞, t ∈ R, and (l ◦ X)(·) is the Fourier transform of l ◦ Z, l ∈ X∗ . So X(·) is weakly continuous. Then the following computation is valid, using the vector integration properties:  Z  Z Z et (λ)Z(dλ) dt f (t)l f (t)X(t)dt = l R R R Z Z Z fˆ(λ)l ◦ Z(dλ) f (t)et (λ)l ◦ Z(dλ)dt = = R R R   Z fˆ(λ)Z(dλ) . (42) =l R

Since l ∈ X is arbitrary, (42) implies Z Z fˆ(λ)Z(dλ) ∈ X. f (t)X(t)dt = ∗

It follows from this and the preceding computations that

Z

f (t)X(t)dt ≤ kfˆku kZk(R) = Ckfˆku , f ∈ L1 (R),

R

(43)

R

R

(44)

2

R where C = kZk(R) < ∞. Hence the following set f (t)X(t)dt : R kfˆku ≤ 1 is bounded for f ∈ L1 (R). The reflexivity of X implies that X is then V -bounded, completing the converse.  Some consequences and extensions of the above result will be presented below both for applications and some of its extensions which also are of interest. It should be observed that the above result (and proof) extends if R is replaced by an LCA (= locally compact abelian) group G, so G = Rn included.

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In particular, if T : L20 (P ) → L20 (P ) is a bounded linear operator and Y (t) = T X(t), t ∈ R, where X is weakly harmonizable, then the abstract analysis so far used implies, for T ∈ B(X), Z  Z itλ Y (t) = T e Z(dλ) = eitλ (T ◦ Z)(dλ) R

R

˜ where Z˜ = T ◦Z is a stochastic measure, kZk(R) ≤ kT kkZk(R) < ∞. Thus, Corollary 2.2.4 The linear span of weakly harmonizable processes forms a module over the class of all bounded linear transformations on X = L20 (P ). As seen already in the earlier discussion that on an orthonormal sequence {Xn , −∞ < n < ∞} which is trivially (weakly) stationary, has its image {Xn , n ≥ 0} by an orthogonal projection need not be stationary again, and not even strongly harmonizable, but is weakly harmonizable by the above corollary; this is of interest in both theory and applications. We have the following approximation, due to Niemi (1975) that is also of interest here. [For another type of standard representation, see Theorem 2.2.10 below.] Theorem 2.2.5 Let {Xt , t ∈ R} ⊂ L20 (P ) be a weakly harmonizable 2 process. Then there exists a sequence {Xn (t), t ∈ R}∞ n=1 ⊂ L0 (P ) of strongly harmonizable processes such that Xn (t) → X(t), in L20 (P ) as n → ∞, uniformly in t on compact subsets of R. The same conclusion holds if R is replaced by an LCA group G and then the sequence is to be replaced by a net of such processes (or fields). Proof. By the preceding (or earlier) analysis the process Xt can be given an integral representation as Z et (λ)dZ(λ), t ∈ R (45) Xt = R

for a stochastic (or vector) measure Z : B(R) → L20 (P ). Let X = 2 sp{X ¯ t : t ∈ R} ⊂ L0 (P ) which is a separable subspace. Hence there exists a complete orthonormal sequence {φn , n ≥ 1} ⊂ X so that Xn (t) =

n X k=1

φk (Xk (t), φk ), t ∈ R.

(46)

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We assert that {Xn (t), t ∈ R, n ≥ 1} is a strongly harmonizable sequence that approximates (strongly), the weakly harmonizable process X(t). It is clear that Xn (t) → X(t) in L20 (P ) = X for each t ∈ R. To see that Xn (t) is strongly harmonizable, let lk : X 7→ (X, φk ), so lk ∈ X∗ , where X∗ is the adjoint space of X. Also since Xt is weakly harmonizable, Z  n n X X Xn (t) = φk · lk (X(t)) = φk · lk et (λ)Z(dλ) =

i=1 n X

R

k=1

Z

Z et (λ) · (lk · Z)(dλ) =

φk R

k=1

et (λ)ζk (dλ), R

P where ζk (dλ) is the sum nk=1 φk lk ◦ Z(dλ) here. Also Gn (A, B) = (ζn (A), ζn (B)) has finite total variation since (lk ◦Z)(·) is a signed measure. It then follows that Xn (t) is strongly harmonizable. Since X(·) is weakly harmonizable, it is seen to be strongly continuous. Also if K ⊂ R is compact then the image X(K) ⊂ L20 (P ) is also (norm) compact. Now Xn (t) in L20 (P ) for each t, and since X(K) ⊂ X = L20 (P ) is compact implies (by the metric approximation property of Hilbert spaces) Xn (t) → X(t) in L20 (P )-uniformly in t ∈ K ⊂ R. This implies all the assertions of the theorem.  It should be noted that the class of weakly harmonizable process forms a proper subset of bounded continuous processes in L20 (P ). We give an example in the complements section below to exemplify this. The following approximation (from above) is thus of interest in our study. Theorem 2.2.6 Let X : R → L20 (P ) be a weakly harmonizable process with Z : B(R) → L20 (P ) as its representing measure given in Theorem 2.2.1 above. Then there exists a sequence of regular Borel measures βn : B → R+ such that for each f ∈ C0 (R), one has:

Z



f (t)Z(dt) ≤ lim inf kf k2,βn . (47)

R

2

n

Proof. By the preceding result there exists a strongly harmonizable sequence Xn → X uniformly on compact subsets of R. If now ζn represents Xn , so both ζn , Z : B → L20 (P ), we have

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Z

Z f (λ)Z(dλ) = lim

n→∞

R

f (λ)ζn (dλ),

(48)

R

for all trigonometric polynomials f (·) which are uniformly dense in C0 (R) and separate point of R. It then follows that (48) holds for all f ∈ C0 (R), by the standard reasoning. Hence for f ∈ C0 (R),

Z

2

Z

2



f (λ)dZ(λ) = lim f (λ)ζn (λ)



n→∞ R 2 2 Z RZ = lim f (λ)f (λ0 )dFn (λ, λ0 ) n→∞

R

R

where Fn (s, t) = (ζn (−∞, s), ζn (−∞, t)) is a covariance of bounded variation for each n. Let |Fn |(·, ·) denote the Vitali variation measure of Fn . Then the hermitian property of Fn gives, |Fn |(A, B) = |Fn |(B, A), and if βn (A) = |Fn |(A, R), it then defines a finite Borel measure, and  Z Z Z 1 f (s) [|Fn |(ds, dt) + f (t)|Fn |(ds, dt)] , f (λ)β(dλ) = 2 R R R (49) and since Fn is positive semi-definite, we have Z Z Z Z f (s)f (t)Fn (ds, dt) ≤ 0≤ |f (s)f (t)||Fn |(ds, dt) 2 2 R R Z Z 1 (|f (s)|2 + |f (t)|2 )|Fn |(ds, dt), since |ab| ≤ (a2 + b2 )/2, ≤ 2 R2 Z = |f (t)|2 βn (dt), by (49). (50) R

It follows from (49) and (50) that

Z

2 Z

f (λ)Z(dλ) = lim

R

2

f (λ)f (λ0 )Fn (dλ, dλ0 ) R2 Z ≤ lim inf |f (λ)|2 dβ(λ). n

n

R

This completes the proof.  To appreciate the significance of the above assertion we now indicate a general result, also of interest in this analysis, in which sequences of measures may be replaced by a single measure.

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Theorem 2.2.7 On a measurable space (Ω, Σ) consider a vector measure ν : Σ → X, a Banach space. Then there exist a finite measure µ : Σ → R+ , a continuous convex function φ : R+ → R+ growing faster than |x|, (i.e., φ(x) ↑ ∞ as x ↑ ∞) and ν has finite φ-semi-variation relx ative to µ, meaning (φ(0) = 0, φ(−x) = φ(x))

 Z 

kνkφ (Ω) = sup (51)

f (ω)ν(dω) : kf kψ,µ ≤ 1 < ∞, Ω X n   o R |f (ω)| where kf kψ,µ = inf α > 0 : Ω ψ µ(dω) ≤ 1 < ∞, and the α integral in (51) is the usual vector (or Dunford-Schwartz) type, where ψ : R+ → R+ is the complementary convex function given as usual, by ψ(x) = sup{|x|y − φ(y) : y ≥ 0}, for φ. The (omitted) proof depends on some standard vector integration results given in Dunford-Schwartz I (1958), and a few Orlicz space ideas. Some aspects of this will be considered below in Sections 3 and 4. Our aim here is to present a much desired extension of Theorem 2.2.6 above so as to include most (weakly) harmonizable and certain other related generalizations of the (weakly) stationary process. Since just about the same time when Lo`eve (1947) introduced the (strongly) harmonizable concept, K. Karhunen intoduced a general concept, still called of Karhunen class (as already done) which is of interest here as it conceptually includes all the harmonizable classes, and it has interest in much of our work. Definition 2.2.8 A process {Xt , t ∈ T } ⊂ L20 (P ), the centered, with covariance r(·, ·), is of Karhunen class if there exists also an auxiliary measure space (S, S, ν), and a set {g(t, ·), t ∈ T } in L2 (S, S, ν) which are again complex valued it follows that r(·, ·) has the representation: Z r(t1 , t2 ) = g(t1 , λ)g(t2 , λ)dν(λ), t1 , t2 ∈ T. (52) S

Here S, T are general sets and if T = R, Z, then S = Tˆ(= [−π, π], or R, etc.). We now establish that the weakly harmonizable process is also of Karhunen class. Theorem 2.2.9 Each weakly harmonizable process {Xt , t ∈ T } is also a Karhunen process, T = R or [0, 2π] (or an LCA group), relative to some finite measure ν on Tˆ and for a suitable (Borel) class of functions {ft , t ∈ T } ⊂ L2 (T, ν).

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Proof. It suffices to outline the argument. Since the given process is weakly harmonizable, by the preceding analysis there exists a station˜ P˜ ) perary process {Yt , t ∈ T } on a larger probability space (Ω, Σ, haps containing the given space such that {Yt , t ∈ T } ⊂ L2 (P ), and Xt = QYt , t ∈ T where Q is the projection operator on L2 (P˜ ) onto L2 (P ), the given space, such that Xt = QYt , t ∈ T . Here Q is the orthogonal projection onto L2 (P ). But by the preceding work, R itλ ˜ ˜ Yt = T e dZ(λ), t ∈ T , where Z(·) is orthogonally valued. If 2 ˆ But ˜ ν(A) = E(|Z(A)| ), then ν(·) is a finite positive measure on (Tˆ, T). by a well-known theorem, due to Kolmogorov (cf. Rozannov (1959), (p. 33) or Masani (1968), Thm. 5.10), there exists an orthogonal projection Q such that Xt = QYt .  Z Z itλ ˜ ˜ π(eitλ )Z(dλ), (53) e Z(dλ) = Xt = QYt = Q Tˆ

Tˆ

˜ is orthogonally valued. Letting f (t, λ) = for each t ∈ T where Z(·) it(λ) ˆ π(e ), λ ∈ T , we get ft (·) ∈ L2 (T, v) and then (53) yields immediately Z ¯ f (s, λ)f (t, λ)ν(dλ). r(s, t) = E(Xs Xt ) = (54) Tˆ

ˆ ν).  This implies that r(·, ·) is of Karhunen class on (Tˆ, T, Remark 10. Although we embed the given measure space in a larger one and construct the desired triple subsuming the given one, the structure of the usable (or applicable) spaces is classified by this enlargement procedure, showing the key part of geometry. The following important extension of the above theorem, based on Bochner’s Berkeley symposium paper (1956), [somewhat simplified], is of interest here: Theorem 2.2.10 Let {Xt , t ∈ T } be a centered harmonizable process admitting an integral representation Z Xt = eitλ Zx (dλ), t ∈ T, (55) Tˆ

where Tˆ = [−π, π] or = R if T = Z (integers) or T = R (the reals) and Zx (·) is σ-additive (in the sense of mean square) set function on

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Borel sets T of T, E(Zx (A)) = 0 and E(Zx (A)Zx (B)) = Fx (A, B) or Fx (A ∩ B), for harmonizable or stationary cases respectively A, B ∈ T. Let {Xt , Yt , t ∈ T } be processes such that ΛXt = Yt , t ∈ T , where Λ is a linear operator (or “factor”) on L20 (P ) into itself, so Xt is the input and Yt the output process. Then there exists a stationary input {Xt0 , t ∈ T }, ΛXt0 = Yt0 , t ∈ T , where the output {Yt0 , t ∈ T } is also stationary whose spectrum coincides with that of {Yt , t ∈ T } outside of Q = {λ : C(λ) = 0}, the zero set for FY (FY = C). The solution is representable as: Z 0 Xt = eitλ C(λ)−1 Zx (dλ). (56) T −Q

Proof. Let Q = {λ : C(λ) = 0}, and FY (·) has the control measure of the output process {Y R t , t ∈ T }, and suppose that [T = Z or R so Tˆ = (−π, π] or = R], T −Q |C(λ)|−2 dFY (λ) < ∞, whence Xt0 of (56) is well-defined, since E(|Xt0 |2 ) is found by the above integral which is finite. That ΛXt0 is, in fact, the above integral, may be seen as follows: Z 0 C(λ)eitλ Zt0 (dλ), Zt0 as the stochastic measure, ΛXt = ˆ ZT 1 = C(λ)eitλ Zy (dλ), by the preceding work, C(λ) Tˆ−Q Z = eitλ Zy (dλ) = Yt , by definition. (57) Tˆ−Q

The spectral measures of Yt and Yt processes agree on the set Tˆ − Q, and it is enough to verify the integrability of Yt on Tˆ − Q. For this, it suffices to show that the condition ΛXt = Yt , t ∈ T , holds. Now for each φ ∈ Ct (R), we have, ! Z Z Z k X j (j) φ(t)Yt dt = (−1) φ (t − u)Ht (du) Xt dt (58) T

T

i=0

Tˆ

where φ ∈ C (k) (T ) of compact supports and k times continuously differentiable. Since Xt is a distributional solution, by assumption, it satisfies (58) for each φ on T , whose Fourier transform is twice continuously differentiable, having compact support. It follows that

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Z Tˆ

itλ

Z

C(λ)e l(Zx (dλ)) =

Tˆ

eitλ l(Zy (dλ)),

since l(·) is a continuous linear functional and commutes with the integral. Hence by the uniqueness theorem for Fourier transforms we deduce that l(C(λ)Zx (dλ)) = l(Zy (dλ)). This easily implies that C(λ)Zx (dt) = Zy (dλ). But C(λ) 6= 0 on Tˆ − Q, and hence the above implies Z Z ψ(λ) ith e Zy (dλ) = ψ(λ)eitλ Zx (dλ), t ∈ T. (59) Tˆ−Q C(λ) Tˆ−Q Since the support of ψ is in Tˆ − Q, this gives for the variances: Z 2 ! Z ψ(λ) 2 Fy (dλ) = E ψ(λ)eitλ Zy (dλ) , ˆ T Tˆ C(λ)

(60)

because Zy (·) has orthogonal increments, E (|Zy (A)|2 ) = Fy (A) and both Zx (A), Zy (A) have zero means. With the inverse H¨older inequality in (61), E(|Zy (A)|2 ) = Fy (A), Zx (·), Zy (·) centered, we get on taking kψ(·)k∞ ≤ 1 arbitrarily (varying), that (60) gives  Z Z 2 −2 2 |ψ(λ)| |C(λ)| Fy (dλ) : kψkx ≤ 1 |C(λ)| Fy (dλ) = sup Tˆ−Q Tˆ−Q n o it(·) 2 ˆ ≤ sup kψ(·)e kkZy k (T ) : kψkx ≤ 1 ≤ kZy k2 (Tˆ) < ∞, where kZy k(Tˆ) is the semi-variation of Zy (·) which is finite.  Remark 11. A useful consequence of this result is that, if the output {Yt , t ∈ T } is stationary, and if there is a weakly harmonizable solution (in the distributional sense), then after deleting some “frequencies” from the spectrum of the Yt -process, one can find a stationary (distributional) solution that satisfies the given filter equation. We shall include some further analysis later on (next chapter) for applications with a filter problem to understand the ramifications of this property. The above argument (last part) contains a useful characterization of weakly harmonizable processes which is separated for reference:

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Proposition 2.2.11 Let {Xt , t ∈ T } ⊂ L20 (P )} be a mean continuous process (T = R or Z). Then it is weakly harmonizable iff it is in a ball (or norm bounded) and for each integrable φ: Z 2 ! ˆ 2, (61) E φ(t)Xt dt ≤ ckφk ω T

ˆ is the Fourier transform of φ and conversely. where φ(·) Proof. First, note that a weakly harmonizable process having the Fourier Transform of an L2 (P )-valued vector measure with finite Fr´echet variation is bounded, i.e., E(|Yt |2 ) ≤ M < ∞, t ∈ T . Regarding the inequality (61), consider with T = R,  Z Z Z itλ e Z(dλ) dt φ(t)Xt dt = φ(t) R R R Z ˆ = φ(λ)Z x (dλ), by interchanging integrals which is valid. R

Hence, E

Z 2 ! ˆ 2 kZx k2 (R) = Ckφk ˆ 2, φ(t)Xt dt ≤ kφk ∞ ∞ R

since Zx (·) has finite Fr´echet variation, and (61) holds. Regarding the converse, it is precisely the V -boundedness condition of Bochner’s, and the process is weakly continuous so that the result is a consequence of Theorem 2.2.3 above, and the proposition follows.  Remark 12. The weak harmonizability concept, calling it V -boundedness was introduced by Bochner in 1956 as a generalization of Lo`eve’s concept, which we now call strongly harmonizable for distinguishing the key differences. Although the class of strongly harmonizable processes (or fields) is contained in the class of weakly harmonizable processes (or fields), it will be of interest to investigate their approximations. This is clarified here. The second part (on approximation) is due to Niemi (1975), and is essentially the same as in Theorem 2.2.5 above and given for comparison.

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Theorem 2.2.12 The linear space of weakly harmonizable processes w over complex numbers forms a module of all bounded linear mappings (b.l.m) on X = L20 (P ), so that (B(X) · w) = w, where B(X) is the space of b.l.m on X. Moreover, each weakly harmonizable process X : R → L20 (P ) is approximable in L20 (P )-norm by a strongly harmonizable sequence Xn : R → L20 (P ), uniformly on compact subsets of R, as n → ∞. Here R can be replaced by an LCA group G where sequences are taken as nets. Proof. If T ∈ B(X) is a bounded linear operator on X and Xt ∈ w is a weakly harmonizable process, then we get: Z  Z itλ Y (t) = T e Z(dλ) = eitλ (T ◦ Z)(dλ) R

R

by a property of the vector (or Dunford-Schwartz) integral, and Z˜ = ˜ T ◦ Z : B(R) → X is a stochastic (or vector) measure, since kZk(R) ≤ kT kkZk(R) < ∞. So Y (·) is weakly harmonizable (Y ∈ w), and this implies the first statement, as well as B(X) · w ⊂ w since the opposite inclusion is clear. The approximation is shown as in (the above) Theorem 2.2.5. The result in the case of R being replaced by an LCA group is also quite simple and is left to the reader.  2.3 Harmonizability and Stationary Dilations for Applications The second concept of stationarity for second order centered processes and its crucial relation with Fourier transforms noted in early 1930s by A. Khintchine and A. Kolmogorov gave birth to an enormous growth of second moment stochastic analysis. The corresponding extensions of stationarity to strong and (later weak) harmonizability concepts by Lo`eve and Bochner (as well as Rozanov) in early 1950s have shown the key roles played by Fourier analysis on LCA groups, with its full force on stochastic analysis and geometry of Hilbert space. We now show some deeper aspects of this interrelationship, with dilations, and some applications. We also discuss the structure of the mean functions of the harmonizable (or stationary) covariance formulas of the first moment classes, extending an early innovation of Balakrishnan’s work.

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Theorem 2.3.1 (Stationary Dilation) Let X : G → L20 (P ) = H be a weakly harmonizable random field, indexed by an LCA group G, on a probability space (Ω, Σ, P ) where as usual L20 (P ) is the centered scalar random variables forming a Hilbert space. Then there exist an extension ˜ Σ, ˜ P˜ ) and a staHilbert space K = L20 (P˜ ) on a probability triple (Ω, tionary random field Y : G → K such that X(g) = QY (g), g ∈ G, where Q : L20 (P˜ ) → L20 (P ) is the orthogonal projection with range L20 (P ). Here if H is determined by X (as a closed linear span) then Y determines K, whence the latter is the minimal superspace for H. Proof. Consider X : G → H(= L20 (P )) to be weakly harmonizable: Z X(g) = hg, siZ(ds), g ∈ G, (62) ˆ G

by the structural representation, and there exists a finite regular Borel ˆ such that measure µ on the Borel σ-algebra of G

Z

2 Z

ˆ

f (t)Z(dt) ≤ |f (t)|2 dµ(t), f ∈ C0 (G). (63)

ˆ G

ˆ G

2

ˆ × G) ˆ → R+ given by Consider the mapping v : B(G ˆ v(A, B) = µ(A ∩ B), A, B ∈ B(G),

(64)

ˆ being the Borel σ-ring, as is B(G ˆ × G), ˆ then v(·, ·) has the class B(G) finite Vitali variation, and concentrates on the diagonal of the product ˆ × G. ˆ Moreover G Z Z Z ˆ × G). ˆ f (s, s)µ(ds), f ∈ C0 (G (65) f (s, t)v(ds, dt) = ˆ G

ˆ G ˆ G×

ˆ G

ˆ × G) ˆ → C be defined by F (A, B) = (Z(A), Z(B)), which Let F : B(G is a bimeasure of finite semi-variation from (62). Using the D-S and MTintegration techniques, one has

Z

2 Z Z

= 0≤ f (s)Z(ds) f (s)f (t)F (ds, dt). (66)

ˆ G

2

ˆ G

ˆ G

Taking f (s, t) = f (s) · f (t) in (66) setting α = v − F one has:

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Z

2

0≤ |f (s)| µ(ds) − f (s)Z(ds)

ˆ

ˆ G G 2 Z Z f (s)f (t)[v(ds, dt) − F (ds, dt)] = ˆ ˆ ZG ZG ˆ = f (s)f (t)α(ds, dt), f ∈ C0 (G) Z

2

ˆ G

(67)

ˆ G

so that α(·) is positive (semi-) definite and = 0 iff v = F whence F concentrates on the diagonals and so X is stationary itself. Excluding ˆ × C0 (G) ˆ → C gives a nontrivial inner product this case, [·, ·] : C0 (G) [·, ·] defined by Z Z ˆ f (s)¯ g (t)α(ds, dt), f, g ∈ C0 (G). (68) [f, g] = ˆ G

ˆ G

n o ˆ , and H1 = C0 (G)/N ˆ If N0 = f : [f, f ] = 0, f ∈ C0 (G) 0 , let [·, ·] be [(f ), (g)] = [f, g]0 , f ∈ (f ) ∈ H1 , g ∈ (g) ∈ H1 .

(69)

This [·, ·] is an inner product on H1 and let H0 be its completion, and ˆ → H0 be the canonical projection (H0 need not be separaπ0 : Cb (G) ble!). That this is always possible is a consequence of Theorem 1.3.5 which was discussed in detail in the context of L´evy’s BM (in higher dimensional time). The analogous complex case is outlined as follows. Let {hi , i ∈ I} ⊂ H0 be a CON set, and let (Ωi , Σi , Pi ) be determined by a complex Gaussian variable, so taking Ωi = C, Σi = Borel σ-algebra of C and   Z −|t|2 −1 Pi (A) = (2π) exp dt, t = t1 + it2 , A ∈ Σi , 2 A let (Ω 0 , Σ 0 , P 0 ) = ⊗i∈I (Ωi , Σi , Pi ), the product space given by the Fubini-Jessen theorem. If Xi (ω) = ω(i), ω ∈ Ω 0 = CI , then the set 2 {Xi , i ∈ I} forms a CON basis of L = sp{X ¯ i , i ∈ I} ⊂ L0 (P ). The correspondence α : hi → Xi sets up a (linear) isomorphism with 2 L = sp{X ¯ i , i ∈ I} ⊂ L0 (P ) and is onto. Also kτ (hi )k22 = E(|X|2 ) = 1 = [hi , hi ], i ∈ I. This implies that τ : hi → Xi is an isometric isomorphism, after polarization, of H0 onto L ⊂ L20 (P ), as desired.

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75

ˆ be the Let π = τ ◦ π0 : f → τ (π0 (f )) ∈ H0 ⊂ L20 (P ), f ∈ C0 (G), 0 composite mapping shown, and let X(t) = π(et (·)) ∈ H , lt : s → (t, s) is a character of G at t ∈ G, (e0 = 1 6∈ N0 ) so π0 (t) can be identified with a constant (say) 0 ∈ C0 (ω). Thus X0 (0) = τ (1),

E(|τ (1)|2 ) = 1.

Let H00 = sp{X(t), ¯ t ∈ G} ⊂ H0 . Then we can find (Ω 00 , Σ 00 , P 00 ), a probability space, such that H00 ⊂ L20 (P 00 ), and set H = H0 ⊕ H00 , the ˜ P˜ ) is the direct π , Σ, direct sum of Hilbert space L20 (P ) and L20 (P 0 ). If (˜ 0 0 0 product (Ω, Σ, P ) ⊗ (Ω , Σ , P ), one can identify K ⊂ L20 (P˜ ). Define Y (t) = X(t) + iX(t), t ∈ G, and we can see that {Y (t), t ∈ G} ⊂ K and if Q : K → H(= H ⊕ {0}) is the orthogonal projection, then X(t) = QY (t), t ∈ G. We assert that the Y (t) is a stationary process to complete the argument. For this, consider the computations (cross products vanishing): r(s, t) = (Y (s), Y (t)) = (X(s), X(t)) + (X1 (s), X1 (t)) + 0 Z Z hs, λiht, λ0 iF (dλ, dλ0 ) = ˆ ˆ G G Z Z + hs, λiht, λ0 iα(ds, dλ) ˆ ˆ G G Z Z hs, λiht, λ0 iv(dλ, dλ0 ), (α = v − F ) = ˆ ˆ ZG G Z = hs, λiht, λiµ(dλ) = hs − t, λiµ(dλ), ˆ G

ˆ G

by the composition of characters and this implies r(s + h, t + h) = r˜(s − t) and so {Y (t), t ∈ G} is stationary, and H = sp{X(t), ¯ t ∈ G}.  An interesting consequence of this result is the following: Corollary 2.3.2 If G is an LCA group, B(G) its Borel σ-algebra and H is a Hilbert space, then a vector measure ν : B(G) → H has an orthogonally valued dilation. R ˆ → H be a mapping given by X(ˆ Proof. Let X : G g ) = G hˆ g , λiν(dλ), as a vector (or D-S) integral. Then X(·) is weakly harmonizable as it is V -bounded. Then by the preceding theorem, there is a super Hilbert

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space K ⊃ H, an orthogonal projection Q : K → H and a stationary ˆ If Z(·) is the stochastic field Y , such that X(ˆ g ) = QY (ˆ g ), gˆ ∈ G. measure representing Y , then we have Z Z ˜ hˆ g , λi(ν(dλ), h) = (X(ˆ g ), h) = (QY (ˆ g ), h) = hˆ g , λi(Q·Z(dλ), h). ˆ G

G

These are scalar (Lebesgue) integrals. Then by the uniqueness of Fourier representation, one has hν(A) − Q ◦ Z(A), hi = 0, A ∈ B(G), h ∈ K. By the uniqueness of Fourier representation ν = Q ◦ Z. Since Z(·) is orthogonally valued, as Y is stationary, the result follows.  Remark 13. The preceding result shows an interesting relation between a positive definite contractive family of operators on a Hilbert space and their restriction to certain subspaces having special geometric properties observed by Sz.-Nagy and Na˘imark in early 1950s. This will be made explicit, as it implies a deep internal equivalence of these authors’ works! The following representation is of interest here: Theorem 2.3.3 Let X : G → L2 (P ) = X, G an LCA group and (Ω, Σ, P ) a probability space, X being a centered (i.e., means zero) weakly harmonizable mapping (or field). Then there is a super (Hilbert) space K = L20 (P ) ⊃ X, on an (possibly) enlarged probability space ˜ Σ, ˜ P˜ ), Y0 ∈ L2 (P˜ ), a set of weakly continuous contractive linear (Ω, 0 operators {T (g), g ∈ G} on K → X (T (0) = id) such that the T (g)family is positive type, and X(g) = T (g)Y0 where g ∈ G. Conversely, every weakly continuous contractive family {T (g), g ∈ G} of the above type from a Hilbert space K ⊃ X into a subspace X, is weakly harmonizable (so X : G → X, X(g) = T (g)X0 , where T (0) = id on X). Proof. For the direct part, let X : G → X − L20 (P ) be weakly harmoniz˜ ⊃ X, and a stationary Y : G → X with able. Then there is a K = L20 (β) X(g) = QY (a), g ∈ G, by Theorem 2.3.1 above. On the other hand Y (g) = U (g)Y (0) for a strongly continuous unitary group {U (g), g ∈ G} on K. If T (g) = QU (g), g ∈ G, then it is asserted that this is the desired class. Clearly T (0) = Q = IX and kT (g)k ≤ kQkkU (g)k ≤ 1.

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The continuity of the U (k)-family implies the weak continuity of the U (t)-class and its positive definiteness is seen from T˜(−g) = (T˜(g))∗ and (T˜(g) = T (g) ) for hi ∈ X: X   T˜(−θ)hs1 , hs2 = (U ∗ (g)hs1 , Qhs2 ) = (hs1 , U (g)hs2 ), since Qhs = hs , U ∗∗ = U = (Qhs1 , U (g)hs2 ), hsi ∈ X, i = 1, 2. It follows from this that n X n  n X n  X X −1 ˜ T (si sj )hsi , hsj = (QU (−sj )U (si )hsi , hsj ) i=1 j=1

i=1 j=1

=

n X

((U sj )∗ U (si )hsi , hsj )

i,j=1

2 n

X

= U (si )ksi ≥ 0.

i=1

The converse depends on Sz.-Nagy’s extension of the Na˘imark result, stating that if T˜(·) = T (·)|X then there is an extension Hilbert space H1 ⊃ H and a weakly continuous group of unitary operators on it such that T˜(g) = Q1 V (g)|H where Q1 is an orthogonal projection of H1 onto H. If x0 ∈ H1 ∩ H then T (g)x0 = T˜(g)x0 = QV (g)x0 = x(g), g ∈ G, and {Y (g) = V (g)x0 , g ∈ G} ⊂ H is stationary, and the earlier result applies. Thus {T (g)x0 , g ∈ G} is weakly harmonizable.  2.4 Domination of Vector Measures and Application to Cram´er and Karhunen Processes Let (Ω, Σ, µ) be a measure space, X a Banach space and ν : Σ → X a vector measure (i.e., σ-additive on Σ), with p-variation (p ≥ 1) denoted |ν|p (·), defined on A ⊂ Σ as: (n ) X |νf |p (A) = sup kν(Ai )k|ai | : Ai ∈ Σ(A), disjoint, kf kq,µ ≤ 1 , i=1

(70)

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P where f = ni=1 ai χAi , q = p/(p − 1) ≥ 1, and f ∈ Lq (Ω, Σ, µ) = Lq (µ), Σ(A) being the trace of Σ on A. If |ν|p (A) < ∞, then ν has finite p-semi-variation on A relative to µ. If p = 1, usually L∞ (µ) is replaced by B(Ω, Σ), the space of bounded measurable scalar functions on Ω with uniform norm, (whatever µ is) and 1-variation is merely termed variation, and (70) becomes ( n ) X |ν|(A) = sup kν(Ai )k : Ai ∈ Σ(A), disjoint . (71) i=1

Note that |ν|(·) is σ-additive if ν is, but |ν|p (·) is not necessarily so if p > 1. We also need p-semi-variation of ν on A:

( n )

X

kνkp (A) = sup ai ν(Ai ) : Ai ∈ Σ(A), disjoint, kf kq,µ ≤ 1

i=1

(72) P where f = ni=1 ai χAi , p−1 + q −1 = 1. If kνkp (A) < ∞, then ν has p-semi-variation finite on A relative to µ. If p = 1, then

( n )

X

kνk(A) = sup ai ν(Ai ) : |ai | ≤ 1, Ai ∈ Σ(A), disjoint .

i=1 (73) It is clear that |ν|(A) ≤ |ν|1 (A),

kνk(A) ≤ kνk1 (A),

and kνkp (A) ≤ |ν|p (A), with (usually) a strict inequality if dim (X) = ∞. Also (72) may be written as:

 Z 



kνkp (A) = sup f dν : kf kq,µ ≤ 1 . (74) A

The technical result of interest here, termed the domination problem, is this: Does ν : Σ → X, σ-additive, have the finite p-semi variation for ¯ + ? A solution of this problem is of importance in a σ-finite µ : Σ → R our analysis. This will be answered ‘yes’ for a class of Banach spaces of great interest here. It is convenient to treat the problem for a general class of spaces that includes the Lebesgue class Lp (µ), p ≥ 1, namely the Orlicz class

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Lφ (µ) where φ : R → R+ is a symmetric convex function with φ(0) = 0 termed a Young function. For each such φ, there exists a similar function ψ, called the complementary function given by ψ(x) = sup{|x|y−φ(y) : p q y ≥ 0}. [If φ(x) = |x|p , then ψ(y) = |y|q , p−1 + q −1 = 1, p, q ≥ 0.] For each measurable f : R → R (or X), one defines the φ-semi-variation of ν : Σ → X, on A ∈ Σ as:

 Z 

kνkφ (A) = sup (75)

f (w)ν(dw) : kf ||ψ,µ ≤ 1 , A

where φ, ψ satisfy, |xy| ≤ φ(x) + ψ(y) and one has, by definition     Z |f | kf kφ = inf α > 0 : φ dµ ≤ 1 , (76) α Ω which when φ(x) = |x|p , p ≥ 1, becomes kf kφ = kf kp , the Lebesgue norm. The problem that we want to solve is that, if ν : Σ → X is a vector measure, is it dominated by a pair (φ, µ) for some φ(x) = |v|p , p ≥ 1? A positive solution is given by the following result: Theorem 2.4.1 On a measurable space (Ω, Σ), a vector measure ν : Σ → X, (a Banach space) is dominated by a pair (φ, µ), where φ(r) ↑ r ∞, φ : R → R+ is a continuous Young function, kνkφ (Ω) < ∞. Proof. Since weak and strong σ-additives of a Banach space valued measure are known to be equivalent, consider for x∗ ∈ X∗ , kx∗ k ≤ 1, let An , n ≥ 1, disjoint sets An ∈ Σn . We have

! n

X [

ν(Ak ) 0 = lim ν An − n→∞

n≥1 k=1 ( ! ) n [ X = lim sup (x∗ ◦ ν) (x∗ ◦ ν)(Ak ) : kx∗ k ≤ 1 . An − n→∞ n≥1

k=1

(77) Thus the signed measures {x∗ ◦ ν : x∗ ∈ S ∗ }, (S ∗ unit ball of X∗ ) are uniformly σ-additive on Σ. Now one invokes a result of Bartle-DunfordSchwartz) (cf. D-S (1), IV.10.5) to conclude that there is a “control measure” µ : Σ → R+ such that x∗ ◦ ν is µ-continuous for each x∗ ∈ S ∗ . By the Radon-Nikodym theorem, gx∗ = d(xk ◦ ν)/dµ exists and one has by (77),

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Z lim µ(A)→0

  ∗ gx∗ (ω)dµ(ω) = 0 = lim |x ◦ ν(A)| , µ(A)→0

A

uniformly in x∗ ∈ S ∗ . Hence {gx∗ : x∗ ∈ S ∗ } ⊂ L1 (µ) is bounded and uniformly µ-integrable. This gives by a classical de la Vall´ee Poussin’s theorem, that there is a convex function φ : R → R∗ of the given description such that Z φ(|gx∗ (ω)|)µ(dω) ≤ K0 < ∞, ∀x∗ ∈ S ∗ . (78) Ω

Let ψ be the complementary function of φ, so ψ : R → R+ , and kνkφ (Ω)

  Z



= sup f (ω)dν(ω) : kf kψ,µ ≤ 1   Ω Z ∗ = sup sup f (ω)gx∗ (ω)dµ(ω) : kx k ≤ 1, : kf kψ,µ ≤ 1 Ω

≤ 2 sup {kg kφ,µ kf kψ,µ ≤ |λ|x∗ ≤ 1} , by H¨older’s inequality for Orlicz spaces, ≤ 2 sup {kgx∗ kφ,µ kx∗ k ≤ 1} ≤ 2K0 < ∞, by (2.4.9).  x∗

[Here we used some elementary analysis of Orlicz spaces, a reference to which is Zygmund (1959, Vol. I) or Rao and Ren [1991].] From the general (and classical) theory of the Lebesgue and Orlicz spaces, it is known, and easy to verify, that each Lφ (µ) ⊂ L1 (µ) for µ(Ω) < ∞, (Lφ (µ) is the Orlicz space), it is not simple to determine if a given vector measure ν : Σ → Y is dominated by a particular pair (φ, µ) to study the useful properties of µ-dominated ν when Y is an Lφ -space (Lφ (µ)). This nontrivial problem was studied and the case φ(x) = x2 , was solved by Linderstrauss and Pelczy´nski (1968) and then some interesting (as well as important) applications for stochastic analysis followed. Here we present their specialization which will be used in the next section(s) for Cram´er and Karhunen classes of second order processes. The problem and the resulting stochastic analysis are intrinsically tied to the Hilbert structure and hence to second order processes! The applications here include most Lp (µ0 )-spaces on a measure space 0 (Ω , Σ 0 , µ0 ) as well as the abstract (M )-spaces such as B(Ω 0 , Σ 0 , µ0 ),

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L∞ (µ) and some others. The L2 (µ0 ) thus includes Banach spaces isomorphic to Hilbert spaces. The desired concept is given as follows: Definition 2.4.2 A Banach space X is an Lp,λ -space, p ≥ 1, λ ≥ 1, if for each n-dimensional space F ⊂ X, there is an n-dimensional subspace F ⊂ X, (E ⊂ F ) such that d(F, lµn ) ≤ n, (lpm is the usual Lebesgue sequence p-space) and d(E1 , E2 ) = inf{kT kkT −1 k : T ∈ B(E1 , E2 )}, the space of bounded linear mappings on E1 to E2 . Then the above defined space Lp is an Lµ,λ -space for some λ > 1. This unmotivated concept was found by these authors (Lindenstrauss and Polczy´nski (1965)) and used it to solve the above problems. The motivation and discovery of this key concept are not obvious. The above concept (and space) is used to prove the following important result: Theorem 2.4.3 Let B(Ω, Σ) be the Banach space of bounded scalar measurable functions f : Ω → C, with uniform norm, and Y be an Lp space 1 ≤ p ≤ 2 introduced above. Then a vector measure ν : Σ → Y is (2, µ)-dominated, so that there is a (finite) measure µ : Σ → R satisfying

Z



f (w)ν(dw) ≤ kf k2,µ , f ∈ B(Ω, Σ) (79)

Ω

Y

and ν has 2-semi-variation finite, relative to µ. Proof. The detail is given in three steps for convenience. We use the standard results from Dunford-Schwartz (Chapter IV, Sec. 10). Now let R T : f 7→ Ω f (ω)ν(dω), so that T : B(Ω, Σ) → Y is a well-defined bounded linear operator. I. Let X = C(S), the real continuous function space on a compact S, and q : s → l0 ∈ X∗ , ls (f ) = f (s), (f ∈ X) the evaluation functional. The set K ⊂ X∗ , the collection of extreme points of the unit ball of X∗ , is, by the (Kre˘in-) Milman theorem, closed and K = q(S) ∪ (−q(S)) where q : s 7→ ls ∈ X∗ , each element of which is = αls , |α| = 1. Thus if Y is an Lµ -space and T ∈ B(C(S), Y), 1 ≤ p ≤ 2, then Lindenstruss and Pelczy´ınski using a key Grothendieck-Pietsch inequality have deduced the existence of a regular probability measure µ0 on K such that

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kT f k2Y

Z ≤ C1

Z

2

|ls (f )| µ0 (dls ) + C2

Zq(S) ≤ C3 |f (s)|2 µ0 (ds), f ∈ X,

|ls (f )|2 µ0 (dls )

−q(S)

S

where S and q(S) are identified, C3 = 2 max(C1 , C2 ). For the complex functions, replace C3 by 2C3 . II. Suppose now X = B(Ω, Σ), and Y is an Lp space, 1 ≤ p ≤ 2. So X is a closed subalgebra of B(Ω). Then there is an isometric and algebraic isomorphism between X and X0 = C(S0 ), S0 being a compact Hausdorff space, preserving all algebraic operations. If I : X → X0 = C(S0 ), and T˜ = T ◦ I −1 : I0 → Y, then T˜ : X0 → Y is continuous and satisfies the hypothesis of Step I. So there is a regular Borel measure µ1 on S0 into R+ such that kT˜f kY ≤ kf k2,µ1 ,

f ∈ X0 .

Since for f ∈ X, we now have f˜ = I(f ) ∈ X0 , the following simplification obtains: kT f k2Y = kT˜f˜k2Y ≤ kf˜k22,µ , f ∈ X = (f¯f˜, µ1 ), since µ1 ∈ X∗0 , (·, ·) is duality pairing, = (I(f )I(f˜), µ1 ) = (I(f f˜), µ1 ), by the algebraic property of I, = (f f¯, I ∗ (µ1 )), I ∗ : X∗0 → X∗ is adjoint, Z = |f |2 (ω)µ2 (dw), Ω

where 0 ≤ µ2 = I ∗ (µ1 ) ∈ X∗ = ba(Ω, Σ), the bounded additive set functions on Σ with total variation norm. It is to be shown that the additive µ2 can be replaced by a σ-additive measure. III. Here we need to use the Carath´eodory procedure to complete in replacing µ2 with a σ-additive measure using this (Carath´eodory) procedure which needs a standard but additional argument. Let Σµ be the class of µ-measurable sets in this procedure. Then Σµ ⊃ Σ and µ(A) ≤ µ2 (A), A ∈ Σ, µ being σ-additive on Σµ , with equality only if µ2 is also σ-additive. Now (79) follows if (70) is valid for µ2 when f is a step function.

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Pm

We show this (for simple functions) as follows. Let f = i=1 ai χAi , Ai ∈ S Σ0 disjoint, ai 6= 0, and for given ε > 0, choose Aεin ∈ Σ, with ε Ai ⊂ ∞ n=1 Ain , and ∞

X ε µ(Ai ) + > µ2 (Aεin ). m|ai |2 n=1

(80)

S Replacing Aεin by Ai ∩ Aεin in Σ, if necessary, we take Ai = n≥1 Aεin , P ε ε ε . Then f in (80). Let fNε = m N ∈ X, fN → f as N → ∞ i=1 ai χ∪N k=1 Aik pointwise and boundedly. Hence Step II simplifies to:

Z

2 Z

ε 2 ε

≤ |fNε (ω)|2 µ2 (dω) kT fN ky = f (ω)ν(dω) N

Ω

=

n X

Ω

y

|ai |2

i=1

N X

µ2 (Aεik ), since µ2 is additive.

k=1

Letting N → ∞ on both sides, and using the usual vector integration properties,

Z

2

2

kT f ky = f (ω)ν(dω)

≤

Ω m X

Zi=1 =

y

ε 2 |ai | µ(Ai ) + , by (80), m|ai |

|f (ω)|2 µ(dω) + ε.

Ω

Since ε > 0 is arbitrary (79) follows for simple functions, and then as observed before, the result holds generally.  An application of the above result to more general classes of processes that include the (weakly/strongly) harmonizable families as introduced by H. Cram´er and K. Karhunen separately and will be recalled here to see the use as well as their potential for some applications. This can complement the analysis included already. Definition 2.4.4 A process {Xt , t ∈ R} ⊂ L20 (P ), (so it is centered) is of Cram´er class (general) if its covariance function r : (s, t) → C is expressible as the strong Morse-Transue integral

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Z Z r(s, t) = R

g(s, λ)¯ g (t, λ0 )F˜ (dλ, dλ0 ), s.t ∈ R,

(81)

R

relative to a class {g(s, ·), s ∈ R} of F˜ -integrable functions, where F˜ has finite Fr´echet variation. If F˜ (·, ·) has finite Vitali variation, then the integrals in (81) become Lebesgue-Stieltjes type, and the concept reduces to the classical (Cram´er) class, as originally introduced and studied above. [This difference turns out to be significant.] Regarding these two general classes of processes, we can present the following type of (quasi-) inclusion relations. Theorem 2.4.5 Let X : R → L20 (P ) be a process, and {g(t, ·), t ∈ R} a family of Borel functions. If X is a Karhunen process for this g(k, ·)family and a σ-finite measure F on B(R), and T : L20 (p) → L20 (p) is a bounded linear mapping, then Y (t) = T X(t), t ∈ R, is a Cram´er process for this g-family and a covariance bimeasure. On the other hand, if {g(t, ·), t ∈ R} is a bounded Borel family and X : R → L20 (P ) is a Cram´er process for this g-family and a covariance bimeasure, then there exists an extension space L20 (P˜ )(⊃ L20 (P )) determined by the given process, a Karhunen process Y : R → L20 (P ) for the same gfamily and a Borel measure on R with X(t) = QY (t), t ∈ R, where Q : L20 (P˜ ) → L20 (P ) is an orthogonal projection. Remark 14. 1. It must be observed that the g-family involved here is not of the character type {eitx , t ∈ R}, and so one cannot conclude that each Cram´er process can be obtained as a projection of some Karhunen process. In fact, Erik Thomas has constructed an example showing that this projection property, which is so true between the stationary and harmonizable classes, need not hold for the Cram´er and Karhunen classes! 2. The preceding remark indicates that the Cram´er class of processes is quite large and some of its members may not be dilated to Karhunen processes. It may be concluded however that the Cram´er class is closed under bounded linear mappings. Also clearly if g(t, λ) = eitλ , a character of R, the Cram´er process reduces to a weakly harmonizable process. The preceding discussion can be given an abstract form for applications and easy reference as follows, without further detail: Theorem 2.4.6 Let G be an LCA group and X : G → L20 (P ) be a (centered) second order random field and T : L20 (P ) → L20 (P ) be a

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bounded linear operator. If X is weakly stationary and Y = T X (or Y (t) = (T X)(t), t ∈ G), then {Y (g), g ∈ G} is weakly harmonizable. On the other hand, if Y : G → L20 (P ) is weakly harmonizable, there exists an extension space L20 (P˜ ) ⊃ L20 (P ), a weakly stationary X : G → L20 (P˜ ) such that Y (t) = QX(t), t ∈ G, where Q : L20 (P˜ ) → L20 (P ) is the orthogonal projection of L20 (P˜ ) onto L20 (P ). The point of this result is that all weakly harmonizable processes or fields are accounted for in this construction. However, the class of these super spaces may only have just L20 (P ) as a subspace. Moreover, this probabilistic result has an immediate operator theoretical consequence which may be of some interest and so we indicate it here. Proposition 2.4.7 A vector measure ν : B(R) → L20 (P ) is derived from a generalized spectral family. R Proof. Let X(t) = R eitλ dν(λ), for the vector measure ν, so that {X(t), t ∈ R} is weakly harmonizable and by the above result there is an extension space L20 (P 0 ) ⊃ L20 (P ), a self-adjoint operator A˜ on it, an X˜0 ∈ L20 (P 0 ) with, for t ∈ R, (cf., Thomas (1970)): Z Z itλ ˜ ˜ 0 , (82) ˜ ˜ eitλ (Q ◦ E)(dλ) X e ν(dλ) = X(t) = (Qg(t, A))X0 = R

R

˜ where {E(t), t ∈ R} is the resolution of the identity of A˜ in L20 (P ). If ˜ E(t) = QE(t), it is the resolution of the identity of A˜ in L20 (P 0 ), and ˜ 0 , and the result follows.  (82) implies ν(·) = E(·)X The above theorem also implies the following self-adjoint dilations of certain (abstract) operators in a Hilbert space. Theorem 2.4.8 Let A be a symmetric linear operator in a Hilbert space H with dense domain, and {gt , t ∈ R} be a family of bounded Borel functions, g0 = 1. Then {Tt = gt (A), t ∈ R} defines a set of bounded linear operators in an extension Hilbert space H1 ⊃ H, a self-adjoint ˜ Q : H1 → H is an operator A˜ on H1 extending A and Tt = Qgt (A), orthogonal projection. Conversely, every densely defined self-adjoint A˜ on a Hilbert space H1 , and a Borel function family {gt , t ∈ R} define closed operators ˜ H = gt (A)|H where A = QA, ˜ H = Q(H1 ), Q is an Tt = Qgt (A)| orthogonal projection on H.

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˜ = eitA˜ = Ut is unitary, and so Tt = Here gt (λ) R= eitλ ⇒ gt (A) ˜ = ˜ Qgt (A) eitλ (Q ◦ E)(dλ) gives a weakly continuous positive defR ˆ 1 . This implies the following reinite contractive operator on H = QH sult due to B. Sz.-Nagy ((1955), Theorem IV) which is obtained from the above result that depends only on Na˘imark’s theorem, and shows that both these are equivalent at a deeper level. Thus Sz.-Nagy’s (1955) theorem can be stated as follows: Theorem 2.4.9 Let {Tt , t ∈ R} be a weakly continuous positive definite contractive set of operators on a Hilbert space H, T0 = id; then there is an extension Hilbert space H1 ⊃ H0 , a unitary group of operators {Ut , t ∈ R} with Tt = QUt , t ∈ R. On the other hand every weakly continuous group {Ut , t ∈ R} of unitary operators defines {Tt = QUt , t ∈ R} a class of positive definite contractive family on H0 = QH1 for each orthogonal projection Q : H1 → H0 . An independent proof of the last theorem was given by Sz.-Nagy who then deduced Na˘imark’s theorem from his. But the above analysis shows that the opposite procedure is also valid so that both theorems are independently proved and deduced from each other implying a deep internal equivalence, of great interest. In the sense of the preceding two results, we include a final characterization of weak harmonizability of a process X : G → L20 (P ) comprising (and completing) the preceding work, the complete details of which are given in the author’s comprehensive paper (Rao (1982), Section 7) which uses several key results from abstract harmonic analysis. The desired result is the following: [details are in the above paper]. Theorem 2.4.10 Let G be an LCA group X = L20 (p) be separable. For a weakly continuous mapping X : G → X, the following statements (i)– (v) are equivalent: (i) X is weakly harmonizable, (ii) X is V -bounded, (iii) X is the Fourier transform of a regular measure on the Borel sets ˆ (the dual group of G) into X, of G ˆ the process Xp (= p · X) : G → L20 (P ) is (iv) for each p ∈ Lˆ1 (G), weakly harmonizable and bounded. Moreover, the above equivalent conditions are implied by the following assertion:

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(v) If X0 = sp{X(g), ¯ g ∈ G} ⊂ X, then there exists a weakly continuous positive contractive set of operators {T (g), g ∈ G} ⊂ B(X) with T (0) = identity, and X(g) = T (g)X(σ), g ∈ G. It will be of interest to restate the work on weakly harmonizable class, in an abbreviated way, for a quick reference with X = L20 (P ) as a separable Hilbert space on a probability space (Ω, Σ, P ): V = weakly continuous V -bounded random fields on G → X, w = weakly harmonizable random fields on G → X, F = the random fields which are Fourier transforms of regular ˆ → X, vector measures on G 1 ˆ ˆ of functions on G → X, that are M = the module over L (G) ˆ X (G), ˆ X (G)}, ˆ i.e., M = {X : G → X|X.Lˆ1 (G) ⊂ M ˆ in M P = the random fields X : G → X, which are projections of stationary fields on G → K, K ⊃ X is some extension (or super) Hilbert space of X. With the above abbreviations, the following comprehensive result holds: Theorem 2.4.11 We have: F = M = P = V = w. This result and the preceding one contain essentially all the known results on the structure of weakly harmonizable random fields (and process). Some applications and adjuncts of these results will be indicated in exercises below which should be of interest for other applications. 2.5 Multiple Generalized Random Fields A random field is a mapping observed at (t, x1 , . . . , xn ), say it denotes the time and an n-space, denoted K, with outcome X(t, x1 , . . . , xn ). The outcome typically depends on chance fluctuations and is abbreviated as Xτ . Thus K is an observation set at (t, x1 , . . . , xn ) with the random outcome at points of K into L2 (P ), so that one has X : K → L20 (P ) if the variables are centered. Then the problem of interest is thus an analysis of the “data” {X(f ), f ∈ K} ⊂ L20 (p), if the set of X(f )’s, termed random fields are assumed to have a natural structure for the index set K. This problem was abstracted and considered by K. Ito (1954) and I. M. Gel’fand (1955) independently and by several others thereafter. A natural space K now is the one introduced by L. Schwartz (1957), and the subject thereafter greatly developed, both in theory and applications.

