Geometry and Martingales in Banach Spacesprovides a compact exposition of the results explaining the interrelations exis

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- Wojbor A Woyczynski

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Geometry and Martingales in Banach Spaces

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Geometry and Martingales in Banach Spaces

Wojbor A. Woyczyński

Case Western Reserve University

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20180816 International Standard Book Number-13: 978-1-138-61637-0 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Woyczyânski, W. A. (Wojbor Andrzej), 1943- author. Title: Geometry and martingales in Banach spaces / Wojbor A. Woyczynski (Case Western Reserve University). Description: Boca Raton, Florida : CRC Press, [2018] | Includes bibliographical references and index. Identifiers: LCCN 2018026561| ISBN 9781138616370 (hardback : alk. paper) | ISBN 9780429462153 (ebook) Subjects: LCSH: Martingales (Mathematics) | Geometric analysis. | Banach spaces. Classification: LCC QA274.5 .W69 2018 | DDC 519.2/36--dc23 LC record available at https://lccn.loc.gov/2018026561

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents Introduction

ix

1 Preliminaries: Probability and geometry in Banach spaces 1 1.1 Random vectors in Banach spaces . . . . . . . . . . 1 1.2 Random series in Banach spaces . . . . . . . . . . . 3 1.3 Basic geometry of Banach spaces . . . . . . . . . . 8 1.4 Spaces with invariant under spreading norms which are finitely representable in a given space . . . . . . 13 1.5 Absolutely summing operators and factorization results . . . . . . . . . . . . . . . 16 2 Dentability, Radon-Nikodym Theorem, and Martingale Convergence Theorem 2.1 Dentability . . . . . . . . . . . . . . . . . . . . . . 2.2 Dentability versus Radon-Nikodym property, and martingale convergence . . . . . . . . . . . . . . . . 2.3 Dentability and submartingales in Banach lattices and lattice bounded operators . . . . . . . . . . . . 3 Uniform Convexity and Uniform Smoothness 3.1 Basic concepts . . . . . . . . . . . . . . . . . 3.2 Martingales in uniformly smooth and uniformly convex spaces . . . . . . . . . . 3.3 General concept of super-property . . . . . . . 3.4 Martingales in super-reflexive Banach spaces .

23 23 29 38

47 . . . 47 . . . . . . . . .

51 61 63

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CONTENTS

4 Spaces that do not contain c0 67 4.1 Boundedness and convergence of random series . . . 67 4.2 Pre-Gaussian random vectors . . . . . . . . . . . . 72 5 Cotypes of Banach spaces 5.1 Infracotypes of Banach spaces . . . . . . . . 5.2 Spaces of Rademacher cotype . . . . . . . . 5.3 Local structure of spaces of cotype q . . . . 5.4 Operators in spaces of cotype q . . . . . . . 5.5 Random series and law of large numbers . . 5.6 Central limit theorem, law of the iterated rithm, and infinitely divisible distributions .

75 75 79 85 92 99

. . . . . . . . . . . . . . . . . . . . loga. . . . 110

6 Spaces of Rademacher and stable types 6.1 Infratypes of Banach spaces . . . . . . . . . . . . . 6.2 Banach spaces of Rademacher-type p . . . . . . . . 6.3 Local structures of spaces of Rademacher-type p . . 6.4 Operators on Banach spaces of Rademacher-type p 6.5 Banach spaces of stable-type p and their local structures . . . . . . . . . . . . . . . 6.6 Operators on spaces of stable-type p . . . . . . . . 6.7 Extented basic inequalities and series of random vectors in spaces of type p . . . . . 6.8 Strong laws of large numbers and asymptotic behavior of random sums in spaces of Rademachertype p . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Weak and strong laws of large numbers in spaces of stable-type p . . . . . . . . . . . . . . . . . . . . . . 6.10 Random integrals, convergence of infinitely divisible measures and the central limit theorem . . . . .

115 115 119 131 140

7 Spaces of type 2 7.1 Additional properties of spaces of type 2 . . . . . 7.2 Gaussian random vectors . . . . . . . . . . . . . . 7.3 Kolmogorov’s inequality and three-series theorem 7.4 Central limit theorem . . . . . . . . . . . . . . . . 7.5 Law of iterated logarithm . . . . . . . . . . . . .

197 197 202 206 208 218

. . . . .

144 153 159

169 178 182

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vii

CONTENTS

7.6 Spaces of type 2 and cotype 2 . . . . . . . . . . . . 223 8 Beck convexity 8.1 General definitions and properties and their relationship to types of Banach spaces . . . . . . . . . 8.2 Local structure of B-convex spaces and preservation of B-convexity under standard operations . . . . . . 8.3 Banach lattices and reflexivity of B-convex spaces . . . . . . . . . . . . . . . . . . 8.4 Classical weak and strong laws of large numbers in B-convex spaces . . . . . . . . . . . . . . . . . . . . 8.5 Laws of large numbers for weighted sums and not necessarily independent summands . . . . . . . . . 8.6 Ergodic properties of B-convex spaces . . . . . . . . 8.7 Trees in B-convex spaces . . . . . . . . . . . . . . . 9 Marcinkiewicz-Zygmund Theorem spaces 9.1 Preliminaries . . . . . . . . . . . . . 9.2 Brunk-Prokhorov’s type strong law rates of convergence . . . . . . . . . . 9.3 Marcinkiewicz-Zygmund type strong lated rates of convergence . . . . . . 9.4 Brunk and Marcinkiewicz-Zygmund laws for martingales . . . . . . . . . .

227 227 236 242 249 258 263 271

in

Banach 273 . . . . . . . . 273 and related . . . . . . . . 276 law and re. . . . . . . . 279 type strong . . . . . . . . 288

Bibliography

297

Index

313

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Introduction In this volume we are providing a compact exposition of the results explaining the interrelations existing between the metric geometry of Banach spaces and probability theory of random vectors with values in those Banach spaces. In particular martingales and random series of independent random vectors are studied. Chapter 1 introduces the basic geometric and probabilistic concepts in Banach spaces. Chapter 2 concentrates on the geometric concept of dentability and provides an exposition of the results originally due to M.A. Rieffel, H.B. Maynard, S.D. Chatterjii, and others. The concept of dentability turns out to be very natural in the context of martingales even though it was originally introduced in a study of the Radon-Nikodym theorem in Banach spaces. The chapter ends with an exposition of the theory of sub-martingales with values in Banach lattices and the related issues of the lattice bounded operators. Chapter 3 deals with the two classical concepts of metric geometry in Banach spaces, namely, the uniform smoothness and the uniform convexity. Here, the works of G. Pisier and P. Assuad (also, see the important 560-page long recent monograph, Martingales in Banach Spaces, by G. Pisier) showed that some of the results obtained earlier by the author for sums of independent random vectors in such Banach spaces carry over to the more general situation of martingales, and even provide a complete characterization of those geometric properties in the language of martingales. Finite tree property and super-reflexivity, the notions introduced by R.C. James, turn out to be the properties that are most intimately related to the martingale theory as shown by results of S. Kwapie´ n, and G. Pisier, which are discussed in this chapter. ix ✐

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Introduction

In Chapters 4 through 9 we concentrate on martingales with independent increments, that is sums of independent random vectors with values in Banach spaces, and the relationships between the geometry of Banach spaces and the asymptotic properties of such martingales. The contents of these chapters may be viewed as an effort to provide different generalizations of the following classical result due to Khinchine1 : Let (ri ) be a sequence of Rademacher functions,i.e., independent, identically distributed real-valued random variables taking on values ±1 with equal probabilities 1/2. Then, for any p, 0 < p < ∞, there exist constants cp , and Cp , such that, for an arbitrary integer n, and any real numbers α1 , . . . , αn , cp

n X I=1

αi2

1/2

n n p 1/p X X 1/2 ≤ Cp αi2 . αi r i ≤ E

i=1

I=1

Spaces that do not contain c0 are discussed in Chapter 4. Spaces of Rademacher cotype, and of Rademacher and stable types, are analyzed in Chapters 5, and 6. More details on spaces of type 2 are provided in Chapter 7, and Chapter 8 explains the concept of Beck convexity and its relationship to the laws of large numbers in Banach spaces. Finally, Chapter 9 provides the proof of the Macinkiewicz-Zygmund theorem in Banach spaces. All the results related to the interplay between the geometry and martingales are provided with full proofs. In contrast, to keep the whole book of manageable length, the purely geometric and purely probabilistic results are provided without proofs which can be found in the cited literature. The reader is assumed to be familiar with the basic facts of functional analysis and probability theory. To keep the size down several topics pertaining to the subject matter of this book have been left out. For instance, the theory of radonifying mappings, operators of type and cotype p, and various notions of orthogonality are not mentioned at all. These results 1

For the best constants in the Khinchine inequality for real-valued random variables see S. Szarek (1976).

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Introduction

xi

can be found in the bibliography listed at the end of the book. Results on extensions of our results to not necessarily locally convex linear metric spaces are also omitted. The formulas, theorems, corollaries, propositions, definitions, and lemmas are numbered using three digits. So, formula (2.3.1) is the first formula in Section 3 of Chapter 2. The author2 is indebted to all the friends who, in the past, were influential in developing the theory: Alexandra Bellow, Michael Marcus, Czeslaw Ryll-Nardzewski, Joel Zinn, Patrice Assuad, Joe Diestel, Tadeusz Figiel, Stanislaw Kwapie´ n and Gilles Pisier, and my first two graduate students, Jan Rosi´ nski and Jerzy Szulga. Also, the benevolent mentorship of the author in his ”salad days” by Kazimierz Urbanik, and Laurent Schwartz, is deeply appreciated.

2

See his website http://sites.google.com/a/case.edu/waw for complete information about his work.

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xii

Introduction NOTATION: := =⇒ |A| Ac a.s. BX Bǫ B(x, ǫ) B(X, Y )

— — — — — — — — —

C d(X, Y )

— —

(ei ) — E — E(.|Σ) — (γi ) H IA i.i.d.

— — — —

Jα (µ) = lp — Lp

—

L(X ) — Λp (Ω, F , P; X) N (Ω, F , P) P(A) Πp,q (X, Y )

— — — — —

defined by implies cardinality of the set A complement of the set A almost surely (with probability 1) Borel sigma-field of the space X ball of radius ǫ with center at 0 ball of radius ǫ with center at x space of bounded operators from X to Y set of complex numbers Banach-Mazur distance between spaces X and Y canonical basis expectation conditional expectation with respect to σ-algebra Σ i.i.d. Gaussian random variables Hilbert space indicator function of the set A independent and identically distributed inf{c : µ{kxk > c} ≤ α} space of sequences summable with p-th power space of functions integrable with p-th power probability distribution of a random variable (or vector) X Lorentz space of random vectors nonnegative integers probability space probability of set A space of (p, q)-absolutely summing operators

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xiii

Introduction πp (U)

—

R — (ri ) — SX — s — σ(X) — U ◦V x, y, . . . X, Y , ... X∗

— — — —

norm of the p-absolutely summing operator U set of real numbers Rademacher i.i.d. symmetric random variables with values ±1 unit sphere of a normed space X space of sequences with finitely many non-zero terms σ-algebra spanned by the random vector X superposition of operators u, and V elements of normed spaces normed, or Banach spaces dual space of X

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Chapter 1 Preliminaries: Probability and geometry in Banach spaces 1.1

Random vectors in Banach spaces

Let X be a real separable Banach space with the dual space, X ∗ , the unit ball BX , and the unit sphere SX . By definition, a random vector X with values in X is a strongly measurable map from the probability space (Ω, Σ, P) (always sufficiently rich) into X equipped with the Borel σ -algebra BX . The set of all random vectors in X will be denoted L0 (Ω, Σ, P; X) or, simply, L0 (X), and will be equipped with the topology of convergence in probability which is determined by the family of gauges, Jα (X, P) := inf c : P(kXk > c) ≤ α , α ∈ (0, 1). (1.1.1) Random vectors on product spaces (Ω1 × Ω2 , Σ1 × Σ2 , P1 × P2 ) satisfy the following Fubini inequality,1 Jγ (Jδ (X, P1 ), P2 ) ≤ Jα (Jβ (X, P2 ), P1),

(1.1.2)

whenever α + β ≤ γδ (see, also, (1.3.1(b)). 1

See L. Schwartz (1969/70).

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Geometry and Martingales in Banach Spaces

By Lp (Ω, Σ, P; X), or, simply, Lp (X), 0 < p ≤ ∞, we shall denote the space of random vectors X in X for which EkXkp := R kX(ω)kpP(dω) < ∞, if p < ∞, and ess supkXk < ∞, if p = ∞, Ω equipped with the corresponding topologies and quasi-norms. lp will denote the analogous spaces on the set of positive integers N, and s will denote the space of real sequences with finitely many non-zero terms. If A ⊂ L0 (Ω, Σ, P; X) then Jα (A) := sup{Jα (X, P) : X ∈ A}.

(1.1.3)

We shall also discuss the Lorentz spaces Λp (Ω, Σ, P; X) := {X ∈ L0 (X) : Λp (X, P)

:= sup cP(kXk > c)1/p < ∞}. c>0

For a random vector X, L(X) denotes the distribution law of X, that is a Borel measure on X such that L(X)(A) = P(X ∈ A), A ∈ BX . For a general finite Borel measure µ on X we shall also introduce gauges, Jα (µ) := inf c > 0, µ(kxk > c) ≤ α , 0 < α < 1.

If X ∈ L1 (X), then EX will stand for the expectation of X in the sense of the Bochner integral. If Y is another random vector then EY X := E(X|Y ) stands for the conditional expectation of X with respect to the σ-algebra, σ(Y ), spanned by Y . A sequence of random vectors (Xn ) in X is said to be exchangeable if, for each n ∈ N, each r ≤ n, and each injection {1, . . . , r} ∋ i 7→ ji ∈ {1, . . . , n} the r-tuples (X1 , . . . , Xr ), and (Xj1 , . . . , Xjr ) are equally distributed. In an analogous definition, a sequence (Xn ) of random vectors is said to be weakly exchangeable if, for each n ∈ N, each r ≤ n, and each order preserving injection {1, . . . , r} ∋ i 7→ ji ∈ {1, . . . , n}

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Preliminaries: Probability and geometry in Banach spaces

the r-tuples (X1 , . . . , Xr ), and (Xj1 , . . . , Xjr ) are equally distributed. Finally, a sequence (Xn ) of random vectors is said to be sign-invariant if, for any n ∈ N, and any choice of εi = ±1, the sequences (X1 , . . . , Xn ), and (ε1 X1 , . . . , εn Xn ) are equally distributed. Weakly exchangeable random vectors satisfy the following maximal inequality. For each p ≥ 1 there exist constants K1 , K2 > 0, such that, for any weakly exchangeable sequence X1 , . . . , Xn with X1 + · · · + Xn = 0, k

p 1/p

X

Xi K1 E sup

≤ K2 inf

1≤k≤n

(1.1.4)

i=1

k

p 1/p o n X

: |k − n/2| ≤ n1/2 ] Xi E i=1

k n

p 1/p

p 1/p X X

Xi ≤ sup E ρi Xi ≤ E 1≤k≤n

i=1

i=1

k

p 1/p

X

, Xi ≤ E sup 1≤k≤n

i=1

where ρi = (1 + ri )/2, i ∈ N, are independent of (Xi ), and (ri ) is the usual Rademacher sequence of independent symmetric random variables with values ±1.

1.2

Random series in Banach spaces

P The series n Xn of independent random vectors with values in a general Banach space X enjoy a number of properties that will be repeatedly used in the following chapters. Ito-Nisio Theorem.2 Let X be a Banach space, and let (Xn ) be a sequence of independent random vectors with values in X. Then the following three conditions are equivalent: 2

See K. Ito and S. Nisio (1968), but part of this result appeared in the earlier work of A. Tortrat.

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4

Geometry and Martingales in Banach Spaces

P (i) The series Pn Xn converges in probability; (ii) The series n Xn converges almost surely; (iii) The probability measures L(X1 +· · ·+Xn ) converge weakly, as n → ∞. Under the additional condition that the random vectors (Xn ) are symmetric, the additional three conditions are equivalent to the conditions (i), (ii), and (iii): (iv) The measures L(X1 +· · ·+Xn ), n ∈ N, are uniformly tight; (v) There exists a random vector X in X such that, for each x∗ ∈ X ∗ , the sequence x∗ (X1 + · · · + Xn ) → x∗ X, almost surely, as n → N; (vi) There exists a random vector X in X such that the characteristic functionals x∗ ∈ X ∗ ,

E exp[ix∗ (X1 + · · · + Xn )] → E exp[ix∗ X], as n → ∞.

As far as convergence in Lp is concerned we also have the following results:

Hoffman-Jorgensen Theorem.3 Let (Xi ) be a sequence P of independent random vectors in a Banach space X such that i Xi converges almost surely. Then: P (i) If E(supi kXi k)p < ∞, then the series i Xi converges in Lq (X) forPevery q ∈ [0, p); (ii) If i Xi ∈ Lp (X) for some p ∈ (0, ∞], then sup kX1 + · · · + Xn k ∈ Lp (X), n

and, if p < ∞, then X1 + · · · + Xn →

X

Xi ,

in Lp (X),

i

as n → ∞.

The next theorem provides a condition on a pair of real random multiplier sequences guaranteeing that the almost sure convergence of one series implies the almost sure convergence of the other series. 3

See J. Hoffmann-Jorgensen (1972/73).

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Preliminaries: Probability and geometry in Banach spaces

5

Comparison Theorem.4 Let (xn ) ⊂ X, and let (ξn ) be a sequence of uniformly nondegenerate real, symmetric, and indeP pendent random variables such that the series n ξn xn converges almost surely. Then, if (ηn ) is another sequence of real, symmetric, independent random variables such that, for some β0 , 0 < α ≤ 1, and all β ≥ β0 , P(|ξn | ≥ β) ≥ αP(|ηn | ≥ β), then the series

P

ηn xn converges almost surely as well.

As a corollary to the above comparison result we shall cite the following, 5 Contraction Let (λn ) ⊂ [−1, 1]. If (xn ) ⊂ X, P Principle. and P the series n rn xn converges almost surely, then the series n λn rn xn converges almost surely as well.

The following two results deal with the Lp behavior of the vector Rademacher series.

Kahane Theorem. Let X be a Banach space. Then, for each p, q > 0, there exists a constant K > 0 such that

q 1/q

p 1/p X X

. rn x n ≤ K E rn x n E n

n

6 Kwapie´ P n Theorem. Let X be a Banach space and (xn ) ⊂ X. If i ri xi converges almost surely then, for each α ∈ R,

X

2

E exp α ri xi < ∞. i

P If the random multipliers θi in the series i θi xi are α-stable, 0 < α ≤ 2, then we also have a number of inequalities and convergence results similar to those above that dealt with Rademacher 4

See N.C. Jain and M.B. Marcus (1975). For extensions of this theorem see M.B. Marcus and W.A. Woyczy´ nski (1979). 5 See J.P. Kahane (1968). 6 See S. Kwapie´ n (1976).

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6

Geometry and Martingales in Banach Spaces

multipliers. Let us recall that, in general, a random vector X in X (or its probability distribution L(X)) is called (symmetric) stable if, for any a, b > 0, and any independent copies X1 , X2 , of X, there exists a c > 0 such that L(aX1 + bX2 ) = L(cX). In the above formula one can always take c = (aα + bα )1/α , for some α ∈ (0, 2], and then X (and L(X)) are called α-stable, or stable with exponent α. Obviously, 2-stable random vectors are Gaussian. The next theorem gives a representation for the characteristic functional of an α-stable random vector. Tortrat Theorem.7 If X is a symmetric α-stable random vector in X, then there exists a finite Borel measure σ on the unit sphere SX such that h Z i ∗ E exp[ix X] = exp − |x∗ x|α σ(dx) . S

X

A random vector Y (or its probability distribution L(Y )) is said to have a Gaussian covariance if there exists a Gaussian random vector X in X such that E(x∗ Xy ∗ X) = E(x∗ Y y ∗ Y ),

x∗ , y ∗ ∈ X ∗ ,

or, equivalently, if E exp[ix∗ X] = exp[−E(x∗ Y )2 ]. For any random vector Y such that, for each x∗ ∈ X ∗ , the real random variable x∗ Y ∈ L2 (R), there exists an operator RY (or, RL(Y ) ) on X ∗ with values in X such that x∗ RY y ∗ = E(x∗ Y y ∗ Y ). The operator RY is called the covariance operator of Y . Any covariance operator admits the factorization, A

A∗

X ∗ 7→ H 7→ X, 7

See A. Tortrat (1976).

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Preliminaries: Probability and geometry in Banach spaces

7

where H is a Hilbert space. The operator A is sometimes denoted by R1/2 . Hoffmann-Jorgensen Theorem.8 Let X be a Banach space, and let (θn ) be a sequence of independent and identically distributed (i.i.d.) α-stable random variables, 0 < α ≤ 2. Then, for each p, q ∈ (0, α), if α < 2, and each p, q ∈ (0, ∞), if α = 2, there exists a constant K > 0 such that, for any finite sequence (xn ) ⊂ X,

p 1/p

q 1/q X X

θn xn . θn xn ≤ K E E n

n

The existence of quadratic exponential moments for Gaussian random series is guaranteed by the following result: Landau-Shepp-Fernique Theorem.9 Let X be a Banach space, (xn ) ⊂ X, and Plet (γn ) be a sequence of i.i.d. Gaussian random variables. If n γn xn converges almost surely then there P exists a constant c > 0 such that E exp[ck n γn xn k2 ] < ∞. Finally, for α-stable random series with 0 < α < 2 we have the following results:

Schwartz Theorem.10 Let (θn ) be a sequence of i.i.d. αstable symmetric random variables with 0 < α < 2. Then: (i) For each p ∈ (0, α) there exists a constant c > 0 such that, for each n ∈ N, and any (a1 , . . . , an ) ⊂ R, n n p 1/p X 1/α X α ai θi =c |ai | ; E i=1

i=1

(ii) For each p ∈ (0, α), and each q ∈ (α, ∞), there exist constants c, and C, such that, for each n ∈ N, and any (a1 , . . . , an ) ⊂ R, c

n X i=1

|ai |α

1/α

n n X p/q 1/p X 1/α |ai θi |q ≤C |ai |α ; ≤ E i=1

i=1

8

See J. Hoffman-Jorgensen (1972/73, 1974). See H. J. Landau and L.A. Shepp (1970), and X. Fernique (1970). 10 See L. Schwartz (1969/70). 9

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8

Geometry and Martingales in Banach Spaces

(iii) For each p ∈ (0, α), there exist constants c, and C, such that, for each n ∈ N, and any (a1 , . . . , an ) ⊂ R, n n o X −1 c inf a > 0 : Φα (a |ai |) ≤ 1 i=1

n X p/α 1/p ≤ E |ai θi |α i=1

n n o X ≤ C inf a > 0 : Φα (a−1 |ai |) ≤ 1 , i=1

where

tα (1 + log(1/t)), for t ∈ (0, 1], t, for t ∈ (1, ∞); P (iv) In particular, the series i |ai θi |α converges almost surely if, and only if, X |ai |α (1 + | log(1/|ai |)|) < ∞; Φα (t) =

i

(v) For each p ∈ (0, α), and each q ∈ (0, α), there exist constants c and C, such that for each n ∈ N, and any (a1 , . . . , an ) ⊂ R, c

X i

1.3

|ai |q

1/q

X p/q 1/p X 1/q ≤ E |ai θi |q ≤C |ai |q . i

i

Basic geometry of Banach spaces

We shall start this section with a discussion of the local properties of Banach spaces. For two normed spaces, X, Y , the BanachMazur distance is is defined as follows: d(X, Y ) := inf kIk · kI −1 k, where the infimum is taken over all ismorphisms I : X 7→ Y .

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“A˙Book” — 2018/8/22 — 11:55 — page 9 — #20

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Preliminaries: Probability and geometry in Banach spaces

9

A normed space X is said to be [crudely] finitely representable in Y if, [there exists λ > 1] for each λ > 1, and for each finitedimensional X 1 ⊂ X, there exists a finite dimensional Y 1 ⊂ Y such that d(X 1 , Y 1 ) ≤ λ. In other words, X is finitely representable in Y if, and only if, for each λ > 1, and for each finitedimensional X 1 ⊂ X there exists an isomorphism I : X 1 7→ Y such that λ−1 kxkX ≤ kIxkY ≤ λkxkX ,

x ∈ X 1.