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We use a few key facts of Schwartz’s theory in our applications, recalling the definitions, but referring the reader to L. Schwartz’s own book (1950), or to that of Gel’fand and Vilenkin (Translation 1964). We now can introduce a generalized random field (g.r.f.) in three ways: Definition 2.5.1 (a)Let K be the Schwartz space of infinitely differentiable (complex) functions on R, with compact supports (with each function). Then a linear mapping F : K → C is a generalized random field (grf) if fn ∈ K, fn → 0 in K (and they all vanish off a bounded set) along with all of their derivatives, implies F (fn ) → 0 in probability. (b)A linear mapping F : K → L2 (P ), is a grf if fn ∈ K, fn → 0 in K ⇒ F (fn ) → 0 in L2 (P ). (c) A map F : (Ω, Σ) → (K∗ , R) is a grf if it is B-measurable i.e., if B is the σ-algebra determined by {l ∈ K∗ : Re (l(f )) < c, Im (l(f )) < c2 , where c1 , c2 ∈ R, and f ∈ K}. It can be verified that a grf F with two moments, and defined as in (a), (b) or (c) above, all agree when compared so any one of these definitions can be used below as needed and there will be no conflicts. These ideas will be applied to the Cram´er class, which is the most general as seen above in this work. Definition 2.5.2 Let K be the Schwartz space on Rn and F : K → C be a second order grf, centered and B(·, ·) = ρ(·, ·) as its covariance functional. Then F (·, ·) is of class (C) relative to ρ and a Borel function g : Rn × Rn → C, if ρ(·, ·) determines a tempered measure ρ˜ on the Borel sets of Rn × Rn , in that (|x| = euclidean length) Z Z |d2 ρ˜(x, y)| (83) k < ∞, Rn Rn [(1 + |x|2 )(1 + |y|2 )] 2 R R g (s, x)d2 ρ(t, s) = b(x) exists, x ∈ for some k ≥ 0 and Rn Rn g(t, x)¯ Rn , b(·) is bounded on bounded sets of Rn , so that Z Z B(u, v) = u˜(t)v¯˜(s)d2 ρ(t, s), u, v ∈ K, (84) Rn

R

Rn

with u˜(t) = R g(t, x)u(x)dx as the g-transform of u, and similarly v˜. Then the g.r.f. F is called a generalized harmonizable random field, and B(·, ·) is of class (C) covariance (and harmonizable when g(t, x) =

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eitx , the complex exponential). The corresponding tempered measure ρ(·, ·) is called the spectral measure of F . We now present an integral representation of the random field F of class (C) which includes the harmonizable class, and this will be of real interest in many applications as well as extensions. Theorem 2.5.3 Let Φ be a test space on Rn as introduced above so that it contains K as a dense set, and F : Φ → L2 (P ) be a generalized random functional with mean zero and covariance B(·, ·) where F is of class (C) relative a g(·, ·) and tempered covariance ρ(·, ·). Suppose the R ˜ g-transforms of members of Φ exist (so f (t) = Rn g(t, x)f (x)dx exists, f ∈ Φ). Then there is a random measure Z(·) relative to ρ and that one has: Z f˜(t)dZ(t), f ∈ Φ (85) F (f ) = Rn

where f˜ is the g(·)-transform of f . Moreover, B(·, ·) is given as: Z Z B(u, v) = u˜(t)v¯˜(s)d2 β(t, s), u, v ∈ Φ. Rn

Rn

The above stochastic integral is defined in the mean-square sense. Conversely, if g(·, ·) and ρ(·, ·) are given with the properties of Definition 2.5.2 above, and Z(·) is a random measure on the Borel algebra of Rn , relative to ρ, and K ⊂ Φ is dense, then F (·) given by (85) is a grf of class (C) on K and has a continuous extension to Φ. Proof. I. We sketch the essential argument in steps for convenience, and take E(F (f )) = 0 and assume B(·, ·) is strictly positive definite so that B(f, f ) = 0 only for f = 0. Thus the inner product in Φ is now Z Z f1 (t)f¯2 (s)d2 ρ(t, s), f1 ∈ Φ. (86) (f1 , f2 ) = B(f1 , f2 ) = Rn

Rn

II. Let K0 = sp {F (f ) : f ∈ Φ} and K be its completion in L2 (P ), where (X, Y ) = E(X Y¯ ) for X, Y ∈ L2 (P ). Then from the equation (F (f1 ), F (f2 )) = B(f1 , f2 ) = (f1 , f2 ), the mapping f 7→ F (f ) defines an isometry of Φ onto K, and the extension onto K0 is unique and the map f 7→ F (f ) gives an isometry from L2 (P ) onto K. If Z(A) and χA , A ∈ B, (bounded) Z(A) ∈

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H, χA ∈ L2 (ρ), then Z : B → H defines a random measure relative to ρ. Since Z(·) is clearly additive, S its σ-additivity is verified as follows. Let Ai ∈ B, disjoint, A = ni=1 Ai , and be bounded. Then with the standard notation, we have:



n n



X X



Z(Ai ) = χA − χ Ai

Z(A) −



i=1 i=1 2 2



S

=

χ Ai i≥n

2

= ρ˜(∪i≥n (Ai × Ai )) ≤ ρ(A × A) < ∞, ρ˜ being the measure determined by ρ. It follows from this that Z(·) is σP∞ additive, and Z(A) = i=1 Z(Ai ) holds in L2 (ρ). For general bounded S1 , S2 in B, we get Z Z ¯ E(Z(S1 )Z(S2 )) = d2 ρ(x, y), S1

S2

so that Z(·) is a random measure relative to ρ(·). III. If Φ˜ = {f¯: f ∈ Φ}, the g-transforms of f , then Φ˜ ⊂ L2 (ρ), and this is a dense subspace, since by (86), (f˜1 , f˜2 ) = B(f1 , f2 ) = (f1 , f2 ). It follows that Φ ⊂ L2 (ρ) R ⇒ f → F (f ) is also an isometry, Pm and F (fn ) = i=1 ami Z(Ai ) = Rn f˜(t)dZ(t), kf − fm k → 0 shows {F (fm ), m ≥ 1} ⊂ H is Cauchy and so the above is well-defined, and Z F (f )n = f˜(t)dZ(t), f ∈ Φ Rn

is uniquely defined. This gives (85). IV. It is to be shown that the random field F (·) of (85) is of class (C). The linearity only its continuity is to be shown. Now R ofR F being clear, 2 ˜ ¯ B(f, g) = Rn Rn f (t)g˜(s)d (t, s), f, g ∈ Φ, so F (·) is of class (C), if it is shown to be continuous. But fn ∈ K ⊂ Φ, fn → 0 in K ⇒ the set {fn , n ≥ 1} is compactly based and tends to zero uniformly there. Since g(·, ·) is locally bounded, it follows that fn → 0 a.e. and boundedly. It then is seen to follow that B(fn , fn ) → 0 so F (fn ) → 0 in H. Thus F (·) is continuous on H. Since K ⊂ Φ, with a stronger topology, F (·) is continuous on K in the topology of Φ also. Now K is dense in Φ, F has a unique extension to Φ, and so is a g.r.f. of class (C). 

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The following consequences of the preceding result are useful. Corollary 2.5.4 Let F : Φ(= K) → L2 (P ) be a g.r.f. of class (C) for a tempered covariance ρ and g(·, ·). Then F admits the representation relative to a random measure Z : B → L2 (P ) so that Z Z Z ˜ F (f ) = f (t)dZ(t) = f (x)g(t, x)dxdZ(t), f ∈ K, (87) Rn

Rn

Rn

uniquely. Conversely, a functional F : K → L2 (p) given by (87) with Z(·), g(·, ·) and ρ(·) is a g.r.f. of class (C) relative to g(·, ·) and ρ. This specializes to the harmonizable class which is given as follows: Corollary 2.5.5 Let the test space K satisfy K ⊂ Φ ⊂ L, the inclusions being both algebraic and topological. If F : Φ → L2 (P ) is a grf which is harmonizable relative to a tempered covariance ρ, then there is a random measure Z : B → L2 (P ) relative to ρ satisfying Z Z Z ˆ F (f ) = f (t)dZ(t) = eitx f (x)dxdZ(t), f ∈ Φ, (88) Rn

Rn

Rn

uniquely. Conversely, F given on Φ by (88) with Z(·) and ρ(·) is a generalized harmonizable random field relative to ρ. With the above discussion, we can present a multivariate Cram´er representation of the corresponding class for reference. Theorem 2.5.6 Let Φ be the (test) space as in Theorem 2.5.3 above. If now F = (F1 , . . . , Fk ) is a k-dimensional grf on Φ of class (C) with respect to a g(·, ·) and ρ = (ρij , 1 ≤ i, j ≤ k) of tempered covariances forming a positive definite matrix, then there exists a random vector measure Z = (Z1 , . . . , Zk ) on the Borel field of Rn such that F (·) is uniquely representable as Z F (f ) = f˜(t)dZ(t), f ∈ Φ, (89) Rn

where f˜ is the g-transform of f , defined before. Conversely, if g(·, ·), ρ, and a random vector measure Z relative to ρ are given, satisfying (89), then F (·) defines a k-dimensional class (C) random field on Φ, relative to g(·, ·) and ρ(·). We next present the corresponding harmonizable form which gives an immediate (extended) version of K. Ito (1954) and A. M. Yaglom (1957).

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Theorem 2.5.7 Let Φ be a complete countably normed space (such as K(Mr )) so that fn ∈ Φ, fn → 0 in Φ, then fn (x) → 0, x ∈ Rn , in the topology of Φ, and let F : Φ 7→ L2 (P ) be a grf, centered, with covariance B(·, ·). If F is ofRclass (C) relative to g(·, ·) and a tempered covariance ρ, let f˜(t) = Rn g(t, x)f (x)dx exist for all f ∈ Φ. Then there is a random measure Z(·), relative to ρ, such that the following unique representation holds: Z f˜(t)dZ(t), f ∈ Φ, (90) F (f ) = Rn

where f˜ is the g-transform of f and we have Z Z u˜(t)v¯˜(s)dρ(t, s), B(u, v) = Rn

u, v ∈ Φ.

(91)

Rn

The stochastic integral in (90) is in the mean square sense. Conversely, if g and ρ are as in Definition 2.5.2, and Z(·) is a random measure, on the Borel sets of Rn , relative to ρ, and K ⊂ Φ is dense, then F given by (90) is a g.r.f. of class (C) on K, with unique extension to Φ. Proof. Let F be a grf of class (C) with covariance B(·, ·), so that we have (E(F (f )) = 0 is assumed, and) Z Z (u, v) = B(u, v) = u˜(t)˜ v (s)d2 ρ(t, s), u, v ∈ Φ. Rn

Rn

The hypothesis on Φ implies by the earlier work, B(u, v) = G(u¯ v ) for some G ∈ Φ2 , where Φ2 = Φ × Φ. Since F has orthogonal values, and (X, Y ) = E(X Y¯ ) for X, Y ∈ L20 (P ), one has (F (f1 ), F (f2 )) = B(f1 , f2 ) = (f1 , f2 )

(92)

so that the relation f1 7→ F (f1 ) defines an isometry of Φ onto H0 = sp{F ¯ (f ) : f ∈ Φ} ⊂ L2 (P ). If A = ∪∞ i=1 Ai , Ai ∈ B, disjoint, then

2

2 n n



X X

2



Z(Ai ) = χA − χAi = χ∪i≥n Ai

Z(A) −



i=1

i=1

= ρ (∪i≥n Ai × Ai ) ≤ ρ(A × A) < ∞ where ρ(·) is the measure determined by “ρ(s, t)”. Since Z(·) is a vector measure (σ-additive!) and so Z(·) is σ-additive, we get

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Z Z E(Z(s1 )Z(s2 )) = (χs1 , χs2 ) = s1

93

d2 ρ(x, y),

s2

implying that Z(·) : B → X is a random measure relative to ρ. Consider the class Φ˜ ⊆ {f˜ : f ∈ Φ} of the g-transforms, so that ˜ Φ ⊂ L2 (ρ) as a dense subset. Indeed we have by (92): Z Z ˜ ˜ (f1 , f2 ) = f˜1 (t1 )f˜2 (t2 )d2 ρ(t1 , t2 ) = B(f1 , f2 ) = (f1 , f2 ) Rn

Rn

and f 7→ f˜ is an isometry, so f˜ ∈ L2 (ρ). Since Φ ⊂ L2 (ρ) is dense so ˜ Thus f˜ ∈ Φ˜ is approximable in L2 (P )-norm and so Φ˜ is dense in is Φ. 2 L (ρ). Hence f˜m =

m X

ami χAi ∈ L2 (ρ),

kf˜ − f˜m k → 0, as n → ∞.

i=1

We can conclude that f 7→ F (f ) is an isometry and ifR fn ∈ L2 (ρ) P ˜ corresponds to f˜m , where F (fn ) = m i=1 ami Z(Ai ) = Rn f (t)dZ(t). Then kf − fn k → 0 so that {F (fm ), m ≥ 1} ⊂ H is Cauchy, and hence Z f˜n (t)dZ(t), f˜n ∈ L2 (ρ), (93) F (fn ) = Rn

is well-defined for step functions, R and then for all f ∈ Φ, by the isometry seen above. Hence F (f ) = Rn f˜(t)dZ(t), f ∈ Φ, is defined uniquely. This establishes (90). It remains to verify that F (·) is of class (C) in order to complete the argument. Thus let Z(·), ρ and g(·, ·) be as in the (converse) hypothesis. Since F (·) defined above is clearly only its continuity has to be shown. R linear, R ˜ It is clear that B(f, g) = Rn Rn f (t)˜ g (s)d2 ρ(t, s), for f, g ∈ Φ so that it is of class (C) if it is shown continuous. Now fn ∈ K ⊂ Φ, fn → 0 in K implies fn ’s are compactly based on a fixed set and converge uniformly there to zero, and are bounded. Interchanging the limit and integral which is permissible, we get B(fm , fn ) → 0 so F (fn ) → 0 in K, and F (·) is continuous on K. Since the topology of K(⊂ Φ) is stronger than that of Φ, we see that F (·) is continuous on K in the topology of Φ, and by the density of the former, F has a unique continuous extension to Φ, and hence is a g.r.f. of class (C) on Φ as desired.  The following specialization is of interest in applications, comparing harmonizable and more general Cram´er classes. Also, as noted before,

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class (C) is the most general class Φ and it includes all the known applications of interest. Corollary 2.5.8 Let K ⊂ Φ ⊂ S(= K(M2 )), the Schwartz space, [i.e., the infinitely differentiable functions on Rn , which are fast decreasing in that |x|n |Dα f (x)| → 0 as |x| → ∞, n ≥ 1] these inclusions being algebraic as well as topological. If F : Φ → L2 (P ) is a g.r.f. which is harmonizable relative to a tempered ρ, then there is a random measure Z : B → L2 (P ), relative to ρ such that we have the unique integral representation for f ∈ Φ as: Z Z Z ˆ F (f ) = f (t)dZ(t) = eitx f (x)dxdZ(x). (94) Rn

Rn

Rn

Conversely, F on Φ defined by (94) with Z(·) and ρ(·), is a generalized harmonizable random field relative to ρ. The result follows from the theorem since Φ ⊂ S and the Fourier transform on S is an onto isomorphism. We now present a general (multidimensional) representation of a set of random fields that are related to the Cram´er class which clarifies the structure of the problem, indicating some extensions. We next consider a somewhat more general class of test spaces K(Mr ), of Gel’fand-Shilov-Vilenkein type in which the weights {Mr (x), r ≥ 1} satisfy a growth condition called nuclearity, also denoted (N), and is defined as: for each r ≥ 1, there is r0 > r so that mr,r0 (x) = Mr (x)/Mr0 (x), is Lebesgue integrable on Rn and mr,r0 (x) → 0 as |x| → ∞. [It can be shown that mr,r0 (x) → 0 as |x| → ∞, and mr,r0 (·) is then Lebesgue integrable on Rn .] This gives a general form of the class (C) g.r.f.’s of use in our study. Proposition 2.5.9 Let Φ be a test space which is a K(Mr ) space where the sequence {Mr , r ≥ 1} satisfies the condition (N) above. If F : Φ 7→ L2 (P ) is a grf of class (C) relative to a g(·, ·) and a tempered covariance ρ and F (·) has orthogonal values so that F (f1 ) ⊥ F (f2 ) if f1 · f2 = 0, then ρ concentrates on x = y and has a tempered measure σ : B(Rn ) → R+ and a random measure Z : B(Rn ) → L2 (P ) relative to σ, with orthogonal values, such that we have: Z Z Z F (f ) = f (x)g(t, x)dxdZ(t) = f˜(t)dZ(t), f ∈ Φ, (95) Rn

Rn

Rn

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uniquely f˜ being the g-transform. Conversely, F on Φ defined by (95) relative to g, Z, and σ is a grf of class (C) with orthogonal values. Proof. For a grf of class (C) with covariance B(·, ·), we have Z Z u˜(t)˜ v (s)d2 ρ(t, s), u, v ∈ Φ, B(u, v) = Rn

Rn

and by hypothesis on Φ, B(u, v) = G(u¯ v ) for some G ∈ Φ∗ . Since F is orthogonally valued and the weights {Mr , n ≥ 1} satisfy the hypothesis (N), it follows from Gel’fand and Vilenkin ((1964), p. 287), that f ∈ Φ × Φ vanishing in a neighbourhood of x = y, implies G(f ) = 0. Thus G is only concentrating on the diagonal x = y and since Gm = G|Km 2 (Mr ), has the same property, one has Z Z Z Z g(t, x)g(s, y)d2 ρ(t, s)dxdy u(x)˜ v (y) Rn Rn Rn Rn X Z Z Mpm (x)Mpn (y)hα,β (x, y)Dα u(x)Dβ u(y)dxdy, = |α|+|β|≤pm

Rn

Rm

for u, v ∈ Km (Mr ). But Gm concentrates on x = y implying that hα,β again vanishes away from x = y, so that ρ(·, ·) in the above concentrates on x = y also as the contrary hypothesis easily leads to a contradiction. Thus Z ρ˜(A × A2 ) = dρ(x, x) = ρ(A1 ∩ A2 ), (96) A1 ∩A2

and the temperedness of ρ˜ implies that of ρ. Thus we are back to the argument of Theorem 2.5.7, and the existence of Z(·) as stated in (95) obtained. The converse follows immediately from the condition (95) as in that theorem, and since Φ is the inductive limit of the spaces Km (Mr ) with orthogonal values, it follows from known properties of such spaces that F is a grf on Φ itself.  As a simple multidimensional extension of Theorem 2.5.7 and the above proposition, we present the following result for ready use. Theorem 2.5.10 Let Φ be the test space as in Theorem 2.5.7 above, and let F = (F1 , . . . , Fk ) be a k-vector g.r.f. on Φ of class (C) relative to some g(·, ·) and a positive definite matrix ρ = (ρij , i, j = 1, . . . , k) of tempered covariances also relative to g(·, ·). Then there is a random vector Z = (Z1 , . . . , Zk ) on the Borel field of Rn relative to ρ such that

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Z F (f ) =

f˜(t)dZ(t),

f ∈ Φ,

(97)

Rn

uniquely where f˜(·) is the g-transform of f . Conversely, if g(·, ·), ρ(·) and a random vector Z relative to ρ, are given, then (97) defines a k-vector g.r.f. on Φ of class (C), relative to g(·, ·) and ρ(·). The P result is established by considering the scalar product a · F (= ni=1 ai Fi ) for an arbitrary real vector ‘a’, and following the familiar arguments. Note that if each ρij (·, ·) concentrates on the diagonal x = y, then the above result reduces to the stationary g.r.f., and the converse can be obtained that includes both K. Ito (1956) and A. M. Yaglom (1957) theorems (cf. next result), but we are concentrating on Cram´er’s extensions. Theorem 2.5.11 Let K be the Schwartz space of infinitely differentiable scalar functions on Rn with compact supports and C˜ be the space of complex random variables with means zero and finite variances. If F : K → C˜ is with mean zero and covariance functional B(·, ·) of compact support, then it can be represented as: Z Z B(f, g) = f (x)g(y)h(x, y)dxdy, f, g ∈ K, (98) Rn

Rn

where h(·, ·) is a continuous (ordinary) covariance function on Rn × Rn based on a compact set, (so h(·, ·) is determined by B). ˜ being a coProof. By definition of the functional B : K × K → C, variance, and the defining property of F , it is continuous and hermitian as well as bilinear. By the famous Kernel theorem of L. Schwartz, (see e.g., Gel’fand and Vilenkin (1964), p. 74) there is a continuous bilinear form G : K2 (Rn × Rn ) → C with compact support and if (f · g)(x, y) = f (x)g(y) so f, g ∈ K 7→ f · g is in K2 that B(f, g) = G(f · g) then one has the integral representation as Z Z f (x)g(y)h(x, y)dxdy, f, g ∈ K, B(f, g) = G(f · g¯) = Rn

Rn

for a unique h(·, ·) with compact support. But B(·, ·) is a positive definite Hermitian functional so that the f · g ∈ K form a dense set, and so it is easily seen that h satisfies (98) as desired. 

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Recall that a random field F : K → C has orthogonal values if f, g ∈ K are disjointly supported then F (f ) ⊥ F (g), i.e. are orthogonal. The following result extends the earlier representations of stationary generalized fields of K. Ito (1954) and A. M. Yaglom (1957) to those of Cram´er class (C) which is of interest in our study. Theorem 2.5.12 Let F : K → C be a generalized random field of class (C) having orthogonal values. Then there exists a random measure Z(·) on the Borel field of Rn with orthogonal increments such that Z F (f ) = f˜(t)dZ(t), f ∈ K, (99) Rn

R where f˜(t) R= Rn f (x)g(x)dx, and the covariance B of F is given by B(u, v) = Rn u˜(t)v¯˜(t)dσ(t), the tempered measure σ on Rn being the spectral measure of F , relative to the g(·) of class (C). Conversely, if Z(·) is the random measure with orthogonal values, relative to σ, then F given by (99) is a generalized random field on K of class (C) with orthogonal values. Proof. Let F : K → C˜ be of class (C) and suppose it is strictly positive definite, for simplicity, so B(f, f ) = 0 only if f = 0. Let L2 (ρ) ⊃ K be the completion of K for the inner product Z Z f˜(x)g¯˜(y)d2 ρ(x, y), f, g ∈ K, (100) (f, g) = B(f, g) = Rn

Rn

R where f˜, g˜ are the g-transforms (i.e., f˜(x) = Rn f (t)g(t, x)dt the g(·, ·) being the defining “g-element” of (C)). If F has, on K, orthogonal values, then B of (100) is concentrated on the set x = y. The proof of this statement is essentially the same as in Gel’fand and Vilenkin ((1964), p. 287). Since F is of class (C), ρ˜ also must concentrate on x = y. If the resulting measure is denoted by σ, then the result is seen to be a simple consequence of Theorem 2.5.11, and further details can now be left to the reader.  In relation to A. V. Balakrishnan’s characterization of the mean functions of a second order process with a given (stationary) covariance function, detailed in Chapter 1, it is of interest to know the structure of the class of all mean functions of a process with a given covariance

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function. Not surprisingly, it is not a linear set in general as the following example due to Skorokhod (1970) shows. Let (Ω, Σ, P ) be a canonically represented probability space for (Xt , t ∈ T ) where Ω = RT , Σ is the σ-algebra of Ω relative to which each Xt (·) : Ω → R, is measurable t ∈ T . If Y (t) = X(t) + f (t), a translate of the X-process, then f (·) is an admissible mean (or translate) of X(·) if Pf (·) = P ◦ τf−1 (·), where τf Xt = Xt + f (t) is P -continuous or Pf  P , (where Ω = RT as noted above). Let f0 ∈ Ω − Mρ where Mp denotes the set of all admissible mean values of P so that f ∈ Mp implies Pf  P where Pf is the measure of the translated process Yt = Xt + f (t), defined above. Assume that the X-process is Gaussian with mean zero, and r(·, ·) as covariance. It is now known that for f9 ∈ Ω − Mp , the Gaussian measures P0 and Pf0 are mutually singular (P0 ⊥ Pf0 ) by the well-known H´ajek-Feldman theorem. Define the mixture measure on Σ as: ∞ ∞ X X αk = 1. (101) αk Pkf0 , αk > 0, Q= k=−∞

k=−∞

Then f0 ∈ MQ but tf 6∈ MQ , 0 < t < 1. Thus MQ is not convex, although it is a semi-group under addition. The following positive result is due to Pitcher (1963), and is stated for comparison and information: Proposition 2.5.13 Let {Xt , t ∈ T } ⊂ L2 (P ), E(Xt ) = 0, r(s, t) = ¯ t ), r(·, ·) being continuous on the compact interval [a, b]. Then E(Xs X 2 2 the = R operator R : L (T, dt) → L (T, dt) defined by1/2(Rg)(s) 2 r(s, t)g(t)dt, is positive definite, compact, and M ⊂ R (L (T, dt)), p T so f ∈ Mp ⇒ f = R1/2 h, for some h ∈ L2 (T, dt). Let {Xt , t ∈ T } be a second order process on (Ω, Σ, P ), a probabil¯ t ), the covariance. ity space, with E(Xt ) = 0 and K(s, t) = E(Xs X Let MP be the set of all admissible means of the X-process, so that the probability measures P and Pf , f ∈ MP , are such that Pf  P . dP The structure of MP is clearly useful and one can ask for dPf for such a class of f and so the space MP will be of interest in applications. If P is Gaussian then this space of “admissible means” MP can be characterized now. The space MP may be seen to be a vector space, but its topological characterization will be useful for applications. In fact an inner product can be introduced and an interesting analysis of

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it may be given. We Pnthus introduce for each covariance K, a space HK = {f : f = i=1 ci K(si,· ), n ≥ 1 and ci ∈ C}. If f, g ∈ HK define a complex number (f, g): (f, g) =

n X m X

K(si , tj )ci d¯j =

i=1 j=1

where f =

Pn

i=1 ci K(si , ·)

n X

ci g¯i (si ) =

i=1

and g =

Pm

j=1

m X

dj f (tj ),

(102)

j=1

dj K(·, tj ). We then have:

Proposition 2.5.14 The space of admissible means (Hk , (·, ·)) of a Gaussian process is a Hilbert space. Moreover for each f ∈ Hk , there is a Z ¯ t ) = f (t) and the likelihood ratio is given by such that E(Z X dPi 1 = exp{Z − E(|Z|2 )}, a.e. [P ]. dP 2

(103)

Proof. The demonstration depends on a basic result that two Gaussian measures in Rn are either mutually absolutely continuous or singular; known as the H´ajek-Feldman theorem [proved in the comparison volume (2nd ed. 2014, p. 226)], and the reader is referred to the book for details which we omit here. Also as a byproduct of the demonstration there, the useful formula (103) emerges, which is also in the above book on p. 230, and we omit the detail referring the reader to it.  The general problem of characterizing admissible means for process with a given covariance function is not easy, in contrast to Balakrishnan’s work noted earlier. We give a solution in Theorem 2.5.15 below which is indicative of the nature of the problem. Using the methods and results of harmonic analysis we can present a general characterization of the set of admissible means of a second order process of a given covariance, supplementing Proposition 2.5.13 above, motivated by an idea of Hida and Ikeda (1967), for second order (not necessarily Gaussian) processes of interest in the general study. So let Ω = RT and D ⊂ Ω be the class of all Rα , α ⊂ T , finite subsets of T . If α, β ∈ D, α < β (to mean α ⊂ β), let Πα : Ω → Rα , and Παβ : Rβ → Rα , (α ⊂ β) so Πα = Παβ ◦ Πβ , and Παα = identity, Παβ ◦ Πβγ = Παγ for P α < β < γ. Then Ω = lim← (Rα , Πα ) the projective limit. We take = σ(∪α Πα−1 Bα ) and P ◦ Πα−1 = Pα on Bα . [The detailed presentation is given in the paper (Rao (1995), p. 546).] Thus τ : L2 (P ) → HC , is an isometric isomorphism onto, where HC is the Aronszajn space of continuous functions in Ω ∗ , the dual of the space

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Ω = lim← (Rα , Πα ) where Ω = RT noted above. We use some simple properties of HC as recalled in Chapter 1 and above. Theorem 2.5.15 Let L2 (P ) be as above on (Ω, Σ, P ) and HC and τ : L2 (P ) → HC be defined by Z eil(φ) φ(ω)dP, φ ∈ L2 (P ), l ∈ Ω ∗ (104) (τ φ)(l) = Ω

is an onto isometric isomorphism, and MP is the set of admissible means for P . Then τ induces a linear onto isometry between L2 (P ) and HC . ˜ P = {f ∈ MP : ρkf ∈ L1 (Pkf ), k ≥ 1} where ρf = dPf so that Let M dP ˜ P if and only if exp {i(·, nf )} ∈ Hc for n ≥ ρkf exists a.e. Then f ∈ M ˜ P is a positive cone (i.e., f ∈ M ˜ P ⇒ αf ∈ M ˜ P , α ≥ 0) 1, and then M iff we have X α (J − I)m ρf (*) n n≥0

exists in L2 (P ) where J k ρf = ρkf , k ≥ 0 and I is the identity. If (*) ˜ , then sp(M ˜ P ) ⊂ MP . In particular holds for all α ∈ R and f ∈ M if for each f ∈ MP , there is a 0 < Kf < ∞ such that Pf (A − f ) ≤ ˜ P = MP . In particular if for each f ∈ MP , Kf P (A), A ∈ Σ, then M there is a 0 < kf < ∞ such that Pf (A − f ) ≤ kf P (A), A ∈ Σ, then ˜ P = MP in the above representation. M Proof. We include the argument for convenience and real feeling. The previous discussion shows that τ of (104) is an onto isometric isomorphism. The fact that MP is a semi-group implies kf ∈ MP , k ≥ 1, ˜ P ⇒ kf ∈ M ˜ P for k ≥ 0. So and so ρkf exists. Thus f ∈ M 2 ρkf ∈ L (P ) and Cnf = C exp(i(·, nf )) ⊂ HC . In the opposite direction, if Cnf ∈ H, n ≥ 1, then ρn,f = τ −1 (Cnf ) ∈ L2 (P ) and Z Z i(`,ω) τ (ρnf )(`) = Cnf (`) = e dPnf (ω) = ei(`,ω) ρnf (ω)dP (ω). Ω

Ω

(105) nl The uniqueness of the Fourier transforms gives ρnf = dP , a.e., and dP 1 ˜P . then ρnf ∈ L (Pnf ), n ≥ 1, so that f ∈ M ˜ P , so ρkf ∈ L2 (P ), and τ (ρkf ) = Ckf . Next let α ≥ 0 and f ∈ M Suppose that the condition (*) holds, so the series converges in L2 (P ). Since τ is a bounded linear map on L2 (P ), we can interchange it with

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the summation of (*) and the series converges in L2 (P ). Since τ is a bounded linear mapping on L2 (P ), and HC is complete, we get, ∞   X α τ ((J − I)k ρf ) ∈ HC . (106) k k=0

Now from J k (ρf ) = ρkf = τ −1 (Ck ), one can define n   n   X X n n n k τ ((J − T ) ρf ) = (−1) Ckf (·) = (−1)k eik(·,f ) (·). k k k=0 k=0 (107) If χf = exp(·, f ) (a character) so χtf = (χf )t , and (106) and (107) ⇒ ∞ ∞  X n X X α k g (n) (1)(χ − 1)n C(·) (−1) χkf C(·) = n n=0

k=0

n=0

= g(χf )C(·) ∈ HC , (108)
dn y where g(χ) = χα and g (n) (1) = dx (1). It then follows that χαf ∈ n −1 2 HC ⇒ ρkf = τ (Cαf ) ∈ L (P ). Thus with α for n there, shows ˜ P ⊂ MP . ραf ∈ L2 (P ) and αf ∈ M ˜ P , then Cα,f ∈ MP , and so (108) holds. For the converse, let α ∈ M −1 Apply τ to both sides of this result. It follows that the series converges in L2 (P ) and (*) holds and ρk f ≤ kf a.e. so ρkf is bounded, and it is ˜ P = MP . If (*) also holds for all α ∈ R and (f, g) ∈ M ˜f seen that M then the pair (αf, βg) also is in it. By the semi-group property of MP , it ˜ P ) = MP as desired.  follows that sp(M Remark 15. In the Gaussian case, the condition (*) above is automatic ˜ P = MP . and M The following result, due to T. S. Pitcher (1963), does not use the Aronszajn space methods and is based on Karhunen’s expansion for second order processes (not restricted to Gaussian) we state the interesting result, referring to the original paper for the proof. Theorem 2.5.16 Let X = {Xt , −∞ < a ≤ t ≤ b < ∞} ⊂ L20 (P ) be ¯ t ). If {λn } a process with a continuous covariance r : (s, t) 7→ E(Xs X and {φn } are the eigenvalues and the corresponding eigenfunctions of r, define Xn and fn by the integrals (T = [a, b])

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Xn =

−1 λn 2

Z X(t)φn (t)dt, fn = T

−1 λn 2

Z f (t)φn (t)dt, n ≥ 1,

(109)

T

so {Xn , n ≥ 1} are orthogonal and {fn , n ≥ 1} ∈ l2 . Let Pn , pn be the distribution and density of {Xi , 1 ≤ i ≤ n} relative to the Lebesgue measure. Suppose the {pn , n ≥ 1} satisfies the conditions: (i) pn > 0, a.e., (ii) limti →∞ pn (t1 , . . . , tn ) = 0, 1 ≤ i ≤ n, for all ti , and (iii) Pn R  ∂ log pn 2 ∂pn , 1 ≤ j ≤ n exists and dP ≤ K0 < ∞, n ≥ 1. j=1 Rn ∂ti ∂ti Then the set M1 = {f ∈ Mp : f = {fn , n ≥ 1} ∈ l1 } is a positive P 1 cone. Further, M1 = MP holds if also n≥1 λ− 2 < ∞. If each pm is symmetric about the origin of Rn , then M1 is also linear. In general MP need not even be a convex set. We omit the proof, referring it to Pitcher (1963). The point of this result is to emphasize that the set of means is not generally linear for non Gaussian processes. The proof of the result itself depends on an approximation result of semi-groups of operators, due to Trotter (1958), in Banach spaces. The point of the result is to emphasize that the structure of the set of means of (second order) processes is involved and exemplifies the earlier discussion of nonlinearity! We next turn to studying some useful functionals based on certain local properties. 2.6 Local Functionals in Probability; Their Integral Representations and Applications The classical probability problems involving sums of independent random variables have been generalized by the Russian mathematicians I. M. Gel’fand, N. Ya. Vilenkin and their associates to the class of (continuous) linear functions on smooth function spaces (e.g., infinitely differentiable functions on Rn ) with independent values. The ‘smooth’ space considered here is denoted by K, the L. Schwartz space of infinitely differentiable real functions (on R) vanishing off compact sets and l : K → F where F is a topological vector space of (scalar) random variables on (Ω, Σ, P ), with l(f1 + f2 ) = l(f1 ) + l(f2 ) for f1 · f2 = 0, (fi ∈ K), `(fi ), i = 1, 2 independent, such a fundamental l(·) is called local, and its characterization under suitable conditions is desired. It is a vast generalization of classical limit laws for sums of independent random variables. Here is a characterization, with some applications, handled by Gel’fand and were obtained by the author.

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Theorem 2.6.1 Let Cc (Ω) be the space of continuous compactly supported real functions on a locally compact space Ω, and Λ : Cc (Ω) → R be a mapping (called a local functional) satisfying (i) (Sequential continuity) If {fn , n ≥ 1} ⊂ Cc (Ω) is a pointwise convergent bounded set, then {Λ(fn ), n ≥ 1} ⊂ R is Cauchy. (ii) (Additivity) Λ(f1 + f2 ) = Λ(f1 ) + Λ(f2 ) if f1 · f2 = 0 for which the quantities are defined. (iii) (Bounded Uniform Continuity) For each ε > 0, γ > 0 there is a δ(= δε,γ ) > 0 with kfi k = γ, fi ∈ Cc (Ω), i = 1, 2, kf1 − f2 k ≤ δ implies |Λ(f1 ) − Λ(f2 )| < ε, where k · k is the uniform norm. Under these conditions, Λ(·) admits an integral representation as: Z Λ(f ) = Φ(f (ω), ω)µ(dω), f ∈ Cc (Ω) (110) Ω

where µ is a finite regular Borel measure on Ω, and Φ : R×Ω → R satisfies the following three conditions: (a)Φ(0, ω) = 0, and Φ(·, ω) is continuous for a · a · ω ∈ Ω, (µ), (b)Φ(x, ·) is µ-measurable for all x ∈ R, (c) for f ∈ Cc (Ω), Φ(f (ω), ω) is bounded for a · a · ω ∈ Ω, and for sequences {fm : n ≥ 1} as in (i) {Φ(fn , ·), n ≥ 1} is Cauchy in L1 (µ). Conversely, if the pair (Φ, µ) satisfies conditions (a)–(c) above, then it is a local functional on Cc (Ω) for which the statements (i)–(iii) hold. Before establishing the theorem we first show, as a consequence, that the classical Riesz–Markov theorem follows immediately. This gives a motivation for applications and other uses later on. Theorem 2.6.2 (Riesz-Markov) Let Ω be a locally compact space and Cc (Ω) be the space of real continuous functions with compact supports. If l : Cc (Ω) → R is a positive linear functional, then there exists a unique regular Borel measure µ on the Borel σ-ring of Ω such that Z l(f ) = f (ω)dµ(ω), f ∈ Cc (Ω). (111) Ω

Proof (of Theorem 2.6.2). This result is a significant generalization of the classical F. Riesz’s representation stating that a positive linear functional l(·) on Cc (Ω), the space of continuous scalar (in real or complex)

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functions on a locally compact space Ω, vanishing off compact sets, corresponds to a finite Borel measure µ ≥ 0, such that Z l(f ) = f (w)dµ(w), f ∈ Cc (Ω), (112) Ω

and we present a quick proof of the general formula (111), containing (112). Since l(·) is a continuous linear functional, |l(f )| ≤ K0 kf k for some 0 ≤ K0 < ∞, and f ∈ Cc (Ω). In particular l(·) is local. So by Theorem 2.6.1, there is a finite signed Borel measure verifying (110). Consider the set Z Z Z C = {A ⊂ Ω : Φ(f +g, ·)dµ = Φ(f, ·)dµ+ Φ(g, ·)dµ}. (113) A

A

A

If lS = l|C(S), for each compact S ⊂ Ω, then lS is continuous and linear satisfying lS1 = lS2 = ls1 ∩s2 on C(S1 ∩ S2 ) for any compact subsets S1 , S2 of S, and µS (·) = µ(S ∩ ·) represents lS in (110). Thus T of (113) contains compact, and then B is a Baire subset of Ω1 and lB (·) is a local functional, satisfying (i)–(iii) of Theorem 2.6.1. It follows from this that T contains each Borel set, and lB (·) is again a local functional satisfying the conditions of Theorem 2.6.1. Hence C ⊃ B, the Borel σ-algebra, and for a · a · (w). Φ(f + g)(w) = Φ(f )(w) + Φ(g)(w),

w ∈ Ω.

(114)

However, one can take Φ(·, w) to be continuous for each w. Since f, g are arbitrary (114) can be identified with the classical Cauchy functional equation. Then the well-known (and familiar) solution is Φ(f (w), w) = β(w)f (w), w ∈ Ω for some Borel function β : Ω → R, which R must be also integrable from the hypothesis of Φ, (we use the fact that Ω f gdµ ∈ R for each g ∈ L∞ (µ) ⇒ f ∈ L1 (µ)). Let dν = βdµ and then the (local but now) linear functional l(·) satisfies Z l(f ) = f (w)dν(w), f ∈ Cc (Ω). (115) Ω

But this is the same as (111), and it also implies as a byproduct that a bounded linear l : Cc (ω) → R is representable as l = l1 − l2 , li positive.  With the preceding important and motivational application, we now outline the main representation theorem for local functionals whose

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quite long proof is only presented in outline here for convenience. [Complete details are in the author’s Measure Theory book (Rao (1987), pp. 456–486 or second Ed. (2004), pp. 676–684).] Proof sketch of Theorem 2.6.1. Here we use a series of steps in establishing the result. 1. Initial simplification: From Cc (Ω), of compactly based scalar continuous functions, consider all bounded pointwise limits from Cc (Ω), denoted by B0 (Ω) which is a vector space of bounded Baire functions, vanishing at infinity, so that each {ω : |f (w)| ≥ ε > 0} is compact. Then C0 (G) is uniformly dense in B0 (G). If A is compact and U open with A ⊂ U (⊂ G), then let for h > 0, phA,U ∈ C∞ (G) with phA,U = h on A = 0 off U , (possible to find by Urysohn’s lemma) to be called a peak function of height h and base A; and phA,U ↓ hχA . So χA ∈ B0 (G). If A, B are compact Baire sets then χA−B , χA∪B are in B0 (G), and T = {A : χA ∈ B0 (G)} is a ring containing all compact Baire sets. Also, T includes the ring R of compact Baire sets. This reduction is needed for the ensuing analysis. 2. Next, we extend the (local) functional l(·) to Bc (G) from Cc (G), the compactly supported continuous function space, on to the locally compact G. If fn ↓ χA , so that {l(fn ), n ≥ 1} is Cauchy in R, its limit ˜ (χA ) = limn M (fn ), and extends. The generalized functional M ˜ (·) is M additive on simple functions on R. ˜ (hχA ). Then µh (·) is 3. For each A ∈ C, h ∈ R, let µh (A) = M additive and can be extended, with some standard work, to be a regular content and with some further work, to the Borel σ-algebra R of G. It is necessary to verify that µh (·) depends on h continuously, and µ0 (·) = 0. 4. If h1 , h2 , . . . is some enumeration of rationals, define µ(·) as ∞ X 1 |µh0 n |(·) µ(·) = 2n 1 + |µh0 n |(G) n=1

and show that µ(·) is a measure and after some further work, we get ˜ |Cn (G) (·) = M (·) M which is well-defined, and one gets after some further work that Z Z h ˜ M (hχA ) = λ0 (A) = a(h)dλ = Φ(hχA )dλ, A

G

where Φ(hχA (t)) = a(h), t ∈ A, and = 0 if t 6∈ A. Finally,

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Z M1 (f ) =

Φ(f (t))dλ(t),

f ∈ C∞ (G),

G

and M1 |C∞ (G) = M in all cases. The many details (omitted) should be obtained from my paper (Rao (1980), pp. 25–29). 5. The converse is simpler. Under the given representation one sees that if {fn , n ≥ 1} ⊂ C∞ (G), then {ψ(fn ), n ≥ 1} is Cauchy in L1 (µ), and one can verify the remaining conditions. [The reader can see all the details from the easily accessible paper (Rao (1980), Theorem 2 and also from the author’s Measure Theory book (2004), second Edition, pp. 676–684).] This completes the outline of the proof.  We now briefly indicate a key probabilistic application of the above theorem as well as its place in extending the L´evy–Khintchine representation in this context as an important motivation here. For this, it is useful to recall a concept, termed a Sazanov topology, on a locally convex vector space. Thus Sazanov’s topology S on a locally convex linear space F is a locally convex one defined by the set of continuous seminorms generated by all the quadratic forms Q(≥ 0) of finite trace on F as follows: If Q and H on P F are such forms, and {ei , i ∈ I} is an orthonormal trace(Q/H) < ∞; let us consider all set in F, and supn ni=1 Q(ei ) = P such pairs in F for H, with supn ni=1 Q(ei ) = trace(Q/H) < ∞. All these pairs define a topology termed Sazanov or (S) topology which is so named and detailed by N. Bourbaki (Livre VI, Integration, Chapitre III). It is locally convex and the neighbourhood system at f0 ∈ F is N (f0 : ε1 , . . . , εn ) = {f : Qj (f − f0 )2 < ε2j , j = 1, . . . , n}.

(116)

This is verified to be a locally convex topology and is coarser than the given one, which coincides with it iff the given one is nuclear. Its importance is exemplified by the following useful result. Theorem 2.6.3 As before let C∞ (G) be the vector space of real continuous compactly supported functions on a locally compact Hausdorff space G, and L : C∞ (G) → C be a mapping. If it is the ch. f of a g.r.f., then one has (i) L(0) = 1, (ii) L(·) is positive definite, and (iii) L(·) is continuous in the topology of C∞ (G). In the opposite direction let L(·) satisfy (i), (ii) and (iii0 ); namely given ε > 0 and k > 0, there is a δ(= δε > 0) such that for kgi k ≤ k, gi ∈ C∞ (G), i = 1, 2, kg1 −g2 k < δ

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⇒ |L(g1 ) − L(g2 )| < ε, so that L(·) is continuous now in the Stopology of C∞ (G). On the other hand, a mapping L satisfying (i)–(ii) and is continuous in the S-topology on C∞ (G) is the ch.f. of a g.r.f. F : C∞ (G) → L0 (P ), on some probability space (Ω, Σ, P ). A proof of this result uses some properties of the space C∞ (G) which is a “bornological space”, and the details are in (Rao (1980), Theorem 7). An interesting consequence of it is a generalized L´evy– Khintchine formula: Since this result has such a distinguished place in the modern studies of Probability Theory, we present it here. As before, let G be a locally compact Hausdorff space and C∞ (G) as a real continuous compactly supported function space, ψ : R × G → R satisfies ψ(0, t) = 0, ψ(·, t) is continuous, ψ(x, ·) is measurable, and Rψ(f (t), t) is bounded for a · a · (t), and for all f ∈ Cc (R) with f (x)dx = 0, then we have R Z Z ψ(x, t)(f ∗ f¯)(x)dxdµ(t) ≥ 0, (117) A

R

whereRf¯(x) = f (−x), any Borel sets A ⊂ G with ‘*’ as convolution. Since R f 0 (x)dx = 0, (117) can be written also as: Z ψ(x, t)(f 0 ∗ f¯0 )(x)dx ≥ 0, a · a · (t), f ∈ K ⊂ Cc (R). R

With this set up the L´evy–Khintchine analog is now given by: Theorem 2.6.4 Let L : Cc (G) → C be the characteristic function of a generalized random field with independent values at each point, and obey conditions (i)–(iii) of the above theorem. Then the functional L(·) can be represented as Z  L(f ) = exp ψ(f (t), t)dµ(t) , f ∈ C∞ (G) (118) G

where µ is a Radon measure on (G, G), Borel couple, ψ(x, ·) is µmeasurable, x ∈ R, and ψ(x, ·) is given by Z ψ(x, t) = [eiyx − α(y)(1 + ixy)]σ(dy, t) + a0 (t) [|y|>0]

+ ia1 (t)x − a2 (t)

x2 , 2

(119)

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where a2 (t) ≥ 0, a1 : G → C Borel measurable, such that; (i) α : R → C is an analytic function of exponential type and α(y)−1 has a zero of order three at y = 0, (ii) σ(·, t) is a Radon measure on R and σ(A, ·) is Borel for each t ∈ G and Borel set A ⊂ R, (iii) for a · a · (t) one has Z Z Z 2 y σ(dy, t)+ σ(dy, t) < ∞; (1−α(y))dσ(y, t) = −a0 (t). 01

|y>0

(120) Conversely, if µ is Radon and ψ satisfies (i)–(iii), then L(·) of (118) with (i)–(iii) defines L(·) of (118) and is the ch.f. of a g.r.f. on C∞ (G) having independent values at each point, and L(·) verifies also conditions of Theorem 2.6.3 above. This general result as well as the preceding ones are detailed in the author’s paper (Rao, (1980), Functional Analysis Journal 39, 23–41) and will be useful for readers to study it carefully and in detail. The above theorem is related to (and is an extension of) a key result due to A. M. Yaglom and N. Ya. Vilennkin, described in the book by Gel’fand and Vilenkin (English translation, 1964, Sec II. 4). The work is included in this section for the purpose of inviting the reader to proceed with this important extended analysis. An interesting question is to characterize functionals L(·) on K of order m that are exponentials, i.e. L(f ) = eM (f ) ,

f ∈K

(121)

where M (·) is a local functional of order m, finite. The next result has a characterization of the L(·) which is an extension of the fundamental L´evy–Khintchine representation, and hence will be of special interest for an advanced analysis in this area: Theorem 2.6.5 Let L : K → C be a functional given by L(f ) = exp{M (f )},

f ∈ K,

(122)

where M (·) is local of order m < ∞. Then L(·) is the characteristic functional of a g.r.f. on K, with independent values if and only if it is representable as, for f ∈ K, Z  α L(f ) = exp Φ(f (t), (Df )(t), . . . , (D f )(t), t)dµ(t) , (123) Rn

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109

where |α| ≤ m, µ is a Radon measure on Rn (and nonatomic for the ‘only if’ part) and Φ : Rv × Rn → R, is given by Z Φ(x, t) = [ei(x,y) − α(ty)(1 + i(x, y))]σ(dy; t)+ |y|>0

a0 (t) +

2 X |k|=1

ak (t)

(ix)k . k!