If P is a property of a normed space then we say that X has the property super P if each Y which is finitely representable in X has the property P . Note that, for each Banach space X, the dual of the dual space, X ∗∗ , is finitely representable in X. This phenomenon is called the Local Reflexivity Principle. Dvoretzky Theorem.11 For any λ > 1, and the integer k ∈ N, there exists an integer N ∈ N such that in every normed space X of dimension greater than N there exist linearly independent x1 , . . . , xk ∈ X such that, for any a1 , . . . , ak ∈ R, λ

−1

k X i=1

|ai |

2

1/2

k k

X X 1/2

|ai |2 . ai xi ≤ λ ≤ i=1

i=1

In particular, this implies that l2 is finitely representable in any infinite dimensional normed space. Now, let (en ) be a sequence of unit vectors spanning X. The norm k.k on X is said to be invariant under spreading for (en ), if for each integer n ∈ N, each a1 , . . . , an ∈ R, and each k1 < k2 < · · · < kn ∈ N, n n

X

X

ai eki = ai ei .

i=1

i=1

The above concept permits us to give an example of the space in which the space lq is also finitely representable. 11

See A. Dvoretzky (1961). For shorter proofs, see T. Figiel (1974/75), and A. Szankowski (1974).

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“A˙Book” — 2018/8/22 — 11:55 — page 10 — #21

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Geometry and Martingales in Banach Spaces

Krivine Theorem.12 Let (en ) be an unconditional basis in a Banach space X equipped with the invariant under spreading norm k.k. Define, 2n

X

n o

n b := inf λ ≤ 1 : lim λ ej = +∞ , n→∞

and

j=1

q = (log2 b−1 )−1 . Then, lq is finitely representable in X. And operator T : X 7→ Y (of norm 1) is said to be [crudely] finitely factorable through a normed space Z if [there exists a λ] for each λ > 1, and each finite dimensional X 1 ⊂ X, there exists a finite dimensional Z 1 ⊂ Z, and an isomorphism I : X 1 7→ Z 1 , such that λ−1 kT xkY ≤ kI xkZ ≤ λkxkX . Clearly, X is finitely representable in Y , if, and only if, the identity mapping of X is finitely factorable through Y . In cases in which that is our main interest, as in the next theorem, we can state more explicitly: The canonical embedding lp 7→ lq , 1 ≤ p ≤ q ≤ ∞, is [crudely] finitely factorable through X if [there exists a λ ∈ (0, 1)] , for each λ ∈ (0, 1), and each n ∈ N, there exist x1 , . . . xn ∈ X such that, for every a1 , . . . , an ∈ R, (1 − λ)

n X i=1

|ai |

q

1/q

n n

X

X 1/p

≤ ai xi ≤ |ai |p . i=1

i=1

In the next theorem we prove that in some cases crude finite factorability implies finite factorability. James Theorem.13 If 1 ≤ p ≤ ∞, and the embedding lp 7→ l∞ (l1 7→ lp ) is crudely finitely factorable through X, then it is finitely factorable through X. 12

See J.L. Krivine ((1976). See R.C. James (1964), but the proof provided above is due to B. Maurey and G. Pisier (1976). 13

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11

Proof. It is easy to notice that lp 7→ l∞ is finitely factorable through X if, and only if, for each n ∈ N, n an := inf a : ∀x1 , . . . , xn ∈ X, n

X p X o

= 1. inf kxi k ≤ a sup bi xi : |bi | = 1

1≤i≤n

i=1

i

On the other hand, an ≤ 1, and the sequence (an ) is nonincreasing and submultiplicative, i.e., ank ≤ an ak ,

n, k ∈ N.

Indeed, let x1 , . . . , xnk ∈ X. For each i = 1, . . . , n, define y i =:=

ik X

bj xj ,

j=ik−k+1

so that ik n X

ky i k = sup

j=ik−k+1

Then

X o

bj xj : |bj |p = 1 . j

ik

X o n X

|b′i |p = 1 inf ky i k = an sup b′i y i : i

i=1

ik nk

X n X o

≤ an sup bj xj : |bj |p = 1 , j=1

j=1

and, on the other hand, by construction, ky i k >

1 ak

inf

(i−1)k 0. Then the submultiplicativity of (an ) implies that an = 1 for each n ∈ N, because if an0 < 1 then ank0 → 0, as k → ∞. Therefore, the embedding lp 7→ l∞ is finitely factorable through X. For the finite factorability of the embedding l1 7→ lp , the proof is similar, but the sequence (an ) should be defined as follows n

an = inf a : ∀x1 , . . . , xn ∈ X, n o

X p X

|bi | = 1 ≤ a sup kxi k . inf bi xi : i=1

QED

i

i

Closed Interval Lemma. Let X be a normed spacee. The set of q’s for which the embedding l1 7→ lq ( lq 7→ l∞ ) is finitely factorable through X is a closed interval. The proof of the above statement is immediate from the definitionPof factorability, the fact that if q > p, then P and (p−q)/pq p 1/p q 1/q n ( |ai | ) ≤ ( |ai | ) .

We conclude this section with the statement that for some spaces the finite factorability through any infinite dimensional space is always possible.

Dvoretzky-Rogers Lemma. Let k ∈ N. For any n ≥ k, and any normed space X of dimension greater than 4n2 , there exist x1 , . . . , xn ∈ X such that, for all a1 , . . . , an ∈ R, n n

X

X 1/2

ai xi ≤ a2i 2−1 sup |ai | ≤ . 1≤i≤n

i=1

i=1

In particular, the embedding l2 7→ l∞ is finitely factorable through any infinite dimensional Banach space.

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“A˙Book” — 2018/8/22 — 11:55 — page 13 — #24

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Preliminaries: Probability and geometry in Banach spaces

1.4

13

Spaces with invariant under spreading norms which are finitely representable in a given space

In this section we will outline a general procedure permitting us to produce spaces with invariant under spreading norms which are finitely representable in a given normed space X. The procedure is based on the following combinatorial lemma: Ramsey Lemma. Let K be the set of k-tuples n = (n1 , . . . , nk ) of different integers. If K = K ′ ∪ K ′′ is a partition of K, then there exists a sequence (in ) such that all n formed from members of (in ) belong either to K ′ , or to K ′′ . Brunel-Sucheston Theorem.14 (i) If (xn ) is a bounded sequence in a Banach space X then, for some subsequence (en ) ⊂ (xn ), there exists a function S ∋ ~a = (a1 , . . . , ak ) 7→ L(~a) ∈ R+ ,

such that, for each n1 < n2 < · · · < nk , (n1 , . . . , nk ) → ∞

X

ai eni → L(~a). i

(ii) If (en ) is not a Cauchy sequence then the mapping X X X ⊃ Φ(S) ∋ Φ(~a) := ai ei 7→ ai ei := L(~a) ∈ R+ , i

i

(1.4.1) is an invariant under spreading norm on Φ(S) ⊂ X (and we can assume that |ei | = 1.) (iii) In the latter case the completion Y of Φ(S) under the norm |.| is finitely representable in X.

Sketch of the Proof. It is is sufficient to show the existence of (en ) ⊂ (xn ), and L : S 7→ R+ such that, for every ~a ∈ S, and every ǫ > 0, there exists an n0 ∈ N such that, for every . . . n2 > n2 > n0 , X

ai ei − L(~a) ≤ ǫ. i

14

See A. Brunel and L. Sucheston (1974, 1975, 1976).

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“A˙Book” — 2018/8/22 — 11:55 — page 14 — #25 ✐

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Geometry and Martingales in Banach Spaces

Assume, without loss of generality, that kxn k ≤ 1, n ∈ N. Let (a1 , . . . , ar ) be a fixed r-tuple of rational numbers. Define r

X

ai xni . Ψ : Kr ∋ ~n = (n1 , . . . , nr ) 7→ Ψ(~n) = i=1

The set {Ψ(~n) : ~n ∈ Kr } is bounded, say, by a constant C > 0. Now, consider the sets, A1 = {~n : Ψ(~n) ≤ C/2},

and B1 = {~n : C/2 ≤ Ψ(~n) ≤ C}.

By Ramsey lemma, either A1 , or B1 , contains all ~n formed from (1) the terms of an infinite sequence, say (in ) ⊂ N. Assume that this is true for A1 (the argument for B1 is analogous), and consider A2 = {~n : Ψ(~n) ≤ C/4},

and B2 = {~n : C/4 ≤ Ψ(~n) ≤ C/2}. (2)

(1)

There exists, again by Ramsey lemma, (in ) ⊂ (in ) such that (2) all ~n formed from the terms of (in ) are either in A2 , or in B2 . An so on, and so on. Let us denote by L(a) the only point conT (n) tained in n An . Then, for the diagonal sequence (in ), we have |Ψ(~n) − L(~a)| < ǫ, for any choice of n1 , . . . , nr , sufficiently far in the sequence. Now, let a(1) , a(2) , . . . , be the sequence of all a ∈ S with rational coefficients. Utilizing again the diagonal argument one obtains a sequence of integers (in ) such that, for every (a(k) ),

X

(k) ai xni = L(a(k) ), lim i

where the limit is taken for n1 → ∞, n1 < n2 < · · · < nk , (in ) ⊃ (ni ). There remains the case of an arbitrary ~a = (a1 , . . . , ar , 0, 0, . . . ) ∈ S. Given an ǫ > 0, and a rational ~a′ = (a′1 , . . . , a′r , 0, 0, . . . ) with k~a − ~a′ k1 ≤ ǫ/2, where k.k1 denotes the l1 norm, we clearly have r r

X

X

′ a − ~a′ k1 ≤ ǫ/2. a x − a x

i ni i ni ≤ k~ i=1

i=1

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“A˙Book” — 2018/8/22 — 11:55 — page 15 — #26

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15

Therefore, if n1 is large enough, say n1 ≥ N,

sup Ψ(~n, ~a) − inf Ψ(~n,~a′ ) ≤ ǫ, n1 >N

n1 >N

so that (i) holds true for an arbitrary ~a. (ii) The fact that (1.4.1) is invariant under spreading is obvious in view of (i). Also, the homogeneity and subadditivity of (1.4.1) follow directly from the definition. So, to complete the proof of (ii) it is sufficient to show that if 0 6= ~a ∈ S, and L(~a) = 0, then the sequence (en ) is Cauchy. Indeed, let q be the first integer such that aq = 6 0. If L(~a) = 0, then, for each ǫ > 0, there exists an n0 ∈ N such that, for all · · · > n2 > n1 > p > n ≥ n0 , and

kaq eq + aq+1 en1 + aq+2 en2 + . . . k ≤ ǫ(2|aq |)−1 ,

kaq ep + aq+1 en1 + aq+2 en2 + . . . k ≤ ǫ(2|aq |)−1 .

Thus, n0 ≤ n ≤ p implies ken − ep k ≤ ǫ, and (ii) is proven. (iii) Let Y 1 be a finite dimensional subspace of Y with basis y 1 , . . . , y p . To prove (iii) we have to find, for each ǫ > 0, an invertible operator V : Y 1 7→ X, such that kV xk − kxk < ǫkxk, x ∈ Y 1.

Because Y is the completion of Φ(S) in the norm |.|, we can find m ∈ N and x1 , . . . , xp ∈ span[e1 , . . . , em ] =: Y 3 , such that the linear extension of the map Uy i = xi is an isomorphism of Y 1 onto Y 2 := span[x1 , . . . , xp ], and kUx′ k − kx′ k ≤ ǫkx′ k/4, x′ ∈ Y 2 . P P On the other hand, the shift, T ( ai ei ) = ai ei+1 , is a linear isometry acting from Y 3 into Φ(S) equipped with k.k. Moreover. m m

X X

n

ai ei , ai ei = lim T n→∞

i=1

i=1

and the convergence is uniform over the compact set of x ∈ Y 3 , with kxk = 1. Therefore, there exists qa q ∈ N such that, for each x ∈ Y 3 with kxk = 1, we have kT xk − 1 < ǫ/4. Finally, for q each x ∈ Y 3 , we have kT xk − 1 < ǫkxk/4, so that V can be taken to be equal to T q U. QED

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“A˙Book” — 2018/8/22 — 11:55 — page 16 — #27

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Geometry and Martingales in Banach Spaces

1.5

Absolutely summing operators and factorization results

Let X and Y be two normed spaces, and let 0 < q < p < ∞. A linear operator U : X 7→ Y is said to be (p, q)-absolutely summing (in short, U ∈ Πp,q (X, Y )) if, there exists a constant C > 0 such that, for each n ∈ N, and all x1 , . . . , xn ∈ X, n X i=1

kUxi k

p

1/p

≤ C sup

n nX i=1

|x∗ xi |q

1/q

o : x∗ ∈ B(X ∗ ) ,

where B(X ∗ ) is the unit ball of the dual space X ∗ . The smallest constant C for which the above inequality holds will be denoted by πp,q (U), and it is a complete norm on the linear space Πp,q (X, Y ). If p = q, then we shall simplify our notation by writing πp,p (U) = πp (U), and Πp,q = Πp . We shall say the U : X 7→ Y is 0-absolutely summing (U ∈ Π0 (X, Y )) if, for each β ∈ (0, 1) there exists an α ∈ (0, 1), and a constant C > 0, such that, for each probability measure µ on X with finite support, Jβ (U(µ)) ≤ sup{Jβ (x∗ µ) : x∗ ∈ B(X ∗ )}, where the gauge Jβ has been defined in Section 1.1. The notions of p-absolutely summing operators coincide for 0 < p < 1 and, moreover, for each 0 ≤ p < q < 1, there exists a constant C such that, for all U, πq (U) ≥ Cπp (U). On the other hand, πp (U) is a decreasing function of p, so that Πp (X, Y ) ⊃ Πq (X, Y ),

if

q < p.

If X, Y are Hilbert spaces then all classes Πp (X, Y ) coincide with the class of Hilbert-Schmidt operators. Another characterization of p-absolutely summing operators is provided by the following result: Pietsch Theorem.15 (i) Let X, Y be normed spaces. A continuous linear operator U : X 7→ Y is p-absolutely summing 15

See A. Pietsch (1967).

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“A˙Book” — 2018/8/22 — 11:55 — page 17 — #28 ✐

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17

if, and only if, there exists a Borel probability measure on B(X ∗ ) such that Z 1/p kUxk ≤ πp (U) |x∗ x|p µ(dx∗ ) . B(X ∗ )

(ii) In particular, a continuous linear operator U belongs to the class Πp (X, L∞ [0, 1]) if, and only if, it can be factored thrugh a space Lp as follows: V

I

W

U : X 7→ C(B(X ∗ )) 7→ Lp (B(X ∗ ), µ) 7→ L∞ [0, 1], where kV k = 1, kW k = πp (U), and I is the canonical embedding. (iii) If U ∈ Π2 (X, Y ), and X, and Y , are Banach spaces, then U can be factored through a Hilbert space H as follows: V

W

U : X 7→ H 7→ Y , where V , and W , are bounded operators. Moreover, if X is a Hilbert space, then V : X 7→ H is a Hilbert-Schmidt operator. In the remainder of this section we shall discuss several results, including more factorization theorems and inequalities related to Lorentz norms which would be useful in the following chapters of the book.

Let n ∈ N, and 1 < p < ∞. Define a Lorentz-like gauge λp on R as follows: 1/p λp ((ai )) = sup cp |{i : |ai | > c}| , n

c>0

where |A| denotes the cardinality of the set A. One can show that λp is equivalent to the quantity sup{i1/p |ai |∗ : 1 ≤ i ≤ n}, where |ai |∗ denote the decreasing rearrangement of |ai |’s. The relationship between the Lorentz norm Λp , and the lq norm, is as follows: X 1/p λp ((ai )) ≤ |ai |p , i

and, if p < q, then X i

|ai |q

1/q

≤

q 1/q λp ((ai )). q−p

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“A˙Book” — 2018/8/22 — 11:55 — page 18 — #29

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Denote by Snp the set of those (ai ) ∈ Rn for which, for some k > 0, |{i : ai = 6 0}| = k, and such that ai = 6 0 =⇒ |ai | = k −1/p . Denote by σp the usual gauge of the convex envelope of Snp . Since, P for each (ai ) ∈ Snp , ni=1 |ai |p = 1, we have, for p ≥ 1, and (ai ) ∈ Rn , X 1/p p |ai | ≤ σp ((ai )). i σp∗

Now, if we denote by the polar gauge of the set {x∗ : |x∗ x| ≤ 1, x ∈ Snp }, then one can check that λp ((ai )) ≤ σp∗∗ ((ai )) ≤ p∗ λp ((ai )),

(ai ) ∈ Rn ,

(1.5.1)

where 1/p + 1/p∗ = 1. By duality, for 1 < p < q < ∞, p∗ 1/p∗ X 1/p p σp ((ai )) ≤ ∗ |a | . i p − q∗ i The next Lemma provides a relationship between the norms of (n) operators from l(n) ∞ 7→ X, and X 7→ l1 , and the gauges λp and σp . Nikishin Lemma.16 Let n ∈ N+ , K > 0, 1 ≤ q < ∞, and let X be a normed space. (i) If an operator U : l(n) ∞ 7→ X satisfies the inequality, n X k=1

k

kU(α )k

q

1/q

≤ K sup

1≤i≤n

n X k=1

|αk (i)|,

(1.5.2)

k k k for any α1 , . . . , αn ∈ l(n) ∞ , α = (α (1), . . . , α (n)), then, for any + C ∈ R there exists an AC ⊂ {1, . . . , n}, such that |AC | ≥ n(1 − C −q ), and, for any (ai ) ∈ Rn ,

X

ai ei ≤ KCn−1/q σq ((ai )).

U i∈AC

(n)

(ii) Dually, if an operator U : X 7→ l1 n X

sup |U(xk )(i)| ≤ K

i=1 1≤k≤n

16

n X k=1

satisfies the inequality

kxk kq

1/q

,

(1.5.3)

This proof of Nikishin’s Lemma is due to B. Maurey amd G. Pisier (1976).

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“A˙Book” — 2018/8/22 — 11:55 — page 19 — #30

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19

for all x1 , . . . , xn ∈ X, then, for each C ∈ R+ , there exists an AC ⊂ {1, . . . , n} such that |AC | ≥ n(1 − C −q ), and such that λq U(x)(i) i∈A ≤ KCn−1/q kxk, x ∈ X, C

Proof. We shall only show Part (i). Part (ii) then follows by a duality argument. So, let C ∈ R+ . Call the set A ⊂ {1, . . . , n} an N-subset if there exist ε1 , . . . , εn = ±1 such that

X

εi ei > KCn−1/q |A|1/q .

U i∈A

Clearly, all N–subsets are non-empty. If there are no N-subsets, then put AC = {1, . . . , , n} and proceed as follows. Assume that there exists at least one N-subset. Denote by A1 , . . . , Am , a maximal system of N-subsets of {1, . . . , n} that are pairwise disjoint. By definition of an N-subset, for each j = 1, . . . , m, there exist εj1 , . . . , εjm = ±1, such that

X

εji ei > KCn−1/q |Aj |1/q .

U i∈Aj

Since m ≤ n, on can apply (1.5.2), and get the inequality, m X q 1/q X X

εji ei ≤ K sup (IAj )(i),

U j=1

1≤i≤n

i∈Aj

1≤j≤m

from which it follows that

KCn1 /q

m X j=1

so that

|Aj |

1/q

≤ K,

m [ Aj ≤ nC −q . j=1

Put AC = {1, . . . , n} \ {∪m j=1 Aj }, and let B ⊂ AC . Because of the maximality of A1 , . . . , Am , for each ε1 , . . . , εn = ±1,

X

εi ei ≤ KCn−1//q |B|1/q ,

U i∈B

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“A˙Book” — 2018/8/22 — 11:55 — page 20 — #31

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20

Geometry and Martingales in Banach Spaces

so that

X

|B|−1/q εi ei ≤ KCn−1/q ,

U i∈B

and, by convexity, for each (ai ) ∈ Rn ,

X

−1/q U a e σq (ai ) .

i i ≤ KCn

QED

i∈AC

Nikishin Theorem.17 Let K be a convex subset of L0 (Ω, Σ, P) formed from all positive functions. Then, for each ǫ > 0, there exists a subset Ωǫ ⊂ Ω, Ωǫ ∈ Σ, with P(Ω \ Ωǫ ) ≤ 2ǫ, such that, for all f ∈ K, Z f dµ ≤ 2Jǫ (K, P ). Ωǫ

Maurey Theorem.18 Let X be a normed space, 0 < p ≤ q ≤ ∞, and let (T, Σ, µ) be a measure space. A continuous operator U : X 7→ Lp (µ) admits a factorization V

Tg

U : X 7→ Lq (µ) 7→ Lp (µ), where V is bounded, and Tg is the opearator of multiplication by a function g ∈ Lr (µ), 1/p = 1/q + 1/r, if, and only if, for each (xi ) ∈ lq (X), Z X p/q kUxi k dµ < ∞. T

i

Maurey-Rosenthal Theorem.19 Let p > 1, X be a normed space, and assume that U : X 7→ L1 does not factor through the space Lp . Then, for each n ∈ N, there exist x∗1 , . . . , x∗n ∈ SX ∗ such that, for each a1 , . . . , an ∈ R, n n

X 1/q X

∗ q ≤ 2 |a | , a x

i i i i=1

17

1/p + 1/q = 1.

i=1

See B. Maurey (1974). See B. Maurey (1974). 19 See B. Maurey (1974). 18

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Preliminaries: Probability and geometry in Banach spaces

21

Finally, we provide a result about unconditional basic sequences that will be used in the analysis of Banach spaces of cotype q in Chapter 5. Johnson Lemma.20 Let k0 ∈ N, k > 1, and ǫ, δ > 0. For each n ∈ N, there exists an N = N(k0 , δ, ǫ, n) ∈ N such that, for each Banach space X, and all x1 , . . . , xN ∈ BX satisfying the property (Pk0 ,δ ) :

∀A ⊂ {1, . . . , N}, |A| ≥ k0 ,

sup (i,j)∈A×A

kxi −xj k ≥ δ,

there exists a subsequence xi1 , . . . , xi2n such that the sequence of increments (xi2j − xi2j−1 ) is an unconditional basic sequence with the constant 21 at most 2 + ǫ, and satisfying the condition kxi2j − xi2j−1 k ≥ δ/2, j = 1, . . . , n.

Proof. Assume, to the contrary, that for given k0 , ǫ, δ, there exists an n ∈ N such that, for all N ∈ N, there exists a Banach N space X N and xN 1 , . . . , xN ∈ BX N which satisfy the condition (Pk0 ,δ ), but do not satisfy the assertion of the Lemma. Now, let U be a non-trivial ultrafilter on N. Define a seminorm on the set S ∋ a = (ai ) via the formula

X

, ai xN |||a||| = lim i U X N i

and define X = S/|||.|||. Let (en ) be the canonical basis in S, and (¯ en ) its image in X. The sequence (¯ en ) is bounded in X, and, by the Brunel-Sucheston Theorem, there exists a subsequence (¯ e′k ) ⊂ (¯ ek ) such that, for every (aj ) ∈ S, X ¯ ′kj ||| L((aj )) := lim ||| aj e k1 β}∩A, with β ∈ R and x∗ ∈ X ∗ . The following result is also due to Davis (1973-74). Proposition 2.1.2. A closed, bounded, convex, and nonempty set A ⊂ X is dentable if, and only if, for any ǫ > 0, there exists a slice of A with diameter less than ǫ. Proof. If A is dentable then, for arbitrary ǫ > 0 we can find an x ∈ A such that x ∈ / conv(A \ Bǫ (x)), so that, for x∗ ∈ X ∗ and α ∈ R, n o x∗ x > α > sup x∗ x : x ∈ conv(A \ Bǫ (x)) . In particular, {x : x∗ x ≥ α} ∩ A ⊂ Bǫ (x), which completes the proof of one implication. The other implication is straightforward. QED A more subtle result6 , using the Ryll-Nardzewski’s Fixed Point Theorem, facilitates checking the dentability of many sets and 5 6

See W.J. Davis (1973-74). Due to E. Asplund and I. Namioka (1967).

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Martingales and Geometry in Banach Spaces

spaces. In particular, it implies that all weakly compactly generated Banach spaces (thus all separable duals) are dentable. Proposition 2.1.3. Weakly compact convex sets in a separable Banach space are dentable. Proof. It is sufficient to show that, for any ǫ > 0, and arbitrary weakly compact and convex K ⊂ X, one can find a closed and convex C ⊂ K with diameter of K \ C less than ǫ. Denote by P the weak closure of the set of extreme points in K, and let ∞ [ P ⊂ Bǫ/4 (xi ), xi ∈ P. i=1

Because P is weakly compact, and thus second category, there is an x ∈ P , and a weakly open neighborhood W of x, such that P ∩ Bǫ/4 ⊃ W ∩ P 6= ∅. Denote K1 = conv(P \ W ),

K2 − conv(W \ P ).