(124)

Here σ(·, t) is a positive tempered measure on Rv for t ∈ Rn , σ(A, ·) is ˜ Borel on Rn for each Borel set A ⊂ Rv , (x, y) is the scalar product in R, α(·) is an entire analytic function of exponential type such that α(y) − 1 has a zero of order 3 at y = 0, and the ai (·) are Borel functions in Rn , determined by L(·) (or Φ), R R |y|2 σ(dy, t) + |y|>1 σ(dy, t) < ∞ 0 0, x > 0, (126) Γ (s) 0 and ˜ α) = √ 1 G(x, 2πα2

Z

x

t2

e− 2α2 dt,

−∞ < x < ∞, α > 0.

(127)

−∞

˜ α), called the gamma and the GausThese two distributions G(·, s), G(·, sian, are both basic in probability theory. They lead to the Riemann problem (or hypothesis) quickly as we show now. R s ∞ It is trivial that G(+∞, s) = 1 so that 1 = Γn(s) 0 e−nt ts−1 dt, and ! Z ∞ Z ∞ s−1 ∞ ∞ X X 1 t s−1 −nt Γ (s) t = e dt. (128) dt = s t−1 n e 0 0 n=1 n=1 If the series on the left is denoted by ζ(s), so ζ(1) = +∞, or the function ζ : (0, ∞) → R, ζ(1) = ∞, and that 1 is a simple pole of ζ. Using complex analysis, ζ(·) can be extended to all of the complex plane C, with (−s)s = exp[s log(−s)] where log Z is defined for complex Z, and using the contour integral from +∞ on going to δ > 0 and back to +∞, using (−t)s = exp[s log(−t)] with log Z for complex Z defined as usual. Thus one obtains Z δ I +∞ Z Z +∞  (−t)s dt (−t)s dt = lim + + t δ↓0 et − 1 t +∞ e − 1 t +∞ |t|=δ δ Z δ s log t−πi Z +∞ (s log t+πi) e dt e dt = lim + +0 t t t↓0 +∞ e − 1 t e −1 t δ Z
∞ ts − 1 iπs −iπs dt + 0 = e −e et − 1 0 = 2i sin(πs)Γ (s)ζ(s), using (128). (129) Now with the classical result sinππs = Γ (s)Γ (1 − s), and Γ (·) extended to R− using the classical Complex Function Theory (cf. Ahlfors (1975), p. 98) the integral for ζ(·) gives

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ζ(s) =

Γ (−s + 1) 2πi

But by the usual Taylor expansion of

I

+∞

+∞

(−t)s dt . et − 1 t

t ,0 et −1

111

(130)

< t < 2π, in (128),

X Bn tn t = , et − 1 n≥0 n!

(131)

where the Bn are the Bernoulli numbers B0 = 1, B1 = − 12 , B2n+1 = 0, n ≥ 1, but B2n 6= 0, so ζ(s) = 0, s = −2n, n ≥ 1, termed the trivial n+1 B x n+1 zeros of ζ(·) and ζ(−n) = (−1)n Bn+1 , ζ(2n) = (2π)n (−1) , n > 1. 2(2n)! The basic question is to find the nontrivial zero of ζ(·), and study its analytical properties. And then Riemann conjuctured that the nontrivial zeros of ζ(·) all lie on the complex line Re (s) = 21 (with the obvious simple pole of Z(·) at s = 1). This is the Riemann hypothesis. In 1975, Norman Levinson has shown that almost all roots of ζ(s) = 0 are near the critical line Re(s) = 21 , s = 12 + it, t ∈ R, and the same is true of the roots of ζ(s) = a. We strengthen this assertion below. To explain the ˜ the Gaussian distribution (127) and problem further, we now consider G get some new insights. To understand the key aspects of the problem let us consider the zeta function, derived from the classical Euler product representation of prime numbers as given by (p denoting primes) the Euler product: ζ(s) = Πp (1 − p−s )−1

(132)

so that ζ1 = +∞ (i.e., there are infinitely many prime numbers), and that s 7→ [ζ(s)]−1 is an entire function on C. The classical work by Khintchine (1923) shows that φσ : t → ζ(σ + it) is a positive definite and non vanishing function for all σ > 1. It is given, more precisely, by the following simple but useful result. Proposition 2.7.1 The mapping φσ : t 7→ ζ(σ+it) , σ > 1, is an infinitely ζ(σ) divisible characteristic function and so never vanishes. Thus s 7→ ζ(s) is a nonzero entire function in the right half-plane determined by σ > 1 of C where s = σ + it, and where ζ(·) is defined by (132). Proof. In the Euler product (132) above, set s = σ + it and let log φσ (t) denote the principal branch (for definiteness), and we get

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log φσ (t) =

X  log(1 − p−σ ) − log(1 − e−σ−it ) p

=

∞ XX p

=

m=1 ∞ XX

p−mσ (p−imt − 1)/m

p

p−mσ (e−imt log p − 1)/m,

m=1

and each of the terms inside of the display above is the logarithm of the characteristic function of a Poisson distribution with the − log p > 0, as its parameter, and hence is infinitely divisible. Therefore φσ (·) is also an indefinitely divisible ch.f ., continuous at t = 0, and never vanishes by the L´evy-Khintchine representation theorem. Thus ζ(·) is an entire function in the right plane.  Remark 16. It was noted by Gram (1903) that there are exactly 15 solutions of ζ( 21 + it) = 0 for 0 < t < 50, and evaluated each for a few decimal places, substantiating Riemann’s conjucture. Now a computer expert Odlysko, found over 1022 zeros of ζ( 21 + it), reported in Derbyshire (2003). For a different approach towards the possible solution of RH, consider the reciprocal of ζ(·), ∞

X µ(n) 1 ζ (s) = = Πp (1 − p−s ) = , ζ(s) ns n=1 ∗

where µ(·) is the M´obius function defined as µ(n) = +1, 0, −1, according as ‘n’ is a product of odd number of distinct primes divisible by a square integer, (+1), or a product of an even number of primes (−1) respectively, or zero otherwise. Although ζ s (1) = 0 was considered by Euler in 1750, the convergence properties were discussed by von Mangoldt (1897) much later. P The Merters’ function M (·) changing values at integers where M (x) = n≤x µ(n), and that n−1

M (n − ε) + M (n + ε) X µ(n) = µ(k) + , 2 2 k=1 whence |M (·)| %, and |M (x)| ≤ x, x > 0, so that one has

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ξ ∗ (s) =

Z 0

∞

x−s dM (x) =

Z

113

∞

M (x)x−s−1 dx,

0

since |M (x)| ≤ |x|α and α < 1. Then x−s M (x) → 0 as x → ∞, σ ≥ 1 with s = σ + it. The following result of Littlewood’s is useful here. Theorem 2.7.2 (Littlewood (1912)) With the Martens’ function M (·) 1 above, for each ε > 0, M (x)x− 2 −ε → 0 as x → ∞ if and only if the RH holds. The details of proof of this theorem are in several books, e.g., see H. M. Edwards (1974). Here the introduction of M (·) brings us back to using the probability method with the occurrence of prime numbers whose appearance is sometimes called the ‘game of chance’, by M. Kac, (1959), and their behavior (in the probabilistic sense) was already used by Denjoy (1931). A usable discussion of the behavior will be included and utilized in our solution. Let Ω = {1, 2, . . .}, the set of positive integers and consider its subset ˜ with square free divisors n, so µ(n) 6= 0. This set is about 1 − 1 Ω ζ(2) 6 ˜ ˜ in proposition, so that by Euler’s formula its value is π2 . Let Ωi (⊂ Ω) be the products of even and odd primes i = 1, 2 on which µ(n) = 1, or −1 respectively. They have the corresponding proportions = π32 . Thus if the set Ω of integers is given the ‘volume’= 1, then the corresponding ˜i , i = 1, 2 will have values 32 , i = 1, 2 and Ω ˜ will have volumes of Ω π 6 the size = 1 − π2 . Thus one may define a probability measure on Ω so that µ(n) = 0, ±1, will become independent random variables with P [µ(n) = 1] = P [µ(n) = −1] = α and P [µ(n) = 0] = 1 − 2α where α = π32 . This describes a probability function on the power set 2Ω . The existence of such a (σ-additive) probability model is to be verified. This can be done nontrivially with the following (standard) probability method. One starts here by using the crucial result that the density of prime numbers is approximately given by (log x)−1 , x > 1. Then the existence of such a P is needed. For this one has to refer to the work of Jensen and his collaborators with extensions (e.g., as detailed in the book by Laurin˘cikas (1996)). (The existence of such a probability measure was assumed and used by Denjoy (1931).) This fact is made explicit here for convenience. With this explanation, we can proceed using the pairwise independence (in the sense of Probability Theory). We can now complete the argument

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by using the classical (Probability Theory) central limit theorem and establish the RH as follows with just the pairwise independence property of the µ(n)’s, and hence for {M (n), n ≥ 1}. It may be restated as P [µ(n) = 1] = P [µ(n) = −1] = π32 and P [µ(n) = 0] = 1 − π62 , where the µ(n) being pairwise independent random Hence E(µ(n)) = 0, V ar(µ(n)) = π62 . Pvariables. n Now M (µ) = i=1 µ(i), the Mertan’s Pnfunction, satisfies E(M (µ)) = 0, σ 2 (µ(n)) = n62 so that M (µ) = i=1 µ(i), and then one verifies. 2 3n E(M (w)) = 0, σ(M (u)) = π2 . Note that σ (Mn (n)) → ∞ which is the appropriate condition for the central limit theorem of Probability Theory for pairwise independent random variables to hold. Hence by such a known central limit theorem for pairwise independent variables we have (cf., Rao (1984), p. 399) the limit relation: " # Z x Mn − 0 1 2 lim P p e−u /2 du, ≤x = √ n→∞ 2π −∞ σ 2 (M (n)) √ so that the sequence {Mn / nα , n ≥ 1}, α = π62 , is bounded in proba1 bility. Hence |M (n)|n− 2 −ε → 0 with probability 1, for each ε > 0. This result and Theorem 2.7.2 above imply: Theorem 2.7.3 Under the preceding conditions leading to pairwise in1 dependence of µ(n), it follows that M (µ)n− 2 −ε → 0 with probability one for each ε > 0 as n → ∞, so that the Riemann Hypothesis holds with probability one, as a consequence of the central limit theorem of probability theory. Remark 17. 1. This is the best conclusion that we can present on the classical RH problem. The only question is to know if the null set is actually the empty set. This problem is not simple to answer and there is no general method to analyze the structure of the sets of measure zero. The details will be omitted again, and referring the reader to the readily available reference to the author [Rao (2012)]. 2. A philosophical discussion about the sets of measure zero (on which the RH can be false) should ideally be empty for a conclusion from the above result for making an absolute assertion. But there is no method available to assert that a Lebesgue null set is empty. On this point, the mathematician J. E. Littlewood (1912), who spent a considerable effort on RH, thinks that all roots of J( 12 + it) are on the

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115

line t ∈ R to be false, this seems to be based in part on the folR x and dt lowing: If L(x) = 2 log t + constant, x > 2, then the great Gauss’s conjecture states that π(x) = L(x), x > 2 where π(x) denotes the number of prime numbers less than x and Littlewood showed that this is true for almost all x (Lebesgue measure). But a South African mathematician by the name of Skews showed that the conjucture fails for many x starting at 10(x) = 1010k−1 (x) , k > 1 where 101 (x) = 10x (and k > 1 ⇒ 10k (x) = 1010k−1 (x) ). See the article by Littlewood (1912), ‘The Riemann Hypothesis’, (ed. I. J. Good et al.), Basic Books, New York. 2.8 Admissible Means of Second Order Processes This chapter will be completed with an analysis of possible mean functions of a given second order process whose covariance functions are subject to the types of conditions considered in the above sections in which the mean functions should obey certain restrictions that will be acceptable for a given covariance. The first steps have been detailed in Section 1.2 in the early result by A. V. Balakrishnan, detailed there. We now include some key facts that enhance and complement the earlier analysis, with the Gaussian as well as some general cases to conclude this chapter. It is convenient to recall the Aronszajn space concept, already used in Chapter 1, for a quick reference here. Thus let K(·, ·) be a Hermitian positive definite function on T × T → C and let Hr (T ) = {f : f = Pn i=0 ai r(si , ·), n ≥ 1, and ai ∈ C}. Define an inner product on H1 (T ) by the equation: (f, g) =

n,m X i,j=1

r(si , tj )ci d¯j =

n X i=1

ci g¯(si ) =

m X

d¯j f (tj ),

(133)

j=1

P P where f = ni=1 ci r(si , ·), g = m ¯(·, tj ), are in Hr (T ). With the i=1 dj r positive definiteness of the covariance r(·, ·), one has the inner product (·, ·) and let k · k be the norm derived from it and Hr be the space completed for (·, ·). Here we note that f ∈ Hr (T ) ⇒ f (t) = (f, r(t, ·)) and (r(s, ·), r(·, t)) = r(s, t) as well as, {r(s, ·), s ∈ T } ⊂ Hr is dense, |f (t)| ≤ kf kkr(t, ·)k, t ⊂ T . The space {Hr , with inner product (·, ·)} is usually called the Aronszajn space (in honor of its author N. Aronszajn (1962).

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We now record a useful property of certain admissible means as: Proposition 2.8.1 The space of admissible means of a Gaussian process, with covariance r, is a Hilbert space. Proof. We use the Aronszajn space technique and the key result that the dP mapping f ∈ MP to Y = dPf > 0 a.e., by the classical H´ajet-Feldman dichotomy theorem on the equivalence of Gaussian measure Pf and P respectively with mean f , and mean zero (cf., e.g., Rao ((2014), 226)): Z Z ¯ t dPf = ¯ t dP = E(Yf X ¯ t ) = L(X ¯ t ). f (t) = X Yf X (134) Ω

Ω

It is seen that L : L → C is a continuous linear mapping, and hence by the Riesz representation, (134) implies that f = u(Y ) of (133) and so f ∈ Hr . Thus MP ⊂ Hr . We need to show the converse inclusion. For the converse, let L(·) be continuous and f ∈ MP so Pf  P . Let {Fi , i ∈ I} be a right order continuous family of σ-algebras from Σ = σ(Xt ), t ∈ T generating it, so Σ = σ(∪i∈T Fi ) and let Pif = R 1 Pf |Fi , and let Hj (P, Pf ) = Ω (dP dPf ) 2 , the Hallinger distance (= R 1 (dPjf /dPj ) 2 dPj ), then it follows from the theorem of E. J. Brady Ω (1971) that limi Hi (Pf , Pi ) = c exists and c = 0 if and only if Pf ⊥ P . Also Hi ↓ c, (0 ≤ c ≤ 1), and c = 0 iff Pf ⊥ P . Since by hypothesis this is not the case, c > 0, because Hi ≥ c > 0 now, choose F0 = σ(Xt )(= Ft0 ) for an arbitrarily fixed t0 ∈ T and then Ht0 using the distributions of Xt and Xt + f (t) to get  21 Z  2 Z −(u−f (t))2 1 u dx − √ c ≤ Hi0 = (dP0 dP0f ) 2 = e 2 − e 2a 2πa Ω   R 1 = exp − f 2 (t) , of (134). (135) 8a If c20 = −8 log c > 0, then (135) implies f (t)2 ≤ c2 a, or f (t) = |L(Xt )| implies that L(·) is continuous, so f ∈ MP , as desired.  The preceding analysis has the following consequence. Corollary 2.8.2 For a Gaussian process {Xt , t ∈ T } on (Ω, Σ, P ) and ¯ t ), f ∈ Mp , the admissible mean, there is a Z = u(f ) ∈ L, f (t) = E(2X and the likelihood ratio is: dPf 1 = exp{Z − E(|Z|2 )}, a.e. [P ]. (136) dP 2

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117

Further analysis in the Gaussian case is clearly possible. Here we present a key result on a regression problem if the process is also a martingale (1975). This includes the Brownian motion case considered by Liptser and Shiryayev (1971). We thus have: Proposition 2.8.3 Let Y = {Yt , Ft , t ≥ 0} be a real continuous Gaussian martingale and X be a random variable depending on Y in that (X, Yt1 , . . . , Ytn ) is a Gaussian vector for any (t1 , . . . , tn ) finite index of times. Then the regression function E(X|Ys , s ≤ t) is representable as Z t K(t, s)dYs , (137) E(X|Ys , 0 ≤ s ≤ t) = E(X|Y0 ) + 0+

where K(·, ·) is a nonstochastic (jointly) measurable and locally integrable function, so that the (stochastic) integral in (137) is well-defined (by the Bochner L2,2 -boundedness principle). Proof. We present the argument in steps for convenience. I. Let E(X) = µx , E(Y ) = µy and the covariance matrices of (X, Y ) and (Y, Y ) be Rxy and Ryy and for simplicity let the vectors −1 be linearly independent so that Ryy exists. Then by the Gram-Schmidt orthogonalization we can find a matrix A so that (Y − µy) ⊥ Z = ([X − µx] − A[Y − µy ]) so that E(Z(Y − µy )∗ ) = 0, i.e., Z ⊥ (Y − µx ). A simplification of this equation gives: −1 0 = E(Z(Y − µy )∗ ) = Rxy − ARyy ⇒ A = Rxy Ryy .

(138)

Thus with this A, Y − µy and Z are (being Gaussian) independent, so that E(Z|Y ) = E(Z) = 0, and E(X|Y ) = µx + A(Y − µy ). We can now obtain the conditional covariance of ZZ ∗ , given Y : E(ZZ ∗ |Y ) = E(ZZ ∗ ) since Y, Z are independent, being Gaussian −1 ∗ = Rxx − Rxy Ryy Rxy ,

(139)

−1 which is nonstochastic. [In case Rxy is singular, if Rxy is taken as a generalized inverse (in the Moore–Penrose sense), the above statements are still valid, but this is not needed here.] II. The argument is a nontrivial extension of that of Lipster and Shiryayev (1977) for the general Gaussian case from their work on Brownian motion. Thus consider the dyadic partition of [0, t], t > 0, as 0 = tn0 < tn1 < . . . < tn2n = t, where tnk = 2ktn , k = 0, 1, . . . , 2n .

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Let Ftn = σ(Ytnk , 0 ≤ k ≤ 2n ), so by the path continuity of the process, Ftn ↑ Ft as n → ∞. If now Xtn = E(X|Fkn ), then E(Xtn+1 |Flm ) = E[E(X|Ftn+1 )|Ftn ] = E(X|Ftn ) = Xtn , a.e.

(140)

and by applying the conditional Jensen inequality E((Xtn )2 ) ≤ E(X 2 ) < ∞, n ≥ 1.

(141)

Hence the uniformly integrable martingale {Xtn , n ≥ 1} converges pointwise and in L2 (P )-mean, Xtn → E(X|Ft ), a.e. III. Since Ftn = σ(Ytnk − Ytk−1 , k = 1, . . . , 2n − 1), we can use the analysis of step II, and get E(X|Ftn )

= E(X|Y ) +

n −1 2X

Ln (t, tn )(Ytnj−1 − Ytnj ),

(142)

j=0

where Ln (·, ·) is a product moment function jointly continuous by the continuity of the Y -process. To see that (142) converges to (137), as n → ∞, consider Kn (·, ·) as (the nonstochastic function) Kn (t, s) =

n −1 2X

Ln (ti tnj )χ[tnj ,tnj∗ ] (s).

j=1

It is (jointly) measurable, integrable on compact sets, so that (142) becomes Z t n Kn (t, s)dYt , a.e.. (143) E(X|Ft ) = E(X|Y0 ) + 0+ Pn But for a simple function f = j=1 aj χ[tj ,tj+1 )×Aj , Aj ∈ Ftj , and a square integrable martingale {Ys , Fs , s ≥ 0} one has by using the Doleans–Dade measure (cf. Rao (1991, p. 66) in the following, which works for general martingale integrals, extending Brownian process): Z t 2 ! X n   E f (s)dYs = a2j E χAj (Ytj+1 − Ytj )2 + 0 0

=

j=1 n X j=1

a2j µ[(tj , tj+1 ]

Z × Aj ] =

|f |2 dµ,

[0,t]∨Ω

(144)

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119

where µ is the above noted Doleans–Dade measure determined by the L1 (P )-bounded submartingale {Yt2 , Ft , t ≥ 0}. By a standard argument (144) can be used to obtain the result for all locally bounded measurable f . Then one can have Z t 2 2 m 2 E(Xt − Xt ) = E [Kn (t, s) − Km (t, s)] dYs Z 0+ = (Kn (t, s) − Km (t, s)]2 dµ(s), (145) [0,t]×Ω

where µ(·) is the Dolean–Dade’s measure of (144). If now, we set µ ˜(0, s) = µ((0, s] × Ω), then (145) becomes for t > 0 Z t n m 2 E(Xt − Xt ) = [Kn (t, s) − Km (t, s)]2 d˜ µ(s). (146) 0+

Then by the early work, {Xtn , n ≥ 1} is Cauchy, so Kn (t, ·) ∈ L2 (R+ , µ ˜), n ≥ 1 and being Cauchy it tends to some K(t, ·) ∈ L2 (˜ µ) for each t. This implies that (143) converges to (137) and completes the argument.  We end this section by presenting a quite general result on the structure of the admissible mean values of a second order process with continuous covariances, to indicate its complex structure and to understand the problem raised by Balakrishnan in its generality. The following comprehensive result on the linearity of the set of admissible means is due to Pitcher (1963) which is based essentially on the Karhunen expansion of a second order process and amplifies the problem showing the nontrivial structure involved. Theorem 2.8.4 Let X = {Xt , t ∈ T } be a centered second order process with a continuous covariance r where T ⊂ R is a bounded closed interval. Let {λn , n ≥ 1} and {φn , n ≥ 1} be the eigenvalues and the corresponding eigenfunctions of r which exist. Let Xn , fn be given by Z Z − 21 − 12 Xn = λn X(t)φn (t)dt, fn = λn f (t)φn (t)dt, f ∈ Mp , T

T

where Mp is the set of admissible mean values of the process X. [Thus {Xn , n ≥ 1} is a sequence of orthonormal random variables and P∞ 2 {fn , n ≥ 1} = f ∈ l , so n=1 |fn |2 < ∞.] Let Pn be the ndimensional distribution of {Xi , 1 ≤ i ≤ n} with densities pn relative

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to the Lebesgue measure, which exist. Suppose that the pn satisfies the following three conditions: (i) pn > 0 a.e, n ≥ 1, (ii) lim|ti |→∞ pn (t1 , . . . , ti , . . . tn ) = 0, 1 ≤ n i ≤ n, for almost all ti , 1 ≤ i ≤ n, i 6= j: and (iii) ∂p ,1 ≤ j ≤ n ∂tj   2 P R px exist, n ≥ 1, and nj=1 Rn ∂ log dPn is bounded by K < ∞. Then ∂tj M1 = {f ∈ Mp : f = {fn , n ≥ 1} ∈ l1 } is a positive cone, and P − 21 M1 = Mp holds if further ∞ n=1 λn < ∞. If also each pn is symmetric about the origin of Rn , then Mp is linear. In general M1 ⊂ Mp only, and Mp need not even be a convex set. The complete detail is to be found in Pitcher’s paper, and we refer the reader to that work to understand the intricacies of the assertion. It may be of interest to end this chapter with a simple note on the relations between weakly harmonizable class of processes and that of the Karhunen’s. This gives a general relationship between them. Theorem 2.8.5 Let {Xt , t ∈ R} be a weakly harmonizable process on a probability space (Ω, Σ, P ). Then it is also a Karhunen process relative ˆ and a suitable Borel set {ft , t ∈ to a finite positive measure ν on R 2 ˆ R} ⊂ L (R, B, ν). Proof. Since {Xt , t ∈ R} is weakly harmonizable, it has a stationary di˜ Σ, ˜ P˜ ), on a larger space as shown lation, denoted {Yt , t ∈ R} ⊂ L2 (Ω, in Section 2.3 above (cf. Theorem 2.3.1). If Q : L2 (P˜ ) → L2 (P ) is the orthogonal projection from L2 (P˜ ) onto L2 (P ), then we have Z  Z itλ ˜ ˜ e dZ(λ) = π(eit(·) )(λ)Z(dλ), Xt = QYt = Q T˜

T˜

and if f˜(t, λ) = π(eit(·) )(λ), then {f˜(t, ·), t ∈ T } ⊂ L2 (T, ν), where ˜ ν(·) is a measure determined by Z(·), an orthogonally valued random set function. Thus, Z ¯ r(s, t) = E(Xs Xt ) = f˜(s, λ)f¯˜(t, λ)dν(λ). T˜

Hence r(·, ·) is of Karhunen class, relative to {f˜(t, ·), t ∈ T }.  To complete this chapter, it is useful to present a multivariate version of a (weakly) harmonizable process which indicates how much

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of the preceding analysis can be given for this extension. Thus let Xt = (Xt1 , . . . , Xtn ), t ∈ T , centered with an n × n matrix covariance r which is representable as Z Z 0 r(s, t) = eius−iu t F (du, du0 ), s, t ∈ T Tˆ

Tˆ

relative to an n × n matrix of (complex) bimeasures F , as in the scalar case which is extended to n-dimensions, making r(·, ·) a positive definite matrix function. The desired analog then can be established and is presented as follows. Theorem 2.8.6 Let {Xt , t ∈ T } be a multivariate harmonizable process (T = R or [0, 2π)), with F (·, ·) as its spectral matrix and   Z Z 2 ∗ 0 0 ˆ L (F ) = f : T → MP | f (λ)f (λ )F (dλ, dλ ) = (f, f ), kf kF < ∞ Tˆ

Tˆ

where kf k2F = trace(f, f ), of the positive definite matrix (f, f ). Then L2 (F ) is a Hilbert space of equivalence classes of matrices with inner product ((f, g)) = trace (f, g), where the linear space is considered with constant matrix coefficients. Although the result is more or less a direct extension of the scalar case of harmonizable processes, some care is needed because of the noncommutative problems with matrix operations. In this connection, we should also mention some early work by J. Kamp´e de Feriet and F. N. Frenkiel (1954) (called class KF) which is continued by them for several years later by extending (weak) stationarity to include some (particularly strong) harmonizability of processes with applications. These results are motivated by the summability methods of classical analysis. They were also related to some work by E. Parzan (1962) and Yu. A. Rozanov (1959) and have interest in several applications. We state this concept and later include an example to show that weakly harmonizable class is not included. (More is in Chapter 3.) Definition 2.8.7 A centered second order process {Xt , t ∈ T }(⊂ L20 (P )) with a continuous covariance r (here T = R or Z), is of class (KF) if the following limits exist: ( R a−|h| lima→∞ α1 0 r(s, s + |h|)ds, if T = R, r˜(h) = Pn−|h|−1 1 limn→∞ n k=0 r(k, k + |h|), if T = Z.

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The point of this extension is to bring in the very useful and powerful summability methods of the general analysis into the popular second order stochastic analysis. It may be verified that r˜(·) is positive definite (and measurable) so that by the well-known Herglotz-Bochner-Riesz theorem r˜ will have a representation relative to a positive measure H(·) as; Z ˜ r˜(h) = eiht dH(t), h ∈ T, (T = R or Z). T˜

It can be verified that stationary, as well as strongly harmonizable classes, are included in class (KF). However, we shall include an example (due to H. Niemi) below to show that weak harmonizability is not always in class (KF). [Cf. Exercise 2 below.] 2.9 Complements and Exercises 1. A weakly harmonizable centered process is norm bounded, but the converse implication is not true. For example, let f ∈ L1 (R, dt), R and µ(·) = (·) f (t)dt ∈ M(R), the space of regular signed measures on R. If Y = {ˆ µ : µ ∈ M(R)}, let f ∈ C(R) − Y, e.g., f (x) = sgn(x)[(log |x|)−1 χ[|x|≥0] + |x| χ ], for x ∈ R so that this e [|x| 0 as above. The sets C, Dn are disjoint, 1 ≤ ak ≤ 2 and the series is finite. The covariances r(k, l) = 0 if k 6= l and n−l−1 n−1 1 X 1X 2 rn (l) = r(k, k + h) = a for h = 0, and = 0, for h 6= 0. n k=0 n k=0 1

Verify that for h 6= 0, limn→∞ rn (h) = 0, and (5 1 − 3·22m−1 , if n = 22m − 1 3 rn (0) = 4 − 3·212m , if n = 22n+1 − 1, 3

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3.

4.

5.

6.

123

and verify that limm→∞ r2m (0) = 53 , limn→∞ r22m+1 (0) = 43 . Conclude that limm→∞ rn (0) does not exist. So {Xn , n ∈ Z} is not in class (KF). Thus class (KF) and weakly harmonizable classes do not include each other although both have weakly stationary and strongly harmonizable classes in their fold. [This is Niemi’s example.] If X : R → L20 (P ) Ris strongly harmonizable so that it is representable as X(t) = R eitλ Z(dλ), for a stochastic measure Z : B(R) → L20 (P ), show that X ∈ class (KF), the class considered above. The same result holds if the weakly harmonizable class is restricted, e.g., if the tensor product Z ⊗ Z is also a stochastic measure. [The preceding exercise implies that this need not hold for all stochastic measures. See the author’s paper (1982), p. 338, the proof there works in this case.] This result deals with a pointwise approximation of weakly harmonizable processes by the strongly harmonizable ones due to H. Niemi (1975). If X : R → L20 (P ) is a weakly harmonizable process then there exists strongly harmonizable Xn : R → L20 (P ) such that Xn (t)χA (t) → X(t)χA (t) as n → ∞, for all compact A ⊂ R, uniformly converging. [Here R can be replaced by an LCA group G.] (In all cases, the convergence is in L2 (P )-norm.) This problem shows that the class of weakly harmonizable processes is a proper subspace of bounded continuous processes in L2 (P ). Identify the Lebesgue space L1 (R) Ras a subspace of M (R), the class of signed measures (f ∈ L1 (R) → (·) f (t)dt(∈ M (R)), as a signed measure). If Y1 = {ˆ µ : µ ∈ M (R)} for f ∈ C0 (R) − Y1 , e.g., f (x) = sgn(x){(log |x|)−1 χ[|x|≥e] + |x| χ } is known to be such e [|x|t Fs , where for the σ-algebras, a ≤ s < t ≤ b ⇒ Fs ⊂ Ft and let Ft = ∩s>t Fs , and all the σ-algebras are P -completed. Let Et = B(Ω, Ft ), the space of bounded (scalar) Ft -measurable functions on Ω, and set ξ = {Et , t ∈ I}, Ω 0 = I × Ω and L(Ω 0 , ξ) of the above Pn type simple functions, so that each element is of the form f = i=1 fi χAi , fi ∈ Einf(Ai ) , Et = B(Ω, Ft ), bounded Ft measurable scalar functions. If k|f |k∞ = supti kf (ti )k∞ , with k · k∞ as the uniform norm on Et , then {L(Ω 0 , ξ), k| · |k∞ } becomes a normed linear space of vector functions f (or vector fields). If A = (ti , ti+1 ] here in the representation of f , we can set Z n X τ (f ) = f (t)dX(t) = fi (Xti+1 − Xti ), f ∈ L(Ω 0 , ξ). I

i=1

All this set up is to define a generalized Bochner L2,2 -(and Lp,q ) type boundedness conditions that include most of the known stochastic integrals due to P. A. Meyer, K. Itˆo, and others, the extension being on Bochner’s ideas. Here we give the generalization based on Orlicz spaces inclusively. So let φ be a Young function (i.e., a symmetric φ convex φ : R → R+ , φ(−x) = φ(x), φ(0) = 0), andf ∈ L (P ) |f | on (Ω, Σ, P ) to scalars, and kf kφ = inf{ε > 0 : E φ ε ≤ 1} < ∞. Then (Lφ (P ), k · kφ ) is a complete metric space including the Lp (P )-spaces for φ(t) = |t|p , p > 0. Then X : I → Lφ (P ) is termed a stochastic integrator if (a) {τ (f ) : f ∈ L(Ω, ξ), kf kφ ≤ 1} is bounded in k · kφ -metrics, and (ii) fn ∈ L(Ω, ξ), |fn | ↓ 0 ⇒ limn τ (fn ) = 0 in Lφ (P ). [If φ(x) = x2 , this was Bochner’s original condition which motivated the extension.] We have the following:

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Let φ1 , φ2 be Young functions, (Ω, Σ, P ) be a probability space and X = {X(t), Ft , t ∈ I ⊂ R} ⊂ Lφ2 (P ), be a process which is Lφ1 ,φ2 -bounded relative to a σ-finite measure α : B(I) × Σ → R+ , as defined above. Then X is a stochastic integrator and the integral τ (f ) extends from the simple functions of L(Ω 0 , ξ) to the class 0 M Φ (L) = sp{L(Ω ¯ , ξ), k · kφ } into Lφ (P ) for which the dominated convergence theorem holds. [The converse holds in a slightly restricted form. An outline of the detail of this result, in its general form, is included in the author’s C. R. Acad. Sci., Paris (tome 314 (1992), pp. 629–633) paper, which should be studied by the probability (functional analysis) students. This leads to interesting general analysis and possible extension of the work given in M. Metivier and J. Pellamail (1980) on the stochastic integrals used at that time with restrictions.] 2.10 Bibliographical Notes The work of this chapter gives a view and substantial enlargement of stochastic analysis and integration, starting with the Khintchine stationarity and going much further into the harmonizable classes introduced from a general point of view by Bochner, with more insights based on the contemporary works of Karhunan and Cram´er. Staying in the powerful Fourier analysis mold, generalized by Bochner to the class called V -boundedness (V stands for variations), presents a great and (by keeping close to Fourier or harmonic analysis) powerful methods of stochastic theory which leads to numerous (new) applications, using the MT-integration crucially. The weak and strong harmonizable classes are the two basic classes, with applications of the sharp Fourier methods pioneered by Bochner, takes a prominent place here and in the theory and some forthcoming applications further some deeper aspects of noncommutative extensions of harmonizable classes will be considered later. The study leads to generalized random processes and fields of Gel’fand and the related deep analysis of L´evy classes, as well as processes with independent increments in the generalized case. There was an open problem on the characterization of real functions for that study. The author had the opportunity of solving the open characterization problem, and the work was handled by Gel’fand himself or his

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associates, published in the Russian Probability Journal and another aspect in the (American) Functional Analysis journal also handled by Gel’fand, have been of special interest and included in Section 2.5. All of this work is dependent on the key concept of positive definiteness of certain function(al)s in multidimensional definitions. But there were serious obstacles. An elaboration of the attending problems as well as their solutions giving rise to a healthy growth of the subject. It may be interesting to briefly describe this here. The key concept of positive definiteness of a function (of two variables) in probability arose in the classical work on Brownian motion, by the botanist Robert Brown, whose studies and experiments gave rise to the analysis, which later got N. Wiener to establish it rigourously hence termed also the Wiener process. This leads to a deeper analysis and extension by P. L´evy who wanted to study {Xt , t ∈ T } if T ⊂ Rn , n ≥ 2, the random field if T is an ‘interval’, n > 2, and found that its covariance must verify (*) cov (s, t) = 21 [ksk + ktk − ks − tk], and its positive definiteness is not easy to establish for a further analysis. L´evy with much difficulty and work was able to prove the result if n = 2, but could not proceed further. According to legend (detailed in P. Cartier (1971)) that P. L´evy announced in his class that he would offer a great prize for anyone who could prove the result in the general case n > 2. Namely, he would offer his daughter’s hand to a successful future mathematician. Thereafter, in a short while, L. Schwartz came up with a complete proof of the positive definitions of (*), for any n > 2, and the promise was fulfilled indeed. [Later I. J. Schoenberg also proved the same result differently.] This appeared to be strange, but I sent two papers in late 1960s to P. L´evy to be communicated for publication in the Paris based C. R. Acad. Sci., and he asked me to spell out my name fully (it is mandatory there, unlike in the U.S. Acad. Sci., where I published some papers as usual with initials only). So I asked P. A. Meyer about Cartier’s account, while spending my sabatical in Strasbourg, France and in all seriousness. Meyer told me that it is a true story! This emphasizes the importance of the property which is also crucial for harmonizable random fields analysis, that occupies a key position in this and coming chapters. The work related to local functional characterization was an open problem at that time and its analysis with solution in Section 2.5 above was handled by I. M. Gel’fand and he was pleased with it and when I

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met him at the Advanced Mathematical Institute in Princeton, he also wanted to have a reprint of it for his files, which was provided. The next problem is on Riemann’s Hypothesis (RH) and the earlier attempts were detailed in H. M. Edward’s book (now Dover) entitled Rieman’s Zeta function gives a readable account along with a short probabilistic attack by A. Denjoy (1931). The current account based on the probabilistic argument with Mertin’s function was recently detailed in my paper (Rao (2012)), and, as noted in that paper, the validity of RH with probability one is proved. The question of the emptiness of the null set cannot be determined with our probability argument. An earlier (different) method by J. E. Littlewood also concludes that RH is valid a.e. who even thinks and asserts that the emptiness of the null set cannot be proven. A similar question about a problem of Gauss was proved by him to be true a.e. but the exceptional null set was shown later by others (as noted in the above work also) to be nonempty. Littlewood (1962) thinks of the same fate for the RH as well. This is somewhat analogous to the following question, attributed to H. Weyl, namely prove or disprove the statement or assertion or conjecture: If the irrational number π is expanded in decimals the natural numbers 0, 1, 2, . . . , 7, 8, 9 occur somewhere in the natural order. Many computer experiments early were unsuccessful. But the Canadian number theorist, M. Borwein (1998), reports that two Japanese computer scientists at the University of Tokyo in 1997 using “massively parallel” Hitachi machine with 210 processors have discovered the first occurrence of the above sequence after 17 billion decimals at 17,387,594,880th digit after the decimal point. The next five occurrences are found at 26,852,899,245; 30,243,957,439; 34, 549,153,953; 41,952,536,161; and 43,289,964,000 places. Thus the RH and Littlewood’s feeling on the emptiness of the null set may be on the same level as H. Weyl’s. Also, the admissible means problem was treated here in some detail, as it was bypassed. We now proceed with harmonizable applications and extensions in the next chapter.

b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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In the preceding chapter, we considered several aspects and applications of harmonizable processes, and found that in some (serious) applications we need to study further nontrivial but useful consequences of bimeasures. Their properties allow extensions of random set functions based on the (Morse-Transue or) MT-type analysis and related theory. This work also leads to key applications of filtering and signal extraction problems. Some properties of the MT-integrals were discussed in Sections 1, 2 (of Chapter 1) and they will be used here. [These are weaker than the usual Lebesgue integrals, and such (resulting) extensions will be needed and employed with details below.] 3.1 Special Classes of Weak Harmonizability It is of interest in applications to consider second order processes with covariance r(·, ·) having special properties, such as: (a) periodically correlated, (b) oscillatory, or (c) some related ones. Definition 3.1.1 A centered second order process {Xt , t ∈ R} ⊂ L20 (P ) is periodically correlated if its covariance r(·, ·) satisfies r(s+α, t+α) = r(s, t) for a fixed α > 0 for all s, t ∈ R (here α is termed a period). A centered second order process {Xt , t ∈ R} is oscillatory if r(·, ·) is representable as Z r(s, t) = ei(s−t)λ as (λ)¯ at (λ)F (dλ), (1) R

for a positive nondecreasing F (·) defining a σ-finite measure and {as , s ∈ R} ⊂ L2 (F ). [If eisλ as (λ) = a ˜s (λ) defines some other function, 131

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more general, then we go to the Karhunan family!] In all cases F (·) is termed a spectral measure of the process. If F (·, ·) is restricted to ¯ (t), then it is termed a factorizable spectral measure F (s, t) = M (s)M ¯ (t) (f.s.m.). [The positive definiteness of F implies that F (s, t) = M (s)N giving N (t) = a · M (t) and so there is no generality gained by taking M, N ‘different’.] We present some structural properties of these special classes and their (abstract) representations, to understand the harmonizabilities better. Proposition 3.1.2 A weakly harmonizable process {Xt , t ∈ R} ⊂ L20 (P ) is of f.s.m. class iff its covariance r(·, ·) is factorizable. As a consequence, such a process (of f.s.m. class) is strongly harmonizable. Proof. Let the process be f.s.m. so that F (A, B) = M (A)M (B). The bimeasure condition of F (·, ·) implies that M (·) is σ-additive on the Borel sets of R and hence is bounded. It follows that Z Z 0 ¯ (dλ0 ) eisλ−itλ M (dλ)M r(s, t) = ZR R Z 0 isλ ¯ (dλ0 ) = e M (dλ) · e−itλ M R

R

ˆ (s)M ˆ (t)). = f (s) · f (t)(= M

(2)

Suppose conversely that r(·, ·) satisfies (2) for some f . Since the process is weakly harmonizable, r is expressible as: Z Z 0 r(s, t) = eisλ−itλ F (dλ, dλ0 ), (3) R

R

as a Morse-Transue integral. This is a well-defined integral. Using the properties of this (nonabsolute) integral discussed in Chapter 1 (see Sec. 1 there), we get with A = (λ1 , λ2 ), B = (λ01 , λ02 ): Z T1 Z T2 iλ2 s 0 0 e − e−iλ1 t eiλ2 s − e−iλ1 t F (A, B) = lim , r(s, t)dsdt, 0≤T1 ,T2 →∞ −T −T −is it 1 2 where A, B are noted above, such that F ({λ1 }, {λ01 }) = 0 = F ({λ2 }, {λ02 }). Putting r(s, t) = f (s) · f (t) here so that (λ1 , λ2 ) and (λ01 , λ02 ) are continuity points of F , this shows on simplification, that F (A, B) =

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M (A)M (B) for a suitable M (·). But F (·, ·) is a bimeasure so that M (·) is σ-additive, and has a unique extension to be a scalar measure on B. Thus the process is f.s.m. From this we can easily conclude, with a standard argument, that the process is strongly harmonizable. Thus the f.s.m. class is contained in the strongly harmonizable class. When F (A, B) = M (A) × M (B), the bounded Borel measure F has a unique extension to be a bounded Borel measure on R × R. Thus the integrals in (1) are in the Lebesgue sense so that the process is strongly harmonizable. Thus the integrals in (2) are also in the Lebesgue sense. In all cases, then the f.s.m. class is a proper subclass of the strongly harmonizable class.  We have just seen the periodically correlated class whose covariance r(s, t) is given by (1) which forms a special case of the Karhunen family representable as: Z Xt = g(t − u)dZ(u), t ∈ R, (4) R

with Z(·) determining F by F (A, B) = E(Z(A)Z(B)), and g(t, ·) ∈ L2 (F ), t ∈ R, E(Z(A)) = 0. This special class of the Karhunen family is said to have a moving average representation, motivated by some applications. Its relation to the harmonizable class itself has the following useful property which is separately stated for some applications as well as reference. It should also be noted here and in this area that H. L. Hurd (cf. e.g., (1992), and in much of his work referred to there and later) has done important work with real applications, and it should be consulted in relation to this area. The following structural analysis is often used in applications. Proposition 3.1.3 Let {Xt , t ∈ R} be a second order process having a moving average representation given by (4) above. Then it is strongly harmonizable and with its spectral measure to be absolutely continuous relative to the planar Lebesgue measure. Proof. Since g = fˆ in (4), hence bounded, it follows that (4) defined as in the Dunford and Schwartz sense can be simplified:

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Z  Z i(t−u)λ ˆ Xt = f (t − u)Z(du) = Z(du) e f (λ)dλ R R R Z Z itλ = e f (λ) e−iuλ Z(du)dλ, using a form Z

R

R

of Fubini’s theorem, Z =

eitλ f (λ)Y¯ (λ)dλ

(5)

R

where {Y (λ), λ ∈ R} is weakly harmonizable, and the symbol becomes R ˜ is a Bochner’s integral. Let Z˜ : A 7→ A Y¯ (t)f (t)dt, A ∈ B so that Z(·) stochastic measure on the Borel field of R, and one gets: Z Z ¯ ˜ ˜ E(Y¯ (s)Y (s)) · f¯(s)f (t)dsdt = ν(A, B) Z(B)) = E(Z(A) A

B

where the covariance ry (s, t) = E(Y (s)Y¯ (t)) is bounded and that ν(·, ·) has finite Vitali variation. Hence (5) can be given as Z ˜ Xt = eitλ Z(dλ), t ∈ R, R

and so {Xt , t ∈ R} is strongly harmonizable. Its covariance r(·, ·) can be calculated as: Z Z 0 ¯ r(s, t) = E(Xs Xt ) = eisλ−itλ ν(dλ, dλ0 ) R R Z Z 0 eisλ−itλ ry (λ, λ0 )f¯(λ)f (λ0 )dλdλ0 , (6) = R

R

so that ν(·, ·) is absolutely continuous relative to the Lebesgue measure.  The following consequence of the preceding discussion is useful: Proposition 3.1.4 The class of oscillatory processes {Xt , t ∈ T } ⊂ L20 (P ) (cf. (1)) and the class of Karhunen processes induced by T = R or Z, coincide. We now present an operator representation of a centered mean continuous process, as a major result also covering several classes studied earlier.

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Theorem 3.1.5 Let {Xt , t ∈ T } ⊂ L20 (P ) be a mean continuous process that is centered. Then it is representable as: Xt = At Ut Y0 , t ∈ T (T = R or Z),

(7)

for some Y0 ∈ H(X) = sp{Xt t ∈ T } ⊂ L20 (P ), where At is a densely defined closed linear operator on H(X) and {Ut , t ∈ T } is a weakly continuous unitary group of operators on H(X), which commute with each At , t ∈ T . If further the process is weakly harmonizable, then there exists a (bigger) Hilbert space K ⊃ H(X), operators At : K → H(X), and a weakly continuous unitary group {Ut , t ∈ T } which forms a weakly continuous positive definite contractive class satisfying A0 = id on H(X). On the other hand, a process given by (7) is always weakly harmonizable whenever {At , t ∈ T } is a weakly continuous positive 2 definite family of operators Pn onPLn0 (P ), with A0 = identity. [Here again positive definite means i=1 j=1 (Ati −tj hi , hj ) ≥ 0, n ≥ 1, as usual.] Proof. The argument below depends on some well-known (but deep) facts and we include the essential details for use as well as completeness. Now, the process being continuous in mean, H(X) is separable, and {Xt , t ∈ T } is of Karhunen class relative to a set {gt0 , t ∈ T } and µ, a Borel measure. Replacing gt0 with eit(·) gt we can express the process as Z Xt = eitλ gt (λ)Z(dλ), t ∈ T, (8) Tˆ

where Z(·) is orthogonally valued measure in H(X) Tˆ), the R from B( ¯ Borel σ-algebra so thatRE(Z(A)Z(B)) = µ(A∩B), T˜ |gt (λ)|2 µ(dλ) < itλ ∞, and hence Yt = Tˆ e Z(dλ) defines a stationary field. So there is a weakly continuous group of unitary operators {Ut , t ∈ T } acting on satisfying Yt = Ut Y0 . Also by the spectral theorem Ut = R H(X) itλ e E(dλ), t ∈ T , the {E(·), B} being the resolution of I, and B T˜ the Borel σ-algebra of Tˆ. Note that Z(A) = E(A)Y0 , A ∈ B here. R Now let At = Tˆ gt (λ)E(dλ). Since µ : B 7→ (E(B)Y0 , E(B)Y0 ) and R |g (λ)|2 µ(dλ) < ∞, it follows that At is a closed densely defined Tˆ t operator in H(X), whose domain contains {Ys , s ∈ T } for each t (from the standard functional analysis results), and At and {E(B), B ∈ B} commute. Then At and {Us , s ∈ T } also commute so that

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Z



itv

At Ut Y0 = At e E(dv)Y0 Tˆ Z  Z itv gt (λ)E(dλ) e E(dv)Y0 , = Tˆ Tˆ Z eitλ gt (λ)E(dλ)Y0 , = Tˆ

(by a property of spectral integrals), Z = eitλ gt (λ)Z(dλ) = Xt , by (8).

(9)

Tˆ

This shows that the representation (7) holds. The converse direction depends on a deep result due to von Neuman and F. Riesz (cf. Riesz-Nagy (1955), p. 35). It follows that Z At = φt (Ut ) = φt (λ)E(dλ). (10) Tˆ

This representation implies Z Z Xt = At (Ut Y0 ) = φt (λ)E(dλ) eitv E(dv)Y0 , ˆ T Tˆ Z eitλ φt (λ)E(dλ)Y0 , (property of spectral integral), = ˆ ZT = eitλ φt (λ)Z(dλ). Tˆ

If now gt0 (λ) = eitλ φt (λ), µ(B) = (E(B)Y0 , E(B)Y0 ), then this implies that gt0 ∈ L2 (Tˆ, B, µ) so that {Xt , t ∈ T } is of Karhunen class. For the last part, let Qt = At Ut then it can be verified that {Qt , t ∈ T } must be positive definite, weakly continuous and contractive, by known results (cf., e.g. Rao (1982), p. 330). Here the Ut -family is a unitary group and weakly continuous. Then the Ai -family must also be positive definite, which is verified as follows. Let hi ∈ H(X), i = 1, . . . , n be any set and ti ∈ T, n ≥ 1 and consider

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0≤ = =

n X

137

(Qti −tj vi , vj ), since the Qt is positive definite,

i,j=1 n  X i,j=1 n  X

Ati −tj Uti −tj Ut∗i hi , Ut∗j hj Ut∗i Ati −tj hi , Ut∗ij , hi





, as Ut∗ and As commute,

i,j=1

as also Ut and Us do, n X n X = (Ati −tj , hi , hj ). i=1 j=1

This shows that the {At , t ∈ T } is positive definite, and since Q0 |H(X) is the identity, we get A0 U0 = A0 as well. To see the opposite direction is also valid, we need to apply a key theorem due to A. Grothendieck so that if A : K → H(X) has the above properties then Qt = At Ut is positive definite and it satisfies the hypothesis of the author’s result (cf. Rao (1982), p. 330), so that the process is weakly harmonizable.  Remark 18. The first part of the result was given in an equivalent form by V. Mandreakar (1972) which refines some work by Glady˘sev (1962– 63). The measure µt (·) defined by Z Z 2 var(Xt ) = r(t, t) = |at (λ)| F (dλ) = µt (dλ), (11) R

R

is called an “evolving spectrum”, by Priestley (1965). We recall that a set Sβ ⊂ R × R is termed the support of a bimeasure β in R × R, if (x, y) ∈ R × R belongs to Sβ if and only if for each neighborhood U1 × U2 of (x, y), |β|(U1 × U2 ) > 0 where |β|(·, ·) is the variation of β, so that Sβ is the smallest closed set outside of which β vanishes. With this concept we can state supports of harmonizable process bimeasures if their covariances are periodic, to indicate that the problem is useful and nontrivial. Proposition 3.1.6 Let {Xt , t ∈ R} ⊂ L20 (P ) be weakly harmonizable with its covariance r being periodic in that r(s + α, t + α) = r(s, t), for

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some α > 0 and (s, t) ∈ R2 . If F (·, ·) is its spectral bimeasure, then its support is contained in SF , where  2σk SF = (λ1 , λ2 ) ∈ R2 : λ1 − λ2 = ,k ∈ Z . α

(12)

In the opposite direction, if S above is the support of a bimeasure F of a weakly harmonizable process then it is periodically correlated. Proof. It is known from the works of Gladysev and Hurd that the periodically correlated discrete parameter processes are always strongly harmonizable and we consider the continuous parameter case here. From the structural analysis above (cf. Niemi’s result on approximating weakly harmonizable processes by strongly harmonizable sequences on compact sets of R), we get limn rn (s, t) = r(s, t) uniformly for (s, t) ∈ K × K, K ⊂ R compact. Since r(·, ·) is periodic, say with period α, we observe that for large n, rn (·, ·) has the same property. If this is not true we must have for some ε > 0, and a point (s, t) ∈ R × R, the inequality, for some k 6= 0, lim inf |rn (s + kα, t + kα) − rn (s, t)| > ε. n

(13)

Since r(s + kα, t + kα) = r(s, t), and moreover we have, with rn → r uniformly on the (compact) set K = {(s, t) : (s + kα, t + kα)}, 0 < ε < |rn (s + kα, t + kα) − rn (s, t)| ≤ |rn (s + kα, t + kα) − r(s + kα, t + kα)|+ |r(s + kα, t + kα) − r(s, t)| + |r(s, t) − rn (s, t)| → 0 as n gets large which gives a contradiction. Hence Xtn must also be periodically correlated. Let Fn and F be the spectral measures of rn and r. We claim that Fn (A, B) → F (A, B) for Borel sets A, B. To show this (there is no Helly-Bray), we consider Xtn with covariance rn . Xtn

=

n X

φk (Xk , φk ),

n ≥ 1,

k=1

{φt , k ≥ 1} a CON system of the separable H(X) of the Xt -process. Then Xtn ∈ H(X), Xtn → X in L20 (P ) as n → ∞, uniformly in t on compacts. Let `k : Y 7→ (Y, φk ), Y ∈ H(X), be a linear mapping. If

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ζm and Z are representing stochastic measures of Xtn , Xt -processes then we get ζn (t) =

n X

φk `k (Z(t)),

Fr (A, B) = (ζn (A), ζn (B)).

(14)

k=1

Since `k (Z(A)) = (Z(A), φk ) we get with Parsevel’s equation that Z(A) = limn ζn (A) in mean, so that F (A, B) = (Z(A), Z(B)) = lim(ζn (A), ζn (B)) = lim Fn (A, B). n

n

Now rn is periodic and thus its support SFn ⊂ SF , n ≥ 1. Hence the above equation implies that F has its support in SF . In the converse direction SFn ⊂ SF for large enough n. The corresponding result of Hurd implies that Xtn is periodically correlated and strongly harmonizable. Since rn → r pointwise and (by Hurd’s work) strongly harmonizable, r is periodic, so Xt is also periodically correlated, as a consequence.  It is perhaps appropriate at this point to consider a large class of second order processes, centered at their mean functions to start with and discuss their integral representations motivated by the Karhunan class, starting with vector harmonizability as indicated by Rozanov (1959), and the consequent (abstract) methods. This will also exhibit the basic structure and general potential of this class, as it was described in Chapter 1. (See Theorem 1.4.2 with full details there.) Theorem 3.1.7 (Consistent Inversion) Let X : R → L20 (P ) be weakly harmonizable with Z : B(R) → L20 (P ) as its (stochastic) representing measure. Writing Z(λ) for Z((−∞, λ)), one has (the inversion) Z T −itλ2 e − e−itλ1 l.i.m. X(t)dt T →∞ −T −it Z(λ2+ ) + Z(λ2− ) Z(λ1+ ) + Z(λ1− ) − , = 2 2 (15) l.i.m. being the L2 (P )-limit. Moreover the covariance bimeasure F (·) of Z(·) is given, for A = (λ1 , λ2 ), B = (λ01 , λ02 ), as:

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T2

0

0

eitλ1 − e−iλ2 t eisλ1 − eisλ2 · r(s, t)dsdt, 0≤T1 0, there is a bounded Borel set Aε ⊂ R, such that sup F (B, B) < ε (bounded-Borel).

(62)

B⊂Aε

If σ0 is defined as σ0 = sup{|x − y| : x, y ∈ Aε }, the diameter of Aε , then for h > σ0 , E(|Xt − Xεt |2 ) ≤

C(t) + 4ε, nt (t1 − σ0 )

(63)

and Xεt is given as: Xεt

=

n X k=−n

 X

kπ h



sin(th − επ) , (th − kπ)

(64)

for some 0 < C(t) < ∞, where C(t) is bounded in bounded t-sets, and if the support S of F is bounded, S ⊂ Aε ×Aε , then (64) is automatically satisfied and ε = 0 can be taken in (63). This result is known in the strongly harmonizable case and the proof extends to the present case with simple modifications which will be left to the reader. Here we include an adjunct to the general operator representation, given as Theorem 3.1.5, which is of interest in several special applications. They are sometimes called a slowly changing class which class is more general than stationary but is included in the (weak) harmonizable class. These were studied by R. Joyeux (1987) and extended by R. J. Swift (1997) much further. If {X(t), t ∈ R} is weakly stationary and A(t) is slowly varying nonstochastic function, then Y (t) = A(t)X(t), t ∈ R, is termed a modulated process which is of interest in applications, when the X(t)-process is (weakly) stationary and more generally (weakly or strongly) harmonizable.

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Definition 3.3.7 Let X : R → L20 (P ) be a mapping and A(t, ·) be a family of measurable scalar functions, that are Fourier transforms: Z A(t, λ) = eitx H(λ, dx), λ ∈ R (65) R

¯ t ) admits a representation: and r(s, t) = E(Xs X Z Z ¯ λ0 )e(iλs−iλ0 t) dF (λ, λ0 ), r(s, t) = A(s, λ)A(t, R

(66)

R

with F (·, ·) as a function of finite Fr´echet variation. Such a mapping X is termed an oscillatory weakly harmonizable process. This class is an abstraction of a class, termed “oscillatory” and some statistical applications are indicated by M. B. Priestley (1981), and followed up by R. Joyeux (1987). They are further extended to the above form by Swift (1997). We include an account to indicate some useful applications of the work. This general class is of interest in applications where stationary and even harmonizable classes should be generalized, as seen in the discussion of Priestly’s analysis, and the consequent extensions. The following general characterization, due to Swift (1997), is of importance in this study. It sharpens Theorem 3.1.5 in some sense: Theorem 3.3.8 The process {Xt , t ∈ R} ⊂ L20 (R) is oscillatory weakly harmonizable iff it is representable as X(t) = A(t)T (t)Y (0), t ∈ R,

(67)

where Y (0) ∈ H(X) = sp{X ¯ t , t ∈ R} and A(t) is a closed densely defined linear operator in H(X), t ∈ R. Here {T (t), t ∈ R} ⊂ H(X) is a weakly continuous family of positive definite contractive maps commuting with {A(t), t ∈ R}. Proof. In the forward direction, the oscillatory weak harmonizability gives Z X(t) = A(t, λ)eitλ dZ(λ), R

with E(Z(B1 )Z(B2 )) = F (B1 , B2 ), giving F (·, ·) to have Fr´echet variation bounded.

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R

Also, the family Y : t 7→ R eitλ dZ(λ) defines a weakly harmonizable process. But by Theorem 2.3.3 above, there is a weakly continuous contractive family {T (t), t ∈ R} of positive type on H(X), the Hilbert space determined by the X(t)-process, such that Y (t) = T (t)Y0 . Now by the spectral representation of the T (t)’s Z ˜ T (t) = eitλ dE(λ), t ∈ R, (68) R

˜ and let Z(A) = E(A)Y 0 , A ∈ B(R), so that Z(·) is well-defined, and let R ˜ a(t) = R A(t, λ)E(dλ), t ∈ R. Then {a(t), t ∈ R} is a densely defined closed set of operators on H(X) whose domain includes {Y (t), t ∈ R}. ˜ ˜ But T (t) ^ E(D), t ∈ D, (commute) we get that a(t) and E(D) families also commute, so that a(t) ^ T (s), s ∈ R, t ∈ R. Consequently, Z  iut ˜ e E(du)Y0 a(t)T (t)Y0 = a(t) R Z  Z Z iwt ˜ e E(dω)Y0 A(t, λ)E(dλ) = R R R Z Z iλt ˜ A(t, λ)eiλt Z(dλ) = X(t). A(t, λ)e E(dλ)Y0 = = R

R

This gives (67) since E˜ and the integral commute. Thus if X(t) is oscillatory weakly harmonizable, then X(t) = a(t)T (t)Y (0) belongs to ¯ H(X) = sp{ ¯ X(t), t ∈ R}, (Y0 = Y (0)), being a closed densely defined operator on H(X), and T (s) ^ a(t), s, t ∈ R, the T (s) family being a positive definite contractive set on H(X), admits an integral representation (von Neumann and F. Riesz) so that we have Z Z X(t) = a(t)T (t)Y (0) = g(t, λ)E(dλ) eiωt E(dω)Y (0) R R Z iλt ˜ = e g(t, λ)E(dλ)Y (0) R Z = eiλt g(t, λ)dZ(λ). (69) R

This is just the representation of oscillatory weakly harmonizable process.  In this connection, it may be of interest to present simple conditions to sample the random field {Xt , t ∈ R} using the above ideas.

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Theorem 3.3.9 Let {Xt , t ∈ R} be a centered Cram´er type process: Z Xt = g(t, λ)dZ(λ), t ∈ R, (70) R

R R E(Z(A)) = 0, E(Z(A)Z(B)) = A B d2 ρ(x, y), A, B ∈ B(R), where ρ(·, ·) is a covariance function of bounded variation on bounded sets of R2 . If L2 (X) ⊂ L2 (P ) and L2 (ρ) the corresponding Hilbert space determined by ρ, let g (n) (·, λ) exist and g (n) (t, ·) ∈ L2 (ρ), n ≥ 1. If for {ti , i ≥ 1} ⊂ R of nonperiodic points, g (n) (t, ·)[∈ L2 (ρ)] are considered 2 and if M = sp{X(t ¯ i ), i ≥ 1 distinct}, then L (ρ) = M. Proof. The hypothesis implies that the covariance r(·, ·) is given by Z Z r(s, t) = g(s, x)g(t, y)d2 ρ(x, y) R

R m+n

and g (n) (t, ·) ∈ L2 (ρ), n = 0, 1, 2, . . .. This implies ∂∂sn tmr exists, and then the covariance r(·, ·) is infinitely differentiable and so also is X(·). ˜ 2 (ρ) ⊂ L2 (ρ) be the subspace corresponding to L2 (X) by the Let L resulting isometry. Since {g(t, ·), t ∈ R} determines L2 (ρ), it follows from the Class C structure, that L2 (ρ) = L˜2 (ρ). Since g(·, ·) is infinitely differentiable, we easily get  Z Z ∂ n r(s, t) ∂ (n−1) 2 = g (s, x)¯ g (t, y) d ρ(x, y) ∂sn s=t0 ∂s R R s=t0 and g (n) (t0 , ·) ∈ N, the closed subspace of L2 (ρ) determined by {g(ti , ·)}. From this, it is easily seen that g (n) (ti , ·) ∈ M. But we also have g(t, x) =

∞ X (t − t0 )n n=0

n!

g (n) (t0 , x).

(71)

This gives from the fact that X(t) ↔ g(t, ·), as is known, we get Xt =

∞ X (t − t0 )n n=0

n!

X (n) (t0 ),

(72)

in the sense of mean-squared convergence. The isometry here leading Xt ↔ g(t, ·) implies that g(t, ·) ∈ M and L2 (ρ) = M, as desired.  As a consequence, we get the statement for the harmonizable case:

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Corollary 3.3.10 Let {Xt , −∞ < t < ∞} be a weakly harmonizable process with ρ as its spectral measure. Suppose its moment generating function exists, then for any sample {X(ti ), i ≥ 1} as in the theorem above, the set {X(ti ), i ≥ 1} determines L2 (ρ). The hypothesis implies that the process must be strongly harmonizable, and ρ(·) has all moments. The result is then a simple consequence of the theorem. We next consider some other conditions that generalize the strongly harmonizable (hence weakly stationary) classes, of some interest in applications. 3.4 Class (KF) and Nonstationary Processes Applications It was pointed out by H. Niemi that there exist weakly harmonizable processes that are not of class (KF), although all strongly harmonizable ones are in it. This is of interest here since the class (KF) is based on summability methods, not completely dependent on Fourier analysis. The positive result is as follows: Theorem 3.4.1 If X : R → L20 (P ) is strongly harmonizable, then it is in class (KF) so that it has an associated spectral function. Proof. Since X is (strongly) harmonizable there exist a stochastic measure Z(·) and a bounded bimeasure F , such that Z
eitλ dZ(λ); F (A, B) = Z(A), Z(B) , (73) X(t) = R

with Z : B(R) → L2 (P ), so F : B(R) × B(R) → C, and K : (s, t) → ¯ E(X(s)X(t)). Then we get for T > 0 Z T −h T −h 1 rT (h) = · K(s, s + h) ds (74) T T −h 0 and we assert that limT →0 rT (h) exists. For this consider (74) as  Z T Z  Z Z 1 T 1 isλ −i(s+h)λ0 0 Z(dλ ) , K(s, s + h) ds = E ds e Z(dλ) e T 0 T 0 R R (75)

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and we note that the right side has a limit as T → ∞. For, let X = Y = L20 (P ) and Z = L1 (P ). Now Z : B → X, Z˜ = Z : B → Y are stochastic measures, we can define a product measure Z ⊗ Z˜ on B × B → X as a Dunford-Schwartz integral which satisfies Z Z Z ˜ ˜ f (s, t)(Z ⊗ Z)(ds, dt) = Z(ds) f (s, t)Z(dt) R×R ZR ZR ˜ = Z(dt) f (s, t)Z(ds) (76) R

R

for f ∈ C0 (R × R), (cf. N. Dinculeanu (1974) and M. Duchoˇn and I. Kluv´anek (1967), for such product measures with finite semi-variation). Here
2 ˜ × R) ≤ kZk(R) < ∞ kZ ⊗ Zk(R implying that Z ⊗ Z˜ is again a stochastic measure. So (76) gives Z Z Z 0 0 −i(s+h)λ0 isλ 0 eis(λ−λ )−ihλ Z ⊗ Z(dλ, dλ0 ) Z(dλ ) = e Z(dλ) e R

R

R×R

(77) and the right side belongs to L1 (P ). Using the same kind of argument one gets with µ : B((0, T )) → R+ as Lebesgue measure, Z T Z Z Z T µ(dt) f (t, λ)Z ⊗ Z(dλ) = Z ⊗ Z(dλ) f (t, λ)µ(dt). 0

R×R

R×R

0

(78) Writing µ(dt) as dt, (76)–(78) yield:   Z T Z 1 is(λ−λ0 )−ikλ0 0 ds e Z ⊗ Z(dλ, dλ ) E T 0 R×R Z  iT (λ−λ0 )   −1 −ihλ0 e 0 e =E χ[λ6=λ0 ] + δλλ0 Z ⊗ Z(dλ, dλ ) . T (λ − λ0 ) R×R (79) But the quantity inside E(·) is bounded for all T , and the dominated convergence for these vector integrals holds, so that we can let T → ∞ and simplify (79) to get Z Z
1 T 0 lim K(s, s + h)ds = E e−ihλ δλλ0 Z ⊗ Z(dλ, dλ0 ) T →∞ T 0 Z R = e−ihλ F (dλ, dλ0 ), [λ=λ0 ]

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159

where F Ris the bimeasure of Z. Hence R limT →∞ rT (h) =0 r(h) exists and −iht r(h) = R e G(dλ), G : A 7→ π−1 (A) δλλ0 F (dλ, dλ ), A ∈ B, is a positive definite measure of X in class (KF), as asserted.  The preceding result implies the following (major) result (an inversion theorem) obtained by Rozanov ((1959), Thm. 3.2) differently. The first part is a type of [Fourier] inversion formula. Theorem 3.4.2 Let X : R → L20 (P ) be a weakly harmonizable process with Z : B → L20 (P ) as its representing stochastic measure. Then for any −∞ < λ1 < λ2 < ∞, with Z(λ) for Z((−∞, λ)), we have Z T −itλ2 e − e−itλ1 l.i.m. X(t)dt T →∞ −T −it Z(λ2 +) + Z(λ2 −) Z(λ1 +) + Z(λ1 −) = − (80) 2 2 where l.i.m. is the L2 (P )-limit. Also, the covariance bimeasure F of Z is given for A = (λ1 , λ2 ), B = (λ01 , λ02 ) as: Z T1 Z T2 −iλ2 s 0 0 e − e−iλ1 s e−iλ2 t − e−iλ1 t lim · r(s, t)ds dt 0≤T1 ,T2 →∞ −T −is −it −T2 1 = F (A, B) (81) where A, B are continuity intervals of F , and r(·, ·) is the covariance function of the X-process. In particular, if r(·, ·) vanishes at RT (±∞, ±∞), and S : R → C is continuous such that T1 0 S(t)dt → a0 , RT then the signal ‘a0 ’, can be estimated as SˆT = T1 0 Y (t) dt → a0 as T → ∞ in that E(|SˆT −a0 |2 ) → 0 as T → ∞, with Y (t) = X(t)+S(t). Sketch of Proof. We reduce this result to the stationary case by using the ˜ Σ, ˜ P˜ ) dilation theorem. Thus there is an enlarged probability space (Ω, 2 ˜ 2 2 ˜ of (Ω, Σ, P ) with L0 (P ) ⊃ L0 (P ), and a stationary Y : R → Lo (P ), such that X(t) = QY (t), t ∈ R where Q : L20 (P˜ ) → L20 (P ), the orthogonal projection, and there is an orthogonally-valued measure Z˜ : B → L20 (P˜ ) such that Z ˜ Y (t) = eitλ Z(dλ), t∈R (82) R

˜ and Z(A) = QZ(A), A ∈ B, with Z : B → L20 (P ) representing the X-process. Since Q is bounded, it commutes with l.i.m. as well as the

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integral. Then the standard familiar argument shows that, applying Q to both sides of (80), the result follows as stated. Regarding (81), consider the left side with (80), and we get  Z T2 −isλ2  e − e−isλ1 L.H.S. = lim E X(s) ds T1 ,T2 →∞ −is −T1   Z T2 −itλ2 e − e−itλ1 X(t) dt · −it −T1      Z(λ1 +) + Z(λ1 −) Z(λ2 +) + Z(λ2 −) − =E 2 2      Z(λ2 +) + Z(λ2 −) Z(λ1 +) + Z(λ1 −) · − 2 2 = F (A, B) (83) by the continuity hypothesis on F , after R T expanding and simplifying. 1 ˆ Finally, let aT = E(ST ) = T 0 S(t) dt, with Y˜ (t) = X(t) + S(t), t ∈ R. Since now Y˜ ∈ class (KF ) and aT → a0 as T → ∞, Z Z 2 T T 2 ˆ E(|ST − a0 | ) = r(s, t)ds dt + 2|aT − a0 |2 T 0 0 Z T 1 rT (h) dh + 2|aT − a0 |2 , = 2T −T R T −|h| where rT (h) = T1 0 K(s, s + |h|) ds → r(h) as T → ∞ since Y˜ ∈ class (KF ). Since Xt is V -bounded, one can invoke the known Cesarosummability, to conclude that r(h) = 0, h ∈ R, (and tends uniformly on compact subsets of R). It follows finally that E(|SˆT − a0 |2 ) → 0 if T → ∞, as asserted.  It is also of interest here to sketch the vector (or multivariate) versions of the preceding analysis. Thus let L20 (P, Ck ) be the space of kvector complex measurable functions (r.v.’s for us) on (Ω, Σ, P ), centered and square integrable, i.e., E(f ) = 0, and kf k2 = (f, f ), for f = (f1 , f2 , . . . , fk ) ∈ L20 (P, Ck ), where Z (f, g) =

((f (ω), g(ω)) dP (ω) = Ω

k Z X i=1

Ω

fi (ω)¯ gi (ω) dP (ω),

(84)

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and let X = L20 (P, Ck ), which becomes a Hilbert
space with (84) as 2 its inner product. Then X : G → X = L (P, C) is weakly or strongly harmonizable (vector) random field if for each a = (a1 , . . . , ak ), ai ∈ C, P the scalar process Ya = a · X = ki=1 ai Xi : G → L20 (P ) is weakly or strongly harmonizable (scalar) random field, a ∈ Ck . In an analogous way, the vector Karhunen or class(C) as well as stationary random fields are defined from the scalar concepts. Letting R(g, h) be the covariance matrix of the vector process X : G → L2 (P, Cn ) we can easily obtain the multivariate version of the random field as follows. Theorem 3.4.3 If G is an LCA group and X : G → X = L20 (P, Ck ) is a weakly continuous centered random field, then it is weakly harmonizable ˜ with X, denoted Z˜ = iff there exists a vector stochastic measure on G k 2 ˆ → L (P, C ), such that (Z1 , . . . , Zk ) : B(G) 0 Z ˜ X(g) = hg, siZ(ds), g ∈ G, (85) ˆ G

ˆ being the dual group of G. Further X is strongly harmonizable if also G the matrix F = (Fij ), i, j = 1, . . . , k with F (A, B) = ((Zj (A), Zk (B)), ˆ the dual i, j = 1, . . . , k is a matrix function of bounded variation on G, of G. The covariance matrix R = (Z, Z) = (Zi (·), Zj (·)) is equivalently given (or representable) as: Z Z R(g, h) = hg, sihh, tiF (ds, dt), g, h ∈ G, (86) ˆ G

ˆ G

where the integral in (86) is the MT-integral, or the LS-integral according as the random field X is weakly or strongly harmonizable. Here F is ˆ according a positive definite matrix function of bounded variations on G as the MT or Lebesgue-Stieltjes concepts apply. Conversely, if R(·, ·) is a positive definite matrix that is representable as (86), then it is the covariance function of an n-variate harmonizable random field. Outline of proof. It is enough to sketch the ideas of proof and the full details can be easily filled in from it by the interested reader. Let Ya = a · X, for a ∈ Ck , as a scalar product. Now Ya is weakly harmonizable when X is, and thus is representable as

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Z Ya (g) =

ˆ G

hg, siZa (ds),

g∈G

(87)

ˆ → H, and that relative to a stochastic measure Za on the dual G k Z(·) (A) : C → H is linear and continuous. By reflexivity of H, ˜ ˆ → Za (A) = a.Z(A) with the ‘dot product’ notation and that Z˜ : B(G) H defines a stochastic measure. Then Z Z ˜ Ya (g) = a · X(g) = hg, sia · Z(ds), = a · hg, siZ(ds), (88) ˆ G

ˆ G

the last integral being an element of X. Since the a is an arbitrary vector, this implies (85). If X is strongly harmonizable, then Ya has the same property, and the covariance bimeasure Fa = a · F where F (A, B) = (Zi (A), Zj (B)), where i, j = 1, . . . , k and the components of F (= Fij ) are of bounded variation. The result follows from the one-dimensional case. Finally the same holds for the weakly harmonizable case, using the MT-integrals, and all the statements hold with this change.  The following result of a Karhunen process, is an easy extension of the above work which is considered already for a useful special case in Gikhman and Skorokhod (1974). We include it here for applications as well as the completeness of the analysis. Proposition 3.4.4 Let X : S → L2c (P ) be a Karhunen process on a locally compact space S relative to a regular or Radon measure F on S with functions {gt , t ∈ S} ⊂ L2 (F ), based on a Radon measure space (S, B, F ). Then there exists a locally bounded regular or Radon stochastic measure Z : B → L2c (P ) where Bc ⊂ B is the δ-ring of bounded (Borel) sets such that (Z is orthogonally valued) ¯ (i) E(Z(A)Z(B)) = F (A ∩ B),

A, B ∈ Bc

and with it one has the D-S integral representations: Z (ii) X(t) = g(λ)Z(dλ), t ∈ S.

(89)

(90)

S

Conversely, if X : S → L2c (P ) is given by (90) for a Z(·) of the above type, with g and F satisfying (89) and (90), then X(t), t ∈ S is of

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Karhunen class. Also the linear hulls HX = HZ iff the gs ∈ L2 (F ) form a dense set. Proof Sketch. This is an extension of the earlier one with the D-S integration given for bounded operators, but is extended (and shown to be valid) for vector measures Z(·) of locally finite type (by Thomas (1970)). Thus for T ∈ B(L20 (P )) Z T X(t) = g(λ)(T ◦ Z)(dλ) S

Z˜ = T ◦ Z is seen to be a locally finite stochastic measure. It is then inferred with our earlier discussion that T X is weakly of class (C). The opposite direction is, however, more restricted. Conversely, let {X(t), t ∈ S} be weakly of class (C) and the ‘accompanying’ g-functions are bounded Borel and M (S) is the uni2 formly closed algebra based R on {gs , s ∈ S}, then each gs ∈ L (FX ), and if T gs = X(s) = S gs (λ)Z(dλ), we extend T linearly to the closed algebra determined by {gt , t ∈ S}, then M (S) ⊂ L2 (Fx ), R and T gt = X(t) = S g(λ)Z(dλ); we extend T linearly to M (S), so T ∈ B(M (S), H), with sup norm for M (S). kT f k ≤ kf k2,µ ,

f ∈ M (S),

for a finite measure µ on S. Then one can follow the dilation theorem’s argument and can complete the proof.  The preceding analysis can be summarized in the following simpler form for reference and extensions. Theorem 3.4.5 Let S be locally compact and X : S → L20 (P ) be a Karhunen field relative to a Radon measure F : B(S) → R∗ and a class {gs (·), s ∈ S} ⊂ L2 (S, B(S), F ), the space of scalar square integrable functions (for F ) on (S, B(S)). Then there is a regular (or Radon) stochastic measure Z : B0 → L20 (P ), with B0 ⊂ B, the δ-ring of bounded sets such that (i) E(Z(A)Z(B)) = F (A ∩ B), A, B ∈ B0 , and (ii) Z X(t) =

gt (λ)Z(dλ),

t ∈ S,

(91)

S

where the integral in (91) is the standard (or DS) vector integral, and {gt , t ∈ S} ⊂ L2 (F ).

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In the other direction, if Q : H → H is a bounded projection op˜ erator, then X(t) = QX(t), t ∈ S, is weakly of class (C), when the X(t)-process is of Karhunen class. Since the gt (·) function is not necessarily gt (λ) = eitλ , for λ ∈ R, the classes of Hilbert spaces containing the Cram´er processes and the Karhunen classes associated with the processes do not necessarily have the familiar inclusions as in the stationary and harmonizable families, and the Sz.-Nagy-Na˘ımark type simple inclusions are not always valid. The Fourier characters {eit(·) } and the general {g(t, ·)}-classes vary differently and in the generalization, the “charm” of harmonic classes is lost! This distinguishes the harmonizable families (weak or strong) from the Cram´er class. These differences make the study of second order processes interesting and this will be illustrated below. 3.5 Further Classifications and Representations of Second Order Processes In this section, we present some useful integral representations of not necessarily stationary or even harmonizable processes that extend our analysis of classes already treated above. Recall that a centered second order Khintchine (or K-) stationary process {Xt , t ∈ R} ⊂ L20 (P ), ¯ t ) as the representation for its covariance r : (s, t) 7→ E(Xs X Radmits i(s−t)λ e dZ(λ), for an orthogonally valued measure Z : B(R) → R L20 (P ), so that (Z(A), Z(B)) = α(A ∩ B), A, B ∈ B(R), and r(s, t) = r˜(s − t). Suppose, more generally, Xs+t , s, t ≥ 0, and for some Pn Vs : Xt → P n linear combination Vs ( j=1 aj Xtj ) = j=1 aj Xs+tj holds. If this is to Pncombination (ai ∈ C) we must have (i) Pnbe valid for each linear a X = 0 =⇒ j=0 aj Xtj +s = 0, n ≥ 2, and for the operj=1 j tj ators {Vs , s ≥ 0} to be bounded on L20 (P ), one must have P P kVs ( nj=1 aj Xtj )k2 ≤ Cs k nj=0 aj Xtj k2 , n ≥ 2, Cs > 0, so that we also will find Vs1 Vs2 = Vs1 +s2 , V0 = I, and that {Vs , s ≥ 0} forms a semi-group with V0 = I, the identity. But for our analysis here, integral representations of these operators is desired. These are called “subnormal” operators on an L20 (P ). That such an extension is possible on enlarging the basic probability space so that L20 (P ) can be assumed

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to carry these processes was established by Bram (1955). Using this, we have ZZ ZZ itλ e E(dλ)X0 = eitλ dZX0 (λ), (92) Xt = V t X0 = ∆

∆

for a Borel set ∆ ⊂ R2 and ZX0 : B(∆) → L20 (P ) is orthogonally valued (vector) measure. Then the covariance r(·, ·) is given by Z Z ¯0 r(s, t) = ei(sλ+tλ ) dβ(λ, λ0 ), (93) ∆

with β(A) = E(|Z(X0 (A))|2 ), (93) being analogous to Khintchine’s form. However, the general operator theory of semi-groups allows us to express Vt = Rt Ut = Ut Rt where the {Ut , t ∈ R} is a unitary group with U−t = Ut∗ , t > 0 and {Rt , t ≥ 0} is a positive self-adjoint semigroup. The Rt , Ut families commuting with all Borel functions of these operators. Thus Vt+s = Vt Vs = Rt Ut Rs Us = Rs+t Us+t ,

(94)

both the Rt and Ut families are strongly continuous semi-groups (the Ut being a group) of operators in L0 (P ) and we then have r(s, t) = (Xs , Xt ) = (Rs Us X0 , Rt Ut X0 ) = (Us X0 , Rs Rt Ut X0 ) = (Ut∗ Us X0 , Rs+t X0 ) = (Rs+t X0 , Us−t X0 ) ¯ s−t Xs+t ) = r˜(s − t, s + t) = E(X

(95)

¯ + ; so r˜ is now defined on the cone {(s, t) ∈ and r is positive definite on R ¯ × R : |t| ≤ s}. Also r(·, ·) and r˜(·, ·), are related by R s + t s − t r˜(s, t) = r , , s > t > 0. (96) 2 2 A characterization of r˜ needs a nontrivial extension of the classical Bochner’s result. This was obtained by A. Devinatz (1954). Thus if C = {(s, t) ∈ R+ × R : |t| < s/2}, then we have the following:

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3 Applications and Extensions of Harmonizable Processes n X

ai a ¯j r˜(si + sj , ti − tj ) =

i,j=1

=

n X i,j=1 n X

ai a ¯j (Rsi +tj X0 , Usi −tj X0 ) ai a ¯j (Rsi Usi X0 , Rtj Utj X0 )

i,j=1

since the Vt = Rt Ut commute, n X = (ai Rsi Usi X0 , aj Rsj Usi X0 ) ≥ 0. i=1

(97) The nontrivial converse implication is given by Devinatz’ result. With these properties and that of his key theorem, we have Z Z 0 t Xt = Vt X0 = Rt Ut X0 = λ dE1λ eitλ dEλ0 X0 , (Rt = R1t ) R+ R Z 0 et(log λ+iλ ) dE˜λλ0 X0 . (E˜λλ0 = E1λ Eλ0 ). (98) = [Re λ>0]

The commutativity of E1λ and Eλ0 is used. We may summarize the above work in the following (general) result: Theorem 3.5.1 Let {Xt , t ≥ 0} ⊂ L20 (P ) be a weakly continuous process admitting a right translation in that τs : Xt → Xs+t is a bounded linear subnormal mapping on sp {Xt , t ≥ 0} ⊂ L20 (P ), and it has an extension to be normal on L20 (P ), possibly enlarging the measure space by adjunction if necessary, so that one has Z g(s, λ)g(t, λ)dα(λ), s, t ∈ T. r(s, t) = (Xs , Xt ) = (99) S

Then there exists a family of commuting set of bounded operators {Bt , t ∈ T } on L20 (P ) such that for a fixed t0 ∈ T , Z Xt = Bt Xt0 = g(t, λ)dZXt0 (λ), t ∈ T, (100) S

holds for a unique orthogonal random measure ZX0 on S into L20 (P ) and the integral (100) is in the Dunford-Schwartz sense. These considerations, utilizing somewhat general but popular and interesting classes of operators from the Hilbert space theory, contribute

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several new avenues of analyzing problems originating from stationarity. Here is another example. Let {Bt , t > 0} be a class of linear weakly continuous semi-group of contractions on L20 (P ) and let Xt = Bt X0 for ¯ t ), Xt ∈ L2 (P ). The process an X0 ∈ L20 (P ) so that r(s, t) = E(Xs X 0 2 {Xt , t ≥ 0} ⊂ L0 (P ) may not admit a shift. It is called conservative if r(s + t, t + h) = r(s, t) for s, t > 0, h > 0 and is dissipative if for each P finite set t1 , . . . , tn and h > 0, the process Yn (h) = ni=1 ai Xti +h has a decreasing variance function in h, i.e., σn2 (s) = E(|Yn (s)|2 ) ≥ σn2 (s + h),

n > 0, s > 0, h > 0.

The existence of such processes can be obtained from the Kolmogorov (Bochner) projective limit theorem. The following property is interesting. (It is given for information and completeness.) Proposition 3.5.2 Let {Xt = Bt X0 , t ≥ 0} be a dissipative L20 (P )valued weakly continuous process. Then it has a stationary dilation in a super L20 (P ) ⊃ L = sp {Xt , t > 0}, and the Xt has the integral representation Z eitλ dZ(λ), t ≥ 0, (101) Xt = R+

where Z(·) is a not necessarily orthogonally valued measure with the vector integral (101) in the Dunford-Schwartz sense. Thus a dissipative process is weakly, not necessarily strongly, harmonizable, but may still be taken (enlarging (Ω, T, P )) as L ⊂ L20 (P ). The following key result in another direction is due to Pitcher (1963), slightly extended, and gives the structure of the set of admissible means of a class of second order processes including Gaussians. Theorem 3.5.3 Let X = {Xt , t ∈ T = [a, b]} ⊂ L20 (P ) be a centered second order process with a continuous covariance function r. If {φn , n ≥ 1} and {λn , n ≥ 1} are the eigenfunctions and eigenvalues of the kernel r, let {Xn , n ≥ 1} and {fn , n ≥ 1} be defined by: Z − 12 Xn = λn Xt φn (t) dt T Z − 21 fn = λn f (t)φn (t) dt, f ∈ MP = Mp (X) (102) T

so that {Xn , n ≥ 1} ⊂ L2 (P ) are orthonormal and f = {fn , n ≥ 1} ⊂ `2 . Let {Pn , n ≥ 1} be the n-dimensional distributions of

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{Xn , n ≥ 1} with densities {pn , n ≥ 1} relative to the Lebesgue measure in Rn , n ≥ 1. Suppose that (i) pn > 0 a.e. (Leb), n ≥ 1, (ii) lim|tj |→∞ pn (t1 , . . . , tn ) = 0, 1 ≤ j ≤ n for almost all ti , i 6= j i (iii) ∂p , 1 ≤ i ≤ n exists for n ≥ 1, and (iv) there exists Kn < ∞ such ∂ti that for n ≥ 1, we have: 2  n Z X ∂ log pn dPn ≤ Kn < ∞. (103) ∂t n j=1 R f ∈ M (X) : f = {fn , n ≥ 1} ∈ `2 is a P −1 positive cone. Further, if n≥1 λn 2 < ∞, then M1 (X) = M (X). If also the Pn are symmetric about the origin of Rn , n ≥ 1, then M1 (X) becomes a linear set. Then the set M1 (X) =



Remark. The result (conclusion here) fails if (103) is replaced by uniform boundedness of the terms. Here we just consider nonstochastic signals of the process, to give a feeling for the subject. Proof. First, observe that X(t) has the L2 (P )-convergent (Karhunen) representation X φn (t) X(t) = Xn √ , t ∈ T (104) λ n n≥1 and by Mercer’s theorem, f = (f1 , f2 , . . .) ∈ `2 since Z Z ∞ X 2 f (t)f (s)r(s, t) ds dt < ∞. |fn | = n=1

T

T

Let f = (f1 , f2 , . . .) ∈ `1 ⊂ `2 and for a ≥ 0, set

aYn2 (ω) = pn X1 (ω)−af1 , . . . , Xn (ω)−afn /pn X1 (ω), . . . , Xn (ω) . Then {aYn , Fn , n ≥ 1}, Fn = σ(X1 , . . . , Xn ) is a positive super martingale, kaYn k22 = 1. Thus by a standard martingale convergence theorem aYn → aY∞ a.e. [P ], as well as in L1 (P ). Let Vn (α) : L2 (P ) → L2 (P ) be defined by the equation, (Vn (α)g)(X1 , . . . , Xn ) = aYn · g(X1 − αf1 , . . . , Xn − αfn )

(105)

where g is a bounded ‘tame’ function — one that depends on a finite number of coordinates. It is seen that Vn (α)h → V (α)h a.e., as well as

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in L1 (P ), for h ∈ n L∞ (Fn ) = M (say), and that {Vn (α), α ≥ 0} is a strongly continuous semi-group of isometries on L2 (Fn ), n ≥ 1, and then the V (α) is defined on L2 (Fn ) as a strongly continuous semi-group of isometries with φ Y∞ in place of α Yn . [We omit the standard (not obvious) computations here, and refer to the details given in Pitcher (1963).] The {V (α), α ≥ 0} is a strongly S measurable semi-group of isometries on L2 (F∞ ), where F∞ = σ( n≥0 Fn ). It can be verified (nontrivially) that for ‘tame’ h: S

khk2 = kV (1)hk2 = kV (α)V (1−α)hk2 ≤ kV (α)hk2 ≤ khk2 , h ∈ M. (106) The crucial step is to verify the semi-group property of the set {V (α), α ≥ 0}. There does not seem to be a simple direct method to see this, and it depends on an approximation theorem ofRTrotter’s (1958) applied ∞ to the resolvent set {Rn (λ), λ > 0}, Rn (λ) = 0 e−λα V (α)dα, λ > 0, which is strongly continuous in L1 (P ). This class satisfies (a) R(λ) − R(λ0 ) = (λ0 − λ)R(λ)R(λ), λ, λ0 > 0, (b) kλn R(λ)n k ≤ K < ∞, n ≥ 1, λ > 0 (c) limλ→∞ λRλ (λ) = I in L2 (F). Then since Rn (λ) → R(λ) strongly, one verifies with some detailed analysis that kλR(λ)g − gk2 = limn kλRn (λ)g − gk ≤ c/λ → 0 as λ → ∞ for g a bounded function defined on Ω, where g(·) depends on only a finite number of coordinate functions (varying with g). Such g’s form a dense set in L2 (F). With this approximation, the result follows.  Remark. The omitted (nontrivial) details are given in the paper (Rao (1975), Inference in Stochastic Processes-VI) and may be consulted for discussions and related computations, as it also answers Skorokhod’s question on Pitcher’s work as well. The preceding result can be stated in a somewhat different form, and it is also due to P¨ucher (1962) which is given here for reference, and this form is useful for the structure theory. Proposition 3.5.4 Let {Xt , t ∈ T } ⊂ L20 (P ) be a centered process with covariance r, and T = [a, b] ⊂ R. ThenR the integral operator R : L2 (T, dt) → L2 (T, dt), given by (Rg)(s) = T r(s, t)g(t) dt is positive definite and compact (so takes bounded sets into compact sets), and the set of means MP of the process, satisfies MP ⊂ R(L2 (T dt)) so that 1 f ∈ MP is of the form f = R 2 h for some element h ∈ L2 (T, dt).

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The processes considered thus far are scalar-valued, and their vector versions are not trivial extensions, as their analysis due to H. Wold (a thesis under H. Cramer’s direction) in the stationary case and its deeper analysis by N. Wiener (and P. R. Masani) later showed. Here we like to indicate possible extensions from our general viewpoint. (1)

(n)

Definition 3.5.5 If Xt = (Xt , . . . , Xt ), t ∈ R or Z, is an ndimensional centered process, it is weakly (or strongly) vector harmonizable whenever for each (complex) vector a = (a1 , . . . , an ), the scalar ˜ t = a · Xt (= Pn ai Xti ) is respectively weakly (or strongly) process X i=1 harmonizable. This entails the covariance matrix R(s, t) of the Xt -vector process with mean zero to have the integral representation Z Z 0 r(s, t) = eisλ−itλ F (dλ, dλ0 ) (107) T ×T


for a (unique) positive (semi-)definite matrix F (·, ·) = Fij (·, ·) , also an n × n [positive (semi-)definite] matrix of bimeasures of Fr´echet (or Vitali) variations on the (product) σ-ring B(T ) × B(T ). The following extensions of the scalar case are of interest, for the vector-valued process, and is given by Mehlman (1992). Theorem 3.5.6 (Wold Decomposition) For a vector process Xt ∈ [L20 (P )]n there exists a unique decomposition as: Xt = Rt + St ,

E(Rt St∗ ) = 0,

t ∈ R,

(108)

where the Rt -process is purely nondeterministic and the St -process is deterministic. Moreover, if the Xt -process is weakly harmonizable, then the Rt and St -processes have the same property, where deterministic means HX (−∞) = HX (+∞) if HX (t) = sp {Xsi , 1 ≤ i ≤ n, s ≤ t} ⊂ L20 (P ), −∞ ≤ t ≤ ∞. Proof. The notations and matrix multiplication complicate the layout more than the theory itself. Thus let It = πXt

and Rt = Xt − St

(109)

− − where π : [HX (∞)]n → [HX (−∞)]n . Now if Xt has a representation as the integral (T = R or (−π, π) for continuous or discrete indexed processes), then

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Z Xt =

eitλ dZ(dλ).

171

(110)

T

R Thus St = T eitλ (π ◦ Z)(dλ), is weakly harmonizable. But this and (110) together imply in (109) that the Rt -process also is weakly harmonizable. Then the Xt -process has the weakly harmonizable decomposition, generalizing the stationary case.  Extending the stationary case of the basic analysis by Wiener and Masani, to a class of harmonizable process Mehlman (1992) has presented some results which may be continued in taking the Wiener’s fundamental ideas to the harmonizable fields and beyond. For now, we leave this (vector) process analysis to future researchers and add some complements based on the above sections. 3.6 Complements and Exercises 1. Recall that by a stochastic process {Xt , t ∈ T } one means that on a probability space (Ω, Σ, P ), for each ω ∈ Ω we observe Xt (ω) at t ∈ T , and in the case of real process one can take Ω = RT and Xt (ω) = ω(t) and f : R → R, Yt = Xt + f (t) a translate of Xt by f (t), Tf X = X + f = Y where f (·) is termed a (nonstochastic) mean and X(t) is the (stochastic) noise. If Pf (= P ◦ Tf−1 ), then f is called an admissible mean of X, if Pf is P -continuous. The class Mp of admissible means of X is needed for an analysis of the process if the covariance behavior is known or can be assumed given. Let {Xt , t ∈ [a, b] ⊂ R} be a second order process with mean zero and continuous covariance r(·, ·). If R : L2 ([a, b], dt) → L2 ([a, b], dt) Rb is given by (Rg)(s) = a r(s, t)g(t) dt then the operator R is posi1 tive definite and compact, show the set Mp ⊂ R 2 (L2 (a, b)), so that 1 f ∈ Mp =⇒ f = R 2 h for some h ∈ L2 ([a, b], dt). [This interesting characterization of Mp is due to Pitcher (1963).] 2. The significance of the linearity of the set of admissible means Mp of the preceding problem is noted by the following (negative) property: Thus if (Ω, Σ, P ) is a Gaussian (function space represented) measure space [P (Ω) = 1], and Mp as in the above problem, let f0 ∈ Ω − Mp then P and Pf0 are singular, i.e., αf0 6∈ Mp , α 6= 0, and Paf0 ⊥ Pbf0 for a 6= b. Consider the mixture Q on Σ:

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Q=

∞ X k=−∞

αk Pkf0 , αk > 0,

∞ X

αk = 1.

k=−∞

So f0 ∈ MQ show tfo ∈ MQ only for integral t and MQ is not convex. [This is a modification of an example of Skorokhod’s (1970).] 3. This problem explains the “admissible means” property further. Let X = {X(t), t ∈ [a, b] = T ⊂ R} ⊂ L20 (P ) with continuous covariance r, and {λn , n ≥ 1}, {fn , n ≥ 1} as its eigenvalues and −1 R the corresponding eigenfunctions. Let Xn = λn 2 T X(t)φn (t) dt; −1 R fn = λn 2 T f (t)φn (t) dt, f ∈ MP , where MP is the set of admissible means of X. This problem gives (good) sufficient conditions in order that MP is a positive cone and even linear, exemplifying the nontriviality of this property. If Pn is the n-dimensional distribution of {Xi , 1 ≤ i ≤ n} with (Lebesgue) densities pn , n ≥ 1, suppose the following conditions hold: (i) pn > 0 a.e., n ≥ 1, n (ii) lim|ti |→∞ pn (t1 , . . . , ti , . . . , tn ) = 0, n ≥ 1, and (iii) ∂p ,1 ≤ ∂tj   2 Pn R pn dPn ≤ K0 < ∞, n ≥ 1. j ≤ n, exist, with j=1 Rn ∂ log ∂tj  Show M1 = f | MP : f = {fn , n ≥ 1} ∈ `2 is a positive cone. P −1 Also M1 = MP if n≥1 λn 2 < ∞. If each pn is symmetric about the origin of Rn , then the M1 is also linear. In general MP need not even be convex as the preceding problem implies. [This result is also due to Pitcher.] 4. In contrast to the preceding two problems, here we present a result that isolates (linear) admissible subspaces (of means) of MP . For this, we need to recall a few (nontrivial and somewhat advanced) properties of (nonnegative) convex functions. Let Φ : R → R+ , Φ(0) = 0, Φ(−x) = Φ(x) convex, |xΦ00 (x)| ≤ c < ∞ for all x ≥ 0, and Φ0 (x) ↑ ∞ as x ↑ ∞. For such a convex Φ, there is (uniquely) another convex function Ψ with similar properties, which is given by Ψ (x) = sup{|x|y − Φ(y) : y ≥ 0}. Such a Ψ is often called the complementary function to Φ, and they obviously satisfy the (Young) inequality, |xy| ≤ Φ(x) + Ψ (y). Suppose also that E(Ψ (βh∗m (·, a))) ≤ C1 < ∞ all β > 0, where h∗a (ω, a) = (∇fn (πn ω), πn a)/f˜(πn ω), ω ∈ Ω, a ∈ Mn

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173

Rt

satisfies 0 h∗n ((ω − sa), a) ds = − log hn (ω, a). The following condition ensures that hn → h a.e. with the desired properties. Thus let (Φ, Ψ ) be the complementary convex pair as defined above with |xΦ00 (x)| ≤ C < ∞, x > 0 but Φ0 (x) ↑ ∞ with the complementary function Ψ : x 7→ sup{|xy| − Φ(y) : y ≥ 0} and that the h∗ satisfies Z (†) sup Ψ (βh∗n (ω, a)) dP (ω) ≤ C1 < ∞, a ∈ MP n

Ω

for some β > 0. Then ta ∈ MP , t ∈ R, (so MP is linear) and dQ0 ∗ ∗ hn → h a.e., t ∈ R. The density h(·, ta) = dP is given by   Z t ∗ h (ω − sa, a) ds , a.e. [P ] (‡) h(ω, ta) = exp − 0

implying that Pta is equivalent to P . [If Φ(x) = |x| log+ |x|, so Ψ (x) = e+|x| − |x| − 1, this result is 2 given by A. V. Skorokhod (1970). If Ψ (x) = ex − 1, then the corresponding Φ cannot be written explicitly, but is also covered now. It is interesting to note that the above conditions yield a result of interest in the analysis which depends on some properties of Orlicz spaces. This and related details are given in the author’s paper (Rao (1977), 311–324) which may be of interest to the reader.] 5. This problem presents conditions in order that the set of all admissible means is a linear space. The result again uses a few properties of the Orlicz spaces for which a convenient reference is the book by Rao and Ren (1991). Let Φ : R → R+ be a twice differentiable symmetric convex function Φ(0) = 0, |xΦ00 (x)| is bounded, x ≥ 0 (and Φ0 (x) ↑ ∞) with Ψ : R → R+ as its conjugate function, i.e., Ψ (x) = sup{|x|y − Φ(y) : y ≥ 0}. Let Pn , Qn be the finite dimensional distributions under the hypothesis and its alternative with densities fn and gn and the likelihood ratio be denoted as h∗n (= fgnn ). R If now supn Ω Ψ (βh∗n (ω, a)) dP (ω) < ∞, for each a ∈ MP and some β > 0, then ta0 ∈ MP for all t ∈ R so that MP is linear. The density h(·, ta) = dQ is given by dP  Z t  ∗ h(ω, ta) = exp − h (ω − sa, a)ds , a.e. [P ]. 0

Hence Pta is equivalent to P . [There are many details to fill, and the reader is referred to Rao (1977); this extends Skorokhod’s (1970) work.]

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6. Here we consider a discrete parameter harmonizable process that satisfies a finite difference equation subject to (or disturbed by a perhaps) different (harmonizable) error process, opening up new avPk enues. Let Ln (·) = n=0 an (·)m−n , be a kth order difference map with an as constant coefficients. If {Yn , ∈ Z} is a (weakly) harmonizable process (observed as an output) P given by the operator Ln called a filter, so that Yk = Ln Xk = ki=0 ai Xi , and the Yn -process is observed and the filter Ln is assumed given, we desire to find the process {Xt , t ≥ 0} ⊂ L2 (P ), that is a unique harmonizable one which when the X-process is strongly (or weakly) harmonizable so is the Yn -sequence and conversely. (This problem was considered by K. Nagabhushanam (1951) if harmonizability is replaced by weak stationarity. Here we present its extension to strong and weak harmonizable processes, and indicate its use.) To utilize the fact that Ln is an nth order difference operator acting on the (weak) harmonizable series, giving a similar process {Yn , n ≥ 1} one uses its spectral properties (representations) and that the Yn process is (centered) harmonizable and hence so is the Xn -process. Thus Z Z Z 0 inλ ¯m) = Xn = e dZX (λ), E(Xn X ei(mλ−mλ ) dµ(λλ0 ) T

T ×T

hold, with ZX (·), as the stochastic and µ(·, ·), the (scalar) spectral measures of Xn ’s. Given a (weakly or strongly) harmonizable process {Yn , n ∈ Z} and a filter Ln such that Ln Xk = Yk , k ∈ Z, there exists a unique process {Xn , n ∈ Z} satisfying the equation above, R R 1 iff (i) |ν|(Q × Q) = 0 and (ii) Q QC F (u)F¯ (ν) d|ν|(u, v) < ∞, where ν(·, ·) is the bimeasure governing the Yn -process and F (·) is Pk 0 the (polynomial) filter determined by Ln (·) = j=0 aj (·)n−j and F (eit ) = P (e−it ), Q = {t ∈ T : F (t) = 0}. [This was first obtained by K. Nagabhushanam (1951) for the weakly stationary case, extended by J. P. Kelsh (1978) for the strongly harmonizable case, and finally by D. K. Chang (1998) for the weakly harmonizable case.] 7. (a) A process {Xt , t ∈ R} ⊂ L20 (P ) with covariance K(s, t) = R ¯ t ) is of class (KF) if K(·, ·) satisfies r(h) = limT →∞ 1 T −|h| E(Xs X T 0 K(s, s + |h|)ds, h ∈ R exists. It is seen that r(·) is positive definite. Verify that all stationary and even all strongly harmonizable

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175

processes are in class (KF). [Even some (but not all) weakly harmonizable ones are also in it.] (b) A process X = {Xt , t ∈ G} ⊂ L20 (P ), G an LCA group, is called almost harmonizable if there is a complex valued measure µ ˆ × B(G) ˆ and a set {g(·, γ), γ ∈ G}, ˆ continuous complex on B(G) functions g(·, γ) being almost periodic in G, such that R 0 (i) G× kg γ kG kgγ 0 kG d(λ, λ ) < ∞, and ˆ G ˆ R (ii) K(t, s − t) = (Xs , Xt ) = G× gγ −1 (t)dµ(γ, γ 0 ) exists. ˆ G ˆ gγ (s)¯ If the process X is almost harmonizable it is asymptotically stationary as well as almost periodically correlated. [For more on this application, see B. H. Schreiber (2004), showing how the second order processes extend with Hilbertian methods and analysis.] 8. With ideas similar to the above, and using the structural analysis of the Karhunen processes detailed in the last part of this chapter establish the following pair of properties: (a) Every weakly harmonizable process {Xt , t ∈ R} ⊂ L20 (P ), is a Karhunen process relative to some Borel family {ft , t ∈ R} and a stochastic measure Z : B(R) → L20 (P˜ ) with orthogonal values, where L20 (P˜ ) ⊃ L20 (P ) can be taken. [Hint: If QR: L20 (P˜ ) → L20 (P ) is ˜ the projection, so that we have X(t) = QY (t) = π(eit(·) )(λ)Z(dλ), R

let ft = π(eit(·) ), and complete the argument.] (b) If X : R → L2 (P ) is a mean continuous process, then on each compact interval it coincides a.e., with a Karhunen process. [Hint: Using the eigenfunction expansion of the covariance on each compact interval, using Mercer’s theorem, apply the preceding part to deduce this assertion.] 3.7 Bibliographical Notes This chapter is devoted to some useful applications of harmonizable processes which often subsume the classical (weakly) stationary processes and fields; the fields part will be treated in depth in the next chapters. Here we treated both oscillatory and periodic classes including an extended analysis of both Cram´er and Karhunen classes; especially their structural aspects are emphasized and detailed. Also, applications to signal extraction from noise and related problems are included.