Evidently, by Krein-Milman Theorem, K is the convex hull of weakly compact sets K1 , and K2 , and, furthermore, K1 6= K because the extreme points of K1 lie within P \ W 7 . Now, define n o Cr = tk1 +(1−t)k2 : k1 ∈ K1 , k2 ∈ K2 , r ≤ t ≤ 1 , 0 ≤ r ≤ 1. Clearly, Cr ’s are weakly compact, convex, and increase as r → 0+, with C0 = K, and C1 = K1 . Finally, note Cr = 6 K, for all 0 < r < 1, because if Cr = K, then each extreme point z of K has the form, z = λx1 + (1 − λ)x2 ,

xx1 ∈ K1 , x2 ∈ K2 , λ ∈ [r, 1].

Hence, z = x1 ∈ K1 , contradicting the statement that K1 6= K. Notice that, if y ∈ K \ Cr , then y is of the form, y = λx1 + (1 − λ)x2 , 7

x1 ∈ K1 , x2 ∈ K2 , λ ∈ [0, r],

See, e.g., J. Kelley and I. Namioka (1963), 15.2.

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Dentability, Radon-Nikodym theorem

27

and ky − x2 k = kλx1 + (1 − λ)x2 − x2 k = |λ|kx1 − x2 k ≤ rkx1 − x2 k, so that y ∈ K \ Cr lies within the distance r · diam(K2 ) of K2 . But diam(K2 ) ≤ ǫ/2, thus, as r → 0+, Cr has the desired property diam(K − Cr ) < ǫ. QED An apparently weaker concept of σ-dentability8 employs the following concept of σ-convexification of a set: ∞ nX o X σ − conv(B) := λi bi : λ≥ 0, λi = 1, bi ∈ B , B ⊂ X. i=1

i

Definition 2.1.3. The set A ⊂ X is said to be σ-dentable if, for each ǫ > 0, there exists an x ∈ A such that x ∈ / σ−conv(A\Bǫ (x)). However, we have the following

Proposition 2.1.4.9 The Banach space X is dentable if, and only if, it is σ-dentable. Proof. Obviously, it is sufficent to show that if X is σ-dentable then it is also dentable. Assume to the contrary that that there exists a set A ⊂ X which is bounded and not dentable. Take x ∈ X so that (x + A) ∩ (−x − A) = ∅. Then B = conv((x + A) ∩ (−x − A)) is a closed, convex, symmetric, and also nondentable by Proposition 1.1, so that we may claim that the unit ball B1 (0) ⊂ X is not dentable. Indeed, were it dentable, the closed convex body B + B1 (0) ⊂ X which generates on X the norm equivalent to the original norm, would also be dentable by Proposition 2.1.1. But it is not since if B is not dentable then one can find ǫ > 0 such that for all x ∈ B, x ∈ conv(B − Bǫ (x)), so that if x + y ∈ B + B1 (0) then also x+y ∈ conv (y+B)−Bǫ (y+x) ⊂ conv (B1 (0)+B)−Bǫ (y+x) . Now, we shall show that non-dentability of B1 (0) implies nonσ-dentability of intB1 (0) what, in turn, would contradict the σdentability of X. 8 9

Introduced by H.B. Maynard (1973). See W.J. Davis and R.R. Phelps (1974).

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Take ǫ > 0 such that, for each x ∈ B1 (0), and x ∈ conv(B1 (0)− Bǫ (x)). If kxk < 1 − ǫ/4, then, for some λ > 0 such that kλxk < 1, we have kx − λxk > ǫ/4, and kx + λxk > ǫ/4. Thus x ∈ conv(B1 (0) − Bǫ/4 (x)). If 1 > kxk > 1 − ǫ/4 then Bǫ/4 (x) ⊂ Bǫ (x/kxk), so that x/kxk ∈ conv(B1 (0) − Bǫ/4 (x). For small ǫ, the origin 0 is an interior point of conv(B1 (0)−Bǫ/4 (x), so that the entire segment [0, x/kxk) is in the interior of that set. In particular, x ∈ conv(intB1 (0) − Bǫ/4 (x), so that intB1 (0) is not dentable . QED The equivalence proven in the above Proposition shows that it is sufficient to know what separable Banach spaces are dentable in order to know what are all dentable Banach spaces. In particular, X is dentable if, and only if, each closed separable subspace of X is dentable. More precisely, we have the following result: Proposition 2.1.5.10 Dentability is a separably determined property, i.e., A ⊂ X is σ-dentable if, and only if, each countable subset of A is σ-dentable. Proof. Assume, to the contrary, that A is not σ-dentable. Then we can find an ǫ > 0 such that, for each x ∈ A, also x ∈ σ − conv(A \ Bǫ (x)). Thus, choosing an arbitrary y ∈ A, we can find a sequence {y n } ⊂ A such that y=

X

λn y n ,

n

and ky − y n k ≥ ǫ.

Now, apply to each y n the same denial of the σ-dentability of A. Reiterating this procedure one gets an infinite tree in A which is countable and, by definition, not σ-dentable. QED 10

Due to J. Diestel (1973).

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2.2

Dentability versus RadonNikodym property, and martingale convergence

It turns out that the dentability property of a Banach space is closely related to certain measure-theoretic and martingaletheoretic theorems in such Banach spaces. Definition 2.2.1. We say that the Banach space X has the Radon-Nikodym Property (RNP) if every X-valued measure m on (Ω, Σ), for which the total variation measure [ X |m|(E) := sup km(Ei )k : Ei ∈ Σ are disjoint with Ei = E ∈ Σ , i

i

is finite, and which is absolutely continuous with respect to a finite positive measure µ, admits with respect to µ a Bochner integrable density. The following illuminating example shows a Banach space without the Radon-Nikodym property. Example 2.2.1.11 The Banach space X = c0 of real sequences convergent to 0 does not have the Radon-Nikodym property. Indeed, let (Ω, Σ, µ) be a finite positive measure space containing no atoms. We’ll construct a measure m : Σ → c0 (the elements of c0 will be denoted (an,i ), n = 1, 2, . . . , 2n ≤ i < 2n+1 ) as follows: m : Σ ∋ E → m(E) = (µ(E ∩ En,i )) ∈ c0 , where (En,i ), n = 1, 2, . . . , 2n ≤ i < 2n+1 , is a sequence of measurable sets such that µ(En,i ) = 2−n µ(Ω), and such that En,i is the disjoint union of En+1,2i and En+1,2i+1 . Such a sequence (En,i ) does exist in view of non-atomicity of µ. Evidently, km(E)k ≤ µ(E) so that m is absolutely continuous with respect to µ and has finite total variation. However, m does not have a Bochner integrable density with respect to µ. Indeed, if such a density, say f : Ω 7→ c0 , existed then, denoting by (en,i ) 11

Due to D.R. Lewis (1972).

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Martingales and Geometry in Banach Spaces

the standard basis in l1 , we would have, for each i = 2, 3, . . . , and E ∈ Σ, the equality, Z f (ω)en,i µ(dω) = µ(E ∩ En,i ). E

Hence, for each i, there would exist a µ-null set Ci ⊂ ΩSsuch that f (ω)en,i = IEn,i (ω), for ω ∈ / Ci . Choose, ω0 ∈ Ω \ i Ci . By the very construction of (En,i ), we have IEn,i, (ω) = 1 for infinitely many indices i, so that limi f (ω)en,i 6= 0, which gives the desired contradiction. Now let’s turn to the issue of martingale convergence in Banach spaces and recall that, given a probability space (Ω, Σ, P), and a sequence of increasing sub-σ-algebras Σ1 ⊂ Σ2 ⊂ · · · ⊂ Σ , an Xvalued (Σn )-martingale is a sequence {Mn } of strongly measurable functions such that Mn meas Σn , and E(Mn+1 |Σn ) = Mn . Definition 2.2.2. Let 1 ≤ p < ∞. We say that the Banach space X has the Lp -Martingale Convergence Property (X ∈ (MCPp )), if for each X-valued martingale (Mn , Σn ), such that supn EkMn kp < ∞, there exists an M∞ ∈ Lp (Ω, Σ, P; X) such that Mn → M∞ , almost surely in norm.

Not all Banach spaces have the Martingale Convergence Property, and here is the counterexample:

Example 2.2.2.12 The Banach space X = L1 (0, 1) does not have (MCP1 ). Indeed, let Σn be the binary Borel algebra in (0,1) generated by the intervals (m/2n , (m + 1)/2n ), 0 ≤ m ≤ 2n−1 , n = 1, 2, . . . , . Define Mn (ω) = 2n I(0,(m+1)/2n ] − I(0,m/2n ] ,

if ω ∈ (m/2n , (m + 1)/2n , and 0 elsewhere. It is easy to check that (Mn , Σn , n ≥ 1) is a martingale with values in L1 (0, 1) and moreover, kMn (ω)k ≡ 1, a.e, 12

Due to S.D. Chatterji (1960).

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Dentability, Radon-Nikodym theorem EkMn (ω)kp = 1,

n ≥ 1,

E sup kMn (ω) − Mn−1 (ω)k = 1, n

M0 = 0.

But, if ω 6= p/2q , then Xn (ω) does not converge to any limit, either weakly, or strongly. The following Theorem is the main result of this chapter: Theorem 2.2.1.13 For a Banach space X the following properties are equivalent: (D) X is dentable, (RNP ) X has the Radon-Nikodym Property, (MCPp ) X has the Lp -Martingale Convergence Property, for any 1 ≤ p < ∞. Proof. We shall prove the theorem in the circular fashion, (D) =⇒ (RNP ) =⇒ (MCPp ) =⇒ (D). (D) =⇒ (RNP ). Assume that X is dentable, and m is an X-valued measure on (Ω, Σ) with finite total variation which is absolutely continuous with respect to a finite positive measure µ14 . The task is to find a Bochner Σ-measurable f : Ω 7→ X such that, for each E ∈ Σ, Z m(E) = f (ω)µ(dω). E

Our First Observation is that under the above assumption m has a locally almost dentable average range with respect to µ, that is for each E ∈ Σ, and arbitrary ǫ > 0, there exists an F ⊂ E such that µ(E \ F ) < ǫ, and the set o n m(F ′ ) ′ ′ : F ⊂ F, µ(F ) > 0 ⊂X AR(F ) := µ(F ′ ) 13

The equivalence of the first and second conditions follows from the work of M.A. Rieffel (1967), H.B. Maynard (1973), and W.J. Davis and R.R. Phelps (1974). A short and direct proof of this equivalence can also be found in R.E. Huff (1974). The equivalence of the second and third conditions has been proved by S.D. Chatterji (1969). 14 The proof is the same in the case of a σ-finite µ.

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is dentable. Indeed, because of deniability of X it is sufficient to prove that AR(F ) is bounded, and the last statement can be verified as follows: The total variation |m| is a finite positive measure on Σ which is also absolutely continuous with respect to µ and, thus by the realvalued Radon-Nikodym Theorem, for some ϕ ∈ L1 (Ω, Σ, µ; R), Z |m|(E) = ϕ(ω)µ(dω), E ∈ Σ, E

so that, given ǫ > 0, there exists a constant K such that ϕ(ω) < K on E0 ∈ Σ, with µ(Ω \ E0 ) < ǫ. Thus, given E ∈ Σ, and ǫ > 0, we take F = E ∩ E0 . Then µ(E \ F ) ≤ µ(Ω \ E0 ) < ǫ, and for each F ′ ⊂ F = E ∩ E0 , with µ(F ′ ) > 0, we shall have

m(F ′ ) |m|(F ′)

≤ K.

≤

′ µ(F ) µ(F ′) Our Second Observation is that, for any ǫ > 0, and E ∈ Σ with µ(E) > 0, one can find an F ⊂ E with µ(F ) > 0, and an x ∈ X such that , for all F ′ ⊂ F , km(F ′ ) − xµ(F ′ )k < ǫµ(F ′ ). Indeed, by first observation, there is an Ed ⊂ E of positive measure µ such that AR(Ed ) is dentable, and, in particular, for the ǫ given above one can find x= such that

m(F0 ) ∈ AR(Ed ), µ(F0 )

F0 ⊂ Ed , µ(F0 ) > 0,

x∈ / Q := conv AR(Ed ) \ Bǫ (x) .