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In this analysis, some extensions related to slowly varying fields are treated. An important aspect is to obtain the (unique) input from the output if the processes are (weakly) stationary, and this was first detailed in his thesis by Nagabhushanam (1951). Conditions for uniqueness are important. This motivated an extension naturally to harmonizable classes. A basic question is the unicity of the solution. It was also given there, but in reviewing the paper, for Math. Reviews, J. L. Doob has constructed an auxiliary process of the same difference equation with zero output, so that the sum of the solutions will satisfy the equation to contradict the key unicity requirement. But the sum of these solutions is not generally stationary, as required, and this was noted by J. Kelsh who then extended Nagabhushanam’s work in his thesis (1978) for strongly harmonizable classes which requires a different set of conditions. It needed a further analysis, and also different techniques to the weakly harmonizable case. This was done in his thesis (1983) by D. K. Chang. Some of these works are included in this chapter and others will be given in the next one. It may be noted that the ‘reviews’ in these review publications are not generally refereed, and contains tentative opinions and views. So they should be considered as a type of mathematical “news reports” and not to be taken generally as refereed facts. [See R. H. Bing’s (1981) advice, and practices on all such works and quick conclusions.] As the reader may have noticed, most of the analysis is usually centered on the covariance properties of the second order processes. But the mean values are important as well. This was analyzed in detail by Pitcher and a few key problems on the nontriviality of the structure of the mean functions was also noted by Skorokhod. We have considered these aspects in this chapter. This shows how simple looking problems on (second order) processes are not to be taken by hunch. Even for the popular Gaussian processes, these are nontrivial but useful. The multiple indexed classes are termed random fields and they will be discussed continuing a new concept called isotropy that enhances the key analysis with further problems and consequences. It will be taken up in the next chapters and the general analysis prompted by it will be continued in the rest of this volume. Thus the analysis of the current problems serves as a concrete illustration and motivation for the rest of the treatment to follow which forms the other half of the projected work.

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We have treated so far various properties of harmonizable random processes and of their structural analyses. But some key applications show that we need to study also various aspects of some related problems if the index is not a linear set as time axis, but is multidimensional such as the one corresponding to the space-time problems in evolution, as in Physics and elsewhere. This, therefore, takes us to considerations of random fields whose index is just a directed set, as in space-time problems mentioned above. It was also found out in early analysis that a stationary field {Xt , t ∈ Rn , n > 1} ⊂ L20 (P ) which satisfies an isotropy condition is the trivial one (i.e., a constant field, with probability one). Such facts lead us to analyze and establish their nontrivial structured and related properties to be used, e.g., in isotropy for harmonizable fields. 4.1 Harmonizability for Multiple Indexed Random Classes If the indexing of random classes {Xt , t ∈ G} where G ⊂ Rn , n > 1, and more generally (including G = Nn or = Zn ), with Xt ∈ L20 (P ), then the structural analysis of the Xt -family presents new nontrivial problems, and their families are called random fields. Generally, these G’s do not have linear ordering, and could be semi-groups. The following example motivates our possible (applicable) analysis to continue. Recall that an orthonormal set {Xn , −∞ < n < ∞} ⊂ L20 (P ), gives a new sequence Yn = ΠXn where Π : L20 (P ) → M = sp{X ¯ n , n ≥ 0} is an orthogonal projection onto M, so that Yn = {Xn , n ≥ 0}, or = 0, n < 0. Thus even though the Xn -set is trivially (weakly) stationary, the Yn -set is not stationary and not even strongly harmonizable! This is 177

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a consequence of a nontrivial extension of a theorem of F. and M. Riesz, by S. Bochner on the vanishing of Fourier coefficients on the half-line. But it (Yn series) is weakly harmonizable, since a weakly harmonizable sequence (or process) remains in the same family under bounded linear transformations. So we need to go beyond stationary classes and thus want to study the structural aspects of such classes again following Bochner, after introducing a new concept called isotopy here. As a motivation, let X : G → L20 (P ), G = Rn , be a mapping with ¯ t ), r(s, t) = r˜(s−t) = r¯(|s−t|), so that it is covariance r(s, t) = E(Xs X invariant under translations and rotations. A random field {Xt , t ∈ Rn } with this property (⊂ L20 (P )) is called isotropic and homogeneous. So what is the structure of r˜? Again Bochner has characterized such r˜(·), and given the fundamental integral representation as: Z ∞ J n−2 (µτ ) 2 r˜(s − t) = an τ = |s − t|, s, t ∈ Rn , (1) n−2 dF (µ), (τ µ) 2 0 n−2

where an = 2 2 Γ ( n2 ), and F (·) is bounded and it is defined by: Z Z F (u) = · · · dG0 (t1 , . . . , tn ), t = (t1 , . . . , tn ) ∈ Rn .

(2)

{y:|y| 0, αn2 = 22ν Γ (n/2)π 2 , ν = n−2 , and 2 + + (iv) µ : B(R ) × B(R ) → C is a positive definite bimeasure of finite Fr´echet (Vitali) variation, the integral for r(s, t) above is in the strict Morse-Transue (Lebesgue) sense, the series for r(·, ·) converging unconditionally. l Sm (·)(1 ≤ l ≤ h(m, n)) =

It should be observed here that, to obtain nontrivial solutions of the Laplacian ∆X = 0, it is necessary not to restrict to the simple and easy looking stationary isotropic fields, because it was found that in this restriction only trivial isotropic fields are found as solutions (i.e., the constants) which will not conform with real applications. (See Yadrenko (1983) on this negative result.) This has been analyzed by Swift (1994) later and found that the harmonizable processes have nontrivial and meaningful solutions. Using the addition formula for Bessel functions, one can simplify (3) with some routine (but not entirely simple) analysis to obtain an equivalent form (cf. Swift (1994) for details) as: Z ∞Z ∞ J(|λs − λ0 t|) r(s, t) = αn dµ(λ, λ0 ) (4) 0 |λs − λ t| 0 0 where µ has finite Vitali variation iff X is strongly harmonizable isotropic. The equivalence asserted above needs a nontrivial proof. For details of this assertion, we refer the reader to Swift (1994) and the properties of the classical special functions (the Bessel class) will be needed. For a clear understanding, it is useful to have a general characterization of weakly as well as strongly harmonizable isotropic covariances. This will also show the fundamental nature of Bochner’s representation (1) above, and its extension in Definition 4.1.1 and Swift’s formula (4). Theorem 4.1.2 Let X be a weakly harmonizable random field in Rn , n ≥ 1, with covariance r : Rn × Rn → C. Then it is isotopic iff r(·, ·) is representable as:  n  Z Z ∗ J (|λs − λ0 t|) ν ν r(s, t) = 2 Γ Φ(dλ, dλ0 ), (5) 0 ν 2 R+ R+ |λs − λ t|

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where Φ is a positive definite function of bounded Fr´echet variation,
Jν (·) is a Bessel function (of the first kind) of index ν = n−2 and the 2 integral is in the strict Morse-Transue sense. The corresponding strongly harmonizable characterization is obtained if Φ of (5) has finite Vitali variation and then the integral is in the standard Lebesgue sense. Thus (4) is an aspect (modification) of this result, due to Swift (1994) and it will also be presented after this proof is detailed. Proof. The direct part is simple since r(gs, gt) = r(s, t), for all g ∈ SO(n), the orthogonal group of matrices on Rn , as |g(λs − λ0 t)| = |λs − λ0 t| in (5), g being an isometry. Thus only the converse is to be established. This is done in four steps as follows: I. Let X : Rn → L20 (P ) be (weakly) harmonizable. Then by the ˜ Σ, ˜ P˜ ) containing the given space so that dilation, there is a larger (Ω, 2 2 ˜ L0 (P ) ⊇ L0 (P ), a stationary field Y : Rn → L20 (P˜ ) and an orthogonal projection Q onto L20 (P ), such that Xt = QYt , t ∈ Rn . Let ρ be the covariance function of Y , so ρ(s, t) = ρ˜(s − t). We assert that when X is isotropic we can take ρ(s, t) as ρ˜(s − t). We assert that when X is isotropic we can take ρ as ρ˜ to be also isotropic. Now Y in L2 (P˜ ) = H ⊗ H0 , is representable as Y = X + X1 , where X1 is stationary, valued in H0 . Although H0 can be realized as L2 (µ) where µ is a finite (Grothendieck) measure, we can also replace it by a standard Gaussian ˜ = ⊗α Bα , as in the proof of Theorem 2.3.1 (on product measure µ ˜, on Σ stationary dilations) and so can omit the detail here. We may arrange things such that µ ˜(gA) = µ ˜(A), which is true for cylinder sets A and g ∈ SO(n), the orthogonal group. Thus the Grothendieck measure generated Hilbert space can be replaced by an isometric Hilbert space based on a Gaussian measure after the initial step. Then the components of X1 = (X1t , t ∈ Rn ) will be invariant under g so X1s and X1t will be identically (Gaussian) distributed. Hence X1 (⊥ X) is stationary and isotropic. [The actual construction of this is based on P and µ ˜.] That the bimeasure F of X is invariant can be verified as follows:

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Z

Z

∗

ei(s·(g

r(gs, gt) = Rn

Z

dF (λ, λ0 )

Rn

Z

∗

= Rn

∗ λ)−t·(g ∗ λ0 ))

˜ ˜0 ˜ gλ ˜0) ei(s·λ−t·λ ) dF (g λ,

Rn

= r(s, t) Z Z ∗ 0 ei(s·λ−t·λ ) dF (λ, λ0 ). = Rn

181

(6) (7)

Rn

Since r(s, t) = r(gs, gt), ∀g ∈ SO(n), s, t ∈ Rn (by hypothesis), F (λ, λ0 ) = F (gλ, gλ0 ) implying invariance so that Ys - satisfies E(Ys Y¯t ) = ρ(s, t) = ρ˜(s − t). This is the key refinement of the dilation (X = QY ) that is needed here. II. Now consider Bochner’s stationary isotropic covariance representation of Y constructed above, and let ρ be its covariance. Then following Yaglom ((1987), p. 353) one has Z Jv (λτ ) ρ(t) = Cn dΦ(λ) (8) ν R+ (λτ ) where t = (τ, u), |t| = τ, u representing the spherical polar of t, J0 the
n ν Bessel function of order ν = n−2 , Φ(·) ↑ bounded and C = 2 Γ . n 2 2 Using the addition formula for Bessel functions (8) can be expressed as ∞ h(m,n) X X

Z

Jm+ν (r1 λ)Jm+ν (r2 λ) dΦ(λ), ν λ2ν (r r ) + 1 2 R m=0 l=1 (9) where s = (r1 , u), t = (r2 , v), h(m, n) = (2m + 2v) (m+2ν−1)! , m ≥ 1, ν!m! l and m = 0, Sm (n) being the spherical harmonics, orthogonal on unit sphere Sn , relative to the surface measure. For each m, there are a total of h(m, n) of them. Using (9), an integral form of Yt can be obtained. For this, we conveniently express (9) as a triangular covariance and apply Karhunen’s result. Again since only the second order properties are considered we may identify the process with a centered stationary Gaussian field with covariance S for (2nd order) computational manipulation. Thus, let ρ(s, t) =

Cn2

l l Sm (u)Sm (v)

·

˜ = {(n, l) ∈ N × N : 1 ≤ l ≤ h(m, n), m ≥ 0} N and P be its power set. Let F˜ : P × P × g(R+ ) → R+ , be given as

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F˜ (A1 , A2 ; B1 ) = J(A1 ∩ A2 )Φ(B1 ),

(10)

R

where Φ(A) = A dΦ(λ), the positive bounded measure by Φ of (9) and ζ(·) as the counting measure on P, so F˜ extends to a σ-finite measure, ˜ × Rn × R+ → R is given by and if g˜ : N l (n) g˜((m, l); t, λ) = Sm

Jm+ν (tλ) , (λτ )ν

l (·), Jν where t = (τ, u) is the spherical polar representation, and ν, Sm are defined earlier. Then g˜ is square integrable for F˜ and one has the ˜ × R+ , formula, with Λ = N Z Z r(s, t) = Cn g˜((m, l); s, λ)g¯˜((m, l); t, λ)dF˜ . (11) Λ

Λ

Now (11) implies that r(·, ·) is a triangular covariance on Rn × Rn relative to F˜ (cf. Rao (1982), p. 313), and so there exists a Gaussian measure Z : P ⊗ B(R+ ) → L20 (P ), giving Yt as: Z Yt = Cn g˜((m, l); t, λ)dZ((m, l); λ) ZΛ Z = Cn g˜((m, l); t, λ)dZ((n, l); λ), (12) ˜ N

R+

¯ 2 , B2 )) = F˜ (A1 , A2 ; B1 , B2 ) = ζ(A1 ∩ A2 ) · where E(Z(A, B)Z(A Φ(B1 ∩ B2 ). If now A1 = {(m, l)}, A2 = {(m0 , l0 )}, singletons, then l (B) and similarly for A2 , B2 , we get for writing Z(A1 , B1 ) = Zm ¯ the above E(Z˜m (B1 )Z˜m0 (B2 )) = δmm0 δll0 Φ(B1 ∩ B2 ). Hence letting l (λ), 0 ≤ p ≤ h(m, n)}, B = (0, λ), the associated processes {Zm m ≥ 0, are orthogonal each with orthogonal increments. Hence in the l case of Gaussians, the Zm (·) become independent, with independent increments. Thus (12) becomes Yt = Y (τ, u) = Cn

∞ h(m,n) X X m=0

l=1

l Sm (u)

Z R+

Jm+ν (τ λ) l Zm (dλ), (τ λ)ν

(13)

the series converging in L2 (˜ p)-mean, the Yt is isotropic and stationary. III. Now let Xt = QYt , with Q : L2 (P˜ ) → L2 (P ), the orthogonal projection, in the isotropic dilated (bigger) space, and we have

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183

Xt = X(τ, u) = (QY )(τ, u) ∞ h(m,n) X X

= Cn

m=0

l Sm (u)

l=1

Z R+

Jm+ν (τ 0 λ) (QZnl )(dλ) (14) (τ λ)ν

the interchange of the integral and the bounded Q is justified by a classical theorem of Hille, and setting Z˜nl (λ) = (QZnl )(λ), (Q being linear), Z˜nl (·) are independent identically distributed (Gaussian) we have l E(Z˜nl (A)Z¯˜m (B)) = δll0 δmm0 F (A, B),

(15)

l , which will not where F is the bimeasure determined by the d.f’s of Z˜m have orthogonal increments now, but F is independent of n, and is the same, and is a bimeasure of finite Fr´echet variation. Thus (13) is the desired integral representation of the field {Xt , t = (τ, u)}. We now compute the covariance of Xs , Xt , where s = (τ1 , u), t = (τ2 , v).

¯ t ) = Cn2 r(s, t) = E(Xs X

∞ h(m,n) X X m=0

Z R+

Z

∗

R+

l l Sm (u)Sm (v)×

l=1

Jm+ν (τ1 λ)Jm+ν (τ2 λ0 ) F (dλ, dλ0 ), (τ1 λ)ν (τ2 λ)ν

(16)

the right side double integral being in strict MT-sense. l l IV. Using the properties of spherical harmonics Sm (u)Sm (v) we can sum the series (16) to obtain h(m,n)

X l=1

l l Sm (u)Sm (v) =

ν (cos(u, v)) h(m, n)Cm , ν ωn Cn (1)

(17)

ν where ωn is the surface area of the unit sphere Sn ⊂ Rn , and Cm (·), m ≥ 0 are the Gegenbauer or ultraspherical polynomials of order ν, for each m ≥ 0. With (16) and (17) and using the addition formula for Bessel functions, indicated below, one gets the asserted representation (4). This gives the converse, hence the result.  The formula that we employed above is obtained on using some properties of spherical harmonics, detailed in M¨uller (1966). The version we need is as follows (using the same notation as above):

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αn2

∞ h(m,n) X X m=0

l=1

Jm+ν (λr1 )Jm+ν (λ l l Sm (u)Sm (v) (λr1 )ν (λ0 r2 )ν νΓ

=2

0

r2 )

Jν (λR(λ, λ0 )) (λR(λ, λ0 ))ν

n 2




  12 0
2 0
where R(λ, λ0 ) = r12 + λλ r22 − 2 λλ r1 r2 cos θ , and cos θ = u · v, the angle between the unit vectors u, v. This formula can be verified using the known properties of the “classical” spherical harmonics. This was given in detail in the paper by R. J. Swift (1994) as Lemma 2.1, and we leave it to the reader, for an independent try. In any case, the simplification is needed above. We now record some equivalent forms of entropy useful in the context of applications. Theorem 4.1.3 Let X = {Xt , t ∈ Rn } be a centered weakly harmonizable random field. Then the following are equivalent: (i) X is isotropic so that the covariance r(s, t) = r(gs, gt), g ∈ SO(n); (ii) the covariance r of X admits the representation (5); (iii) the covariance r of X admits a series representation (16) with the MT-integration relative to a bimeasure F of finite Fr´echet variation; (iv) X is representable as an L2 (P )-convergent series (32) with the stochastic integral representation in the standard vector (or Dunford-Schwartz) sense. The following approximation of a weakly harmonizable field on Rn with the one whose spectral measure lives on bounded Borel sets of Rn . Proposition 4.1.4 Let X : Rn → L20 (P ) be weakly harmonizable (perhaps not isotropic), with µ(·, ·) as its spectral measure that is essentially contained in a bounded Borel set, i.e., for an ε > 0, there is a bounded Borel set Aε ⊂ Rn such that kµk(Rn × Rn − Aε ) = 0. Then there is a weakly harmonizable field Xε : Rn → L20 (P ) with spectrum in Aε × Aε , and kX(t) − Xε (t)k2 < ε, where t ∈ Rn . (18)

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Proof Sketch. Since X(t) has the integral representation with Z(·), as ¯ its stochastic measure, and E(Z(A)Z(B)) = µ(A, B), kZk2 (A) = kµk(A, A), choose Borel Aε satisfying kZk(Aε ) < ε, and define Z1 , Z2 as Z1 (·) = Z(Aε ∩ ·) and Z2 (·) = Z(Acε ∩ ·) so that Z Z itλ e Z1 (dλ) + eitλ Z2 (dλ) = X1 (t) + X2 (t). X(t) = R

R

Then letting Xε (t) = X1 (t), one has kX(t) − Xε (t)k2 = kX2 (t)k2 ≤ kZ2 k(R) = kZ2 k(Acε ) < ε.



4.2 A Classification of Isotropic Covariances It is interesting to consider classes of random fields relative to their second order structures in the analysis of their isotropic covariances. From our preceding work, it is clear that the following proper inclusions between the stated classes obtain: Karhunen fields ⊃ isotropic fields ⊃ weakly harmonizable and isotropic fields ⊃ strongly harmonizable isotropic ones ⊃ stationary isotropic fields

Our analysis also gives the following integral representation in a series from: Theorem 4.2.1 Let X : Rn → L20 (P ) be a random field with a continuous covariance r. Then we have: (i) X is also isometric implies X(τ, u) =

∞ h(m,n) X X m=0

l l ξm (τ )Sm (u),

(19)

l=1

l where {Sm : 1 ≤ l ≤ h(m, n), m ≥ 0} are the spherical harmonics on the unit sphere Sn ⊂ Rn , that are orthonormal for the normalized surl 0 ¯l0 0 face P∞ measure 0µm of Sn , and E(ξm (τ )ξm (τ )) = δm,m0 δll0 bm (τ, τ ) with n=0 bm (τ, τ )h(m, n) < ∞, all the other symbols as defined before. (ii) X is isotropic and harmonizable implies that the series (19) holds l in which the variables ξm can be represented as: Z Jm+ν (τ λ) ˜ ρ l ξm (τ ) = cν dZm (λ), (20) (τ λ)ν R+

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l where ν = n−2 : R+ → L20 (P ) is a stochastic measure satisfyand Z˜m 2 l l0 0 0 ˜ ˜ ing E(Z˜m (dλ)Z¯˜m 0 (dλ )) = F (dλ, dλ )δmm0 δll0 with F as a bimeasure, ν m Cν = 2 νΓ (ν)i , and the integral in (20) is in the Dunford-Schwartz sense. Further bm (τ, τ 0 ) is given as: Z Z ∗ Jm+ν (τ λ)Jm+ν (τ 0 λ0 ) ˜ 0 2 bm (τ, τ ) = |Cν | F (dλ, dλ0 ) (21) ν 0 0 ν (τ λ) (τ λ ) R+ R+

the integral now being in the strict M T -sense. The random field is strongly harmonizable iff F˜ in (21) has finite Vitali variation whence the integral in (21) becomes one in the Lebesgue-Stieltjes sense. l (iii) X is isotropic and stationary implies Z˜m (·) are orthogonally ˜ valued for all 1 ≤ l ≤ h(m, n), m ≥ 0 so F of (21) reduces to a bounded Baire measure, say Φ, and we have: Z Jm+ν (τ λ)Jm+ν (τ 0 λ) 0 2 bm (τ, τ ) = |Cν | dΦ(λ), (22) (λ)2ν (τ τ 0 )ν R+ and the integral now is in the standard Lebesgue sense. In the converse direction each of the stated representations, gives the corresponding random field. Proof. (i) This is a slightly revised version of Yadrenko’s (1983) analysis. By the classical results, the isotropic covariance r(·, ·) of X can be expressed as r(s, t) = r˜(τ1 , τ2 , cos(u, v)) with s = (τ1 , u), and t = (τ2 , v) in spherical polar coordinates (cf., e.g., Vilenkin (1968), Ch. XI). Thus r˜ can be expressed as: r˜(τ1 , τ2 , cos(u, v)) =

∞ h(m,n) X X

l l bm (τ1 , τ2 )Sm (u)Sm (v),

(23)

m=0 k=1

where bm (τ1 , τ2 ) ≥ 0 are eigenvalues of r˜ so that by the well-known Funk-Hecke formula (C. S. M¨uller (1964), p. 20), we have: Z l l r˜(τ1 , τ2 ; cos(u, v))Sm (u)dµn (u) = bm (τ1 , τ2 )Sm (v) (24) Sn

where µn is the surface measure of the unit sphere Sn of Rn . [Recall that 2π this is, µn : Sn → n−1 µ (S ), n ≥ 2, µ2 (S2 ) = 4π, µ1 (S1 ) = 2π.] R n−2 n−2 l l Letting ξm : τ 7→ Sn X(τ, µ)Sn (u)dµn (u), we get for each τ ∈ R+ , l the r.v.’s ξm (τ ) are orthogonal with the asserted properties. The series

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(19) converges in mean, and then (23) follows from the addition formula for spherical harmonics. (ii) Replacing the general isotropic fields with the weakly harmonizl able class, the processes ξm just defined, can be given a sharper representation. Thus comparing the two forms of X(τ, u) given above, one has Z ∞ h(m,n) ∞ h(m,n) X X X X Jm+ν (τ λ) ˜ l l l Sm (u) Sm (u)ξm (τ ) = cn Zm (dλ). (τ λ)ν R+ m=0 l=1 m=0 l=1 0

l Multiply both sides by Sm 0 and integrate with the surface measure µm , using the orthogonality relations, then we find Z Jm+ν (τ λ) ˜ l l Zm (dλ); τ ∈ R+ (25) ξm (τ ) = cn ν (τ λ) + R l0 l (τ1 )ξ¯m which is (20), and (21) follows from the fact that E(ξm 0 (t2 )) = l ˜ δll0 δmm0 · bm (τ1 , τ2 ), since Zm are orthogonal in M, and have the same second moments in general. This will be more difficult to show generally. l (iii) This part is immediate now since each Zm has orthogonal increments in (l, m) so F (A, B) = Φ(A ∩ B) in the above work, and (22) obtains, the integral now is in the usual Lebesgue sense. In the converse direction, if the process (or field) satisfies the conditions (i)–(iii) one can construct a process (or field) by Kolomogorov’s basic existence theorem, and then by Karhunen’s representation (on a probability space) it is isotropic as well as harmonizable or stationary. This construction is standard and can be left to the reader (it is also in Chapter 1) of the first volume of this trilogy). 

A weaker form of the preceding result, admitting a series representation, has still a useful application potential, and we include the result. Proposition 4.2.2 Let X : Rn → L20 (P ) be a weakly harmonizable isotropic field. Then it is representable (t = (τ, u)) as a series: X(τ, u) ν

= Γ (ν)ωn 2 ν

∞ X m=0

h(m,n)

i

m

X l=n

l Sm (u)

Z R+

Z S∗

Jm−ν (τ λ) l Sm (v)Z(dλ, dv) (τ λ)ν (26)

ωn being the surface area of the unit sphere Sn .

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Proof. Given that X is weakly harmonizable, we can express it as  Z  its X(t) = X(τ, µ) = e Z(ds) Rn Z Z = eitλ cos(u,v) Z(dλ, dv), (27) R+

Sn

with s = (λ, v), t = (τ, u) as polar representations. Now using the known form (cf. Vilenkin (1968), p. 957) and expanding eitx in a series with orthogonal ultraspherical polynomials cνn (·) on Sn from the above Vilenkin volume, we have for |x| ≤ 1, t = (τ, u), e

itx

∞ X

= Γ (ν)

im (m + ν)

m=0

Jm+ν (τ ) ν c (x). (τ /2)ν m

(28)

Putting (28) into (27), letting x = cos(u, v), and using the formula, h(m,n)

cνm (cos(u, v))

=

X

l l Sm (u)Sm (v)

l=1

wn cνm (1) , h(m, n)

we get X(τ, u) ν

= Γ (ν)wn 2 ν

∞ X m=0

h(m,n)

i

m

X l=1

l Sm (u)

Z R+

Z Sn

Jm−ν (τ λ) l Sm (v)Z(dλ, dv). (τ λ)ν (29)

Since the series in (28) is L2 (Sn , µn )-convergent, the valid substitution of cωm in eitx series is valued and use it in (27), to deduce (26), as desired.  We now present a characterization of isotropic weakly harmonizable fields with a parameter set in a Hilbert space. This will give a kind of completeness of the problem considered mainly here. Proposition 4.2.3 Let X : H → L20 (P ) be an isotropic weakly harmonizable random field, H being a separable Hilbert space. Then the ¯ y ), x, y ∈ H, admits the representation covariance r : (x, y) 7→ E(Xx X (as a Morse-Transue integral) as:

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Z

Z

∗

r(x, y) = R+

189

exp{−(λx − λ0 y, λx − λ0 y)}F (dλ, dλ0 ), x, y ∈ H,

R+

(30) where F : R+ × R+ → C is a positive definite bimeasure of finite Fr´echet variation, with (·, ·) as a scalar product in H. Proof. We start with the stationary case, so r(s, t) = r˜(s − t), and by isotropy r˜ is invariant under the rotation group so that r˜(x) = r˜(kxk), which depends only on the length of x. Then by a key representation theorem of Schoenberg ((1958), Thm. 2) one has Z ¯ y ) = r˜(kx − yk) = E(Xx X exp{−λ(x − y, x − y)}dΦ(λ), (31) R+

for a bounded nondecreasing (left continuous) Φ : R+ → R+ . We now replace H by its isometric image l2 , the square summable sequence space with each x ∈ H. Let t = tx = τ (x) ∈ l2 , and consider exp{−λ(s − t, s − t)} = exp{−λ[(s, s) + (t, t) − 2(s, t)]} = exp{−λ(ksk2 + ktk2 )} ∞ X X (2λ)m × (si1 , t1 )k1 . . . (sim , tm )km , k! · · · km ! m=0 j∈l m

where lm = {j = [(im , km )], im ≥ 0, km ≥ 0, k1 + · · · + km = m}. Consider ψj defined by m

(2λ) 2

−kxk2

ψj (s, λ, m) = e

(k1 ! . . . km !)

1 2

sk11 . . . skmm ,

with s = (si1 , . . . , sim ) as a vector of reals, associated with j, so that (13) gives ∞ XZ X r(s, t) = r˜(s − t) = ψj (s, λ, m)ψj (t, λ, n)dΦ(λ). (32) m=0 j∈lm

R+

This is a triangular covariance relative to Φ, if we set ψ(x, λ) = (ψj (x, λ, m), j ∈ lm , m ≥ 0) R so that r(s, t) = R+ ψ(s, λ)ψ(t, λ)∗ dΦ(λ), ψ(·, λ)∗ is the adjoint ψ(·, λ). Hence there exists a stochastic measure (can be taken Gaussian) so that we have the representation:

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Z ψ(t, λ)dZ(λ) =

Xt = R+

∞ XZ X m=0 j∈lm

R+

j ψj (t, λ, m)dZm (λ),

(33)

0 j j where Zm (λ) ∈ L20 (P ) and satisfying E(Zm (A)Z¯nj (B)) = δjj 0 δmn Φ(A ∩ B). The procedure now is similar to the earlier cases. We proceed to the harmonizable case use the dilation which is valid in this case also. Thus there exists L20 (P˜ ) ⊃ L20 (P ) and a stationary Y · H → L20 (P˜ ) such that Xt = QYt , t ∈ H(≡ l2 ), Q an orthogonal projection onto L20 (P ). With the familiar procedure, we get ∞ XZ X j Xt = QYt = ψj (t, λ, m)QZm (dλ), (34)

m=0 j∈lm

R∗

j j = Z˜m where QZm 0 is the centered stochastic measure such that j0 j E(Z˜m (A)Z¯˜m 0 (B)) = δjj 0 δmm0 F (A, B) j the same F (·, ·) for all Z˜m -processes, and r(s, t) = r(gs, gt) for all relations g, and then ∞ XZ X ψj (s, λ, m)ψj (t, λ0 , m)F (dλ, dλ0 ). (35) r(s, t) = m=0 j∈lm

R+

Interchanging the (legitimate) sum and the (strict) MT-integral, we get Z Z ∗ 0 0 e−λ(s,s)−λ (t,t)+2λλ (s,t) F (dλ, dλ0 ) r(s, t) = + + ZR ZR 0 0 = e−(λs−λ t,λs−λ t) F (dλ, dλ0 ) R+

R+

giving (30).  The preceding argument (proof) also implies the following consequence. Corollary 4.2.4 If X : l2 → L20 (P ) is an isotropic harmonizable random field, then it admits the representation as: ∞ XZ X j Xt = (dλ), (36) ψj (t, λ, m)Z˜m m=0 j∈lm

R+

j where E(Z˜m ) satisfies (34), the series converging in L2 (P )-mean.

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191

Thus we have a complete extension of stationary isotropic fields to the (weakly) harmonizable class, using the (strict) Morse-Transue integration in place of the Lebesgue method. Thus much of the classical stationary field analysis can be extended. There is, however, considerable noncommutational analysis of the stationary fields given by A. M. Yaglom (1961) that needs to be and can be extended (see the next chapter) following the above work. There are many specializations and corresponding structural analysis of the forms, most of which depend on properties of spherical harmonics and various classes of spherical functions. A considerable part of the resulting analysis was given by R. J. Swift (1994) and later, and much of it includes Yadrenko’s (1983) basic (stationary) work. We present a few of the resulting properties to show how many of the familiar questions have been solved and useful conclusions drawn. The following is a first extension to isotropic fields that can be considered obtained from the preceding work, and the Karhunen series representations. This is stated for comparison as follows. Theorem 4.2.5 A random field X : Rn → L20 (P ) is weakly harmonizable isotropic iff it is representable as Z ∞ ∞ h(m,n) X X Jm+ν (λr) l l Sm (u) X(t) = αn dZm (λ), (37) ν (λr) 0 m=0 l=1 l0 l (B2 )) = δmm0 δll0 F (B1 B2 ), Bi ⊂ Rn Borel and (B1 )Z¯m where E(Zm F (·, ·) is a function of bounded Fr´echet variation, r = ktk, ν = rt , δmn 2 is the Kronecker
2ν+1 delta with the Dunford-Schwartz integral in (37), αn = n π π2Γ 2 2 .

The preceding representation can be given a different formulation which leads to various applications, noted by Swift (1994). This can be presented as follows, along with some consequences. Theorem 4.2.6 A random field Y : Rn → L20 (P ) is weakly harmonizable as well as isotropic if it is representable as Z ∞ ∞ h(m,n) X X Jm+ν (λr) l l dZm (λ), t ∈ Rn , (38) X(t ) = an Sm (u) ν ∼ ∼ (λr) 0 m=0 l=1 l0 where E(Znl (B1 )Z¯m (B2 )) = δmm0 δll0 F (B1 , B2 ), r = ktk, u = t /r, an = ∼ ∼ p n 2ν+ · Γ ( n2 )·π 4 , δmn being the Kronecker delta with Jm (·) as the Bessel

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function of order m and the vector integral is in the Dunford-Schwartz sense, with F (·, ·) having finite Fr´echet variation, and the functions J(·) being Bessel’s. The proof is somewhat similar to the analysis used above and we shall omit it here. The reader can reconstruct with the same procedures as before, or can also refer to Swift’s (1990) paper. If Yml (r) = R∞ (λr) l αm 0 Jm+ν dZm (λ), so E(Yml (r)) = 0, Yml (r) ⊥, and E(Yml (r)Y¯nl (r)) (λr)ν = F (r, r), the integral is a standard D-S symbol. The point is that the above result implies the following key representation obtained by M. I. Yadrenko (1985) differently based on several other results. It is a consequence of the preceding result, and the random field is just isotropic, but not necessarily harmonizable. This was also in Swift (1994), which is of interest and so we include it. Theorem 4.2.7 (i) A random field X : Rn → L20 (P ) is just isotropic if it admits the (spectral) representation (n-fixed) ∞ h(m,n) X X

X(t ) = ∼

m=0

l Sm (u)Yml ,

Yml ∈ L20 (P ),

(39)

l=1

and the Yml are orthogonal with E((Yml )2 ) = bm , l ≥ 1, m ≥ 1. (ii) Also, r(·, ·) is the covariance of an isotropic field on S n (unit sphere on Rn ) iff it is representable as r(s , t ) = r˜(cos θ) = ∼ ∼

with bm ≥ 0,

P∞

m=0 bm h(m, n)

∞ ν 1 X C∞ (cos θ) bm h(m, n) ν (1) ωn m=0 Cm

< ∞.

A consequence of this complicated looking format is the next representation that is of real interest in many applications of the present analysis: Theorem 4.2.8 A random field X : Rn → L20 (P ) is isotropic (not necessarily harmonizable) iff its covariance is representable as: ∞ 1 X C ν (cos θ) h(m, n) m ν bm (r1 , r2 ) (40) ∼ ∼ ωn m=0 Cm (1) P where bn (0, r) = 0, m 6= 0, and ∞ m=0 h(m, n)bm (r1 , r2 ) < ∞, with ν Cm (·) being the ultraspherical polynomials. With this set up we have

r(s , t ) =

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X(t ) = ∼

∞ h(m,n) X X m=0

l=1

193

l Sm (u)Yml (τ ) ∼

where 0 Yml (·), m = 0, 1, . . . , h(m, n) and E(Yml (r)Y¯ml 0 (r)) = 0 P except for m = m0 , and E(Yml (τ )Y¯ml (τ )) = bm (τ, τ ), ∞ m=0 bm (τ, τ ) · h(m, n) is finite for each l ≥ 1. Also r(·, ·) is the isotropic covariance of X(·) iff it is given by

∞ 1 X C ν (cos θ) h(m, n) m ν bm (τ1 , τ2 ), ∼ ∼ ωn m=0 Cm (1) (41) P h(m, n)b (τ , τ ) < ∞, b (0, τ ) = 0 if m 6= where bm satisfies ∞ m 1 2 m m=0 ν (·) are the ultraspherical polynomials. 0. Here Cm

r(s , t ) = r˜(τ1 , τ2 , cos θ) =

This result was obtained by Yadrenko (cf. his book 1983). It is given here for understanding the involved nature of the isotropy property. But it is also true for harmonizable (weakly) when the bm (·, ·) above are restricted there. Theorem 4.2.9 A random field X(t), t ∈ Rn is isotropic and weakly harmonizable iff the bn , n ≥ 1, of (40), (or (41)) admit the integral representation as: Z ∞Z ∞ Jm+ν (λr1 )Jm+ν (λ0 r2 ) 2 2 bm (r1 , r2 ) = αn d t(λ, λ0 ) ν (λ0 r )ν (λr ) 1 2 0 0 where F
is a function of bounded Fr´echet variation, (2π)−p/2 αn2 = 22νµ Γ m . n Proof. If X(t) is weakly harmonizable isotropic, we have r(s , t ) ∼ ∼

=

αn2

∞ h(m,n) X X m=0

l=1

l l Sm (u)Sm (v)

Z 0

∞

Z 0

∞

Jm+ν (λr1 )Jm+ν (λ0 r2 ) dF (λ, λ0 ) (λr1 )ν (λ0 r2 )ν

which on comparison with the earlier result on isotropic fields gives

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bm (r1 , r2 ) =

αn2

Z 0

∞

Z 0

∞

Jm+ν (λr1 )Jm+ν (λ0 r2 ) dF (λ, λ0 ). (λr1 )ν (λ0 r2 )ν

Conversely, if this formula holds for bm (r1 , r2 ), then r(s , t ) is obtain∼ ∼

able as above, with Theorem 4.2.3, so that X(t) obeys the weakly harmonizable isotropic condition, which completes the sketch.  Remark 23. There are a few other forms that one can obtain with the above work, and we shall also sketch a few more properties, as complements since they can be completed by the reader with analogous computation and include a few other properties that augment the above work. We conclude this section with an extension of the harmonizable isotropy to the vector case whose consequences may be studied by researchers in generalizing this work and applying it. Definition 4.2.10 A random vector field Xt = (Xt1 , . . . , Xtm ) ∈ L20 (P, Cn ), is weakly (or strongly) harmonizable isotropic if for each complex m-vector α = (α1 , . . . , αm ) the scalar field m X Yα = α · X = αi Xti , (Rn → L20 (P )) i=1

is a weakly (respectively strongly) harmonizable random field as defined. One can see that this extension implies that the various covariance ¯ j (t)), 1 < i, j ≤ n are harmonizable and sets Bij : (s, t) → E(X(s)X isotropic, and the matrix increment (∆Bij (s, t), 1 ≤ i, j ≤ n) has the earlier studied integral representation relative to (matrix) bimeasures. Thus the vector-valued case is direct, but not entirely simple. It is also of interest to note another extension suggested by Yaglom in his research monograph (Yaglom (1987)), as follows. Namely, each permutation matrix T of order m, acting on the random field X, should have the same type of harmonizable isotropic covariance and study its structure. The work needed for its analysis is more intricate as it depends on some aspects of group representations, to be discussed in the next chapter, that will show the deeper analysis of isotropic fields (extending the locally harmonizable isotropic class) and is also of interest in this analysis. We note that some (early) aspects of this problem restricted to stationary classes had already been pioneered by Yaglom (1987). This leads to an interesting analysis. Thus the next section will be devoted to an outline of this extension.

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195

4.3 Representations of Multiple Generalized Random Fields The primary task here is to obtain integral representations of Cram´er class [or class (C)] random fields through the analysis of generalized random functions, using the Schwartz theory of such classes. [On the latter subject see Schwartz (1957) or Gel’fand and Vilenkin (1964) for basic results with details.] There are at least three ways of introducing such fields and it will be useful to use any one of them for analysis, since they can be shown to be equivalent, although this fact needs some further analysis. In the following, K denotes the Schwartz space of infinitely many times differentiable scalar functions on Rn , vanishing off compact sets, varying with each function. Let C˜ be the space of complex random variables on (Ω, Σ, P ). If F : K → C˜ is linear it is termed as before a generalized random field (g.r.f.), provided for fn ∈ K.fn → 0 in K (so all the fn live on a fixed compact set and fn → 0, n → ∞, uniformly) then the random variables F (fn ) → 0 in probability, [or fn → 0 in K, and F (fn ) → 0 in L2 (P )] or if F ∈ K∗ (adjoint of K), and F (f ) is measurable. In all the different definitions, useful in computations, F is termed a g.r.f. and their equivalence is verified by G. Y. H. Chi (1969), and we use some of these properties according to convenience below. Let us formally introduce: Definition 4.3.1 On the Borel σ-field B(= B(Rn )) of Rn , a mapping Z : B → L20 (P ) is called a generalizedSrandom measure if (i) E(Z(A)) P = 0, A ∈ B, (ii) A, An ∈ B, A = ∞ n=1 An , An disjoint ∞ 2 ⇒ Z(A) = n=1 Z(An ), series existing in L0 (P ), and (iii) there is a tempered measure ρ : B × B → C such that for A, B ∈ B, Z Z ¯ E(Z(A)Z(B)) = d2 ρ(x, y). (42) A

B

The measure Z(·) so defined has orthogonal values if ρ(·, ·) concentrates ¯ on the diagonal x = y whence E(Z(A)Z(B)) = σ(A ∩ B); σ(·) is an + R valued measure. The following standard property of σ(·), defined above, is used below. Lemma 4.3.2 If σ : B → R+ is a tempered measure then it is bounded on bounded sets. If σ(·) has bounded variation on each bounded set of

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Rn , and has a polynomial growth (i.e., σ((0, t))/|t|k → 0, as |t| → ∞ for some k ≥ 0), then σ(·) induces a tempered measure on B. Proof. If σ(·) is a tempered measure, then by hypothesis there is a k > 0 R dσ(t) so that µ : A 7→ A (1+|t| 2 )k defines a finite measure on B. This shows dµ (t) = (1 + |t|2 )−k ≤ 1. Also dσ (k) = (1 + |t|2 )k on A, so that that dσ dµ R σ(A) = A (1 + |t|2 )−k dµ ≤ const · µ(A) < ∞. R dσ(t) Also σ(t)/|t|k → 0(|t| > 1) for some 0 < k < ∞ ⇒ Rn (1+|t| 2 )1/p < ∞, for some p > 0, so that σ(·) determines a tempered measure on B.  This shows that the tempered measure here also depends on k > 0 and hence the g.r.f. of a Cram´er class depends on such a k ≥ 0 (or ≤ 0). Here we shall obtain an integral representation of a Cram´er class which is the most general one used, and it includes the harmonizable classes. We then present a simple extension to the isotropic class also here. Proposition 4.3.3 Let K(N ) be the Schwartz space of infinitely differentiable functions on a compact rectangle N ⊂ Rn and F be a g.r.f. on K(N ), centered, with covariance B(·, ·) and F : K(N ) → L20 (P ) be a random field. Then there is a continuous positive definite hermitian function h : N × N → C and an integer m > 0 such that Z Z h(x, y)(Dα g)(x)(Dα g)(y) dxdy, f, g ∈ K(N ) B(f, g) = N

N

< ∞, where Dα =

δ |α| α1 n δx1 ...δxα n

(43) , αi ≥ 0,

Pm

i=1

αi ≤ m and α = (αn , . . . , αm ).

Proof. Since B(·, ·) is, by definition, a continuous bilinear functional on K(N ) × K(N ), by the Kernel theorem of Schwartz (cf. Gel’fandVilenkin (1964)) there exists a continuous linear G on K(N ×N ) satisfying B(f, g) = G(f g¯) [(f g)(x, y) = f (x)g(y)] so that f g ∈ K(N × N ), and linear combinations of such sums being dense. But then it is known (cf. Friedman (1963)) that there is a bounded measurable h : N × N → C and m ≥ 0 such that (43) holds. By increasing m, slightly if necessary, or integrating by parts again, h(·, ·) can also be taken as continuous. Since B(·, ·) is Hermitian positive definite (and letting f, g polynomials), it follows that h(·, ·) is a positive definite function, as needed. 

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Remark 24. This is essentially known but motivates the general case of class (C) to be considered. Theorem 4.3.4 Let Φ be a test space on Rn continuous, each compactly supported, real functions, and F : Φ → L2 (P ) be a centered grf of class (C), with covariance B(·, ·). Suppose F is of class (C) relative to g(·, ·) and a tempered covariance ρ.R Let the g-transform of all elements of Φ exist (i.e. f ∈ Φ ⇒ f˜(t) = Rn g(t, x)f (x)dx holds). Then there is a random measure Z relative to ρ, such that we have the representation: Z F (f ) = f˜(t)dZ(t), f ∈Φ (44) Rn

where f˜ is the g-transform of f and Z Z B(u, v) = u˜(t)v¯˜(s)d2 ρ(t, s), u, v ∈ Φ. Rn

(45)

Rn

The stochastic integral in (44) is defined in the mean square sense. On the other hand if g(·, ·) and ρ are given with properties noted earlier and Z(·) is a random measure, on B (of Rn ), and K ⊂ Φ is dense, then F (·) of (44) defines a g.r.f. of class (C) on K and has a unique extension to Φ. Proof. We present the argument in steps, shortening later some (similar) arguments. I. Let F : Φ → L20 (P ) be of class (C), so E(F (f )) = 0, and B(f1 , f2 ) = E(F (f1 )F¯ (f2 )). We assume, for convenience, that B(·, ·) is strictly positive definite, and introduce the inner product in Φ as: Z Z (f1 , f2 ) = B(f1 , f2 ) = f1 (t)f¯2 (t)d2 ρ(t, s), f1 , f2 ∈ Φ. (46) Rn

Rn

II. Let K0 = sp {F (f ) : f ∈ Φ} ⊂ L2 (P ) and let H be its completion. It follows that (F (f1 ), F (f2 )) = B(f1 , f2 ) = (f1 , f2 ),

(47)

so that fi 7→ F (fi ) gives an isometry of Φ onto K0 . If L2 (ρ) is the completion of Φ in the norm induced by (46) then the map f → F (f ) has an extension uniquely form L2 (ρ) onto H. The correspondence χA → F (χA )(= Z(A)) gives Z : B → H as a random measure relative to ρ, using the standard simplification

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kZ(A)−

n X

Z(Ai )k = kχA −

i=1

n X

χAi k = kχ∪i>n Ai k ≤ ρ(A×A) < ∞

i=1

where P˜ P is the measure determined by P . Letting n → ∞, we get ∞ Z(A) = i=1 Z(A). This easily implies that Z(·) is a random measure relative to ρ(·). III. If Φ˜ = {f˜ : f ∈ Φ}, then Φ˜ ⊂ L2 (ρ) and is dense in it since Z Z ˜ ˜ (f1 , f2 ) = f˜(t)f¯˜2 (s)d2 ρ(t, s) = B(f1 , f2 ) = (f1 , f2 ), Rn

Rn

by the preceding computations showing f → f˜ to be an isometry. The density of Φ in L2 (ρ) implies that of Φ in L2 (P ). It also follows in a standard way that Φ˜ in L2 (ρ) has the same property. Hence Z m X F (fn ) = ami Z(Ai ) = fn (t)dZ(t), Rn

i=1

and kf − fn k → 0 so that {F (fn ), n ≥ 1} ⊂ K is a Cauchy sequence in the latter, and we get with the usual notations: Z F (fn ) = fˆn (t)dZ(t), fˆn ∈ L2 (ρ), Rn

for simple and then for all functions in Φ, so that Z F (f ) = fˆ(t)dZ(t), f ∈ Φ, Rn

giving (44). IV. For the opposite direction, only the continuity is to be verified to see that it defines a class (C) random R For this it2 suffices to show R field. the continuity, since then B(f, g) = Rn Rn f˜(s)g¯˜(t)d ρ(t, s) will be of class (C). But g(·, ·) is bounded on bounded sets, it follows that B(fn , fn ) → 0 as fn → 0 a.e. also, boundedly, B(fn , fn ) → 0 in K, so that F is continuous on K. But K ⊂ Φ with a stronger topology, hence F is continuous in the topology of Φ in which K is dense. This implies easily that F on Φ is of class (C), as asserted.  This analysis implies the following consequence:

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Corollary 4.3.5 Let K ⊂ Φ ⊂ L be a test space, the inclusions being also topological. If F : Φ → L2 (P ) is a g.r.f. which is harmonizable w.r.t. a tempered covariance ρ, then there is a random measure Z : B(Rn ) → L2 (P ) such that Z Z Z ˆ F (f ) = f (t)dZ(t) = eitx f (x)dxdZ(t), f ∈ Φ, (48) Rn

Rn

Rn

uniquely. Conversely, F given by (48) with Z(·) and ρ, is a generalized harmonizable random field relative to ρ. Since Φ ⊂ L and the Fourier transform on L is isomorphism onto, the hypothesis holds and the result is a consequence of the theorem. We now present the Cram´er classes with conditions imposed on their local behavior. This was shown by A. M. Yaglom (1987) for vectorvalued stationary random fields, and he appreciated (in a letter) the author’s extension to the class (C) g.r.f. We thus present the result here. Theorem 4.3.6 On the test space Φ, let F : Φ → L2 (P ) be a grf, locally class (C), relative to g(·, ·), so that the g-transforms exist. Suppose ∂g g(t, x − y) = g(t, x)g(t, x), for all t, x, y in Rn , and that (i) ∂x (s, 0) 6= 0 ∂g for s 6= 0, (ii) ∂sk (s, r) = αk (s, r)rk and αk (0, r) = αk 6= 0 the αk being independent of r. [sk , rk are kth elements of vectors s, r.] Then there is a tempered covariance ρ on B(Rk −0)2 such that for some p (the index of temperedness of ρ) one has: Z Z |x||y||d2 ρ(x, y)| 1, r˜(h) = lim r(p) (h) = lim T1 R0T −|h| (63) T →∞ T →∞ r(s, s + |h|)dh, p = 1, T 0 exists, so that if p = 1 it becomes the original class (KF ). The early study for p = 1, has been developed by several people and the generalized class KF (p), p > 1, due to Swift (1992) does unify and extend the previous work. The point of studying the above classes, with summability methods, is to analyze (in a general way) the processes introduced by Karhunen

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and Cram´er subsuming the harmonizable families employing the same (summability) analysis which already became standard. This enhances the basic ideas of nonstationary analysis bringing it into the Fourier fold in general, and especially the applicational potential is enhanced. The following representation leads to the generalized analysis: Theorem 4.5.3 Let X : T → L20 (P ) be a process (T ⊂ R) of weak Cram´er class relative to {g(t, ·), t ∈ T } real Borel functions so that the covariance r(·, ·) of X is given by Z Z r(t1 , t2 ) = g(t1 , λ)g(t2 , λ0 )d2 F (λ, λ0 ), (64) S

S

with respect to a positive definite function F (·, ·) of locally bounded variation on S × S, (S, S) being a measurable set and g(t, ·) satisfies Z Z (65) 0≤ g(t, λ)g(t2 λ01 )d2 F (λ, λ0 ) < ∞, t1 ∈ T. S

S

Then there exists a vector measure Z : S → L20 (P ) such that Z X(t) = g(t, λ)dZ(λ), t ∈ T,

(66)

S

¯ E(Z(A)Z(B)) = F (A, B), A, B ∈ B = σ(S).