Now, either F0 can be taken as the desired F , or not. If it can, we are done, and if not, we can find E1 ⊂ F0 such that µ(E1 ) ≥ 1/k1 (and let k1 be the smallest integer for which such an E1 exists), and m(E1 ) ∈ Q. µ(E1 )

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Next, either F1 = F0 \ E1 is the looked for F , or not. If yes, then we are done again. If not, we can find E2 ⊂ F1 as above. So, either we find our F in a finite number of steps, or, by induction, we choose a sequence (Ei ) of pairwise disjoint subsets of F0 , and a sequence of (minimal in the above sense) integers ki ↑ ∞(because µ(F0 ) < ∞), such that µ(Ei ) ≥

1 , ki

m(Ei ) ∈ Q, µ(Ei )

and

and such that if ′

E ⊂ F0 \

n [

Ei ,

and

i=1

m(E ′ ) ∈ Q, µ(E ′ )

then µ(E ′ ) < 1/(kn − 1) (remember the minimality of kn !!). But, in this case we can surely take F = F0 \

∞ [

i=1

Ei ⊂ F0 \

n [

Ei ,

n = 1, 2, . . . ,

i=1

because, if for any F ′ ⊂ F we have µ(F ′ ) > 0, m(F ′ )/µ(F ′) ∈ Q, then for each n = 1, 2, . . . , we would also have µ(F ′) < 1/(kn − 1) and that would imply µ(F ′ ) = 0, a contradiction. If F were of µ measure 0, then also m(F ) = 0 and, by a convexity argument, we would have that S ∞ m(F0 ) m( Ei ) X m(Ei ) µ(Ei ) S x= = = · S ∈ Q, µ(F0 ) µ( Ei ) µ(E ) µ( Ei ) i i=1 yielding another contradiction.

Third Observation: For each ǫ > 0, one can find a sequence (xn ) ⊂ X, and an (at most countable) partition (Ei ) ⊂ Σ of Ω such that F ⊂ Ei , µ(F ) > 0

=⇒ km(F ) − xi µ(F )k ≤ ǫµ(F ).

(∗)

Indeed, using repeatedly the Second Observation either we exhaust Ω in a finite number of steps or else we find a sequence

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(Ei ) of pairwise disjoint subsets of Ω, a sequence xi ⊂ X, and a nondecreasing sequence ki ↑ ∞ of (minimal, as in the Second Observation) integers such that, for each i = 1, 2, . . . , (*) holds, µ(Ei ) ≥ 1/ki , and if, for some n, E ⊂ Ω\

n [

Ei

i=1

is such that, for some x ∈ X, and all F ⊂ E, µ(F ) > 0, and km(F ) − xµ(F )k S < ǫµ(F ), then µ(E) < 1/(kn − 1)., But, in the latter case, µ(Ω \ Ei ) = 0 because, otherwise, using the Second S Observation, we could find E ⊂ Ω \ Ei , µ(E) > 0, as above, and that would mean that, for each n, µ(E) < 1/(kn − 1), i.e., µ(E) = 0, a contradiction. Now, we are ready to complete the proof of the implication (D) =⇒ (RNP ) and construct the density f as follows: Let Π(∋ π) be the directed family of finite partitions of Ω into sets of positive measure, and put fπ =

X m(E)

E∈π

µ(E)

IE .

If we manage to show that limπ fπ exists in L1 (Ω, Σ, µ; X) then, clearly, f = limπ fπ is the desired density because then, in particular, for each E ∈ Σ, µ(E) > 0, Z Z f dµ = lim fπ dµ, π∈Π

E

E

and because, for π ≥ {E, Ω \ E}, Z fπ dµ = m(E). E

So the proof will be complete as soon as we show that (fπ ) satisfy the Cauchy condition in L1 (Ω, Σ, µ; X). Take ǫ > 0. Because |m| is a finite positive measure, absolutely continuous with respect to µ, we can choose E ⊂ Ω such that

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|m|(Ω \ E) < ǫ/6, and a δ > 0 such that |m|(F ) < ǫ/6 whenever µ(F ) < δ. Now, S employing the Third Observation, in the decomposition E = Ei (and using ǫ/6 instead of ǫ) take n such that n [ µ E\ Ei < δ, i=1

and

n o n [ Ei , Ω \ E . π0 = E1 , . . . , En , E \ i=1

Then, for arbitrary π ≥ π0 , which is evidently of the form, n o n o n o π = Fij : 1 ≤ i ≤ n, 1 ≤ j ≤ m ∪ G1 , . . . , Gl ∪ F1 , . . . , Fk , with

k [

Fij = Ei ,

j=1

1 ≤ j ≤ n,

we have Z

ki n X

X

m(Fij ) m(Ej ) − kfπ (ω) − fπ0 (ω)k µ(dω) =

µ(F ij)

µ(F ) µ(E ) ij j i=1 j=1

Sn l X

m(Gi ) m E − i=1 Ei Sn + −

µ(Gi ) µ(G E µ E − i i i=1 i=1

≤

ki n X X i=1 j=1

k X

m(Fi ) m Ω \ E + −

µ(Fi ) µ(Fi µ Ω\E i=1

km(Fij ) − xi µ(Fij )k + +2

l X I=1

km(Gi )k + 2

n X i=1

k X i=1

km(Ei ) − xi µ(Ei )k

km(fi )k

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≤

ki n X X i=1 J=1

n

X ǫ ǫ µ(Fij ) + µ(Ei ) σµ(E) σµ(E) i=1

n [ +2|m| E \ Ei + 2|m|(Ω \ E) i=1

ǫ ǫ ǫ ǫ + + 2 + 2 = ǫ, 6 6 6 6 which completes the proof of the implication (D) =⇒ (RNP ). ≤

(RNP ) =⇒ (MCPp ). Assume that X has the RadonNikodym property, p > 1 and that (Mn , Σn ) is an X-valued Lp bounded martingale on (Ω, Σ, µ), where µ is a probability measure. Now, the martingale property implies that X-valued, countably additive measures Z mn (E) := Mn (ω) µ(dω), E ∈ Σn , n = 1, 2, . . . , E

extend to a finitely additive set function, m, on the algebra S∞ Σ . i=1 i The total variation of m on any set E ∈ Σi , say E ∈ Σn , |m|(E) =

Z

1 q

E

kMn k dµ ≤ [µ(E)] · sup n

q=

Z

Ω

kMn kp

1p

< ∞,

p < ∞. p−1

Moreover, S the above inequality shows that |m| is countablySadditive on Σn , and thus extends to a measure on Σ∞ = σ( Σn ) which is absolutely continuous with respect to µ. And, since km(E)k ≤ |m|(E), E ∈ Σn , m is also countably additive on Σn , extends to Σ∞ , and is absolutely continuous with respect to µ.15 Thus, by the RNP, one can find a function g ∈ L1 (Ω, Σ∞ , µ; X) such that Z m(E) = g dµ, E ∈ Σ∞ . E

15

For all the basic properties of vector set functions and their variation, see N. Dunford and J.T. Schwartz (1958), Chapter III, and Chapter IV.10.

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Evidently, for n = 1, 2, . . . , E(g|Σn) = Mn , because , for each E ∈ Σn , Z Z g dµ = m(E) = mn (E) = Mn dµ, E

E

and, by the general theorem on conditional expectations,16 as n → ∞, a.s., and in Lp ((Ω, Σ, µ; X),

Mn = E(g|Σn ) → E(g|Σ∞) = g,

which completes the proof of the implication (RNP ) =⇒ (MCPp ). (MCPp ) =⇒ (D). Suppose the space in not dentable, i.e., by Proposition 2.1.4, it is not σ-dentable. Then there exists a bounded, closed and convex set A ⊂ X which is not σ-dentable, that is,P for some ǫ > 0, and each x ∈ A, there are positive numbers αi (x), i αi (x) = 1, and ai (x) ∈ A, i = 1, 2, . . . , such that inf kx − ai (x)k > ǫ, i

and x=

X

αi (x) · ai (x).

Now, we shall construct a bounded martingale Mn with values in A which diverges almost surely (a.s.), thus contradicting (MCPp ). We proceed by induction, and construct a tree as in the the scheme pictured below: M0 = x M1 (0, 1),

M1 (02),

M1 (0, 3),

...

M2 (0, 1, 1), M2(0, 1, 2) . . . M2 (0, 2, 1)M2 (0, 2, 2) . . . M2 (0, 3, 1), M2 (0, 3, 2) . . . ...

...

...

...

...

...

Take as the probability space Ω = N × N × ..., 16

Σ = σ(2N × 2N × . . . ),

P = lim Pn ,

See J. Neveu (1972), Prop. V-2-6.

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where Pn are probability measures on the increasing sequence of σ-algebras Σn = |2N × 2N {z × · · · × 2N} ×N × N × . . . , n−times

and define

Mn (ω) = Mn (i0 , i1 , i2 , . . . ) = Mn (i0 , . . . , in ), where M0 is any point x ∈ A a.s., and inductively, given Mn , and Pn , we define Mn+1 (i0 , i1 , . . . , in , in+1 ) = ain+1 (Mn (i0 , . . . , in )), and Pn+1 ({i0 , i1 , . . . , in+1 }) = Pn+1 ({i0 , i1 , . . . , in }) · αin+1 (Mn ({i0 , i1 , . . . , in })). The construction implies that (Mn ) is a martingale which has values in the bounded set A ⊂ X, and which is evidently divergent since, for any ω = (i0 , i1 , . . . ) ∈ Ω, kMn+1 (ω) − Mn (ω)k > ǫ,

n = 1, 2, . . . .

This completes the proof of the implication (MCPp ) =⇒ (D), and of the Theorem 2.2.1 itself. QED

2.3

Dentability and submartingales in Banach lattices and lattice bounded operators

In this section17 we will discuss behavior of submartingales with values in Banach lattices, that is, Banach spaces (X, k.k) endowed with a partial order ≤ satisfying the following properties: 17

The submartingale results of this section are due to J. Szulga and W.A. Woyczy´ nski (1974).

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39

(i) Translation invariance; x ≤ y implies x + z ≤ y + z; (ii) Positive homogeneity: For any scalar α ≥ 0, x ≤ y implies αx ≤ αy; (iii) For any pair of vectors x, y ∈ X there exists a supremum, x ∨ y ∈ X with respect to partial order ≤; (iv) The norm is monotone, that is |x| ≤ |y| implies kx ≤ kyk. In the above notation |x| := x+ +x− , where x+ := x∨0, x− := −x ∨ 0. The norm dual X ∗ of X is also a Banach lattice with the natural ordering, and by X + , and X ∗+ we denote the non-negative cones in X, and X ∗ , respectively. Definition 2.3.1. Let (Ω, Σ, P ) be a probability space, and let Σ1 ⊂ Σ2 ⊂ · · · ⊂ Σ be a sequence of sub-σ-algebras. The sequence (Xn , n ∈ N) of Banach lattice X-valued random vectors with Xn ∈ L1 (Ω, Σn , P ; X) is said to be a sub-martingale if E(Xn+1 |Σn ) ≥ Xn ,

n = 1, 2, . . . , a.s.

For real-valued submartingales the following results is classical : 18

Proposition 2.3.1. If (Xn , n ∈ N) is a real-valued submartingale, and supn EXn+ < ∞, then there exists an X∞ ∈ L1 such that Xn → X∞ , a.s. The results presented below show how the above Proposition (which can also be dually formulated for supermartingales) carries over to the case of submartingales with values in Banach lattices.19 Definition 2.3.2. The set A ⊂ X is said to be order bounded if there exists an x0 ∈ X such that, for all y ∈ A, |y| ≤ x0 . The linear operator T from a Banach space Y into a Banach lattice X is said to be lattice bounded if it maps the unit ball of Y into an order bounded subset of X. 18

See J. Neveu (1972). The initial work was motivated by L. Schwartz’s (1973) extensive work on supermartingales that have measures as their values. The results had an elegant application to the problems of disintegration of measures. His model fits into our general framework. 19

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40

Martingales and Geometry in Banach Spaces The following Proposition is well known20 :

Proposition 2.3.2. (a) Let C be the Banach space of real continuous functions on the unit interval, and let X be a separable Banach lattice with an order continuous norm. Then T : C → X is lattice bounded if, and only if there exists a function g : [0, 1] → X of bounded 0-variation, X |g(ti+1) − g(ti )| ∈ X, ess var g(t) := sup such that Tf =

Z

1

f (t)dg(t), 0

f ∈ C,

where the integral is understood as an order limit of Stieltjes sums; (b) The operator T : lq 7→ lp (q > 1, p ≥ 1) is lattice bounded if, and only if it is of the form, Ty =

∞ X

aik yk

k=1

where

∞ hX ∞ X i=1

k=1

|aik |

i∈N

,

q/(q−1)

y = (yk ) ∈ lq , i(q−1)p/q

< ∞.