(67)

Conversely, if X(·) is defined by (66) with (67) holding, then it is of weak class (C). The proof is quite standard and we leave it as an exercise for the reader. [The details can also be found in the paper, Chang and Rao (1986, Thm. 1 in Sec. 7). They are indeed standard using the MTintegrals in place of the Lebesgue integrals.] Motivated by these ideas of summability analysis Swift (1997) introduced an extension of class (KF ) as follows. Definition 4.5.4 A process X : R → L20 (P ) with a continuous covariance r is termed of class {(KF, p), p ≥ 1} if for h ∈ R, we have (p)

r¯(h) = lim rT (h), T →∞

existing for each h, where

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( (p) rT (h)

=

1 T 1 T

R T (p,1) r (t)dh, p>1 R0T r(s, s + |h|)dh, p = 1. 0

[Thus (KF, 1) is the original class (KF ).] The classical summability analysis implies that these are inclusive with p i.e., class (KF ) ⊂ class (KF, p) ⊂ class (KF, p + 1). Also, we have the following result obtained by Swift (1997). Theorem 4.5.5 The (C, p)-summable Cram´er processes are contained in the class (KF, p), p ≥ 1, relative to a class {g(t, ·), t ∈ R} of Borel functions with g(t, ·) as in Theorem 4.5.3. If Q : L20 (P ) → L20 (P ) is a bounded linear operator, then Y (t) = QX(t), t ∈ R is (C, p) summable (p ≥ 1) weak class (C) relative to the functions g(t, ·) and F (·, ·) as in (64), (65). There are numerous extensions and alterations of the above classes with the corresponding analysis, nontrivially. We invite the interested workers in the subject to study, Swift’s extended analysis with details given as noted above. The profitable applications of the well-developed summability analysis results, in the stochastic theory, are of immense use in extensions as well as in applications. It is possible to go further and use almost periodic analysis in this context and advance the subject. This will be indicated in the complements section below. We treat another aspect now. 4.6 Prediction Problems for Stochastic Flows There is always a deep interest to study the structure of stochastic flows governed by certain (stochastic) differential equations with orthogonal or (naturally) harmonizable noise processes. An early work by Dolph and Woodbury (1953) and the latter one extended by Dym (1966) are useful contributions for analysis and discussion here. Consider a linear differential operator Lt defined by: (Lt g)(t) =

n X i=0

α(t)

dn−1 g(t) , α0 (t) 6= 0, t ≥ t0 . dtn−2

Then the process {Y (t), t ∈ T } satisfying

(68)

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(Lt Y )(t) = Z(t), t ∈ T = [t0 , t],

207

(69)

is termed a stochastic flow driven by the (harmonizable) noise process, {Z(t), t ∈ T }, where α(t)’s are (n − 1) times continuously differentiable real functions. Here (69) is “interpreted” as satisfying (formally let dZ(t) = Z 0 (t)dt) Z Q(t)(Lt Y )(t)dt T

Z =

 Z  Q(t)Z (t)dt = Q(t)dZ(t) , 0

T

(70)

T

for all Q(·) with compact supports and the right side is the Bochner (or a D-S) integral. Theorem 4.6.1 Let {Y (t), t ≥ t0 } be a process defined by Z t Y (t) = R(s, t)dZ(s) t0

where Z(t) is a centered weakly harmonizable (noise) process as in (69). Then it belongs to a weak Cram´er class so that its covariance r satisfies r(s, t) = (X(s), Y (t)) Z Z ∗ = g(s, λ)g(t, λ0 )dF (λ, λ0 ) ˜ R

(71)

˜ R

˜ is the dual group of R, and r is 2n − 2 times for (s, t) ∈ R × R, where R continuously differentiable with (2n−1)st derivative having a (possible) jump at t = s, given by: lim[(D12n−1 r)(t, s + ε) − (D12n−1 r)(t, s − ε)] ε↓0

= (−1)n (α(t))−2 . Proof. Since Z(·) and Z 0 (·) are centered, one has for r(s, t):

(72)

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r(s, t) = E(Y (s)Y¯ (t)) Z sZ t ¯ t (v)E[Z 0 (w)Z¯ 0 (v)]dudv, Rs (w)R = t0

t0

the interchange of E(·) and integral being valid, Z Z ∗ Z sZ t 0 ¯ Rt (w)Rt (v) eiuλ−ivλ dG(λλ0 )dudv, = t0

t0

R

R

with dG(λ, λ0 ) = λλ0 dF (λ, λ0 ), F being the spectral bimeasure of Z(·) and Z 0 (·), which is weakly harmonizable,  Z  Z Z ∗ Z iuλ ¯ ivλ0 ¯ e R(u)du e R(v)dv dG(λ, λ0 ), = R

R

R

˜ where R(u)(χ (t,s) , Rs )(u) and the integral interchange being justifiable, Z Z ∗ ¯ˆ 0 ˆ R(λ )dG(λ, λ0 ). (73) = R(λ) R

R

ˆ s is the Fourier transform of R ˜ s , well-defined by the properties Here R of Rs . Since G is a bimeasure, (73) implies that the (covariance and hence) Y -process is of (weak) Cram´er class. The other differentiability properties are deduced from (73) and the fact that Lt and the integrals in (73) commute, so that (Lt rs )(t0 ) = 0 and (Lt rs )(t) = 0 as well as the remaining assertions are verified by simple computations. P Since (Lt Rt )(t0 ) = 0, one has (Dtn R)(s, t) = − nt=1 α(t)(D1n−k R) ×(s, t), so that D1k r exists, 0 ≤ k ≤ n − 1, and then D2k r, 0 ≤ k ≤ n also exist. Finally Z sZ s 2n−1 k ¯ v)dudv + (−1)n−1 δk , lim(D1 r)(s − t, s) = D1k R(s, u)R(s, ε↓0 α02 (t) t0 t0 since (Dtn−1 R)(s, t) = α0 (t)−1 where δk is the Kronecker symbol above. If the αi (·) are C ∞ -smooth, then r(·) will be also C ∞ -smooth. This property plays a basic role in Dym’s (1911) analysis of stationary measures, in differential equations governed by white noises. This completes the essentials of proof of this result.  The following result extends a basic fact on stochastic flows observed by Dolph and Woodbury (1952) which played a key motivational role in the theory of unbiased linear least squares prediction:

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Theorem 4.6.2 Let X(t) = Y (t) + Z(t), a ≤ t ≤ b be a signal plus noise model, with the Y, Z processes having two moments. Suppose the covariances and cross covariance of Y, Z are known and are smooth in that they are n-times continuously differentiable. If the process {X(t), a ≤ t ≤ b} is observed, an unbiased linear weighted leastˆ 0 ) of Y (t0 ) of the form squares predictor X(t n Z b X ˆ 0) = X(t X (k) (t)dpk (t, t0 ) (74) k=0

a

relative to a set of complex weight functions pk (·, t0 ), k = 0, 1, . . . , n, of bounded variation is possible if and only if the pk minimize J0 (p): n Z bZ b X 0 ≤ J0 (p) = [D1k D2l K(s, t)dpk (s, t0 ) + rk (t0 , t0 ) k,l=0 a n X

λ(t0 )gj (t0 ) − 2

+2 +

a

j=0 m X

n0 Z X k=0

b

[Re (D1k K1 )(t, t0 )

a

(k)

λj (t0 )gj (t)]dpk (t, t0 )

j=1

with λj as Lagrange multipliers. Thus the minimum of J0 yields the best solution. The many details, some of which are technical, are given in the author’s paper (1994) which improves and generalizes the basic work by Dolph and Woodbury (1952). The reader is advised to read through both these papers to understand the nontrivial mathematical analysis that is basic to this type of applications. Some related extensions are also given there and should be of interest to the workers now. An extended form of the preceding result, useful is a number of applications, given in a Karhunen form that contains both the stationary and harmonizable classes, is defined by: Z ˜ u)dZ(u) Y (t) = R(t, (75) R

for a suitable kernel R(·, ·) so that we have (with centered Z)

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Z

s∧t

¯ u)dH(u) r(s, t) = E(Y (s)Y¯ (t)) = R(s, u)R(t, Z ¯˜ u)dH(u) ˜ u)R(t, = R(s,

(76)

R

˜ u) = (χ(−∞,u) R(s, ·))(u). This is Karhunen’s form. For where R(s, such processes we have the following general result of interest. Theorem 4.6.3 Let X(t) = Y (t) + Z(t), a ≤ t ≤ b, be a second order process with Y (t)-process as signal and Z(·), the noise, with orthogonal increments having H(·) as the variance function, and Pm E(Y (t)) = i=1 ai gi (t), ai ∈ R. Suppose that the noise process Z(·) also has orthogonal increments with the variance function H(·). Assume that the covariance function of Y (·) and Z(·)-processes are ry (s, t), ryz (s, t)(= cov (Y (s), Z(t))) as given. Then an unbiased linˆ 0 ) of Y (t0 ) of the form ear weighted least-squares predictor X(t Z t0 ˆ X(t)dp(t, t0 ) (77) X(t0 ) = a

relative to p(·, t0 ), t0 > b of bounded variation exists when p(·, t0 ) is a solution of the integral equation Z b m X K1 (t, t0 ) + λi (t0 )gj (t) = K(s, t)dp(s, t0 ) (78) j=1

a

with K1 (t, t0 ) = (ry + ryz )(t, t0 ), K(s, t) = ry (s, t) + H(s ∧ t) + Rb 2ryz (s, t), the λi (t0 ) being Lagrange multipliers and a gi (t)dp(t, t0 ) = ˆ 0 )− gj (t0 ), i = 1, . . . , m. The minimum mean-squared error σ 2 = E(X(t 2 Y (t0 )) is given by: Z b m X 2 K1 (t, t0 )dp(t, t0 ). (79) σ = ry (t0 , t0 ) + λj (t0 )gj (t0 ) − i=1

a

Remark 25. In the case that the signal and noise are uncorrelated, whence ryz = 0 and so K = ry and K = ry + H, the above equations simplify with K = ry + H. If Y is nonrandom, then ry = 0 = ryz Rb P and (79) reduces to m H(s ∧ t)dp(s, t0 ). But j=1 λi (t0 , gj (t0 )) = a 0 if no signal is sent (so Y = 0), then with p (·, t0 ) existing, we have

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Rb

dp(t, t0 ) = 0, and other simplifications result. If n = 0 and m = a 2, r(s, t) = exp[−β(t − s)] with β > 0 Z-Gaussian, the result was detailed in Dolph and Woodbury (1952). Specializations for Cram´er and Karhunen classes can now be obtained and they are useful. The preceding analysis motivates the following extension of interest and the related analysis for people desiring to go further. Proposition 4.6.4 Consider a second order process X(t) = Y (t)+Z(t) on the interval of observations Pma ≤ t ≤ b where the signal process Y is satisfying: E(Y (t)) = i=1 ai gi (t), ai ∈ R, the noise Z(·) has orthogonal increments with variance function H(·) and the covariances of Y and Z are given, so that ry : (s, t) → cov (Y (s), Y (t)), ryz : (s, t) → cov (Y (s), Z(t)) are also given along with aj , 1 ≤ j ≤ n. Then ˜ 0 ) of Y (t0 ) of an unbiased weighted linear least squares predictor X(t the form: Z b

˜ 0) = X(t

X(t)dp(t, t0 )

(80)

a

relative to a real weight p(·, t0 ) for t0 > b exists provided p satisfies: Z b n X K1 (t, t0 ) + λi (t0 )ai (t) = K(s, t)dp(s, t0 ) (81) a

i=1

with K1 (t, t0 ) = (ry + ryz )(t, t0 ), K(s, t) = ry (s, t) + H(s ∧ t) + 2ryz (s, t), the λi (t0 ) being the Lagrange multipliers and Z b gj (t)dp(t, t0 ) = aj (t0 ), j = 1, . . . , m. (82) a

The minimum mean-squares error σ 2 is given by ˆ 0 ) − Y (t0 ))2 = ry (t0 , ts ) + σ = E(X(t 2

m X

λi (t0 )g0 (t0 )

j=1

Z −

b

K(t, t0 )dp(t, t0 ).

(83)

a

Remark 26. If the signal and noise are uncorrelated, so ryz = 0 and K = ry and if the signal Y (·) is also deterministic so ry = ryz = 0 then (83) reduces to

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Z

b

H(min(s, t))dp(s, t0 ).

λj (t0 )gj (t0 ) =

(84)

a

j=1

Rb If no signal is sent, so Y = 0, one gets a dp(s, t0 ) = 0 and σ2 = 0. If m = 2, n = ∞ and r(s, t) = exp[−β|(t − s)|]β > 0 with Z-Gaussian, an example is detailed in Dolph-Woodbury (1952), which may be of interest to the reader who wants to study with more details, in order to appreciate the problems in applications. We now discuss (briefly) the spectral function estimation of a (weakly) harmonizable process. Thus if a process {Yt , −T ≤ t ≤ T } is assumed weakly harmonizable, centered, with a covariance kernel R, so that Z t Yt = Rt (s)dZ(s) (85) t0

whose covariance ry (·, ·) is represented by Z Z ¯ ry (s, t) = E(Ys Y¯t ) = Ry (u)R(v)dG(u, v) R

(86)

R

where dG(λ, λ0 ) = λλ0 dF (λ, λ0 ), the F (·, ·) being the spectral function of Z in (85) and dG(λ, λ0 ) = λλ0 dF (λ, λ0 ). We now give an estimator ˆ 1 by the above) when the process is observed on FˆT of F (so it gives a G −T ≤ t ≤ T , with the following properties: (i) FˆT is unbiased in limit as T → ∞ (i.e., asymptotically unbiased). (ii) FˆT is consistent, i.e., (FˆT − FT ) → 0 in probability as T → ∞. (iii) g(T )−1 (FˆT − FT ) → a random variable (in distribution), as T → ∞. (iv) The speed of convergence in these limits is given. The following brief discussion on the spectral estimation may be of interest for research workers in this area. It is a consequence of Theorem 4.6.3 above and is given to explain the subject for applications. Proposition 4.6.5 In a signal plus noise model X(t) = Y (t) + Z(t), a ≤ t ≤ b where the observed process X(·) is a signal process Y (·), disturbed by the noise Z(·), a process of orthogonal increments, aj ∈ R, with a variance function H(·), and covariance ryz (s, t) be that of Y, Z processes, and similarly ryy , ryz are defined for the Y and Y, Z

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213

and ai , i ≤ j ≤ m given. Then a weighted linear unbiased least squares ˆ 0 ) of X(t0 ) of the form predictor X(t Z b ˆ X(t)dp(t, t0 ) X(t0 ) = a

relative to a weight function p(·, t0 ) for t0 > b exists when p(·, t0 ) solves Z b m X K1 (t, t0 ) + λi (t0 )aj (t) = K(s, t)dp(s, t0 ) (87) a

j=1

where K1 and K are given by K1 (t1 t0 ) = (ryy + ryz )(t, t0 ), λj (t0 ) being Lagrange multipliers, K(s, t) = ry (s, t) + H(s ∧ t) + 2ryz (s, t), and Z b gj (t)dp(t, t0 ) = gj (t0 ), j = 1, . . . , m. (88) a

The minimum mean squared error σ 2 is given by ˆ 0 ) − Y (t0 ))2 σ 2 = E(X(t Z b m X K1 (t, t0 )dp(t, t0 ). = ry (t0 , t0 ) + λi (t0 ) −

(89)

a

i=1

Note: If the noise and signal are uncorrelated so that ryz = 0 and K1 = ry , K = ry + H, the above equations slightly simplify. When Y is deterministic, so that ry = ryz = 0, then we have Z b m X H(s ∧ t)dp(s, t0 ). (90) λi (t0 )g(t) = a

j=1

Thus this reduces for m = 1, Y = a (constant) so g = 1 and (90) becomes with the unbiasedness constraint Z b Z b λ1 = H(s ∧ t)dp(s, t0 ), dp(s, t0 ) = 1. (91) a

a

Thus, in this case, the minimum value of the estimator is λ1 and the values of λ1 and p are obtainable from the equations (91). R b If Y = 0 (no signal is sent), then assuming for p(·, t0 ) = a dp(s, t0 ) = 0, and

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Z

b

Z

0

H(s)p (s, t0 )ds → H(t)

0= a

b

p0 (s, t0 )ds,

a

˜ 0 ) = 0 so σ 2 = 0 the error variance. one concludes that p(·) = 0 and X(t We conclude this section with a few results on generalized harmonizable, and also of Cram´er (Karhunen) type, random fields. The following technical result will clarify in studying the structural properties of these classes. Theorem 4.6.6 Let r : Rn × Rn → C be a bounded continuous covariance function. Then it is (weakly) harmonizable if and only if there exist a sequence rm , m ≥ 1 of uniformly bounded continuous covariances such that (i) rm → r uniformly in Rn × Rn as m → ∞; (ii) the support Em of rm is compact, Em ⊂ Em+1 , and ∪m≥1 Em = Rn × Rm and (iii) for each m, if {φm k , k ≥ 1} denotes a complete set of eigenfunctions of rm for some {Gm } of bounded variation on Rn . k Remark 27. If r(s, t) = r˜(s − t) then we can take rm = φm k , k ≥ 1, m when the rm are suitable restrictions of r and the Gk are then positive finite measures as in Bochner’s classical characterization of covariance generally. We sketch a proof of the result depending on a theorem of A. D. Alexandriff (cf. Dunford-Schwartz (1958) p. 366) and present its extension which is of interest here. Proof. In the forward direction, if r(·, ·) is harmonizable then there is a covariance function of bounded variation, ρ, such that Z Z e((i(s,x)−i(t,y)) d2 ρ(x, y). (92) r(s, t) = Rn n

Rn n

If now En ⊂ En+1 ⊂ R × R , is a sequence of compact rectangles covering the whole space let ρn = ρ|En and rn be the resulting integral of (92) with ρn for ρ there. That the sequence {rn , n ≥ 1} of (92) satisfies (ii) and (i) follows from the classical A. D. Alexandroff’s result (cf., Dunford-Schwartz (1958, p. 316). About (iii) let ρn = α, rm = h and by Mercer’s theorem h(x, y) =

∞ X

µi φi (x)φ¯i (y),

µi ≥ 0,

(93)

i=1

where {µi }, {φi } are its eigenvalues and the normalized complete set of eigenfunctions, Rthe series (93) converging uniformly and absolutely. Also if µj φj (x) = Rn h(x, y)φj (y)dy so that (92) gives (for h):

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Z φj (x) =

215

ei(t,x) dGj (t)

Rn

R where Gj (t) = µ1n Rn φˆj (s)ds α(t, s), φˆj being the Fourier transform of φj . So Gj is of bounded variation, whence (iii) holds. For the converse, let (i)–(iii) hold and consider an rm . Then an exm pansion of the type (93) holds for rn with some Gm j . If Fj (s, t) = n m m then from the (uniform) convergence of F (s, t) = G Pj∞(t)Gmj (s), m j=1 µj Fj (s, t) one easily obtains, with the convergence of (93), Z X Z ∞ m rm (x, y) = uj ei[(s,x)−(t,y)] d2 Fjm (s, t) Rm j=1

Z

Z

= Rm

Rn

ei[(s,x)−(t,y)] d2 F n (s, t)

(94)

Rn

and that the F m is a covariance of bounded variation, so that rm is a harmonizable covariance, and (iii) already characterizes continuous harmonizable covariance on a compact set. Now conditions (i) and (iii) can be used to infer harmonizability of r(·, ·). Indeed, since F m determines a finite positive regular measure on Rn × Rn , (i) implies that the integrals in (94) converge when Fm is considered as determining a regular measure and then invoke the converse part of Alexandroff’s theorem as used above, so that F m → ρ, a covariance of bounded variation on Rn × Rn . This gives the converse and the theorem follows.  This result admits extensions to Cram´er and Karhunen processes. We include that result, of interest here, to complete this set of ideas. It also admits an easy characterization of generalized harmonizable fields, such an extension is of interest for this analysis as well as applications. Recall that K is the L. Schwartz space of infinitely differentiable compactly supported functions (complex) on Rn , defined and discussed in Chapter 2 (and used there for the Gel’fand representation theorem on local functions). We use some properties here without repeating the concepts and related details. It will be motivational but also useful to present the development of the result and summarize the work in a proposition as it also gives the motivation. Thus X : R → L20 (P ) is a (measurable) mapping, centered, and its covariance r : (s, t) → ¯ E(X(s)X(t)) is of Karhunen class where we set r(·, ·) : (s, t) → ¯ E(X(s)X(t)) = (X(s), X(t)), with inner product notations. It admits an (LS)-representation as:

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Z r(s, t) =

g(s, λ)¯ g (t, λ)F (dλ),

s, t ∈ R,

(95)

R

relative to a class of Borel functions {g(s, ·), s ∈ R} and a σ-finite Borel measure F : B(R) → R∗ . Then the process is shown to be representable as Z g(t, λ)Z(dλ), t ∈ R, (96) X(t) = R 2 L0 (P ),

such that (Z(A), Z(B)) = F (A ∩ B), where Z : B0 (R) → for A, B ∈ B0 (R) of bounded Borel sets of R, so that Z(·) has orthogonal increments, and this has an extension, to be called a Cram´er process if Z(·) has non-orthogonal increments, but covariance is to be representable as Z Z r(s, t) = g(s, λ)g(t, λ0 )F˜ (dλ, dλ0 ), s, t ∈ R, (97) R

R

relative to Borel functions {g(s, ·), s ∈ R} and a given covariance bimeasure F , satisfying Z Z 0≤ g(s, λ)g(t, λ0 )F (dλ, dλ0 ) < ∞, (98) R

R

of finite Fr´echet variation on B0 (R) × B0 (R). Evidently, the Cram´er class includes the Karhunen family, somewhat analogous to the harmonizable class includes the stationary family. Here MT integration is used. [A complete account of the (restricted) MT-integration is detailed in a chapter by D. K. Chang and the author (1986) and will be used here. The reader can fill in the omitted details easily from it.] Using this we now discuss here how the Cram´er and Karhunen classes of process that extend the stationary and harmonizable classes to some extent, and the limitations of the method as compared with the harmonizable case will be made explicit. Our guidance (and real motivation) here is that, on an extended L2 (P )-space, we consider a (weakly) stationary process whose orthogonal projection gives a (weakly) harmonizable process, and we wish to extend these ideas to some nonstationary classes, especially Karhunen and Cram´er classes on suitably enlarged L2 (P )spaces. We want to make this idea precise and get a possible extension of classes involving these two families which also show the limitations of this method. The motivation here is the classical de la Val´ee Pussion’s criterion of uniform (Lebesgue) integrability of a bounded set of (real) functions on

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the line. This is needed for a vector measure ν : Σ → X relative to a ¯ + so that we have σ-finite measure µ : Σ → R Z k f dνkX ≤ kf kf,µ , f ∈ Lf (µ) (99) Ω

for some 1 ≤ f < ∞, and then ν is termed dominated by µ. A solution of this problem is needed for us in our ensuring analysis. Here we need to recall a generalized variation of a vector measure and obtain a solution, and the L1 (µ) spaces should be replaced by more general Orlicz spaces to understand the problem. Thus if φ : R → R∗ is a symmetric convex function, φ(0) = 0, its conjugate ψ : R → R∗ is defined as ψ(x) = sup{|x|y − φ(y) : y ≥ 0}, so that ψ(·) is also a symmetric convex function, called the conjugate of φ and satisfies |xy| ≤ φ(x) + ψ(y), x, y ≥ 0, and the norm, (Lφ (µ), k · kφ ) is given as     Z f ψ kf kψ,µ = inf α > 0 : dµ ≤ 1 . (100) α Ω With this preparation, the φ semi-variation of ν : Σ → X, a Banach space, is defined as: Z kνkφ (A) = sup{k f (ω)ν(dω)k : kf ||ψ,µ ≤ 1}, (101) A


R where the norm of f is given by kf kψ,µ = inf{α > 0 : Ω ψ αs dµ ≤ 1} which is the usual Lebesgue integral (kf kψ,µ ) is the norm of f ∈ Lψ (µ), ψ(·) a convex Young function, called the conjugate of the convex φ. In the classical case when φ(x) = |x|p /p, p ≥ 1, then the conjugate of φ is ψ(·) given by ψ(y) = (y)q /q, p−1 + q −1 = 1. For us, the pair (φ, ψ) is quite general. With this introduction we have the following: Theorem 4.6.7 Let (Ω, Σ) be a measurable space, X is a Banach space with ν : Σ → X, σ-additive. Then there is a measure µ : Σ → R+ , a ↑ ∞ as x ↑ ∞, with kνkφ (Ω) < ∞ Young function φ : R → R+ , φ(x) x in (101) and so ν is dominated by the couple (φ, µ) [µ is often termed a 2 2 control measure of ν. If φ(x) = x2 , so ψ(x) = x2 also, then ν is said to have 2 semi-variation finite]. Proof. Since the weak and strong σ-additives are known to be the same, for a disjoint {An , n ≥ 1} ⊂ Σ, and x∗ ∈ X∗ , kx∗ k ≤ 1, we have

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0 = lim kν( n→∞

∞ [

An ) −

n=1

ν(Am )k

m=1

= lim sup{|x∗ (ν)[ n→∞

n X

[

n≥1

An ] −

n X

(x∗ (ν))(Ak )| : kx∗ k ≤ 1}

(102)

K=1

and thus the scalar measures {x∗ (νn ), kx∗ k ≤ 1} are uniformly σadditive on Σ. Then by a well-known result (Dunford-Schwartz (1958), IV.16.5) there is a finite positive dominating measure µ : Σ → R+ such that x∗ ◦ ν  µ, kx∗ k ≤ 1. Hence by the classical Radon-Nik´odym ∗ theorem gx∗ = dxdµ)(ν) exists and (102) implies: Z ∗ gx∗ (t)µ(dt), |x∗ | ≤ 1. (103) 0 = lim |x ◦ ν(A)| = lim µ(A)→0

µ(A)→0

A

Thus {gx , x∗ ∈ S ∗ } ⊂ L1 (µ) is bounded and uniformly integrable, since µ(r) < ∞, by the classical de la Vall´ee Poussin’s theorem there exists a convex φ of the desired type such that Z φ(|gx∗ (t)|)µ(dt) ≤ K0 < ∞, kx∗ k ≤ 1. (104) Ω

To simplify further, let ψ : R → R∗ be the conjugate of φ so that (104) gives: Z kνkφ (Ω) = sup{k f (ω)ν(dω)k; kf kψ,µ ≤ 1} Ω    Z = sup sup f (ω)gx∗ (ω)µ(dω) : x∗ εX∗ , Ω  kf kψ,µ ≤ 1 ≤ 2 sup {sup kgx∗ kφ,µ kf kψ,µ ; x∗ ∈ S ∗ , kf kψ,µ ≤ 1} , by H¨older’s inequality for the Lφµ -spaces, ≤ 2 {kgx∗ kφ,µ : x∗ εS ∗ } < 2k0 < ∞,

by (104).

This completes the proof. Remark 28. Since kνk(Ω) = kνk1 (Ω) in this notation with φ(x) = |x|, and µ(Ω) < ∞, it is clear that Lφ (µ) ⊂ L1 (µ) and then kνkφ (Ω) ≤

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C0 kνk(Ω) < ∞, and φ(·) can grow exponentially. Thus if p > 1, φ(x) = |x|p , it is nontrivial to find a good 0 < C < ∞, with kνkφ (Ω) bounded by a simple verifiable bound C in terms of k · kp for immediate applications. The following special result, established by a particular method, due to J. Lindenstrauss and A. Pelczy´nski (1968), will be given for some immediate applications towards Cram´er and Karhunen processes, showing open-ended extension of our theory. The special result originally obtained by these authors is as follows. Theorem 4.6.8 [Lindenstrauss and Pelczy´nski] Let (Ω, Σ) be a given measurable couple with B(Ω, Σ) as the Banach space of scalar bounded measurable functions and Y be an Lp -space, 1 ≤ p ≤ 2 (i.e. an Lφ with φ(x) = xp in the above). Then a vector measure ν : Σ → Y is (2, µ)dominated, so that there is a finite positive µ on Σ such that

Z



f (ω)ν(dω) ≤ kf k2,µ , f ∈ B(Ω, Σ), (105)

Ω

y

whenever ν has 2-semi-variation relative to µ finite. The proof of this result with full details is in the author’s graduate text (cf. Rao (2004)), Measure Theory and Integration pp. 527–529) and need not be reproduced here. The interested reader may study there the details and applications as well as some extensions. Here we consider only applications to Cram´er and Karhunen processes. Recall that a centered second order process {Xt , t ∈ R} with covariance r(s, t) is of Cram´er class if r(·, ·) has the Morse-Transue integral representation: Z Z r(s, t) = g(s, λ)¯ g (t, λ0 )F (dλ, dλ0 ) (106) R

R

relative to {g(s, ·), s ∈ R}, a Borel class, and F of finite Vitali variation, and it is of Karhunen class if F (·) concentrates on the diagonal s = t which is σ-finite (on the diagonal). These are extensions of stationary and harmonizable processes studied earlier. We now discuss how the earlier analysis extends to this generalized class taking over the stationary and harmonizable families in which the Fourier analysis is desired to be extracted and find its possible limitations. Thus let X : R → L20 (P ) be a Karhunen process relative to a σ-finite F on B(R), as in (95) above and if T : L20 (P ) → L20 (P ) is a bounded linear operator, consider Y (t) = T X(t), t ∈ R so that we have

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Z Y (t) = T

Z g(t, λ)(T ◦ Z)(dλ),

g(t, λ)Z(dλ) = R

(107)

R

by classical vector analysis, as extended by Thomas ((1970), p. 79) where Z˜ = T ◦ Z : B(R) → L20 (P ), and the Y -process becomes a ˜ ˜ Cram´er-process relative to a bimeasure F˜ : (A, B) 7→ (Z(A), Z(B)), A, B ∈ B(R), so that Y (t) is indeed of the Cram´er class. We next discuss the opposite (a kind of converse) inclusion. Thus consider {X(t), t ∈ R}, a Cram´er process, so that Z ˜ X(t) = g(t, λ)Z(dλ), t∈R (108) R

˜ ˜ ˜ relative to Z-integrable {g(t, ·), t ∈ R}, and F˜ : (A, B) 7→ (Z(A), Z(B)) 2 ˜ and that it is well-defined with Z : B0 (R) → L0 (P ) as a vector measure. We need to use (a form of) the Lindenstrauss-Pelczy´nski theorem specialized from the above and proceed to get: Theorem 4.6.9 Let X : R → L20 (P ) be a process and {g(t, ·), t ∈ R} be a family of Borel functions such that X is a Karhunen process relative to the g(t, ·)-family and a σ-finite F on B(R), and T : L20 (P ) → L20 (P ) be a bounded linear mapping. Then {Y (t) = T X(t), t ∈ R} is a Cram´er process relative to the g(t)-family and a suitable bimeasure. Conversely, if {g(t, ·), t ∈ R} is a bounded Borel set and X : R → L20 (P ) is a Cram´er process relative to this family and a suitable covariant bimeasure, then there is an extension space L20 (P˜ )(⊃ L20 (P )) determined by the given process, a Karhunen process Y : R → L20 (P˜ ) with the same g family and a suitable Borel measure on R such that ˜ → L2 (P ) is an orthogonal X(t) = QY (t), t ∈ R, where Q : L20 (D) 0 projection onto the range. The details are extensions of the stationary to harmonizable case and they are spelled out in the author’s paper (cf. Rao (1981) pp. 304–308), and they will not be reproduced here, as they are easily available. But the following consequences are interesting. We have seen earlier that each harmonizable process is obtainable as a (an orthogonal) projection of a weakly stationary process on a super Hilbert space. The work depended on the fact that the class g(t, λ) = eitλ so that the classical harmonic analysis results and methods are employable. But now the {g(t, ·), t ∈ R} is more general and just (uniform) boundedness is not sufficient. In fact, on attending the presentation of this paper by the author, Erik

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Thomas whose (Radon) vector measure theory of the 1970s is important to our work, has constructed a counter example to show that there exist processes of Cram´er class that are not projections of a Karhunen class on a larger (or super) Hilbert space so that the harmonic analysis is crucial here. The preceding discussion and the result show that the class of Cram´er processes is quite large and some of its members may not be obtained from a (fixed) Karhunen class membership. Also, the Cram´er class is closed under continuous (or bounded) linear mappings, and thus this class is closed under such transformations, following from a result of E. Thomas (1970), p. 79). Note that in all the work above the index set R can be replaced by a locally compact group with the (spectral) measure F being Radon on these spaces. We end this chapter with an operator characterization of the Cram´er process as it contains many others, considered above. Definition 4.6.10 Let X be a Banach space A : X → X be a (not necessarily bounded) linear operator with spectrum σ(A) ( C. Let F(A) = {f : C → C analytic in a neighborhood of σ(A) and A}. The neighborhood can depend on f and not connected. For f ∈ F(A), define Z 1 f (A) = f (∞)I + f (λ)R(λ, A)dλ, R(λ, A) = (A − λI)−1 . 2πi Γ (109) The operator f (A) is well-defined, and assume that g(·, ·) = 1 so that r(0, 0) = k < ∞ and F (·) of (106), for the Karhunen process, a finite measure. We now establish the following representation: Theorem 4.6.11 Let X : R → L20 (P ) be a Cram´er process relative to {g(t, ·), t ∈ R} of bounded Borel functions with g(0, ·) = 1. Then there is an extension space L20 (P 0 ) ⊃ L20 (P ), some Y0 ∈ L20 (P 0 ) and an unbounded linear operator A : L20 (P 0 ) → L20 (P ) such that A|L20 (P ) is symmetric with dense domain sp ¯ {X(t), t ∈ R} ⊂ L20 (P ), and g(t, A)’s are as above and X(t) = g(t, A)Y0 ,

t ∈ R.

(110)

Conversely, if A is symmetric and densely defined in L20 (P ), X0 ∈ L20 (P ) and the g(t, ·) are as above, then Y (t) = g(t, t)X0 , t ∈ R is always a Cram´er process relative to the given g-family.

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Proof. Let X : R → L20 (P ) be a Cram´er process relative to the given g-family. Then by Theorem 4.6.9 above, there is L20 (P 0 ) ⊃ L20 (P ) and a Karhunen process Y : R → L20 (P 0 ) with X(t) = QY (t), t ∈ R and Q as the orthogonal projection onto L20 (P ). Since g(0, ·) = 1, the representing measure is finite. However, the Karhunen process can also ˜ (0) for an A, ˜ be given in operator theoretic form as: Y (t) = g(t, A)Y 2 0 with dense domain sp ¯ {Y (t), t ∈ R} ⊂ L0 (P ). [This version has been verified by Getoor (1956), Thm. 3.4.] Hence we have Z ˜ ˜ g(t, λ)E(dλ)Y (0) g(t, A)Y (0) = R

˜ ˜ It now follows with {E(t), t ∈ R} as the resolution of the identity of A. from the extended spectral calculus of Thomas (1970), that Z ˜ (dλ)Y (0) = g(t, A)Y (0), X(t) = QY (t) = g(t, λ)(QE), R

˜ where A = R λE(dλ) · [E(λ) = QE(λ), λ ∈ R] [E(λ) is a generalized spectral family in that the increments are only positive]. It is verified easily that A|L20 (P ) is symmetric and densely defined, g(0, A) = Q and so X(0) = QY (0). The converse depends on Na˘imark’s theorem. Thus, if A is symmetric and density defined, Na˘imark’s theorem ˜ a self-adjoint operator onto the extenimplies that it extends to A, 0 2 ˜ and g(t, A) = Qg(t, A). ˜ But Y (t) = sion L0 (P ) so that A = Q(A) 0 2 ˜ g(t, A)Y0 , t ∈ R, is a Karhunen process on L0 (P ) relative to the gfamily. So by the representation X(t) = QY (t), t ∈ R, is a Cram´er process in L20 (P ) for the g-family, completing the proof.  R

Remark 29. Each vector measure on (R, B(R)) into a Hilbert space is derived from a generalized spectral family. We close this section with the following interesting result on self-adjoint dilations of certain operators useful in abstract analysis. Theorem 4.6.12 Let A be a densely defined symmetric operator in a Hilbert space H and {gt , t ∈ R} be a class of bounded Borel functions with g0 = 1. Then the family {Tt = gt (A), t ∈ R} defines a class of ˜ ⊃ H, (H ˜ a Hilbert space) on which there bounded operators on H ˜ where Q : is a s.a. operator A˜ extending A such that Tt = Qgt (A) ˜ H → H is an orthogonal projection. Conversely, every densely defined

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self-adjoint A˜ on H and a family {gt , t ∈ R} defines a class of closed ˜ H = gt (A)|H where A = QA˜ and H = Q(K) operators Tt = Qgt (A)| ˜ with Q as orthogonal projection on H, A = QA. The preceding discussion and analysis easily allow one to complete the details of proof of this result and thus will be left to the reader, thereby concluding the section. As before we include some complements and problems to round out the discussion of the section (and the chapter). 4.7 Complements and Exercises 1. Let K be the L. Schwartz space of compactly based real C ∞ functions on Rn , and F : K → C be a second order generalized random field with B(·, ·) as its covariance functional. Then there exists a continuous positive definite compactly supported hermitian function h : Rn × Rn → C, such that Z Z B(f, g) = f (x)g(y)h(x, y)dxdy, f, g ∈ K. Rn

Rn

[Thus h(·, ·) is the covariance function of an ordinary L2 (P )-field.] 2. The following class was introduced by Cram´er (1964). If K is the L. Schwartz space on Rn and Φ is a separable Hausdorff space F : R×Φ → C˜ is a random functional, F (t, f ) = Ft (f ) ∈ L2 (P ) which is continuous in t and f separately, let Kt = sp ¯ {Fs (f ) : s ≤ t} and ∩t∈R Kt = {0}. Then Cram´er terms F purely nondeterministic. In Rt this case one can show that F (tf ) = Ft (f ) = −∞ G(t, u)dZ(u) for a process Z(·) of orthogonal increments, and G is an operator valued function taking Kt into K∞ . [Cram´er (1961) has shown this if Φ is one dimensional, and Kallianpur-Mandrekar (1963) extended this result for the general Φ as given. The reader should follow these works for related matters. See the author’s (1967) PNAS note for details and related analysis.] 3. The following result was given by Sz.-Nagy, as an appendix to the Second Edition of the classical Riesz-Sz.-Nagy book Functional Analysis under the title: “Extensions of Linear Transformations in Hilbert Space Which Extend Beyond This Space”. Verify that the result can be obtained from Theorem 4.6.12 (the last result of this section) which depended only on Na˘imark’s theorem. The result is:

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If {Tt , t ∈ R} is a weakly continuous positive definite set of contractive operators on a Hilbert space H, T0 = id., then there is a super (or extension) Hilbert space K ⊃ H, a unitary group of operators {Ut , t ∈ R} on K such that Tt = QUt , t ∈ R, holds. On the other hand, every weakly continuous group of unitary operators {Ut , t ∈ R} defines a weakly continuous positive definite contractive set of operators {Tt = QUt , t ∈ R} on H = Q(K) for each orthogonal projection Q on K. [This shows that the Na˘imark and Sz.-Nagy extensions are abstractly equivalent. Here Q : K → H is ⊥ projection.] 4. Using the concepts of Exercise 1 above, let F : K → C be a generalized random field of class (C) with orthogonal values for functions of disjoint supports in K. Then there is a random measure Z : B(Rn ) → L2 (P ), onR the Borel field of Rn , with orthogonal values such that, if f˜(t) = Rn f (x)g(t, x)dx, we have Z f˜(t)dZ(t), f ∈ K, F (f ) = Rn

R and the covariance B : (y, y) 7→ Rn g(t)¯ y (t)dσ(t), a tempered mean sure σ in R is the spectral measure of F with g(·), related to class (C). On the other hand (i.e., conversely) if Z(·) is an orthogonally valued random measure relative to a tempered σ(·), and F is given by the integral above, then the resulting F (·) is a generalized random field on K of class (C) with orthogonal values. [If g(t, x) = eitx here, the representation gives the earlier formulas of A. M. Yaglom (noted in Gel’fand-Vilenkin (1964)), and the first case of stationary fields of K. Ito (1954).] 5. This problem gives an extension of Bochner’s characterization of the classical continuous stationary covariance to a larger class. Thus r(·, ·) is a harmonizable covariance on Rn × Rn → C if and only if there exist uniformly bounded covariances rm : Rn × Rn → C such that (i) rm (s, t) → r(s, t), uniformly in (s, t) ∈ Rn × Rn as m S → ∞, (ii) rm is supported on a compact rectangle Em ⊂ Em+1 , ∞ m=1 En = n R , (iii) if {φm k , k ≥ 1} is a complete set of eigenfunctions of rm and m for each m, {Qm k } is the set of eigenfunctions of rm , then φk is m n the Fourier transform of some Gk of bounded variation in R , for

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m, k ≥ 1. [If r(s, t) = r˜(s − t), the result reduces to Bochner’s classical theorem.] 6. A weakly stationary centered random field X : Rn → L20 (P ) is isotropic if its covariance r : (s, t) → C is invariant under (not only translations but also) relations of the (s, t)-coordinates. In this case, Bochner (introduced this concept and) characterized the resulting representation of r(·, ·). The following formula extends his representation to weakly harmonizable classes. Thus Bochner’s characterization of isotopy of a (weak) stationary random field X : Rn → L20 (P ) with covariance r(s, t) = r˜(s − t) is given by (cf. GikhmanSkorokhod (1969), 37–39) the representation:  n  Z J (λ|s − t|) ν ν r(s, t) = r˜(s − t) = 2 Γ dG(λ), s, t ∈ Rn , 2 Rn (λ|s − t|)n with ν = n−2 , G : R−1 → R−1 is a bounded Borel measure and 2 2 |s − t| = (s − t, s − t), n ≥ 1 the (squared) Euclidean distance. Here Jν is a Bessel function (of the first kind) of order ν. Extending this, we say that X is weakly harmonizable isotropic (centered) if there is a positive definite bimeasure µ : B(Rn ) × B(R+ ) → C and n-variate covariance r(·, ·) given by r(s , s ) = ∼ ∼

αn2

∞ h(m,n) X X m=0

l l Sm (u)Sm (v ) ∼

l=1

Z

Z

× R+

R+

∼

Jm+ν (λs)Jm+ν (λ0 r) dµ(λ, λ0 ) ν 0 ν (λs) (λ r)

where (i) g = (s, u), t = (r, v ) are the spherical polar coordinates ∼

∼

of s , t in Rn , (ii) slm (·), 1 ≤ l ≤ h(m, n) = (lm + 2ν)(m + ∼ ∼

2ν − 1)![(2ν)!m!]−1 , 1 ≤ m, are the spherical
harmonics on the n unit sphere of Rm , (iii) αn > 0 αn2 = 22ν Γ n2 π 2 . The integrals are in the (strict) Morse-Transue sense. If the integrals are strongly MT-sense, then the integrals here can be strengthened to Lebesgue’s. With this explanation, we can assert the following integral representation of isotropic harmonizable random fields extending the stationary case. Thus let X : Rn → L20 (P ) be a weakly harmonizable isotropic field. Then it admits an integral representation as:

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4 Isotropic Harmonizable Fields and Applications ∞ X l X l X(t ) = X(r, v ) = αn (m, ν)Sm (v) ∼

∼

m=0 l=1

Z × R+

Jm+ν (λr) l Zm (dλ) (λr)ν

l l0 with Zm (·) verifying E(Zm (B1 )Z¯m 0 (B2 ) = δmm0 δll0 F (B1 , B2 ), and l0 l (B1 )Z¯m F (·, ·) a bimeasure of Znl (·) in that E(Zm 0 (B2 )) = δmm0 δll0 ×F (B1 , B2 ), the F (·, ·) being a bimeasure of finite Fr´echet variation. [The result is important and a nontrivial extension of Bochner’s classical representation of the weakly stationary and harmonizable cases that we have discussed earlier. The detailed analysis of the problem is given in the author’s basic extension in (Rao (1991), Thm. 4). [This and much of the related extensions are in Swift (1997) with references to earlier works motivated by these problems.] 7. This problem presents an approximation of a general weakly harmonizable random field by one whose spectral mass is essentially on a bounded Borel rectangle of Rn × Rn . Thus let X : Rn → L20 (P ) be weakly harmonizable (may not be isotropic) with µ(·, ·) as its spectral bimeasure. Then for each ε > 0 there is a weakly harmonizable Xε : Rn → L20 (P ), with spectrum in Aε × Aε satisfying kX(t) − Xε (t)k2 < ε, t ∈ Rn . [This result implies that the spectrum of the Xε (t) field is contained in a compact rectangle. The result also gives analogs of the sampling theorems of the Kotehnikov-Shannon type that are useful in applications.] 8. The dilation properties obtained earlier for harmonizable fields have analogs for isotropic harmonizable families also, as shown by the work of Swift (1994). Here is a key extension. Thus let X : Rn → L20 (P ) be a weakly harmonizable isotropic field. Then there is an extension space L20 (P˜ ) ⊃ L20 (P ) and a stationary isotropic random field Y : Rn → L20 (P˜ ) with X = QY, Q : L20 (P˜ ) → L20 (P ) being the orthogonal projection onto. If X is strongly harmonizable isotropic, then the covariance function is m times continuously differentiable in the representing co n−1  ordinates τ1 , τ2 , θ where m = , the integral part, and θ is 2 arccos(s , t ). For the strongly harmonizable isotropic case [so r(s, t) ∼ ∼

= r(gs, gt) for any unitary mapping g on Rn ], the µ with Vitali variation

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4.8 Bibliographical Notes

Z

∞Z

r(s, t) = αn 0

0

∞

227

Jν (|λs − λ0 t|) dµ(λ, λ0 ). |λs − λ0 t|ν

[Many of these representations have analogs if we replace the index T by an LCA group G, and one has to use deeper properties of the resulting harmonic analysis (see Swift (1994) for details). Some of this extension will be discussed in the next chapter indicating the interplay of the probability and harmonic analysis at a deeper level, and showing the prospects as well as some interesting applications.] 4.8 Bibliographical Notes This chapter contains some crucial aspects of second order random fields, their structural analysis and the impact of the new property of isotropy that arises when the index is the (vector) group including the n-dimensional space Rn , n > 1. The structural problems are deeper but are useful for many real applications. When the index set is Rn , n > 1, the new concept of isotropy appears and has several interesting (deep) applications but the corresponding mathematical analysis is also deeper. It is fortunate that the great harmonic analyst S. Bochner has taken interest and obtained key integral representations and also extended the Fourier analysis as needed. The Russian Probabilists, Yaglom (1987), Yadrenko (1988) and others have used these formulas for stationary random fields. Serious limitations were noted for stationary isotropic analysis and even though there existed problems that are more general, the methods and existing analysis were not adequate. The early real extension for more general random fields, starting with the harmonizable class, Swift (1994) was able to generalize the classical stationary analysis, and made several useful applications and extensions thereafter. The problems are such that one needs to use the analysis of spherical harmonics. We have presented the main ideas without detailing the many aspects of these results but giving the necessary references. The main thrust here is to present the analysis not only on a (weak/ strong) harmonizable class that uses Fourier analysis for the most part, but we tried to show how its extensions are formulated by H. Cram´er (1951) and K. Karhunen (1947) as well as the works of Bochner (1956), Lo`eve (1947) and Rozanov (1959), among others.

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In some sense, the material presented in this chapter is to be regarded as an introduction and a preview of what harmonic analysis contributes to this area of stochastic process and fields. This work will be a good motivation for random fields on LCA and related groups some to be considered in the next chapter, opening up new ways that many problems can be formulated and solved with a full use of (abstract) harmonic analysis on (LCA) groups as well as on some generalized classes, termed hypergroups, that appear to have potential both for interesting applications as well as theory. This will be discussed in the next chapter.

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5 Harmonizable Fields on Groups and Hypergroups

This chapter is devoted to a comprehensive treatment of harmonizable random fields on LCA groups and some results extended to hypergroups which are of interest in applications as well as in extending the subject to some new directions. We start with a brief (but abstract) account of bimeasures to use it in the ensuing analysis freely, mostly abstracted from the basic weakening of the Lebesgue analysis by M. Morse and W. Transue rendering and enabling it for the (weaker) Fr´echet integrals which play a significant role in a study of (weakly) harmonizable processes. This class lies between the Riemann and Lebesgue integral analyses. Here we include a necessary outline to appreciate the general subject. Then the work proceeds with LCA groups and hyper groups. 5.1 Bimeasures and Morse-Transue (or MT-) Integrals A systematic study of bimeasures and their integrals started with the work of Fr´echet (1915) and detailed by Morse and Transue (1949–56) as well as P. L´evy (1946). This is sketched as follows. If (Ωi , Σi ), i = 1, 2 are measurable spaces and β : Σ1 × Σ2 → C is a mapping with β(A, ·) and β(·, B) as complex measures for A ∈ Σ1 , B ∈ Σ2 , then β(·, ·) is called a bimeasure, which may not have an extension to be a (complex) measure on the product σ-algebra Σ1 ⊗ Σ2 . We present an alternative form to explain and extend the concept so as to define integrals for applications below, and that β(·, ·) may have an extension to a (complex) measure on the generated product σ-algebra Σ1 ⊗ Σ2 , and then using it to get the desired integrals for analyzing the (stochastic) theory of processes and fields. To appreciate the subject 229

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better, we need to briefly recall tensor products of Banach spaces with their cross-norms, clarifying our general approach. A few of the results needed here will utilize some work of Voropoulos (1967), and then we apply it to the MT-integration which is prominent in our work here. If X1 , X2 are Banach spaces, Pn let X1 ⊗ X2 denote the vector space of formal sums f ⊗ g = i=1 fi ⊗ gi , fi ∈ X1 , gi ∈ X2 . A norm α on this product space satisfying kf ⊗ gk = kf kkgk, is called a crossnorm α(·), and X1 ⊗α X2 denotes the completed space so obtained for α(·). There (clearly) exist several (such) norms, but the following two denoted k · kγ , k · kλ called the greatest and least cross-norms are needed and defined as: ( n ) n X X kf ⊗ gkγ = inf kfi kX kgi kX : f ⊗ g = fi ⊗ gi , n ≥ 1 , (1) i=1

i=1

and if X∗i is the adjoint of Xi , let ( n ) X kf ⊗ gkλ = sup x∗1 (fi )x∗2 (gi ) : x∗i ∈ X∗i , kx∗i k ≤ 1, i = 1, 2 . i=1

The corresponding completed spaces are denoted by X1 ⊗γ X2 , X1 ⊗λ X2 ˆˆ X , and termed the tensor ˆ 2 and X1 ⊗ and sometimes simply as X1 ⊗X 2 product spaces. For the spaces of continuous linear functions denoted X∗i , i = 1, 2, the following are true and well-known relations. Proposition 5.1.1 If X1 , X2 are Banach spaces, then we have: ˆ 2 )∗ ∼ (a) (X1 ⊗X = L(X1 , X∗2 )(∼ = L(X2 , X∗1 )), ˆˆ X )∗ ,→ L(X∗ , X ) (b) (X1 ⊗ 2 2 1 where “∼ is an isometric isomorphism in (a) and the symbol in (b) is =” an isometric imbedding of the first into the second space. The point of recalling these results is that the property (a) gives an alternative definition of bimeasures and illuminates the MT-integrals. Thus if Ωi is locally compact, X0 = C0 (Ωi ) as the continuous function space with functions vanishing at “∞”, that V (Ω1 , Ω2 ) = C0 (Ω1 ) ⊗ C0 (Ω2 ), then one verifies that V (Ω1 , Ω2 )∗ = L(C0 (Ω1 ), C(Ω2 )∗ ) and the correspondence is given by B(f, g) = F (f ⊗ g) = T (f )g, ∀f ∈ C0 (Ω1 ), g ∈ C0 (Ω2 ),

(2)

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with F ∈ V (Ω1 , Ω2 )∗ , T : C0 (Ω) → C0 (Ω1 )∗ , bounded and linear. Now B : C0 (Ω1 ) × C0 (Ω2 ) → C is a bounded bilinear form of F . Since C0 (Ω2 )∗ = M (Ω2 ), bounded regular Radon measures, we get by the Riesz-Markov theorem the representation: Z (T f )(·) = f (ω1 )µ(dω1 ) ∈ M (Ω2 ). (3) Ω1

Here we use the facts that T is weakly compact, M (Ω2 ) is weakly sequentially complete (cf. Dunford-Schwartz (1958), VI.7.3, IV.29). Letting µf (·) = (T f )(·) in (3), (2) is simplified as: F (f ⊗ g) = B(f, g) = (T f )(g) Z = g(ω2 )µf (dω2 ) Ω Z  Z 2 = g(ω2 ) f (ω1 )µ(dω1 , ·) (dω2 ) Ω2 Ω1 Z Z = (f, g)(ω1 , ω2 )µ(dω1 , dω2 ). Ω2

Ω1

Thus B(·, ·) can be identified with µ(·, ·), a bimeasure. Also kF k = kBk = sup {|B(f, g)| : kf k∞ ≤ 1, kgk∞ ≤ 1}  Z Z = sup (f, g)(ω1 , ω2 )µ(dω1 , dω2 ) : Ω1 Ω2 ) kf k∞ ≤ 1, kgk∞ ≤ 1 = kµk, the semi-variation of µ.

(4)

For a bimeasure µ, kµk is the Fr´echet variation. Since V (Ω1 , Ω2 )∗ ⊃ 6=

M (Ω1 × Ω2 )(= C0 (Ω1 × Ω2 )∗ ), µ is generally not a restriction of a scalar measure on [C0 (Ω1 × Ω2 )]∗ . Thus we need to consider a generalized (called an MT) integral for our work. This is perhaps the reason that Varopoulos calls the members of V (Ω1 , Ω2 )∗ just bimeasures. This distinction is needed. We next sketch the bimeasure integrals with a slight update and modifications, using some works of Kluvanek (1981), and Vrem (1979), as modified and used from Morse-Transue theory. The discussion is meant

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to explain the Morse-Transue (vector) type integration as a Riemanntype (nonabsolute in real distinction with the Lebesgue type extensions in the literature) and the Kluvanek modification is included which still retains the nonabsolute conditions. ThusPlet fP n : Ω = Ω1 × Ω2 → C be a simple function given as kn kn n n n n n fn = i=1 j=1 aij χAi χBj , Ai ∈ Σ1 ; Bj ∈ Σ2 (disjoint) and let β : Σ1 × Σ2 → C be a measure; now set Z fn dβ = Ω

kn X

aij β(Ani , Bjn ).

(5)

i=1,j=1

R If fn → h, pointwise with limit R and { Ω fn dβ, n ≥ 1} ⊂ C is Cauchy R α0 , define α0 = Ω hdβ. It can be verified that h 7→ Ω hdβ, is welldefined and linear. [An equivalent form is given by Kluvanek (1981), valid even if C is replaced by Banach spaces with some extension.] We now designate a subclass called strict integrals to use. If (Ωi , Σi ) is a measurable pair, fi : Ωi → C is measurable (Σi ) and β : Σ1 ×Σ2 → C is a bimeasure, let f1 be β(dω1 , ·) and f2 be β(·, dω2 ) integrable with β : Σ1 × Σ2 → C as a bimeasure, and suppose that f1 is β(dω1 , ·), f2 is β(·, dω2 )-integrable (DS) as vector measures. Then the complex measures β˜1F , β˜2F given by Z Z F F β˜1 (A) = f (ω2 )β(A, dω2 ), β˜2 (B) = f1 (ω1 )β(dω1 , B) F

F

for A ∈ Σ1 , B ∈ Σ2 , and for each E ∈ Σ1 , F ∈ Σ2 clearly define complex measures. The pair (f1 , f2 ) is termed strictly β-integrable if one has Z Z F ˜ f1 (ω1 )β1 (dω1 ) = f2 (ω2 )β˜2E (dω2 ). (6) E

F

R R∗

The common value is denoted E F f1 (ω1 )f2 (ω2 )β(dω1 , dω2 ). In the earlier case, this was demanded only for a single pair E = Ω1 , F = Ω2 . This strengthening renders the dominated convergence valid as well as the absolute continuity of the integral; but it is still weaker than the Lebesgue theory, and e.g., the Jordan decomposition is still not valid here. It should be noted that these extended concepts are interesting even in integral representations of bilinear operators having analogs in multilinear theory. Some aspects of these results with applications to abstract

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analysis have been given by Dobrakov (1987). For instance, a vector bimeasure β has finite Fr´echet variation if kβk(Ω1 , Ω2 ) < ∞ where

( n

X

kβk(Ω1 , Ω2 ) = sup ai bj β(Ai , Bj ) : Ai ∈ Z1 , Bj ∈ Σ2 ,

i,j=1 X ) disjoint, |ai | ≤ 1, |bj | ≤ 1, scalars, n > 1

< ∞.

(7)

This follows from the fact that l ◦ β is a scalar bimeasure, l ∈ X∗ . With a scalar extension method, Ylinen (1978) has obtained the following version of the Riesz-Markov representation stated for reference: Theorem 5.1.2 Let (Ωi , Bi ), i = 1, 2 be Borelian l.c spaces, and C0 (Ωi ) as the continuous real function space with functions vanishing at “∞”. If X is a reflexive Banach space and B : C0 (Ω1 ) × C0 (Ω2 ) → X is a bounded bilinear form, then there is a regular (in each component) bimeasure β : B1 × B2 → X, such that Z Z B(f1 , f2 ) = f1 (ω1 )f2 (ω2 )β(dω1 , dω2 ), fi ∈ C0 (Ωi ), Ω1

Ω2

satisfying the bound kBk = sup {kB(f1 , f2 )kX · kfi k∞ ≤ 1, i = 1, 2} with kBk = kβk(Ω1 , Ω2 ). For nonreflexive X, the same holds provided ˆ 0 (Ω2 ) into relatively B maps bounded sets of V (Ω1 , Ω2 ) = C0 (Ω1 )⊗C compact sets. The problem receives a complete and comprehensive view when the ¯t) mapping X : R → L20 (P ) with covariance r : (s, t) → E(X X is weakly stationary provided (by the classical Bochner theorem) as, R r(s, t) = r˜(s − t) = R ei(s−t)λ µ(dλ), for a bounded Borel µ on LCA groups, and for (weakly) harmonizable case if Z Z r(g1 , g2 ) = hg1 , λihg2 , λ0 iβ(dλ, dλ0 ). (8) ˆ G

ˆ G

When β(·, ·) has finite (Fr´echet) variation, then it was generalized by Cram´er (and Karhunen) from the harmonizable concepts. All of this is better understood if G is an LCA group, as we now put it in perspective for harmonizability in the present context.

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5.2 Harmonizability on LCA Groups This section is denoted to some abstract extensions of the harmonizability concepts when the indexing set is a general locally compact group with some extensions in the next section to hypergroup indexing. Only then the real potential of the subject and its uses are appreciated. This study opens up essentially the full scope of the subject for analysis. As a motivation let us restate the Cram´er and Karhunen classes when ˆ is the dual they are given with index G = Rn or Zn , n > 1. Thus if G 2 group, and X : G → L0 (P ) is weakly (strongly) harmonizable, then the ¯ g2 ) is representable as (hg, ·i ∈ G ˆ is a covariance r(g1 , g2 ) = E(Xg1 X character of G): Z Z r(g1 , g2 ) = hg1 , r1 ihg2 , r2 iβ(dr1 , dr2 ), (9) ˆ G

ˆ G

ˆ × B(G) ˆ of Fr´echet (or with β as a positive definite bimeasure on B(G) finite Vitali) variation of the random field {Xg , g ∈ G}. ¯ t ) as Let {Xt , t ∈ T } ⊂ L20 (P ) be a family with r(s, t) = E(Xs X its covariance. If B is a σ-algebra of T , then the Xt -family is said to be of class (C) if there is a measurable pair (S, S) and a positive definite bimeasure β : S × S → C with finite Vitali variation such that Z Z gs (λ)gt (λ0 )β(dλ, dλ0 ), s, t ∈ T, r(s, t) = (10) S

S

relative to a β-integrable collection {gs , s ∈ T } of scalar functions such that r(s, t) < ∞, s ∈ T , and if β is only of finite Fr´echet variation, then we have a weak class (C). If β in (10) concentrates on S = T one has the Karhunen class and then (10) becomes Z r(s, t) = gs (λ)gt (λ)µ(dλ), s, t ∈ T (11) S

and this was introduced by Karhunen in 1947. We can have the Karhunen field, extending the original stationary and harmonizable classes: Proposition 5.2.1 Every weakly or strongly harmonizable random field X : G → L20 (P ), G an LCA group, belongs to a Karhunen class. In fact, if the family is weakly harmonizable then there is a finite Borel measure ˆ and a family {gs , s ∈ G} ⊂ L2 (G, ˆ µ) such that (11) holds with µ on G ˆ T = G and S = G.

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We specialize the analysis now for harmonizable fields on LCA groups and also present some analogs for the nonabelian case, as this reveals an internal structure of the problem better. It will be useful to recall the D-S integral of a scalar function relative to a vector measure. Thus if (Ω, Σ) is a measurable space, f : Ω → C is (Σ) measurable, and Z : Σ → X, a Banach space, is a vector measure, then f is D-S integrable for Z if the following pair of conditions holds: (i) there is a sequence fn : Ω → C of simple (Σ-measurable) Pkn n memn bers such that f → f pointwise, and (ii) if f = n n i=1 ai χAi , and R R n n f dZ = Σai Z(E ∩ Ai ) ∈ X ⇒ { E fn dZ, n ≥ 1}R is Cauchy in X C n for each E ∈ Σ. Then the unique limit is denoted by E f dZ, E ∈ Z. It may be verified that the D-S integral above is a well-defined element of X, is linear, and the dominated convergence theorem is valid for it. R However, the evaluation of E f dZ, as a Stieltjes integral is generally false. Note that the convergence in (i) is pointwise and not uniform. Now let L1 (Z) be the space of scalar (D-S) integrable functions for Z0 , and L20 (β), the collection of strictly β-integrable (MT-sense) functions f : S → F when β : (A, B) 7→ E(Z(A)Z(B)) is a bimeasure for Z, with X = L20 (P ), on (Ω, Σ, P ). Here Z(·) is termed a stochastic measure and β its spectral bimeasure of the second order process, we then have: Theorem 5.2.2 Let (S, S) be a measurable space β : S × S → C be a positive definite bimeasure. Then there is a probability triple (Ω, Σ, P ) and a stochastic measure Z : Σ → L20 (P ) such that (i) E(Z(A)Z(B)) = β(A, B), A, B ∈ S, and (ii) L1 (Z) = L2∗ (β), equality between sets of functions. This result can be obtained quickly using the (Aronszajn) reproducing kernel Hilbert space techniques. A general form of the latter is as follows. If (S, S) is a Borelian space, (S-topological) a bimeasure β : S × S → C is said to have locally finite Fr´echet (or Vitali) variation if β : S(E) × S(E) → C has finite Fr´echet (or Vitali) variation on each bounded Borel set E ⊂ S. [See Edwards (1955) on the distinctions.] With these ideas, the following general representation, to be used later, holds: Theorem 5.2.3 Let S be locally compact and (S, S) be Borelian pair, and {Xt , t ∈ T } ⊂ L20 (P ), on a probability space (Ω, Σ, P ), be a locally weakly class (C) relative to a positive definite bimeasure β :

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S0 × S0 → C of locally finite Fr´echet variation and a family gt : S → C, each gt being locally strictly β-integrable where S0 is the δ-ring of bounded sets of S. Then there exists a vector measure Z : S0 → L20 (P ) such that for the index set T : Z (i) Xt = gt (λ)Z(dλ), t ∈ T, (D-S integral) (12) S

(ii)

¯ E(Z(A)Z(B)) = β(A, B), A, B ∈ S0 .

(13)

In the opposite direction, if {Xt , t ∈ T } is given by (i), then the process is of local weak class (C) for a bimeasure β, given by (ii) and the gt of (i) being locally strictly β-integrable. The process is of (local) Karhunen class iff (i) and (ii) hold with β(A, B) = µ(A∩B) for a σ-finite measure µ on S. Sketch of Proof. If K ⊂ S is compact, consider S(K) of S restricted to K and β : S(K) × S(K) → C is a positive definite bimeasure and the theorem applies for it. If now Z˜ : S(K) → L20 (P ) is the corresponding representing stochastic measure, then one will have: Z  Z ˜ gt (λ)Z(dλ) =j gt (λ)β(dλ, ·) ∈ L20 (P ), (14) K

K

where j is the mapping above between Z˜ and β. With the local compactness we can define a measure Z : S → L20 (P ) and extend the above representation using the standard procedure (cf. Hewitt-Ross (1963), pp. 133–134). Without local compactness, this piecing together can fail. From now on we will follow the standard, but not completely trivial, and which can be given (see e.g. Chang and Rao (1986), p. 53), but the details will not be reproduced here.  To understand the type of functions gt , it may be of interest to specialize the class for harmonizable and stationary representations. We present the special result for reference. Theorem 5.2.4 Let {Xt , t ∈ G} ⊂ L20 (P ) be a class with G as an LCA group. Then the set is weakly (or strongly) harmonizable relative to a ˆ × B(G) ˆ → C (of finite Vitali positive definite bimeasure β on B(G) ˆ → L2 (P ) such variations) iff there is a stochastic measure Z : B(G) 0 that Z (i)Xt = ht, λiZ(dλ), t ∈ G, (D-S integral) (15) ˆ G

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ˆ (ii)E(Z(A)Z(B)) = β(A, B), A, B ∈ B(G),

237

(16)

where ht, ·i is a character of G. When these conditions hold the mapping t 7→ Xt is strongly uniformly continuous on L20 (P ); and the random field {Xt , t ∈ G} is weakly stationary iff (i), (ii) hold with β(A, B) = µ(A ∩ B) for a bounded Borel measure µ so that Z(·) has orthogonal increments. It is natural to ask if the corresponding result obtains for G that may not be abelian. The nonabelian G shows the potential on the use of ideas and methods of Fourier that are also at the base of the deep and satisfying applications of harmonic analysis. Thus we discuss the harmonizable extension of X : T → L20 (P, X) for X-valued strongly measurable fields so that l(X) : G → L20 (P ), is its scalar correspondent. Then l(X) : G → L20 (P ), l ∈ X∗ , is a harmonizable field so that l(X) : G → L20 (P ) admits a representation as: Z l(Xt ) = ht, λiZl (dλ), t ∈ G, (17) ˆ G

where Zl is a stochastic measure. The mapping l 7→ Zl is linear (Zl is ˆ < ∞, a regular vector measure with finite semi-variation) so kZl k(G) where ˆ = sup kl(Xt )k2 ≤ klk sup kXt k2 < ∞, kZl k(G) (18) t

t

ˆ since X(G) is bounded. By the uniform boundedness principle kZt k(G) ∗∗ ˜ ˜ ≤ K for some K < ∞, and there is Z with Zl = l(Z(·)) ∈ X = X here. ˜ also reflexivity and By reflexivity of X, there is Z˜ with Zl = l(Z), ˜ uniform boundedness imply E(Z(A)) ∈ X so that we have (for l ∈ X∗ ) Z  Z ˜ l(Xt ) = ht, λil(Z)(dλ) = l ht, λiZ(dλ) . ˆ G

ˆ G

It follows from this that Z Xt =

˜ ht, λiZ(dλ)

t ∈ G.

(19)

ˆ G

This gives us the representation as: Theorem 5.2.5 Let G be an LCA group, X a reflexive B-space and X : G → L20 (P, X), such that X(G) is relatively weakly compact,

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(equivalently norm bounded). Then X is weakly harmonizable iff there is a stochastic measure Z˜ such that (19) obtains. This implies that it may be possible to characterize weakly harmonizable fields without using bimeasure integration. If G = R, Niemi (1975) considered some special representations, and certain others were given in Chang and Rao (1988). The above result unifies all these cases. To appreciate and study the potential of harmonizable random fields, it is desirable to consider these objects also on a few nonabelian groups G as well. But these involve Fourier analysis on nonabelian groups containing G, since then there is no dual group of G, one needs to invoke now some ideas and results of C ∗ -algebras, and this was done by Ylinen (1975, 1984, and 1987) using certain related techniques of Edwards (1964) which can be simplified if the LC group G is restricted to be separable and unimodular. We present a little of this analysis simply to show the real potential of the subject and for future extended analysis. Here we include a brief account to be used in our analysis essentially following Mautner (1955); especially Tatsuma (1967) and Segal (1950). [See also Na˘imark (1964), Chapter 8.] ˆ denotes the set of all irreducible strongly continuous repre(i) If G ˆ a topology sentations of G into a Hilbert space, then we can endow G with which it becomes a locally compact Hausdorff space. Thus one can ˆ a topology making it also a locally compact Hausdorff space. endow G If now µ is a Haar measure on G, then there is a unique Radon measure ˆ so that (G, ˆ ν) becomes a dual gauge of (G, µ) and the Plancherel ν on G formula holds for it. (ii) The representation Hilbert space H can be taken as L2 (G, µ) × 2 (L (G)) and then H = ⊕y∈Gˆ Hy , the direct sum, where Hy is the repˆ If Ay ⊂ L(Hy ) is the weakly closed resentation space for y in G. self-adjoint subalgebra of L(Hy ), generated by the strongly continuous unitary family {Uy (g), g ∈ G}, then Ay is of ‘type I or II’, and Z ⊕ 2 L (G) = Hy ν(dy), direct integrals. (20) ˆ G

Further, if (La f )(x) = f (a−1 x), x ∈ G, then the weakly closed selfadjoint algebra a generated by {La , a ∈ G} ⊂ L(H), gives a direct sum ˆ and for each f ∈ L1 (G) ∩ L2 (G), the decomposition of Ay , y ∈ G, following (Bochner) integral exists:

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fˆ(y) =

Z

ˆ Uy (g)f (g)µ(dg), Uy (g) ∈ Ay , y ∈ G,

239

(21)

G

and defines a bounded linear mapping on H. Moreover, fˆ(y) may be extended uniquely to a dense subspace of H containing L1 (G) ∩ L2 (G) so that it is closed and self-adjoint. This extended form y 7→ fˆ(y), is the generalized Fourier transform of f, and this general setup is needed for us here. (iii) There is a trace functional τy : Ay → F which is positive, linear, normal, semi-finite and faithful in terms of which there is the Plancherel formula, that we need for fi ∈ L2 (G), i = 1, 2, with fˆi∗ as the adjoint of fˆ, so that one has: Z Z f1 (g)f2 (g)µ(dg) = τy (fˆ1 (y)fˆ2∗ (y))ν(dy). (22) ˆ G

G

The desired measurability of fˆi and y → τy (fˆ1 (y)fˆ2∗ (y)) for ν are not obvious, but can be proved in this work, and f → fˆ is one-to-one. These are established in Mautner (1955). ⊂ A, y 7→ A(y) is measurable, norm bounded, and R (iv) If A(y) ∗ 2 ˆ ˆ τy (A(y)A (y))dy < ∞, then there is f ∈ L (G) such that f (y) = G 2 ˆ Also if h ∈ L (G) and h = f ∗ f for some f ∈ L2 (G), A(y), y ∈ G. then one has the inversion, (‘*’ being the convolution product): Z ˆ τ (Ug (g)∗ h(y))ν(dy), g ∈ G. h(g) = ˆ G

With this abstract set up on the generalized Fourier transform we can present the needed formulation of Bochner’s classical notion of V boundedness, and the (long awaited) characterization of weakly harmonizable random fields. Set forth as follows: Definition 5.2.6 Let G be a separable locally compact unimodular group and X = {Xg : g ∈ G} ⊂ L20 (P ) be a random field. Then X is weakly harmonizable if it is weakly continuous, and Z  1 2 ˆ ∞ = sup kφ(y)k ˆ Xg φ(g)µ(dg) : kφk ≤ 1, φ ∈ L (G) ∩ L (G) G

y∈G

is bounded in L20 (P ), φˆ being the above defined generalized Fourier transform of φ.

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With this preparation, the general integral representation of a weakly harmonizable random field can be given as follows. Theorem 5.2.7 Let X : g → Xg ∈ L20 (P ), g ∈ G, be a weakly harmonizable random field. Then there exists (i) a weakly σ-additive regular ˆ operating on Hy into operator valued measure M(dy) defined on G, ∼

L20 (P ), vanishing on ν-null sets, and (ii) a trace functional τy : Ay → C such that Z trace(Ug (y)M(dy)), g ∈ G (Bartle’s integral) (23) Xg = ∼

G

and X(·) is uniformly continuous in the strong topology of L20 (P ). Conversely, a weakly continuous X : g → Xg given by (23) is weakly harmonizable. Further, the covariance function r of the weakly harmonizable X, of (23), has its covariance represented as: Z Z τy1 ⊗ τy2 (Ug1 (y1 ) ⊗ Ug2 (y2 ))β(dy1 , dy2 ), r(g1 , g2 ) = (24) ˆ G

ˆ G

ˆ × B(G), ˆ with B(G) ˆ where β is an operator valued bimeasure on B(G) ˆ as the Borel σ-algebra of G.

Proof. If f ∈ L1 (G) ∩ L2 (G), let fˆ be defined by (21) above which is R ⊕a measurable (operator) function. It is also bounded since H = ˆ Hy ν(dy) and Hy can be embedded in H, and may treat it as a closed G subspace. So Uy (g) = U (g, y) is in L(Hy ) and be extended as U˜ = U on Hy and as identity on Hy⊥ so that {U˜ (g, y), g ∈ G} again unitary in L(H) with {U˜ (g, ·) ∈ L(H), g ∈ G} as a unitary family in L(H) with U˜ (g, ·) ∈ L(H), g ∈ G. If the resulting operators of (21) obtained by replacing U with U˜ , denoted by fˆ, then it is measurable and a(H) = {fˆ(y) ∈ L(Hy ), y ∈ ˆ is identifiable with a subalgebra of L(H). G} If T : f → fˆ is the mapping, then it is one-to-one and is a contraction, the former property is a consequence of the general theory, and its contraction is verified as follows: (k · kop denotes operator norm)

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241

Z

ˆ ˜

kf (y)kop = f (g)U (g, y)µ(dg)

G op Z ≤ |f (g)|kU˜ (g, y)kop µ(dg), as a vector integral, G Z ≤ |f (g)|µ(g) = kf k. G

Hence supy∈Gˆ kfˆ(y)kop ≤ kf k1 < ∞, so that T : L1 (G) ∩ L2 (G) → a(H) is a contraction. Since X is weakly harmonizable, we have for each f ∈ L1 (G) ∩ L2 (G) the following: Z T1 (f ) = f (g)Xg µ(dg) ∈ L2 (P ). (25) G

Clearly, T1 is bounded. Let T˜ = T1 ◦ T −1 so that T˜(fˆ) = T1 (T −1 (fˆ)) = T1 (f ), f ∈ L1 (G) ∩ L2 (G), and T˜ is well-defined. Also, kT˜(fˆ)k2 ≤ ckf k2 ,

(c, a constant).

(26)

Thus T˜ can be expressed as a direct sum of bounded operators from ˆ by the general theory. Since the range of T˜ is a(Hy ) into L20 (P ), y ∈ G, 2 ˜ in L0 (P ), T is weakly compact. Applying a form of the Riesz-Markov theorem (e.g. see, Dinculeanu (1967), p. 398, Thm. 9) and using the theory of direct integrals for which the separability conditions of G are needed, (cf., Na˘imark (1964), Ch. 8, Sec. 4), one gets a regular weakly ˆ into the bounded operator class σ-additive operator measure M on B(G) ∼

L(a(H), L2 (P )) such that (tr = trace) Z   T˜(fˆ) = tr fˆ(y)M(dy) , fˆ ∈ a(H), ˆ G

∼

(27)

where the integral is in the Bartle-Dunford-Schwartz sense. Here M(·) : ∼ ˆ → L2 (P ) is σ-additive and regular for each x ∈ a(H). It follows B(G) 0 now from (18) and the preceding analysis that the ensuing computations are valid.

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Z

Z

  tr fˆ(y)M(dy) ∼ ˆ  ZG Z ˜ = tr f (g)U (g, y)µ(dg)M(dy) ∼ ˆ G G Z Z h i ˜ = f (g) tr U (g, y)M(dy) µ(dg)

f (g)Xg µ(dg) = G

G

∼

ˆ G

since f is scalar and the trace is linear and commutes with the integral over G, so that Fubini’s argument applies. The above can be rearranged to obtain:  Z Z   f (g) Xg − tr U˜ (g, y)M(dy) µ(dg) = 0. (28) ∼

ˆ G

G

Since f ∈ L1 (G) ∩ L2 (G) is arbitrary and this product set is dense in both L1 (G) and L2 (G), we get that the function in [ ] = 0. Now replacing U˜ with U in (28), which is valid, (28) implies (23). The converse is similar. In fact, if (23) holds and φ ∈ L1 (G)∩L2 (G), then by the standard arguments one has  Z Z Z tr(U (g, y)M(dy) µ(dg) Xg φ(g)µ(dg) = φ(g) ∼ ˆ G G G Z Z h i = tr φ(g)U (g, y)M(dy) µ(dg) ∼ ˆ  ZG GZ φ(g)U (g, y)µ(dg) M(dy) = tr ∼ ˆ G G Z h i ˆ = tr φ(y)M (dy) . ˆ G

∼

From this we conclude:

Z



ˆ ˆ

Xg φ(g)µ(dy) ≤

φ kMk(G), G

op

∼

(29)

where kMk(·) is the semi-variation of M (cf. Dinculeanu (1967) Sec. 19). ∼ ∼ ˆ < ∞ in (29) it follows that X(·) is weakly Letting C = kMk(G) ∼

harmonizable since it is clearly weakly continuous. Finally, to obtain (24), we can calculate the covariance r of X(g), using a few other properties of the MT-integral as follows:

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243

¯ g2 ) r(g1 , g2 ) = E(Xg1 · X Z  Z =E try1 (U˜ (g1 , y1 )M(dy1 ) · try2 (U˜1 (g2 , y2 )M(dy2 ) (30) ∼ ∼ ˆ ˆ G G Z h i ˜ ˜ =E try1 ⊗ try2 U (g1 , y1 ) ⊗ U (g2 , y2 )M(dy1 ) ⊗ M(dy2 ) ˆ G ˆ G⊗

∼

∼

using the tensor product properties of trace functionals (cf., Hewitt-Ross (1970), Appendix D), Z Z try1 ⊗ try2 [U˜ (g1 , y1 ) ⊗ U˜ (g2 , y2 )]E(M(dy1 ) ⊗ M(dy2 )) = ∼ ∼ ˆ G ˆ G Z Z = try1 ⊗ try2 [˜ u(g1 , y1 ) ⊗ u˜(g2 , y2 )]β(dy1 , dy2 ) (31) ˆ G

ˆ G

whereβ(·, ·) is the operator valued positive-definite bimeasure. This is just (24) written in a different form. Thus the result follows.  Remarks. 1. The representation (23) can be used to solve filter equations ˆ is also a group, and Hy = C. in other applications. If G is abelian, so G Thus the result reduces to a previously known case (cf. Rao (1982)). 2. The simpler representation for stationary random fields was first obtained by Yaglom (1960, 1961). It is possible now to extend his work for homogeneous spaces as well as multidimensional fields also to the weakly harmonizable case. 3. The measure M(·) in (27) need not be σ-additive in the uniform ∼ operator topology. In the LCA case, this difficulty disappears, since then H is C and the classical Pettis theorem implies stating that weak and strong σ-additives agree. 4. If G is not a separable group, the decomposition, runs into difficulties, and one may have to use the C ∗ -algebra approach, as was initiated by Ylinen (1975). The representation (23) is not valid in general. 5. If the group G is compact, then the Fourier transforms can be derived through the Peter-Weyl theory. Thus a representation similar to (23) can be obtained through it. (See Hewitt and Ross (1970) for the general Fourier analysis on compact groups.) We close the main part of this section with an application to linear filters: ΛX = Y . Let G be an LCA group, and X, Y : G → L2 (P, Ck ), be k-vector random fields. Then Λ : X → Y is assumed to satisfy Th (ΛX)(g) = Λ(Th X)(g) for all g, h ∈ G. Here (Th X)(g) =

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X(g + h), g ∈ G, (translation) and (ΛX)(g) = Y (g), g ∈ G, is the filter equation. Thus Λ is called a linear filter and we discuss its solution if Λ is of the form, Z Y (g) = (ΛX)(g) = Λ(s)X(g − s)ds. (32) G

In the context of a harmonizable process, one has: Theorem 5.2.8 Let the output Y be a k-dimensional weakly harmonizable random field with βy as its k × k matrix spectral bimeasure. For the filter equation (32) given above, there exists a weakly harmonizable solution iff Z Z ∗ (I − F F −1 )(λ)βy (dλ, dλ0 )(I − F F −1 )∗ (λ0 ) = 0 (33) (i) D

D

ˆ where F = Λ, ˆ the Fourier transform of Λ, for all Borel sets D ⊂ G −1 with F denoting the generalized inverse of F and ‘*’ for the adjoint of a matrix, the integral in (i) being in the strict MT-sense, and where the following integral similarly exists: Z Z ∗ F (λ)−1 βy (dλ, dλ0 )(F (λ0 )−1 )∗ . (34) (ii) ˆ G

ˆ G

Under these conditions, the solution X of the problem is given as: Z Xt = ht, λiF −1 (λ)Zy (dλ), t ∈ G, (35) ˆ G

where Zy (·) is the stochastic measure representing Y . The solution is ˆ unique iff F (λ) is nonsingular for each λ ∈ G. Here F (λ) in (34) is usually called the spectral characteristic of the filter Λ. Under further restrictions on A(·) one can obtain the following simpler result: Proposition 5.2.9 Let F be the spectral characteristic of the filter Λ in (32). If (i) and (ii) of the above theorem hold, and if there is a k×k matrix function f , which is Lebesgue integrable, whose Fourier transform fˆ 1 satisfies kF −1 − fˆ∗ k2,βy = 0, with the norm kf k = kT 2 f k2 , then the solution is given by: Z Xt = f (s)Y (t − s)ds, t ∈ G. (36) G

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245

When G = Rn , similar problems were considered by Chang and Rao (1986), and their methods, extend to the present case of an LCA group G. The further conditions under which the solution X is stationary have been discussed by Bochner (1956). These results were extended by the author (1984) to include the above considerations, and the interested readers are referred to it. 5.3 Harmonizability on Hypergroups In some statistical analyses, for example, sample means of stationary or harmonizable sequences, we are often, confronted with ‘new’ processes that are not of the original form. They are of second order classes which are not easily compared and classified with the original processes. But still many of these form a generalization which is now termed ‘hypergroups’, and the subject has potential for an extended analysis. It has applicational prospect and potential for developments, as seen from the detailed and lucid exposition by Heyer (2014), we discuss harmonizability for such classes (on hypergroups) and present some results on their representation. The abstract concept is as follows: Definition 5.3.1 A locally compact space K is called a hypergroup if the following conditions are satisfied for it: There exists an operation, termed convolution, ∗ : K × K → M (K) such that for x, y ∈ K, (x, y) → δx ∗ δy (δx is Dirac measure) where M (K) is the set (or space) of Radon probability measures on K, endowed with the weak*-(also termed vague) topology and so M (K) can be regarded as the dual space of C0 (K), such that: (i) δx ∗ (δy ∗ δz ) = (δx ∗ δy ) ∗ δz , (ii) δx ∗ δy has compact support, (iii) there is an involution denoted “∼”, on K such that (x∼ )∼ = x and (δx ∗ δy )∼ = δx˜ ∗ δy˜, x, y ∈ K where for a measure µ on K, µ ˜(A) = ˜ ˜ µ(A) with A = {˜ x : x ∈ A}, and there is a unit element ‘e’ in K so that δe ∗ δx = δx ∗ δe = δx , (iv) ‘e’ ∈ sup(δx ∗ δy ) iff x = y and the mapping (x, y) → supp(δx ∗ δy ) is continuous when the space 2K is given the Kuratowski topology. Note: If (iv) is not assumed, then the space K having properties (i)–(iii) is called a weak hypergroup. A standard reference for hypergroups is the book by Bloom and Heyer (1995).

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A hypergroup satisfying conditions (i)–(iii) is termed a weak hypergroup. A number of examples and details of these objects are given by Lasser (1983), and a general theory is detailed in the above noted reference volume of Bloom and Heyer (1995). Also, there exist some further extensions and recent research on the subject seen in a nice exposition by Heyer (2010). It was shown by Spector (1978) that an abelian hypergroup K admits ˆ is its dual (so K ⊂ K) ˆ it is an invariant (or Haar) measure and if K ˆ termed a strong hyper group provided K = K. A considerable amount of classical analysis extends to such hypergroups. In the probabilistic context, we say that X : K → L20 (P ), with covariance ρ : (a, b) → ¯ b ), is hyper-weakly stationary if ρ is representable as: E(Xa X Z ρ(a, b) = ρ(x, 0)(δa × δb )(dx), a, b ∈ K, (37) K

and then X is hyper-weakly stationary on the commutative hypergroup K. We now introduce hyper weak harmonizability as a useful concept. Definition 5.3.2 Let K be a commutative hypergroup with its dual obˆ and X : K → L20 (P ) be a centered random field. If ρ : (a, b) 7→ ject K, E(Xa X b ), a, b ∈ K, denotes its covariance function, then X is termed hyper-weakly (strongly) harmonizable if ρ admits a representation: Z Z ∗ α1 (a)α2 (b)β(dα1 , dα2 ), (38) ρ(a, b) = ˆ K

ˆ K

ˆ × B(K) ˆ → C is a positive definite bimeasure (of where β : B(K) finite Vitali variation) and the integral (with ‘star’) is in the MT- (or Lebesgue-Stieltjes) sense. It is now known that β has always a finite Fr´echet variation on the ˆ The Fourier transform is well-defined and the Borel σ-algebra B(K). strict M T -integral will be used (as detailed in the article by Chang and Rao (1986)). This is one-to-one and the Fourier transform is also oneto-one. With these properties, the following result can be established. Theorem 5.3.3 Let X : K → L20 (P ) be a hyper-weakly harmonizable random field defined above. Then there exists a stochastic measure Z : ˆ → L2 (P ) such that B(K) 0 Z Xa = α(a)Z(dα), a ∈ K (39) ˆ K

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247

¯ 2 )) = β(A1 , A2 ) defining a bimeasure β. Indeed a with E(Z(A1 )Z(A second order weakly continuous random field on a commutative hypergroup K admits the representation (39) so that it is hyper weakly harmonizable, iff the following set is bounded in L2 (P ): Z  1 2 ˆ ∞ ≤ 1, φ ∈ L (K, µ) ∩ L (K, µ) ⊂ L2 (P ) φ(a)Xa dµ(a) : kφk 0 K

(40) ˆ is the Fourier transwith µ being a Haar measure carried on K and φ R ˆ ˆ form, given as φ(α) = K α(a)φ(a)dµ(a), α ∈ K. The result is obtained from the works of R. Lasser and R. C. Verm noted above. There is also much activity by these authors on related results. Further discussion on the problems will be omitted here. 5.4 Remarks on Strict Harmonizability and V -Boundedness Instead of restricting to the second moment properties, some extensions will be useful for comparison as well as for some applications. Hence we consider some related problems based on the complete distributional properties for their structural and probabilistic features when Fourier analysis is still available. Definition 5.4.1 On a measurable couple (S, S) let Z : S → Lp (P ), 0 < p ≤ 2, be a mapping. Then Z(·) is termed an independently valued random measure of exponent p if: (i) Ai ∈ S, i = 1, . . . , n, n ≥ 1, disjoint, {Z(Ai ), 1 ≤ i ≤ n, n ≥ 1} is a mutually independent set of random variables, (ii) for each A ∈ S, Z(A) is a stable random variable of exponent pi , P∞ (iii) An ∈ S, disjoint, A = ∪n≥1 An ⇒ Z(A) = n=1 Z(An ), the series converging in probability (hence also with probability one). The above concept allows us to present an integral representation of certain strictly stationary processes with p moments, 1 ≤ p ≤ 2 that admit integral representation relative to a stochastic measure, with stable independent values, opening up some new directions than before. This shows how new types of integrals arise in such classes of studies. Theorem 5.4.2 Let X : Z → Lα (P ), 1 ≤ α ≤ 2 be a (discrete) process. Then there exists an independently valued stable random measure

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Z of exponent α, which is isotopic on the Borel field S of the interval (−π, π) such that Z π Xn = eint Z(dt), n ∈ Z, (41) −π

iff {Xn , n ∈ Z} is strictly stationary and V -bounded in the sense that



( n ) n



X X



ak Xk ≤ C0 sup (42) ak e−2πiαk : t ∈ (−π, π) ,



K=1

α

k=1

P where C0 is constant, and if α = 1, { nk=1 ak Xk , n ≥ 1, ak ∈ C} is relatively weakly compact in L1 (P ) in addition. When these conditions hold we obtain their characteristic functions as: " k #! X φn1 ,...nk (u1 , . . . uk ) = E exp inXnj j=1

( Z = exp −

π

−π

|

k X

) ui eiλui |α dG(λ) ,

(43)

j=1

for a bounded increasing function G ≥ 0. Some Remarks. It may be noted that the integral representation (41) is obtained using (42) and Bochner’s V -boundedness condition for processes valued in Banach spaces. The calculations, giving the representation (42) are due to Hosoya (1982), who also extended it thereafter for 0 < α < 1 and, for Xt of (41). There are some related extensions of the integral formula (41) by several authors, including Combanis (1983), Weron (1985), Urbanik (1968), Kuelbs (1993), Okazaki (1979), Rosinski (1986), Marcus (1987), and perhaps some others. We end this section with a strict sense version of the Karhunen class due to Kuelbs (1973) which is an extension of Schilder’s (1970) result who considered this for a symmetric stable class. This is given for comparison and reference here. Theorem 5.4.3 Let {Xt , t ∈ T } be a random field indexed by a separable Hausdorff space T whose finite dimensional distributions form a symmetric stable class of index α, 0 < α ≤ 2. Then there exists an independently valued stable random measure Z(·) of index α ∈ [− 21 , 12 ]

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and a family {ft , t ∈ T } contained in Lα (− 12 , 21 ), the Lebesgue space, such that Z 1 2

Wt =

ft (λ)dZ(λ), t ∈ T,

(44)

− 12

and the processes {Xt , t ∈ T } and {Wt , t ∈ T } have the same finite dimensional distributions, the integral in (44) being a Wiener (type) integral. Here ft (λ) cannot be replaced by eitλ in general. Also, the argument is based on an analog of V -boundedness. Further analysis and extensions as well as sample path studies are detailed in Okazaki (1979), see also Marcus (1987), which should be of interest for related analysis and extensions of these works. 5.5 Vector-Valued Harmonizable Random Fields To complete our discussion, it will be of interest to consider briefly the vector and (operator) valued harmonizable fields also, for the following studies. If {Xg , g ∈ G} ⊂ L20 (P ) is a centered random field with r as its ¯ g2 ), then there are a right, left, and a covariance, r(g1 , g2 ) = E(Xg1 X two-sided stationarity concept and one needs to consider the dilation problem in each class. Thus X is left (or right) stationary if r(gg1 , gg2 ) = r˜(g2−1 g1 ), [r(g1 g, g2 g) = r˜(g1 g2−1 )],

(45)

and it is two-sided stationary if it is both left and right stationary. Then for dilation problems, each class has to be studied separately. Now X is termed hemi-homogeneous, as defined by Ylinen (1996), if its covariance r(·, ·) can be expressed as: r(g1 , g2 ) = ρ1 (g2−1 g1 ) + ρ2 (g1 g2−1 ),

g1 , g2 ∈ G,

(46)

for a pair of positive definite covariances ρ1 , ρ2 on G. Then Ylinen’s result gives the following using the early terminology. Theorem 5.5.1 Let G be a separable unimodular locally compact group, and X : G → L20 (P ), a continuous random field in L20 (P ). Then X is weakly harmonizable iff it has a hemi-homogeneous dilation Y , Y : C → L20 (P˜ ) ⊃ L20 (P ) so that X(g) = (QX)(g), g ∈ G, where Q is the orthogonal projection of L20 (P˜ ) onto L20 (P ).

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This result, due to Ylinen (1987), is actually proved for all locally compact groups using the Fourier transform method of Eymard’s (1964) which employs results from C ∗ -algebras. The dilation problem has additional difficulties for vector random fields. The result is presented here, to indicate the general patterns in the noncommutative case. 5.6 Cram´er and Karhunen Extensions of Harmonizability Compared We now have two types of extensions of (weak) harmonizability due first to Karhunen (1947) and later to Cram´er (1951). In both extensions, the mappings are more general than the group characters t 7→ eit(·) , t ∈ R which are replaced by a class of smooth functions t 7→ g(t, ·). Here we show that these interesting extensions retain many useful properties. However, the key dilation property may be lost, and that the mapping t → g(t, ·), being a group character, is special. The extension itself utilizes a new technical result, on a (generalized) domination of measures, due to Lindenstrauss and Pelcyzi´nski (1968). The desired Cram´erKarhunen general result is then given as follows: Theorem 5.6.1 Let X : R → L20 (P ) be a process and {g(t, ·), t ∈ R} be a set of Borel functions with the following properties: If X is a Karhunen process relative to this g-family, and a σ-finite measure F on B(R) and T : L20 (P ) → L20 (P ) is a continuous linear map, then the process {Y (t) = T X(t), t ∈ R} is a Cram´er process relative to the same g-family and a (suitable) covariance bimeasure. In the opposite direction, if {g(t, ·), t ∈ R} is of bounded Borel class, X : R → L20 (P ) is a Cram´er process relative to this g-family and a suitable covariance bimeasure β, then there is an extension space L20 (P˜ )(⊃ L20 (P )), determined by the given process, namely a Karhunen process Y : R → L20 (P˜ ), relative to some g-family and an appropriate finite Borel measure on R such that X(t) = QY (t), t ∈ R, with Q : L20 (P˜ ) → L20 (P ) as a projection (contained as a subspace) onto it. Remarks. 1. This is a generalization of the harmonizable class, but is weaker in that l2 (P˜ ) spaces change with each given (Cram´er) process! Indeed the construction of the spaces depends on the particular vector integrals and applications (see E. Thomas (1970), which is his thesis).

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This implies the differences and weaknesses between the Fourier analysis and its generalizations. The problem appears in the fact that L2 (Z) ˜ where Z˜ = QZ for orthogonal projection Q can have the and L2 (Z) spaces so that the former may not be including the latter, in contrast to the harmonizable case in which the vector measures Z(·) are more restricted. [An example of this effect was constructed by E. Thomas (unpublished).] 2. The preceding observation and result show that the set of Cram´er classes is quite large, and some cannot be easily dilated to Karhunen processes. But it is seen that the Cram´er class is closed under transformation T : ν → T ◦ ν, when appropriate integrability conditions are imposed on T . This will not be detailed here. We end this chapter with some complements and in the next (and final) chapter some extensions of the subject, motivated by applications, will complete the work of the volume. 5.7 Complements and Exercises 1. Let X = {Xt , t ∈ T = [a, b]} be a Karhunen process, E(Xt ) = 0, t ∈ T . If A : L20 (P ) → L20 (P ) is a bounded linear operator, let Yt = AXt , t ∈ T . Show that the {Yt , t ∈ T } is a Cram´er process if the spectral measure of Xt is finite. Verify that the converse holds when the representing measure of the Xt -process is finite and the corresponding class g(t, ·) of the Xt -process is bounded. 2. Verify that a weakly harmonizable process X = {Xt , t ∈ R} ⊂ L20 (P ) is automatically of Karhunen’s class relative to a suitable Borel measure µ and some Borel family {ft , t ∈ R} ⊂ L2 (R, µ). 3. Let {Xt , t ∈ T = [a, b]} ⊂ L20 (P ) be a process with continuous covarianceRr. Then the operator R : L20 (P ) → L20 (P ) defined by (Rg)(t) = T r(s, t)g(s)ds is positive definite and compact and 1 if Mρ = {E(Xt ), t ∈ T }, then we have Mρ ⊂ R 2 (L2 (dt)) i.e., 1 f ∈ Mp implies f = R 2 h for some h ∈ L2 (T, dt). [This result is essentially due to T. S. Pitcher (1963) and is useful in studying measures and the process Xt .] 4. Let G be an LCA group and X : G → L20 (P ) be a random field which is weakly harmonizable. Verify that there is a finite Borel ˆ of G and a fammeasure µ on the Borel σ-algebra of the dual G 2 ˆ ily {gs , s ∈ G} ⊂ L (G, µ) such that we have the representation of

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r as:

Z r(s, t) = ˆ G

gs (λ)gt (λ)µ(dλ), s, t ∈ G

ˆ so that t ∈ G, and g ∈ G. [Regarding this and some related results, one can find the details and extensions in Rao (1985).] 5. On an LCA group G, let X : G → L20 (P ) be a weakly harmonizable random field. Then show that there is a finite Borel measure µ on ˆ and a family {gs , s ∈ G} ⊂ L2 (G, ˆ µ) for a σ-finite measure µ B(G) ˆ on the Borel sets of G such that Z r(s, t) = cov (Xs , Xt ) = gs (λ)gt (λ)dµ(λ), s, t ∈ G. ˆ G

Thus each harmonizable field X : G → L20 (R) belongs to a Karhunen class. 6. A centered second order random field X : Rn → L20 (P ) on a probability space with covariance r(s, t) = r˜(s − t), (so it is stationary), is termed isotopic (concept due to Bochner) if r˜(·) is invariant under rotations and reflexions, in addition to translation, and he has characterized it as:  n  Z J (λ|s − t|) ν n r(s, t) = r˜(s − t) = 2 Γ dG(λ), 2 Rn (λ(|s − t|)ν ) where ν = n−2 , G(·) is a unique bounded Borel measure on R+ = 2 [0, ∞) and |s − t|2 = (s − t) · (s − t), n ≥ 2. Here J(·) is the Bessel function of the first kind of order ν. Using the MT-integration we can now present an extension (of isotropy) to harmonizable fields and outline its consequences giving a real opening here. [We need to consider the weakly harmonizable isotropic class as it appears in applications and restricting it to a weakly stationary class excludes many interesting practical problems as pointed out in the monograph by Yadrenko (1983)]. We have to employ the move complicated notation, to present a key extension of Cram´er’s class (C) for illustration. Recall (at least here) that a bimeasure F : S × S → C (S a semi-ring of a set T ) with S0 = {A ∈ S : kF k(A, A) < ∞}, S(A) = {A ∩ B : B ∈ S}, with kF k(·, ·) denoting the Fr´echet variation introduced earlier. The covariance r : T × T → C is then of weak class (C) if it is representable as:

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Z Z

gs (λ)g t (λ0 )F (dλ, dλ0 ), (s, t) ∈ T × T,

r(s, t) = Λ

253

Λ

the family {g(·), s ∈ T } verifies the conditions of the above strict MT-integral. We then have the integral representation of X(·) as: Z X(t) = gt (λ)dZ(λ), t ∈ T (Dunford-Schwartz type) A

and A, B ∈ S = σ(T ).

E(Z(A)Z(B)) = F (A, B),

The converse of the above statement is also true. [More details are in Rao (1991), and extensions and application of these are useful.] 7. This problem indicates how an integral representation of a second order general Cram´er type random field looks. Thus let (A, S) be a measurable space, F : S × S → C a positive definite bimeasure, and let X : T → L20 (P ) be a random field with covariance r admitting a representation Z Z gs (λ)g t (λ0 )F (dλ, dλ0 ), s, t ∈ T, r(s, t) = A

A

relative to a class gs (·) so that the above is an MT-integral. If X : T → L20 (P ) with covariance r given by the above integral, then verify that there is a stochastic measure Z : S → L20 (P ) such that, as a Dunford-Schwartz integral, we have Z X(t) = gt (t, λ)Z(dλ), t ∈ T, A

E(Z(A)Z(B)) = F (A, B),

A, B ∈ S.

The converse statement is also true. As a consequence, we have an explicit representative of X X(t, r) = α

∞ h(m,n) X X n=0

l=1

l gm (0)

Z R+

Jm+ν (λr) l Zm (dλ) (λr)ν

l with Zm (·) satisfying the orthogonal relations. [See Rao (1991) for more.]