(c) The operator T : Y 7→ Lp [0, 1], p ≥ 1, where Y is a separable Banach space, is lattice bounded if, and only if, it is of the form, (T y)(t) = f ∗ (t)y, where y ∈ Y , t ∈ [0, 1], and f ∗ : [0, 1] 7→ Y is ∗ - weakly measurable and such that kf ∗ k ∈ Lp [0, 1]. Now we shall formulate a result on dentable Banach lattices that will be used later on. Proposition 2.3.3. If X is a dentable Banach lattice and the sequence x0 ≤ x1 ≤ x3 ≤ . . . in X is norm bounded then it is convergent. 20

See, e.g., L.V. Kantorovich, B.Z. Vulikh and A.G. Pinsker (1950).

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41

Proof. Because X is dentable it does not contain isomorphic copies of c0 (see, Example 2.1.2, and Theorem 2.1.1), and in every such Banach lattice monotone norm-bounded sequences are convergent.21 QED Now we turn to the investigation of analogues of Proposition 2.3.1 for X-valued submartingales. Note, that the classical Doob’s condition, supn EXn+ < ∞, for real-valued random variables has two different versions for Banach-lattice-valued random vectors, namely, order boundedness of EXn+ , n ∈ N, and norm boundedness, sup kXn+ k < ∞. Both boil down to the Doob’s condition in the real case. However, as we shall see below, in general, neither is sufficient to assure a.s. convergence of a submartingale (Xn ), n ∈ N. It is not difficult to check that for both real- and vector-valued sub-martingales, the set (EXn+ , n ∈ N) is order bounded if, and only if, (E|Xn |, n ∈ N) is such. However, even for vector-valued martingales it might happen that supn EkXn+ k < ∞, and still supn EkXn− k = ∞, so that it will not be surprising to see that the condition supn EkXn+ k < ∞ does not imply, in general, the a.s. convergence of the submartingale (Xn ) even in dentable Banach lattices. On the other hand, the condition supn EkXn+ k < ∞ is stronger than order boundedness of (EXn+ , n ∈ N) for any sequence (Xn ) of random vectors with values in the Banach lattice X because, for each x∗ ∈ X ∗+ , sup x∗ E(Xn∗) ≤ supn EkXn+ kkx∗ k < ∞, and because the set A ⊂ X is order bounded if, and only if, for each x∗ ∈ X ∗+ , the set x∗ A is bounded on the real line. The last statement follows from the fact that in a Banach lattice X, x ≥ 0 if, and only if, for each x∗ ∈ X ∗+ , x∗ x ≥ 0. It is not hard to see that if X is a dentable Banach lattice then in order to produce examples of, (1) a martingale (Mn ) with values in X such that supn EkMn+ k < ∞ and, at the same time, supn Ekm− n k = ∞, and supn EkMn k = ∞, and (2) a submartingale (Xn ) with values in X such that 21

See, e.g., Theorem 14 in L. Tzafriri (1972).

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supn EkXn+ k < ∞, and Xn diverges a.s., it is sufficient to find (3) an a.s. divergent sequence (Yn ) of non-negative in1 dependent random vectors in PnL (Ω, Σ, P ; X) such that both, supn EkYn k < ∞, and supn k i=0 EYi k < ∞. Indeed, given such a sequence (Yn ) it is enough to take Σn = σ(Y0 , . . . , Yn ), Z0 = 0, n−1 X Yi , Zn = i=0

Mn = −Zn+1 + EZn+1,

and

Xn = Mn + Zn = EZn+1 − Yn ,

n ≥ 1.

Now, (Mn ) defined in such a way is a zero-mean martingale such that supn EkMn+ k ≤ supn kEZn+1 k < ∞ (because Mn+ ≤ EZn+1 ) but, at the same time, EkMn k = EkZn+1 − EZn+1 k ≥ EkZn+1k − kEZn+1 k is unbounded. Also, (Xn ) is evidently a submartingale that is divergent a.s., and for which sup EkXn+ k ≤ sup kEZn+1k < ∞. n

n

Below, we provide an example of sequences (Yn ) of random vectors with values in certain classical Banach lattices that satisfy the condition (3). Example 2.3.1. Let X = lp , p > 1 (the reason why p = 1 is excluded is provided in the Corollary at the end of this Section), Ωi = [0, 1), Σi be the family of all Borel subsets of [0, 1), and λi be the Lebesgue measure on Ωi , i ∈ N. Define Y Y Y Ω= Ωi , Σ= Σi , P = λi , i∈N

i∈N

i∈N

Y0 (ω0 , ω1 , . . . ) = 0, and Y2n−1 +k (ω0 , ω1 , . . . ) = I[k/2n−1 ,(k+1)/2n−1 ) (ω2n−1 +k )e2n−1 +k ,

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Dentability, Radon-Nikodym theorem

where n = 1, 2, . . . , k = 0, 1, . . . , 2n−1 − 1, IA is the indicator function of the set A, and (en , n ∈ N) is the standard basis in lp . By definition Yn , n ∈ N, are independent, non-negative and in 1 L (Ω, Σ, P ; X), sup EkYn k ≤ 1,

and

n∈N

n∈N

because EY0 = 0,

n

X

sup EYi < ∞, i=1

EY2n−1 +k = 21−n e2n−1 +k ,

for n = 1, 2, . . . , k = 0, 1, . . . 2n−1 − 1, so that ∞ ∞ n

X

X

X 1/p

2i(1−p) , EYi = EYi = sup

n∈N

i=0

i=0

i=0

but, at the same time, (Yn ) is divergent for each ω ∈ Ω because, for each ω ∈ Ω there exist sequences (ni ), (n′i ) ⊂ N such that kYni (ω)k = 1, and kYn′i (ω)k = 0.

Now, in the next two theorems22 we discuss positive results concerning convergence of sub-martingales in Banach lattices. We start with the observation that the Doob’s decomposition of real martingales survives in Banach lattices. Namely, if (Xn , Σn , n ∈ N) is a submartingale with values in a Banach lattice X, then Xn = Mn + Zn , where (Mn , Σn ) is a martingale, and the sequence (Zn , n ∈ N) is predictable, i.e., Zn ∈ L1 (Ω, Σn−1 , P ; X). As in the real case, to prove the above statement it is sufficient to define, Z0 = 0, M0 = X0 , and n X Xi − E(Xi |Σi−1 ) , Mn = X0 + i=1

and Zn =

n X i=1

22

E(Xi − Xi−1 |Σi−1 ),

n = 1, 2, . . . .

Due to J. Szulga and W.A. Woyczy´ nski (1974).

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Martingales and Geometry in Banach Spaces

The above Example 2.3.1 shows that, in general, the condition supn EkXn+ k < ∞ for the submartingales Xn = Mn + Zn does not imply its a.s. convergence. However, we have the following result: Theorem 2.3.1. For a separable Banach lattice X the following conditions are equivalent: (i) X is dentable; (ii) For each X-valued submartingale (Xn = Mn + Zn , n ∈ N) satisfying the conditions, sup EkXn+ k < ∞,

and

n

sup EkXn− k < ∞, n

there exists an X∞ ∈ L1 (Ω, Σ, P ; X) such that Xn → X∞ , a.s.; (iii) For each X-valued submartingale (Xn = Mn + Zn , n ∈ N) satisfying the conditions, sup EkXn+ kp < ∞, n

and

sup EkXn− kp < ∞, n

for some p ∈ (1, ∞), there exists an X∞ ∈ Lp (Ω, Σ, P ; X) such that Xn → X∞ , a.s., and in Lp . Proof. (i) =⇒ (ii)[(i) =⇒ (iii)]. Since Xn = Mn + Zn ≥ Mn , a.s., we also have that Xn+ ≥ MN+ , a.s., and the monotonicity of the norm implies that sup EkMn kp ≤ 2p (sup EkXn+ kp + sup EkMn− kp ).

n∈N

n∈N

n∈N

Hence, by Theorem 2.1.1, there exists M∞ ∈ L1 [M∞ ∈ Lp ], such that Mn → M∞ , a.s. [ Mn → M∞ , a.s., and in Lp ] . Because Zn = Xn − Mn , we have that Zn ≤ Xn+ + Mn− , so that supn∈N EkZn kp < ∞, 1 ≤ p < ∞. Utilizing again the monotonicity of the norm, and the Lebesgue Monotone Convergence Theorem, we get that E supn∈N kZn kp < ∞, so that supn∈N kZn k < ∞, a.s. However, because of Proposition 2.3.3, there exists a random vector Z∞ such that Zn → Z∞ , a.s. The Fatou Lemma yields that EkZ∞ kp ≤ lim inf EkZn kP ≤ sup EkZn kp < ∞, n∈N

n∈N

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Dentability, Radon-Nikodym theorem

45

so that Z∞ ∈ Lp , 1 ≤ p < ∞, and letting X∞ = M∞ + Z∞ , completes the proof of the first two implications. (ii) =⇒ (i)[(iii) =⇒ (i)]. If Xn = Mn , then the conditions in (ii)[(iii)] boil down to supn∈N EkMn k < ∞ [supn∈N EkMn kp < ∞], and Theorem 2.1.1. gives the dentability of X. QED In the next theorem we impose weaker assumptions on the submartingale (Xn ), but then the convergence takes place only for a transformed submartingale. Theorem 2.3.2. Let X be a separable Banach lattice, Y a dentable separable Banach lattice, and T : X 7→ Y be a linear bounded positive operator such that its transpose T ∗ : Y ∗ 7→ X ∗ is lattice bounded. If (Xn , n ∈ N) is a submartingale with values in X such that (E(Xn+ ), n ∈ N) is order bounded, then there exists a Y∞ ∈ L1 (Y ) such that the submartingale T Xn → Y∞ , a.s., as n → ∞.

Proof. Let Xn = Mn + Zn , as before, and let E(Xn+ ) ≤ x0 ∈ X + , for all n ∈ N. We will show that under the above assumptions, sup Ek(T Xn )+ k < ∞, n∈N

and sup Ek(T Xn )− k < ∞,

n∈N

what, in view of Theorem 2.3.1, would give the desired result because the Doob’s decomposition for the sub-martingale T Xn is T Mn + T Zn . Indeed, sup Ek(T Xn )+ k ≤ sup EkT (Xn+ )k n∈N

n∈N

= sup E sup y ∗ T (Xn ) : ky ∗ k ≤ 1, 0 ≤ y ∗ ∈ Y ∗ n∈N

= sup E sup (T ∗ y ∗ )Xn+ : ky ∗ k ≤ 1, 0 ≤ y ∗ ∈ Y ∗ . n∈N

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Martingales and Geometry in Banach Spaces

However, the transpose T ∗ of a positive T is also positive, and thus the fact that T ∗ is lattice bounded implies the existence of an x∗0 ∈ X ∗+ such that, for each y ∗ with ky ∗ k ≤ 1, we have |T ∗ y ∗ | ≤ x∗0 . Thus, we get that sup EkT Xn )+ k ≤ E(x∗0 Xn+ ) = x∗0 E(Xn+ ) ≤ x∗0 x < ∞.

n∈N

Proceeding as above, and utilizing the inequality E(T Mn )− = E(T Mn )+ − E(T Mn ) = E(T Mn )+ − E(T M0 ) ≤ E(T Xn )+ − E(T M0 ) we get that sup Ek(T Mn )− k ≤ x∗0 x0 + |x∗0 E(T M0 )| < ∞,

n∈N

because M0 ∈ L1 . This ends the proof. QED

Because l1 is a dentable Banach lattice (see, Proposition 2.1.3, and the preceding comments), and because the operator [Id(l1 , l1 )]∗ is lattice bounded in l∞ , we also obtain the following, Corollary 2.3.1. If (Xn , n ∈ N) is a sub-martingale with values in l1 such that (E(Xn+ ), n ∈ N) is order bounded, then there exists an X∞ ∈ L1 (Ω, Σ, P ; l1 ) such that Xn → X∞ , a.s. as n → ∞.