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8. We now present an integral (series) representation of a weakly harmonizable isotropic random field X(·) : Rn → L20 (P ), useful in some applications. Thus X(·) is representable as a series given by: X(t ) = X(r, v ) = αn ∼

∼

∞ h(m,v) X X m=0

l=1

l Sm (v)

Z Rn

Jm+ν (λr) l Zm (dλ), (λr)ν

l l0 with E(Zm (B1 )Z¯m 0 (B2 )) = δmm0 δll0 F (B1 , B2 ), F (·, ·) being of finite (bimeasures of) Fr´echet variation. In the converse direction, the above representation gives X(·) to be an isotropic harmonizable random field. Moreover in the above representation X(·) is weakly stationary iff Z(·) is orthogonally valued. [More details on this representation and several applications can be found in this author’s paper (Rao (1991)) along with some additional results on sampling of such random fields with some related extensions.] 9. If H is a Hilbert space let T(H) denote the collection of trace class operators on H, so that with Z τ ([f, g]) = hf, gidP, kf k22,τ = τ ([f, f ]), Ω

we see that (L20 (P ), H), becomes a Hilbert space with inner product τ ([·, ·]) and hf, gi(ω) ∈ H. Thus for H, a Hilbert space, T(H) ⊂ L(H), is the set of trace class operators, the mapping [·, ·] : X0 × X0 → T(H) has the properties: (i) [x, x] ≥ 0, [x, x] = 0 iff x = 0, (ii) [x + y, z] = [x, z] + [y, z], (iii) [Ax, y] = A[x, y] for A ∈ L(H), (iv) [x, y]∗ = [y, x] with ‘*’ for the adjoint operation on L(H). The mapping [·, ·] is the Gramian on X0 . If kxk2τ = τ ([x, x]) = trace([x, x]) let X be the completion of X0 for the norm k · kτ and X is termed the normal Hilbert L(H) module if [x, y] = xy ∗ and A · x = Ax, x, y ∈ HS(K, H), and A ∈ L(H). Now let X : G → H be a mapping on an LCA group G, into a normal L(H)-module X. Then X is weakly harmonizable if we have Z X(t) = ht, λiZ(dλ), t ∈ G G

ˆ → the integral being in the Dunford-Schwartz sense, and Z(·) : B(G) ˆ ⇒ H being a vector measure of finite semi-variation, i.e., A ∈ B(G) ˆ F ⊂ A ⊂ O with there is a compact F and open O of G,

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kZk(O − F ) < ε, for the semi-variation k · k of Z. With this elaboration, we have “A random field” X : G → X, which is a normal L(H)-module, is weakly harmonizable iff it is V -bounded and continuous in the norm topology of X. [Some technical machinery is needed to complete the proof and the reader will find the work of Kakihara (1986) useful here.] 10. We present a simple extension with its characterization, of class (C) random fields of interest in several applications. Thus let (Λ, B) be a measurable space and F : S × S → C be a positive definite bimeasure. Suppose X : T → L20 (P ) is a random function (centered) with covariance r of weak Cram´er class hence representable as (for r): Z Z r(s, t) = gs (λ)gt (λ0 )F (dλ, dλ0 ), s, t ∈ T, Λ

Λ

where the bimeasure F has finite Fr´echet variation on S, to mean ( n X kF k(A, B) = sup ai bj F (Ai , Bj ) : |ai | ≤ 1, |bi | ≤ 1, {Ai }, i,j=1 ) {Bj } are disj. collections from S(A), S(B), n ≥ 1 . If X : T → L20 (P ) whose covariance r is as above then it has a 2 representation R relative to a stochastic measure Z : S → L0 (P ): (i) X(t) = Λ g(λ)Z(dλ), t ∈ T (Dunford-Schwartz integral). (ii) E(Z(A)Z(B)) = F (A, B), A, B ∈ S. Conversely, if X( ) is given as above relative to a Z(·) and F , then X is a random field of weak class (C). 11. This problem gives a series representation of an isotropic weakly harmonizable random field and indicates the progress of work. Thus let X : Rn → L20 (P ) be a centered weakly harmonizable isotropic random field. Then it is representable as: X(t ) = αn ∼

∞ h(m,n) X X n=0

l=1

l Sm (v)

Z R+

Jn+ν (λr) l Zm (dλ), (λr)ν

l l0 where the Zm (·) are orthogonal and satisfy E(Zm (B1 )Z¯m 0 (B2 )) = 0 0 δmm δll F (B1 , B2 ). In the opposite direction, X(·) given by the

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above series is a weakly harmonizable isotropic random field. Furl (·) is just orthogonally ther, the above given X(·) is stationary iff Zm valued so that the measure F (·, ·) concentrates on the diagonal of Rn × Rn . [For further details and extensions of this problem, the reader may see the author’s paper in the Journal of Combinatorics, Information and System Sciences 16 (1991), 207–220.] 5.8 Bibliographical Notes The work here begins by outlining the necessary bimeasure integration, weaker than the standard one based on Lebesgue’s ideas, the weakening is due to M. Morse and W. Transue who developed it around the 1950s, and these are utilized here. Then our main thrust is in obtaining extensions of the harmonizability on LCA groups. A reasonably detailed account of the analysis on locally compact groups following some classical methods and the work due to Mautmer (1955) are utilized and the corresponding extensions are detailed in our long Section 2. Actually, the paper by Mautmer is easily adapted for our cases, but this opens the way to utilize and extend the theory for the noncommutative cases leading the way to consider Harish-Chandra, A. Borel, A. Weil and other great harmonic analysts’ works. Also, K. Ylinen’s method is briefly indicated to show the possibility of this analysis for more general random fields. Here Ylinen’s work indicated how there is another direction and further treatment. A different procedure is to replace the LC groups as indexed with hyper groups. This is somewhat new to the subject but Heyer has indicated the possibility and had discussed directions in a recent survey, Heyer considers random fields on such objects as hyper groups which is of special interest for workers in other fields as well. This leads to vector harmonizability and some new problems appear, as indicated in Section 5.5. We only sketched the work of Ylinen and the need to employ C ∗ -algebra technique, and this may be considered by the readers later on. An interesting aspect here is an extension of (weak) harmonizability by Karhunen (1942) followed by H. Cram´er (1951). These are of special interest from both applicational point of view as well as theory. This usually goes beyond harmonic analysis, and the corresponding theory is not yet well developed, compared to the one using only harmonic (or Fourier) methods. There are some additional ideas of interest

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257

indicated in the complements section, but we end the abstract treatment here. The problem of isotropy is important and we only indicated its analysis briefly. Much work is available in Swift’s publications and his monograph on the subject to be completed soon. It would have the details to study more of this part of analysis, and we end this general survey here. On the hypergroup extension, we mention two theses that have some detailed applications of hypergroup structure in B. Englehardt (2002) and some another thesis from a student under Heyer at the University of Tubingen since the book appeared from Bloom and Heyer (1995).

b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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6.1 Introduction This chapter concludes our treatment of harmonizable random fields whose extensions include the Karhunen and Cram´er classes as well as some results on Lp -valued processes (or fields) when 1 < p < ∞, with some applications. To understand the problem better, we restate the (equivalent) Na˘imark-Sz.-Nagy theorem connecting the harmonizable and stationary processes, the former being a projection of a suitably located (super) Hilbert space. The type or form that we use is as follows: Theorem 6.1.1 Let {Yt , t ∈ T } ⊂ L20 (Ω, Σ, P ) or L20 (P ) be a weakly (or strongly) harmonizable process (T ⊂ R). Then there is an enlarged ˜ Σ, ˜ P˜ )(⊃ (Ω, Σ, P )) so L2 (P˜ ) includes L2 (P ) probability space (Ω, 0 0 with an orthogonal projection Q : L20 (P˜ ) → L20 (P ), so that there is a stationary process {Xt , t ∈ T } ⊂ L20 (P˜ ) with Yt = QXt , t ∈ T , the Xt -process being termed a dilation of the Yt -process. On the other hand, each continuous linear mapping A : L20 (P ) → L20 (P ) defines Yt = AXt , t ∈ T to be a weakly harmonizable process in L20 (P ). This result is established without difficulty from the previous methodology and the proof is left to the reader. The following specializations (and modifications) will lead to several applications. To consider this in some detail and depth we introduce the key concept of isotropy for harmonizable fields and include their integral representations, with related discussion, along with some additional developments of our subject. Further, a few consequences (or adjuncts) will be indicated for some possible extensions and new applications, along with some multidimensional indications of these classes. 259

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6.2 Harmonizability, Isotropy and Their Analyses We recall that a centered second order random field X = {Xt , t ∈ T ⊂ ¯ t ) is weakly stationary Rn } ⊂ L20 (P ), with covariance r(s, t) = E(Xs X if r(s, t) = r˜(s − t) and Z r(s, t) = r˜(s − t) = ei(s−t)λ F (dλ) (1) Tˆ

relative to a bounded increasing positive F (·) and is weakly harmonizable if r(·, ·) is just representable as: Z Z ∗ 0 eisλ−itλ F (dλ, dλ0 ) (2) r(s, t) = Tˆ

Tˆ

where F (·, ·) is a (positive definite) function of bounded Fr´echet variation. Hence X n ˆ ˆ kF k(T × T ) = sup ai a ¯j F (λi , λj ) : λi ∈ Tˆ, ai ∈ C, i,j=1

 |ai | < 1, 1 ≤ i ≤ n, n ≥ 1 < ∞,

(3)

the ‘star’ on the integral indicates that it is a Morse-Transue integral. A detailed exposition of this (weaker) integral is in Chang and Rao (1986), and it will be needed and used here. [Strongly harmonizable obtains if var(F ) < ∞ and then we use Lebesgue integral in (2).] A centered random field {Xt , t ∈ T } ⊂ L20 (P ) is isotropic if the covariance r verifies r(gs, gt) = r(s, t), s, t ∈ T ⊂ Rn , and g ∈ SO(n), the set of all orthogonal matrices on Rn . The above classes of (centered) random fields have the (proper) inclusion relationships: stationary isotropic X ⊂ strongly harmonizable isotropic X ⊂ weakly harmonizable isotropic X ⊂ isotropic X. Characterizations of the first (i.e. stationary isotropic) was given originally by S. Bochner, and the last one by M. I. Yadrenko, but the characterizations of the others were not done. All these cases are included here to understand the problem better. We first present a basic characterization of each of the weak and strong harmonizable isotropic covariances: Theorem 6.2.1 Let X be a centered weakly harmonizable random field on Rn , n > 1, with covariance r : Rn × Rn → C. Then r is isotropic iff, using the strict Morse-Transue integral, we have:

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 ν  Z Z ∗ J (|λs − λ0 t|) ν ν r(s, t) = 2 Γ Φ(dλ, dλ0 ), 2 R+ R+ |λs − λ0 t|ν

261

(4)

where Φ is a positive definite function of bounded Fr´echet
variation, Jν , and the inis a Bessel function (of the first kind) of index ν = n−2 2 tegral is in the strict Morse-Transue sense. The corresponding strongly harmonizable characterization obtains if Φ in (4) has finite Vitali variation and then the integral is in the standard Lebesgue sense. Proof. We present the proof in a series of steps for convenience. The direct (or the “if”) part follows easily since r(gs, gt) = r(s, t) for g ∈ SO(n). This is because |g(λs − λ0 t)| = |λs − λ0 t| since ‘g’ gives an isometry in (4) and only the converse is nontrivial which is detailed in the following four steps. I. Let X : Rn → L2 (P ) be a harmonizable random field so that by the ˜ Σ, ˜ P˜ ), with dilation theorem there is an enlarged probability space (Ω, 2 ˜ 2 2 ˜ L0 (P ) ⊃ L0 (P ), a contractive (onto) projection Q : L0 (P ) → L20 (P ), and a stationary random field Y : Rn → L20 (P˜ ) satisfying X = QY and that Xt = QYt , t ∈ Rn . Let ρ be the covariance of Y so that ρ(s, t) = ρ˜(s − t). We assert that ρ and ρ˜ can also be taken isotropic when X is isotropic. ρ) = K = H ⊕ H0 is representable Now the random field Y in L20 (˜ as Y = X + X1 , where X1 ∈ H0 is stationary, and the space H0 is obtained from L2 (µ), µ being a finite measure (due to Grothendieck). But now H0 may be realized as a subspace of some L2 (˜ µ) on a mea˜ Σ, ˜ µ sure space (Ω, ˜) (e.g. as a Gaussian probability space with a standard (Kolmogorov-Bochner) method whose construction gives even µ ˜(gA) = µ ˜(A)). This shows that the above type generated Hilbert space can be replaced by such an isometric space based on a Gaussian measure in this construction after the initial step. This modification makes the components of X1 = (X1t , t ∈ Rn ) to have invariant distributions under g so that X1t and X1gt to be identically (Gaussian) distributed under g so X1gt and X1t to be identically distributed. Consequently, the random field X1 is stationary, isotropic and orthogonal to X. Thus our P˜ is determined in the final construction by P and µ ˜. [The calculation here is the same in detail as in Rao (1982), the L’Enseignment Math., 28, 295–351, paper and will not be reproduced here.] The invariance of the bimeasure F of X is verified simply as follows.

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Z

Z

∗

0

ei(s·(gλ)−t·(gλ ) dF (λ, λ0 )

r(gs, gt) = n

n

ZR ZR∗

˜ ˜0 ˜ gλ ˜0) ei(sλ−tλ ) dF (g λ, n n R R Z Z ∗ 0 ei(s·λ−t·λ ) dF (λ, λ0 ). = r(s, t) =

=

Rn

(5)

Rn

Since r(s, t) = r(gs, gt), g ∈ SO(n), s, t ∈ Rn by hypothesis, it follows from the properties of the strict MT-integrals that F (λ, λ0 ) = F (gλ, gλ0 ) implying the invariance. Thus the isometries of X and X1 reflect in the same property of Y , and thus E(Ys Yt ) = ρ(s, t) = ρ˜(s − t) = ρ(gs, gt) = ρ˜(g(s − t)). This is the needed refinement of the dilation theorem, so X = QY , which is used here. II. Next, the representation of stationary isotropic covariance. ρ(·) of Y constructed above is representable as (cf., e.g., Yaglom (1987), p. 353): Z Jν (λτ ) ρ(t) = Cn dΦ(λ), (6) 2 R+ (λτ ) where t = (τ, u), |t| = τ , with u as the spherical polar of t, Jν being the Bessel function of order ν =
(n − 2)/2, with Φ as a bounded increasing n ν function and Cn = 2 Γ 2 . Now using the addition formula for Bessel functions, (6) above can be expressed as: ρ(s, t) =

Cn2

∞ h(m,n) X X

l l Sm (u)Sm (v)×

m=0

Z l=1 Jm+ν (r1 λ)Jm+ν (r2 λ) dΦ(λ), (r1 r2 )ν λ2ν R+

(7)

where s = (r1 , u), t = (r2 , v), h(m, n) = (2m + 2ν) (m+2ν−1)! , m ≥ 1, ν!m! l (u) being the spherical harmonics which are and = 1 if m = 0, the Sm orthogonal on the unit sphere Sm relative to the surface measure. For each m, there exist h(m, n) of them in number. [This series representation originates from the classical Mercer theorem for positive definite kernels.] With (7) an integral representation of Yt and then Xt is obtained. For this, it is convenient to express (7) as a triangular covariance so that we can invoke Karhunen’s theorem. III. Since only the second order properties of the process are considered, by the classical Kolmogorov theorem, we can find a centered

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263

stationary Gaussian random field with covariance function ρ. Here Y can be taken to be Gaussian with such a ρ for the ensuing computation. ˜ = {(m, l) ∈ N × N : 1 ≤ l ≤ h(m, n), m ≥ 0}, and P be its Let N power set. Next, let F˜ : P × P × B(R+ ) → R+ be defined as F˜ (A1 , A2 ; B1 ) = ζ(A1 ∩ A2 )Φ(B1 ) (8) R where Φ : A 7→ A dΦ(λ) is the positive bounded measure given by the Φ of (7), ζ(·) as the counting measure on P, so that F˜ extends to a ˜ × Rn × R+ → R by: σ-finite measure. If g˜ is defined on N l (u) · g˜((m, l); t, λ) = Sm

Jm+ν (τ λ) , (λτ )ν

(9)

l (·), Jν where t = (τ, u) is the spherical polar representation, and ν, Sm ˜ are defined earlier. Then g˜ is square integrable for F and Z Z r(s, t) = Cn g˜((m, l); s, λ)g¯˜((m, l); t, λ)dF˜ , (10) Λ

Λ

˜ × R+ . But (10) implies that r(·, ·) is, on Rn × Rn , a where Λ = N triangular covariance relative to F˜ , (cf., e.g., Rao (1982) and Chang-Rao (1986), p. 53). From these facts we can conclude that there is a Gaussian measure Z : P ⊗ B(R+ ) → L2 (P ) that represents Y as follows: Z Yt = C0 g˜((m, l); t, λ)dZ((m, l); λ) ZΛ Z = C0 g˜((m, l); t, λ)dZ((m, l); λ), (11) ˜ N

R+

¯ 2 , B2 )) = F˜ (A1 , A2 ; B1 ∩ B2 ) = ζ(A1 ∩ A2 ) × with E(Z(A1 , B1 )Z(A Φ(B1 ∩B2 ). If now we set A1 = {(m, l)}, a singleton and similarly A2 = l {(m0 , l0 )}, then with Z(A1 , B1 ) = Zm (B) and likewise for A2 , B2 , the 0 l l ¯ last relation gives E(Zm (B1 )Zm0 (B2 )) = δmm δll0 Φ(B1 ∩ B2 ). Now takl ing B = (0, λ), interval, the associated processes {Zm (λ), 0 ≤ l ≤ h(m, n), m ≥ 0} are orthogonal with orthogonal increments. Hence in the case that the Y -process is Gaussian, as it can be and so assumed, each l of the Zm (·)-process becomes independent of the others, and are identically (Gaussian) distributed with independent increments. Thus (11) becomes:

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Yt = Y (τ, u) = Cn

∞ h(m,n) X X m=1

l Sm (u)

l=1

Z R+

Jm+ν (τ λ) l Zm (dλ), (τ λ)ν

(12)

which is the integral representation of the isotropic stationary random field Y , the series (12) converging in L2 (P˜ )-mean. IV. Next let Xt = QYt with Q : L2 (P˜ ) → L2 (P ) as the orthogonal projection in the (isotropic) dilation result of Step I above. Now with the mean convergence of the series in (12), it is possible to interchange Q and the sum in the integral to obtain: X(τ, u) = (QY )(τ, u) = Cn

∞ h(m,n) X X m=0

l (u) Sm

l=1

Z · R+

Jm+ν (τ λ) l (QZm )(dλ), ν (τ λ)

(13)

the interchange of Q and the sum as well as the integral is permissible l l )(λ) are (λ) = (QZm here based on a theorem due to Hille. Since the Z˜m also Gaussian and Q linear, so that these are independently and identically distributed, we get l0 l (B)) = δll0 δmm0 F (A, B), E(Z˜m (A)Z¯˜m

(14)

where F is the bimeasure determined by the common distribution of l Z˜m , which now does not have orthogonal increments any longer. The key point is that F is independent of m, which is necessary for our work here. Also, note that F is just a bimeasure of finite Fr´echet variation. Since our interest is only on the second order properties of the processes, the Gaussian assumption is convenient. Thus (13) is the integral representation of the isotropic harmonizable field Xt , t = (τ, u). Let s = (τ1 , u), t = (τ2 , v) be the spherical polar coordinates, so that (13) and (14) give the covariance r of X as: ¯t) = r(s, t) = E(Xs X

Cn2

Z R+

∞ h(m,n) X X

l l Sm (u)Sm (v)×

Z m=0 l=1 Jm+ν (τ1 λ)Jm+ν (τ2 λ0 )F (dλ, dλ0 ),

(15)

R+

the integrals being in the strict MT-sense. l l V. The spherical harmonics Sm (u) · Sm (v) can be summed, and they satisfy the relation,

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6.2 Harmonizability, Isotropy and Their Analyses h(m,n)

X l=1

l l Sm (u)Sm (v)

ν h(m, n)Cm (cos(u, v)) = 0 ωn Cm (l)

265

(16)

ν (·), m ≥ where ωn is the surface area of the unit sphere Sn ⊂ R∗ and Cm 0, are the Gagenbeuer or ultraspherical polynomials of order ν for each m ≥ 0. Putting (16) into (15) and using the addition formula for Bessel functions one gets the conclusion of the theorem, since the converse is established by the above computation. (See also Swift (1994), Lemma 2.1.) 

We next present an equivalent form of isotropy which is useful for applications. Theorem 6.2.2 Let X = {Xt , t ∈ Rn } be a centered (or mean zero) weakly harmonizable random field. Then the following four statements are equivalent: (i) X is isotropic, so the covariance satisfies r(s, t) = r(gs, gt), g ∈ SO(n), (ii) the covariance r of X admits the representation (14), (iii) the covariance r of X is given by (15) relative to a positive definite bimeasure F of finite Fr´echet variation, (iv) X is representable as an L2 (P )-convergent series (12) where the stochastic integral is in the Dunford-Schwartz sense. Sketch of Proof. (i)⇔(ii) is shown by the above theorem, and (iii)⇒(iv) follows on using Karhunen’s theorem suitably. That (13)⇒(15) is established above. The rest is obvious and completes the proof.  Remark. A weaker form of the representations of X(τ, u) can be established without invoking the dilation theorem. This will be indicated below which may be of interest in the same applications. [A form of this result was established by A. Grothendieck, and so is also referred to as Grothendieck’s theorem.] The following slightly weaker version is also useful in applications. So we include it with a sketch here for reference as well as for some applications. Proposition 6.2.3 Let X : Rn → L20 (P ) be a weakly harmonizable isotropic random field. Then it is representable as t = (τ, u):

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ν

X(τ, u) = Γ (ν)ωn 2 ν

∞ X

h(m,n)

X

m

i

m=0

l Sm (u)

Z

Z

l=1

R+

Sn

Jm+ν (τ λ) × (τ λ)ν

l Sm (v)Z(dλ, dv)

(17)

where ωn is the surface area of the unit sphere Sn of Rn . Proof. Since X is weakly harmonizable, it is representable as: Z X(t) = X(τ, u) = eit·s Z(ds) n ZR Z = eiτ λ cos(u,v) Z(dλ, dv) R+

(18)

Sn

with s = (λ, v) and t = (τ, u) as polar coordinates. Using the orthogoν nal ultraspherical polynomials Cm (·) on Sm expand eitx in a series (cf. Vilenkin (1968), p. 557), we get for |x| ≤ 1, e

itx

= Γ (ν)

∞ X

im (m + ν)

m=0

Jm+ν (τ ) ν C (x). (τ /2)ν m

(19)

Putting this into (18) with x = cos(u, v) and the relation h(m,n) ν Cm (cos(u, v))

=

X

l l Sm (u)Sm (v)

l=1

ν ωm Cm (1) h(m, n)

we get X(τ, u) ν

= Γ (ν)ωn 2 ν

∞ h(m,n) X X m=0

l=1

l Sm (u)

Z R+

Z Sn

Jm+ν (τ λ) l Sm (v)Z(dλ, dv). (τ λ)ν

Since the series in (19) converges in L2 (Sn , µn ) the simplification interchange of integral here is valid. This gives (17).  We include an adjunct to the above, with a Hilbert space parameter. Proposition 6.2.4 On a separable Hilbert space H, let X be an isotropic weakly harmonizable random field X : H → L20 (P ). Then its contin¯ y ) admits the representation for uous covariance r : (x, y) 7→ E(Xx X x, y ∈ H.

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Z

Z

∗

r(x, y) = R+

267

exp{−(λx − λ0 y, λx − λ0 y)}F (dλ, dλ0 ),

(20)

R+

with F : R+ × R+ → C as a positive definite bimeasure of finite Fr´echet variation, (·, ·) being the scalar product in H and the above integral is in the strict MT-sense. Proof. If the random field is stationary, so r(s, t) = r˜(s − t) and by isometry r˜ is invariant under the rotation group, so that r˜(x) = r˜(kxk) which depends just on the length of x. Under these conditions Schoenberg’s result (cf. Schoenberg (1935), Thm. 2) gives the representation: Z ¯ y ) = r˜(kx − yk) = E(Xx X exp[−λ(x − y, x − y)]dΦ(λ), (21) R+

for a bounded increasing left continuous function Φ : R+ → R+ . We now assert that r˜ is also a triangular covariance, so that the Karhunen integral representation argument of Theorem 6.2.1 can be used. Now employing an idea from Yadrenko (1983), one can consider an isometric image of the (separable) space H by l2 , the sequence Hilbert space. The desired isometry τ : H → l2 is obtained by the Fourier series expansion of each element relative to an orthonormal basis of H, and then r ◦ τ and r have the same property and thus they can be identified in our analysis. Then with x ∈ H, let t = tx = τ (x) ∈ l2 , and consider exp{−λ(s − t, s − t)} = exp{−λ[(s, s) − (t, t) − 2(s, t)]} = exp{−λksk2 } · exp{−λktk2 }× m X X (2λ)m (s1 , t1 )k1 · · · (sm , tm )km k ! · · · k ! 1 m m=0 j∈l m

where lm = {j = [(i1 , k1 ), · · · (im , km )] : im ≥ 1, km ≥ 0, k1 + · · · + km = m}. Next consider ψj (·) defined by (2λ)m/2 2 ski11 . . . skimm ψj (s, λ, m) = e−kxk p (k1 ! · · · km !) where s = (si1 . . . sim ) is the vector of reals associated with j, so that (21) becomes

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r(s, t) = r˜(s − t) =

∞ XZ X

ψj (s, λ, m)ψj (t, λ, m)dΦ(λ). (22)

R+

m=0 j∈lm

This is a triangular covariance relative to ψ if we set ψ(x, λ) = {ψj (x, λ, m) : j ∈ lm , m ≥ 0} so that

Z

ψ(s, λ)ψ ∗ (t, λ)dΦ(λ),

r(s, t) = R+

ψ ∗ (·, λ) being the transpose of the vector ψ(·, λ) which is similar to (10). Hence there exists a stochastic measure such that the following representation obtains: Z ∞ XZ X j ψj (t, λ, m)dZm (λ), (23) Xt = ψ(t, λ)dZ(λ) = R+

m=0 j∈lm

R+

j0 j j (A)Z¯m (λ) ∈ L20 (P ), and E(Zm where Zm 0 (B)) = δjj 0 δmm0 Φ(A ∩ B). The remaining analysis is similar to the earlier case. Thus to proceed to the harmonizable case, we use the dilation theorem which is also valid in this case. Thus there is a Hilbert space L20 (P˜ ) ⊃ L20 (P ) and a stationary isotropic random field Y : H → L20 (P˜ ), such that Xt = QYt , t ∈ H (∼ = l2 ), where Q is the orthogonal projection onto L20 (P ). Replacing Xt of (23) by Yt and since the series converges in L20 (P ), one obtains that Xt = QYt , so that ∞ XZ X j ψj (t, λ, m)(QZm Xt = )(dλ). (24) m=0 j∈lm

R+

j j = QZm satisfies the conditions: E(Z˜m (A)) = 0, and Here Z˜m 0 j j E(Z˜m (A)Z¯˜m (B)) = δjj 0 δmm0 F (A, B), where F (·, ·) is the same for j all processes Z˜m which are i.i.d. in the Gaussian case. Now one can conclude that for the harmonizable covariance r(s, t) = r(gs, gt) for all rotations g and then Z ∗ ∞ XZ X r(s, t) = ψj (sλ, m)ψj (tλ0 , m)F (dλ, dλ0 ). (25) m=0 j∈lm

R+

R+

Interchanging the sum and the strict MT-integral, which is valid (cf. Chang and Rao (1986)), one obtains (H ∼ = l2 )

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Z

Z

∗ +

ZR ZR∗ = R+

0

e−λ(s,t)−λ (s,t)+2λλ (s,t) F (dλ, dλ0 )

r(s, t) = +

0

269

0

0

e−(λs−λ t,λs−λ t) F (dλ, dλ0 )

R+

which establishes (20) and the proposition follows.  A consequence of the above representation is stated separately for a reference. Corollary 6.2.5 If X : l2 → L20 (P ) is an isotropic harmonizable random field, then it is representable as: Z ∞ XZ X j Xt = ψ(t, λ, m)Z˜m (dλ), (26) m=0 j∈lm

R+

R+

j where E(Z˜m ) verifies the same conditions as in (24) above and the series (26) converges in L20 (P )-mean.

Thus there is an analogous spectral representation of isotropic weakly harmonizable random fields using the (strict) MT-integration and the dilation analysis. Extension of Yaglom’s (1961) analysis may now be treated. 6.3 Some Moving Averages and Sampling of Harmonizable Classes Let us introduce precisely a moving average representation of a process that is (strongly or weakly) harmonizable and give some results on their structural representations to be used for sampling and related problems on this class. Definition 6.3.1 Let {Xt , t ∈ T } ⊂ L20 (P ) be a harmonizable process where T = Z or R. It is termed a (strongly or weakly) moving average represented, if it has the form: P ei(j−t) ξt , T = Z, Xt = R t∈Z (27) ei(λ−t) ξt dt, T = R. R ˆ The integral in the continuous case is in Bochner’s sense. Here C(·), as 2 usual, is the Fourier transform of some C(·) ∈ L (dλ), and cov (ξs , ξt ) is the covariance ρ(s, t), with ρ as a scalar covariance. In case the ξt process is stationary or (strongly/weakly) harmonizable, then the moving average is termed stationary or harmonizable accordingly.

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Since the moving average property is important in many applications, we point out a useful specialization for some key applications, termed a virile moving average, defined now. Definition 6.3.2 A weakly (or strongly) harmonizable moving average process {Xt , t ∈ T } introduced above in (27) is a virile (moving average) process if the (matrix valued) function C(·) falls off “to zero fast at infinity” in the sense that if   Z X iθλ ijλ  ˆ ˆ CN (λ) = C(θ)e dθ or = , C(j)e |θ|>N

then

Z lim

N →∞

ˆ D ˆ (λ,λ0 )∈D×

|j|>N

CN (λ)CN∗ (λ0 )µξ (dλ, dλ0 ) = 0.

(28)

This somewhat involved concept is useful in understanding the moving average processes. This property is detailed for n-dimensional strongly harmonizable processes as follows. Theorem 6.3.3 Let {Xt , t ∈ T } ⊂ L20 (P ) be an n-dimensional strongly harmonizable process. Then it has a virile moving average representation with full rank say m, so that P ˆ − t)ξt , t ∈ Z, (discrete) C(i (29) Xt = R t∈Z ˆ − t)ξt dt, t ∈ R, (continuous) C(λ R RR itλ−itλ0 where rξ (s, t) = µξ (dλ, dλ0 ), iff it is a strongly harmoˆ D ˆ e D× nizable process. Moreover, the Xt -process has spectral characteristic R ˆ C(·) so that Xt = Dˆ eitλ C(λ)Z ξ (dλ). This interesting representation is detailed in the paper by Mehlman (1991) and we refer the reader for the detailed proof and related results (e.g., one-sided moving averages and specializations for the stationary case and factorizable spectral measure where r(s, t) is expressible as: Z Z 0 r(s, t) = ei(λs−λ t) C(λ)C ∗ (λ0 )µ(dλ, dλ0 ) (30) ˆ D ˆ D×

with C(·) as an n × p-matrix valued function (for both weak and strong harmonizable processes). Also, a centered process is called splitting if

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271

its covariance r(s, t) = g(s)g ∗ (t) for suitable matrices g(·) which are detailed. A study of Mehlman’s (1992) paper is helpful here. In this connection it is useful to include a general nonstationary class of L2 (P )-valued processes that extend the main Cram´er class (hence also of Karhunen’s) for further study and applications, calling it a weak class (C) as it automatically includes the weakly harmonizable class: Definition 6.3.4 A process {Xt , t ∈ R} ⊂ L20 (P ) with covariance r is termed of weak class (C) [C for Cram´er], if for a covariance bimeasure F : B(R) × B(R) → C of class (C), there is a covariance bimeasure F : B(R) × B(R) → C of locally Pnbounded semi-variation to mean ¯ that (i) F (A, B) = F (B, A), (ii) i,j=1 ai a ¯i F (Ai , Aj ) ≥ 0, ai ∈ C, and (iii) kF k(A × A) < ∞, A, A , B ∈ B(R), i j bounded, where  Pn ¯j F (Ai , Bj ) : |ai | ≤ 1; Ai , Bj ∞ > kF k(A × A) = sup i,j=1 ai a disjoint , then r(·, ·) is representable as an MT-integral written as r(s, t)(= I(gs , g¯t )): Z Z r(s, t) = gs (λ)¯ gt (λ0 )F (dλ, dλ0 ), s, t ∈ R. (31) R

R

We now present a general characterization of the classical weak Cram´er class: Theorem 6.3.5 Let X : R → L20 (P ) be a weakly class (C) process relative to a bimeasure F and a family {gs , s ∈ R} of MT-integrable class for F , as in the above definition. Then there exists a stochastic measure Z : B0 → L20 (P˜ ), where B0 is the δ-ring of bounded Borel sets ˜ Σ, ˜ P˜ ) is some enlargement of (Ω, Σ, P ) so that L20 (P˜ ) ⊃ of R and (Ω, 2 L0 (P ), such that (i) (ii)

E(Z(A)Z(B)) = (Z(A), Z(B)) = F (A, B), A, B ∈ B0 Z X(t) = g(t, λ)Z(dλ), t ∈ R (32) R

where the integral is in the Dunford-Schwartz sense for B0 . Conversely, if {X(t), t ∈ R} is defined by (32) relative to Z : B0 → L20 (P ) with {gt , t ∈ R} a D-S integrable class, and where ¯ F : (A, B) → E(Z(A)Z(B)), with the gs -class MT-integrable, and Hx = sp{X(t), ¯ t ∈ R} with Hz = sp{Z(A), ¯ A ∈ B0 } ⊂ L20 (P ), then Hx = Hz when and only when, the functions {gt , t ∈ R} satisfy the following equation:

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Z Z

0

Z Z

0

f (λ)¯ gt (λ )F (dλ, dλ ) = 0 ⇐⇒ R

page 272

R

R

f (λ)f¯(λ0 )F (dλ, dλ0 ) = 0

R

(33)

both being MT-integrals. Proof. We keep the basic format of Cram´er’s, but now the integrals are in the Morse-Transue sense, and the details are spelled out for appreciating this general case. First let the (centered) process be of class (C) so that, using the MTintegrals to represent r(·, ·), we have Z Z ¯ r(s, t) = E(Xs Xt ) = gs (λ)¯ gt (λ0 )F (dλ, dλ0 ). (34) R

R

L2F -space

for the bimeasure F , defined by  Z Z 2 LF = f : f (λ)f¯(λ0 )F (dλ, dλ0 ) = (f, f )F R R  with 0 ≤ (f, f )F < ∞ and f is MT-square integrable for F .

Consider the

Then L2F is a semi-inner product space. Define T : L2F → HF by (T gs ) = X(s) and extend it linearly so that (T gs , T gt ) = (gs , gt ), s, t, ∈ R, and that T is an isometric mapping of Λ2F = sp {gt , t ∈ R} ⊂ L2F and is onto HF (the space as in the theorem). Now the MT-integration implies that each bounded Borel set A is integrable and this gives for T : g → Xg (T gs , T gt )Hx = (gs , gt )F , s, t ∈ R,

(35)

so that T is an isometric mapping of Λ2F = sp ¯ {gt , t ∈ R} onto Hx , the space defined in the theorem. This implies if T χA = ZA then for bounded Borel sets A, B, we have E(ZA Z¯B ) = (T χA , T χB ) = (χA , χB ) = F (A, B), and for disjoint A, B (Borel), we get E(|ZA∪B − ZA − ZB |2 ) = (χA∪B − χA − χB , χA∪B − χA − χB ) = 0 since F is additive in these components. So Z(·) : B0 → L20 (P ) is again additive. S Its σ-additivity follows from continuity, so Ai ∈ B0 , disjoint, A0 = ∞ i=1 Ai ∈ B0 , implies

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E((ZA −

n X

2

ZAi ) ) = E(|Z∪i≥1 Ai −

i=1

n X

273

ZAi |2 )

i=1

= E(|Z∪i>n Ai |2 ) = F (∪i>n Ai , ∪i>n Ai ) → 0 (36) as n → ∞ since F is suitably continuous. So Z(·) is also σ-additive. Thus Z(·) is σ-additive on B0 , and is a stochastic measure of finite semi-variation on compact sets, and KZ ⊂ KX . From this a standard approximation argument that, since Pshows n 2 {gt , t ∈ R} ⊂PLF is dense, we get with g˜n = i=1 ai g(ti ) → χA in hL2 ⇒ E[| ni=1 ai X(ti )i− Z(A)|2 ] → 0 as n → ∞ and hence P 2 E | ni=1 ai X(ti ) − Z(A)| → 0 as n → ∞. It follows that {ZA , A ∈ Bn } is a dense subspace in KX and each ¯ 2 , the completion of element in KX corresponds uniquely to one in L F 2 LF . If now Y (t) is defined that Z gt (λ)Z(dλ), (∈ KZ = KX ), (37) Y (t) = R

then it can be verified as Z Z (Y (s), Y (t)) = R

gs (λ)gt (λ0 )F (dλ, dλ0 )

R

which holds if gt is B0 -step function and then generally by the MTanalysis. It also can be shown from this that Z(A) = T (χA ) = l · i · m · (T g˜n ) = l · i · m ·

n X

ai T (gti )

i=1

=l·i·m·

n X i=1

Then the L2 (P )-limits imply

˜ m (say). ai X(ti ) = l · i · m · X

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¯˜ ) ¯ E(X(s)Z(A)) = lim E(X(t)X m

(38)

n

= lim n

= lim n

n X i=1 n X

Z Z ai R

i=1

Z Z = R

ai E(X(s)X(ti )) gs (λ)¯ gt (λ0 )F (dλ, dλ0 )

(39) (40)

R

gt (λ)χA (λ0 )F (dλ, dλ0 ).

(41)

R

P ˜ n = P n bj X A ∈ By isometry, if ζ˜n = nj=1 bj Z(Aj ), one has when h j j=1 L2F , Z Z ¯ gs (λ)g¯˜n (λ0 )F (dλ, dλ0 ). E(X(s)ζ˜n ) = R

R

Since gs (·) is MT-integrable, this gives on letting n → ∞, Z Z ¯ gs (λ)gt (λ0 )F (dλ, dλ0 ). E(X(s)Y (t)) = R

R

From this, it follows that E(|X(t) − Y (t)|2 ) = 0. Hence X(t) = Y (t), a.e., t ∈ R. So if ΛF is dense in L2F , this shows that (31) is true in the case that Y (t) is given by (37). The general case follows from this result easily as shown below. In the general case let L2T = Λ˜2F , and {ht , t ∈ F˜ } be a basis of ˜˜ = R+ ˜ (a disjoint sum) as a new index, {hi , i ∈ ˙ R, the space Λ˜lF . If R ˜ forms a basis of Λ˜F . By the preceding case, on extending T from R} ˜ Σ, ˜ P˜ ) is an enlargement of (Ω, Σ, P ) by an L2F (P ) to L2F (P˜ ) where (Ω, adjunction procedure with T χA = ZA ∈ L2 (P˜ ), one finds Z ˜ Y (s) = g˜s (λ)Z(dλ), (∈ L20 (P˜ )). (42) R

Hence all g˜ are Borel and MT-integrable. So as before, T˜(s) = X(s), s ∈ R, and (31) holds again and so the inclusion HZ ⊃ HX holds. For the converse, P let X(t) be given by (31) and F (A, B) = (Z(A), Z(B)). If gn = ni=1 ai χAi (Ai , A, B ∈ B0 , Ai disjoint), then

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( F (A, B) = sup

n X

275

) ai a ¯j F (Ai , Aj ) : Ai ∈ B(A), |ai | ≤ 1

i,j=1

( n )

X

≤ sup ai Z(Ai ) : |ai | ≤ 1, Ai ∈ B(A) ≤ kZk2 (A) < ∞.

i=1

Thus if Xgn =

R

gn (λ)dZ(λ), then by standard arguments Z Z ¯ m (λ0 )F (dλ, dλ0 ). ¯ gn (λ)h E(Xgn Xhn ) = R

R

(43)

R

2 Next using the obvious properties of the MT-integrals, if {gn }∞ 1 ⊂ LF with gn → gs boundedly a.e., one also gets similarly with g˜n → gt ,

I(gn , g˜n ) → I(gs , gt ), I(|gs |, |gt |) < ∞. Hence, Z Z R

R

˜ (dλ, dλ) ¯ = lim g(λ)¯ g (λ)F

Z Z

n

R

gn (λ)g¯˜n (λ0 )F (dλ, dλ0 )

R

= lim(Xgn , Xg˜n ) n  Z Z gn (λ)Z(dλ), g˜n (λ)Z(dλ) = lim n R R  Z Z gs (λ)Z(dλ), gt (λ)Z(dλ) , = R

R

since the dominated convergence holds for the D-S integrals = (Xs , Xt ) = r(s, t). Thus {Xt , t ∈ R} is of weak class (C). The last assertion is easy since {gs , s ∈ R} is a basis for L2F iff I(f, gt ) = 0, t ∈ R ⇒ I(f, f ) = 0 so that KZ = KX . This implies all the assertions, and the result holds on stated.  If kF k(R × R) < ∞, then each bounded measurable function is MTintegrable for F . So taking gt (λ) = eitλ , the above theorem gives the following result due to Rozanov (1959), and another proof is in Niemi (1975). We state it for comparison.

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Theorem 6.3.6 Let X : R → L20 (P ) be kX(t)k2 ≤ M < ∞, t ∈ R, and weakly continuous. Then it is weakly harmonizable relative to some covariance bimeasure F of finite semi variation iff there is a measure Z : B → L20 (P ), (B is Borel field) with F (A, B) = (Z(A), Z(B)), and Z Xt = eitλ Z(dλ), t ∈ R, (44) R

the integral in D-S sense and kZk(R) < ∞. Further, X is strongly harmonizable iff the covariance bimeasure F of Z is of bounded variation, in R2 . In either case the harmonizable process X on R2 is uniformly continuous, and is represented as the integral (20). If Z(·) is orthogonally valued so that Z F (A, B) = (Z(A), Z(B)) = F˜ (A ∩ B), r(s, t) = g(λ)¯ g (λ)dF˜ (λ), R

(45)

this reduces to a Karhunen process. The following result gives a kind of summary statement on the class of weakly harmonizable processes on an LCA group and will be useful as a reference of this account. Theorem 6.3.7 Let G be an LCA group and X : G → X be a weakly continuous mapping where X = L20 (P ), separable. Then we have the following equivalent assertions: (i) X is weakly harmonizable. (ii) X is V -bounded (in Bochner’s sense). ˆ into (iii) X is the Fourier transform of a regular vector measure on G X. ˆ 1 (G), ˆ the process Yp = pX : G → L2 (P ) is weakly (iv) For each p ∈ L 0 harmonizable and bounded. Further the following statement implies all the four assertions above: (v) If K = sp ¯ {X(g) : g ∈ G} ⊂ X, then there is a weakly contractive positive type of operators {T (g), g ∈ G} ⊂ B(X), such that T (0) = id, and X(g) = T (g)X(0), g ∈ G. In view of the desirable properties of harmonizable processes, let us consider some general classes of processes covered by it that are not necessarily stationary. One such class was introduced and extensively

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277

studied by the well known French probabilists J. Kamp´e de F`erier and F. N. Frankiel (also called class (KF)) which includes the weak stationary as a subclass. It is related to a class called asymptotically stationary by E. Parzen (1953), but is somewhat more general. We discuss this class as an adjunct of weakly harmonizable family that is being studied above. [We actually consider it easily and recall.] Definition 6.3.8 If X : R → L20 (P ) is a (centered) random process with ¯ covariance r : (s, t) → E(X(s)X(t)), then it is of class (KF ) (or of Kamp´e de F´eriet and Frankiel (1995) class), if the limit Z 1 T −|h| r(h) = lim K(s, s + h)ds = lim rT (h), h ∈ R, (46) T →∞ T 0 T →∞ exists. [Note rT (·) as in (46): hence r(·) is positive definite.] By the classical Bochner-Riesz theorem there is a unique increasing bounded function F , termed an associated spectral function of the process X. It can be verified that every strongly harmonizable process is in class (KF). However, all weakly harmonizable processes are not necessarily in class (KF). The following example due to H. Niemi verifies the negative condition, in order to present a simple (sufficient) condition for a weakly harmonizable X to be in class (KF) and obtain a positive result. Example 1. Let {an , n ∈ Z} be a sequence such that a0 = 1, an = a−n , and for k < ∞ let ak be given by ∞ X ak = (χCn + 2χDn )(k),

(47)

n=0

where we take Cn = [22n , 22n+1 ) and Dn = [22n+1 , 22n+2 ). So Cn ∩ Dn = ∅, and for each k ≥ 0, 1 ≤ ak ≤ 2, let {εk , −∞ < k < ∞} ⊂ l2 be a complete orthonormal sequence, and let A : εn 7→ an δn , |an | ≤ g. Let an = a−n , a0 = 1. We define for k > 0, ak as above, so 1 ≤ ak ≤ 2, k ≥ 0. The covariance r(k, l) = 0 for k 6= l. Then  n−k−1 1 X 0 if h 6= 0 rn (h) = r(k, k + h) = 1 Pn−1 2 n k=0 k=0 ak , if h = 0. n

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Thus limn→∞ rn (h) = 0 for h 6= 0 and (5 − 3·21n−1 , if n = 22m − 1 3 rn (0) = 4 − 3·212m , if n = 22m+1 − 1. 3 Hence limm→∞ r2m −1 (0) = 53 , and limm→∞ r22m+1 −1 (0) = 34 . So limn→∞ rn (0) does not exist, and {Xn , n ∈ Z} is not in class (KF ). Thus n → ∞ the {Xn = Aεn , n ∈ Z} is a weakly harmonizable process, and is not in class (KF) so not all harmonizable processes are in class (KF). In view of this example, one may ask under what (further) conditions does a (weakly) harmonizable process belong to class (KF) that generalizes the (weakly) stationary classes. We include a family that is more general than (weakly) stationary but a somewhat restricted class of (weakly) harmonizable processes which will be shown now to belong to class (KF). Thus this class is also large. Definition 6.3.9 A stochastic measure Z : B(R) → L20 (P ) is said to be of second order if Z˜ = Z ⊗ Z : B ⊗ B → L20 (P ) is also σ-additive in L20 (P ) and Z˜ is of the same type, so that kZ⊗Zk(R×R) ≤ [kZk(R)]2 < ∞. The corresponding harmonizable process X is termed second order weakly harmonizable. This (strengthened) property of Z(·) for the stationary case, implies that Z(·) has finite Vitali variation so that Z ⊗ Z and even higher products are included. The positive result, for applications, is as follows. Theorem 6.3.10 Let X : R → L20 (P ) be a second order weakly harmonizable process in the sense of Def. 6.3.9. Then X ∈ class (KF) so that it has a well-defined associated spectral function. Proof. The hypothesis implies that we have Z X(t) = eitλ Z(dλ), t ∈ R,

(48)

R

for a stochastic measure Z of the type given in Definition 6.3.9 F (A, B) = (Z(A), Z(B)),

(49)

so that F : B × B → C is a bounded bimeasure. Also with K(s, t) = ¯ E(X(s)X(t)), let

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rT (h) =

T −h 1 · T T −h

279

T −h

Z

k(s, s + h)ds.

(50)

0

To see that limT →∞ rT (h) exists, consider   Z T Z Z Z 1 t 1 itλ −i(s+h)λ0 0 Z(dλ ) . k(s, s + h)ds = E ds e Z(dλ) e T 0 T 0 R R (51) We assert that the right side has a limit as T → ∞. Now let X = Y = L20 (P ), and Z = L1 (P ). So Z, Z˜ : B → Y are stochastic measures, using the bilinear mapping (x, y) → x · y of X × Y → Z, the product measure Z ⊗ Z˜ : B × B → Z is defined and satisfies the D-S integrals: Z Z Z Z ˜ ˜ f (s, t)(Z ⊗ Z)(ds, dt) = Z(ds) f (s, t)Z(dt) R×R ZR ZR ˜ f (s, t)Z(ds) (52) Z(dt) = R

R

for f ∈ C0 (R × R), (cf. Dinculeanu ((1974), p. 388), and also see (Ducho˜n and Kluv´anek (1967)) for some relaxations that we use to get ˜ kZ ⊗ Zk(R × R) ≤ kZ(R)k2 < ∞, so the measure function Z ⊗ Z˜ is stochastic. 0 This gives for fs,t (λ, λ0 ) = eisλ · e−i(s+h)λ , f ∈ C1 (R × R), Z Z Z 0 isλ ˜ f (s, t)(Z ⊗ Z)(dλ, dt) = e Z(dλ) e−i(s+h)λ Z(dλ0 ) R×R R ZR 0 0 eis(λ−λ )−ihλ Z ⊗ Z(dλ, dλ0 ) = R×R

(53) and the right side is in L1 (P ). With the same type of calculations and argument applied to Z ⊗ Z : B(R × R) → X and µ : B([0, T ]) → R+ with µ as Lebesgue measure, and (x, a) → ax on Z × R → X, we can define µ ⊗ (Z ⊗ Z) : B(0, T ) × B(R × R) → X. Now writing λ for (λ, λ0 ), we get Z Z T Z µ(dt) f (t, λ)Z ⊗ Z(dλ) = 0

R×R

R×R

Z Z ⊗ Z(dλ)

T

f (t, λ)µ(dt). 0

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6 Some Extensions of Harmonizable Random Fields

This gives easily   Z T Z 1 is(λ−λ0 )−ihλ0 0 ds e Z ⊗ Z(dλ, dλ ) E T 0 R×R Z  iT (λ−λ0 ) −1 −ihλ0 e 0 =E e Xλ±λ0 ) + δλλ0 Z ⊗ Z(dλdλ ) . (54) iT (λ − λ0 R×R But the quantity inside the integral on Rthe right is bounded for all T and we can let T → ∞ in (54) to get = [λ=λ0 ] eihλ F (dλ, dλ0 ) where F is the bimeasure of Z. Hence limT →∞ rT (h) = r(h) exists. R ihλ This implies that R limT →∞ rT (h) =0 r(h) exists and = R e dG(λ) where G : A 7→ π−1 (A) δλλ0 F (dλ, dλ ), A ∈ B, is positive definite and thus is an associated spectral measure of X in class (KF). This finishes the proof of the theorem.  As an application of the preceding result, we can give a slight extension of a useful application obtained originally by Yu. A. Rozanov (1959) that has some engineering as well as Fourier analytic interesting consequences. Theorem 6.3.11 Let X : R → L20 (P ) be a weakly harmonizable process with Z : B → L20 (P ) as its representing stochastic measure. Then for (λ1 , λ2 ) ⊂ R with Z(·) as a stochastic measure where Z(A) is Z((−∞, λ)), we have Z T −itλ2 e − eitλ1 l · i · m. X(t)dt 0