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Chapter 3 Uniform Convexity and Uniform Smoothness 3.1

Basic concepts

We begin this chapter by describing the concept of uniform convexity of a Banach space.1 Definition 3.1.1. Let X be a Banach space of dimension ≥ 2. The modulus of convexity of X is defined by the formula: δX (ǫ) := inf 1−k(x+y)/2k : kxk = kyk = 1, kx−yk = ǫ , 0 ≤ ǫ ≤ 2.

X is said to be uniformly convex if δX (ǫ) > 0, for ǫ > 0. X is said to be q-uniformly convex if there exists a constant C such that δX (ǫ) ≥ Cǫq , q ≥ 2. Example 3.1.1. The LP space, p ≥ 1, is max(p, 2)-uniformly convex.2 Definition 3.1.2. Let X be a Banach space of dimension of at 1

Introduced by J.A. Clarkson (1936). Consult V.D. Milman (1972) for a detailed exposition of metric geometry of Banach spaces. More information about moduli of convexity and smoothness of Orlicz and other spaces can be found in T. Figiel (1976) and M.M. Day (1973). 2

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Geometry and Martingales in Banach Spaces

least 2. The modulus of smoothness of X is defined by the formula: ρX (τ ) := sup k(x+ y)/2k + k(x−y)/2k −1 : kxk = 1, kyk = τ .

X is said to be uniformly smooth if ρX (τ ) = o(τ ), τ → 0. X is said to be p-uniformly smooth, 1 < p ≤ 2, if there exists a constant C such that ρX (τ ) ≤ Cτ p .

It is evident that the above notions are not invariant under equivalent renorming of the Banach space. So, we say that the Banach space (X, k.k) is uniformly convexifiable (q-uniformly convexifiable, uniformly smoothable, p-uniformly smoothable) if it admits a norm equivalent to k.k that is uniformly convex (q-uniformly convex, uniformly smooth, p-uniformly smooth). The Banach space X is uniformly convex if, and only if, the dual space X ∗ is uniformly smooth, and the relationship between their moduli of convexity and smoothness is given by the following result:3 Proposition 3.1.1 For any Banach space X the modulus of smoothness of the dual space X ∗ is the function conjugate in the sense of Young to the modulus of convexity of X, that is, τǫ ρX ∗ (τ ) = sup − δX (ǫ) : 0 ≤ ǫ ≤ 2 , 2

τ > 0.

Proof. First, note that for every positive ǫ, and τ , τǫ δX (ǫ) + ρX ∗ (τ ) ≥ . 2

(3.1.1)

Indeed, if x, y ∈ X, kxk = kyk = 1, kx−yk = ǫ, and x∗ , y ∗ ∈ X ∗ are such that kx∗ k = ky ∗ k = 1, and x∗ (x + y) = kx + yk, y ∗ (x − y) = kx − yk , then 2ρX ∗ (τ ) ≥ kx∗ + τ y ∗ k + kx∗ − τ y ∗ k − 2 ≥ x∗ x + τ y ∗ x + x∗ y − τ y ∗ y − 2 3

Due to J. Lindenstrauss (1963).

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49

Uniform Convexity and Uniform Smoothness = x∗ (x + y) + τ y ∗ (x − y) − 2 = kx + yk + τ ǫ − 2 so that 2 − kx + yk ≥ τ ǫ − 2ρX ∗ (τ ),

which gives (3.1.1). Now, let x∗ , y ∗ ∈ X ∗ satisfy the conditions, kx∗ k = 1, ky ∗ k = τ , and let α > 0. Then, there exist x, y ∈ X such that kxk = kyk = 1, and x∗ x + y ∗ x ≥ kx∗ + y ∗ k − α,

x∗ y − y ∗ y ≥ kx∗ − y ∗ k − α.

Therefore, kx∗ + y ∗ k + kx∗ − y ∗ k ≤ x∗ x + y ∗ x + x∗ y − y ∗ y + 2α = x∗ (x + y) + y ∗ (x − y) + 2α ≤ kx + yk + τ kx − yk + 2α ≤ 2 + 2 sup{ǫτ /2 − δX (ǫ) : 0 ≤ ǫ ≤ 2} + 2α.

In view of the arbitrariness of α, we get our Proposition. QED Corollary 3.1.1. For every Banach space X the modulus of smoothness satisfies the following inequality: ρX (τ ) ≥ (1 + τ 2 )1/2 − 1,

τ > 0,

where the right-hand side of the inequality is the modulus of smoothness of the Hilbert space. Its asymptotics at 0 is, obviously, of the order τ 2 . Example 3.1.2. It is easy to see that the space Lp , p > 1 is min(p, 2) uniformly smooth. The next result4 shows that p-uniform smoothness is equivalent to being on “one side of the parallelogram equality”. Proposition 3.1.2. A Banach space X is p-uniformly smooth if, and only if, there exists a constant C > 0 such that, for all x, y ∈ X, kx + ykp + kx − ykp ≤ 2kxkp + Ckykp . 4

(3.1.2)

Due to P. Assuad (1974), and J. Hoffmann-Jorgensen (1974).

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Proof. Let us start with the observation that X is p-uniformly smooth if, and only if, there exists a constant K such that, for all x, y ∈ X,

p

x + y x − y kyk

2 + 2 ≤ kxk 1 + K kxkp . Now, assume that X is p-uniformly smooth. Then,

1 kykp . kx + yk − (kxk + kyk) + kx − yk − (kxk − kyk) ≤ K 2 kxkp If kyk ≤ kxk, then max(kx + yk, kx − yk) ≤ 2kxk. Given the following two elementary inequalities for real numbers u, and v, up − v p ≤ pup−1 (u − v), 1 |u + v|p + |u − v|p ≤ |u|p + |v|p , 2 we obtain the following estimate:

u, v ≥ 0, p ≥ 1, u, v ∈ R, 1 ≤ p ≤ 2,

1 kx + ykp + kx − ykp 2 ≤

p−1 kykp 1 kxkK (kxk + kyk)p + (kxk − kyk)p + p 2kxk 2 kxkp

≤ kxkp + kykp + p2p−1Kkykp = kxkp + (1 + p2p−1 K)kykp .

On the other hand, if kyk ≥ kxk, then max(kx+yk, kx−yk) ≤ 2kyk, so that 1 kx + ykp + kx − ykp ≤ kxkp + 2p kykp, 2

and consequently we get the desired inequality with C = max 2p , [1 + p2p−1 K] .

Conversely, if the inequality (3.1.2) is satisfied, then X is puniformly smooth because ρX (τ ) := sup k(x + y)/2k + k(x − y)/2k − 1 : kxk = 1, kyk = τ ✐

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51

p ≤ sup k(x + y)/2k + k(x − y)/2k − 1 : kxk = 1, kyk = τ n1 o ≤ sup k(x+y)kp +k(x−y)kp −1 : kxk = 1, kyk = τ ≤ cτ p , 2 which completes the proof of the proposition. QED Remark 3.1.1. Another way to look at uniform smoothness is related to the concept of the (Gα )-type of Banach space5 which is important to the study of random series in Banach spaces, and is defined as follows: A Banach space X is said to be of (Gα )-type, X ∈ (Gα ), 0 < α ≤ 1, if there exists a mapping G : x ∈ X : kxk = 1 7→ x∗ ∈ X : kx∗ k = 1 ,

such that

kG(x)k = kxkα ,

G(x)x = kxk1+α ,

and there exists a constant C > 0, such that, for all x, and y, of norm one, kG(x) − G(y)k| ≤ Ckx − ykα .

It turns out6 that X ∈ (Gα ) if, and only if, X is (1+α)- uniformly smooth which provides an alternative descriptions of p-uniform smoothness.

3.2

Martingales in uniformly smooth and uniformly convex spaces

Uniform smoothness. Now, we turn to the investigations of interrelations between uniform smoothness, and uniform convexity of Banach spaces, and the behavior of martingales in such spaces. Theorem 3.2.1.7 A Banach space X is p-uniformly smooth if, and only if, there exists a positive constant K such that, for 5

See W.A. Woyczy´ nski (1973). See J. Hoffmann-Jorgensen (1974). 7 Due to P. Assuad (1974). 6

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any two random vectors X1 , X2 ∈ Lp (Ω, Σ, P ; X), such that X1 mes Σ1 ,

X2 mes Σ2 ,

Σ1 ⊂ Σ2 ⊂ Σ,

(3.2.1)

and k2X1 − E(X2 |Σ1 )k ≥ kX1 k,

a.s.,

we have the inequality

E kX2 kp − kX1 kp Σ1 ≤ KE kX2 − X1 kp Σ1 .

(3.2.2)

Proof. Assume that X is p-uniformly smooth. Then, by Jensen’s Inequality and (3.2.1), E k2X1 − X2 kp Σ1 ≥ k2X1 − E(X2|Σ1 )kp ≥ kX1 kp ,

so that, by Proposition 3.1.2, substituting x = X1 , y = X2 − X1 , and averaging conditionally given Σ1 , we see that E kX2 kp + kX1 kp Σ1 ≤ E kX2 kp + k2X1 − X2 kp Σ1 ≤ 2E kX1 kp |Σ1 + CE kX2 − X1 kp Σ1 ,

which yields (3.2.2). Now, conversely, assume that (3.2.2) holds true for any X1 , X2 , defined above. So, in particular, given x, y ∈ X, we can take X1 = x, X2 = x + εy, where ε is a Bernoulli random variable, and Σ1 is a trivial σ-algebra. Then (3.2.1) is clearly satisfied because E(X1 |Σ1 ) = X1 , and for such X1 , X2 , the formula (3.2.2) becomes the inequality, E(kx + εykp − kxkp ) ≤ KEkεykp ,

x, y ∈ X,

which, in view of Proposition 3.1.2, means that X is p-uniformly smooth. QED Definition 3.2.1. A (Σn )-adapted sequence of random vectors in X, Xn mes Σn , Σn ⊂ Σn+1 ⊂ · · · ⊂ Σ,, n = 1, 2, . . . , is said to be a norm supermartingale if , for each n = 1, 2, . . . , k2Xn − E(Xn+1 |Σn )k ≥ kXn k.

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Example 3.2.1. Obviously, each martingale with values in any Banach space is a norm supermartingale. Also, if (Mn , Σn ) is a martingale with values in any Banach space, and (Xn ) is a sequence of positive, decreasing, and predictable (Xn+1 mes Σn ) real-valued random variables, then the sequencee (Xn Mn ) is a norm supermartingale (an application of this fact will be given later on in this section). Finally, any positive real-valued supermartingale is also a norm supermartingale, and some “rapidly” growing submartingales are norm supermartingales as well. The following result is a direct consequence of Theorem 3.2.1. Corollary 3.2.1. The Banach space X is p-uniformly smooth if, and only if, there exists a constant K > 0 such that, for any Xvalued norm supermartingale (Xn , Σn ), n = 0, 1, 2, . . . , in Lp (X), sup EkXn kp ≤ EkX0 kp + K n

∞ X n=1

EkXn+1 − Xn kp .

The above Corollary, Theorem 3.1.1, and the fact that every p-uniformly smooth space is reflexive (and thus dentable8 ), yield the following Theorem 3.2.2. If X is a p-uniformly smooth Banach space, 1P < p ≤ 2, and (Mn ) is an X-valued martingale such that p n EkMn+1 − Mn k < ∞, then there exists a random vector p M0 ∈ L (Ω, Σ∞ , P ; X) such that Mn → M∞ a.s. and in Lp . Applying the above theorem and the Kronecker Lemma to the martingale n X Mi − Mi−1 ′ Mn = , i i=1 we get the following Strong Law of Large Numbers for Martingales in Banach spaces: 8

See, e.g., V.D. Milman (1971), p 79.

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Corollary 3.2.2. If (Mn , Σn ) is a martingale with values in a p-uniformly smooth Banach space such that M0 = 0, and ∞ X EkMi − Mi−1 kp i=1

then

ip

Mn = 0, n→∞ n lim

< ∞,

a.s., and in Lp .

Remark 3.2.1. Theorem 3.2.2 and Corollary 3.2.2 generalize results for sums of independent Banach space valued random vectors.9 It is worth noticing that p-uniform smoothness is not uniquely characterized by the behavior of general martingales. Indeed, from the proof of Theorem 3.2.1, and Corollary 3.2.1, we immediately get the following characterization: Banach space X is p-uniformly smooth if, and only if, there exists a positive constant C such that, for any n = 1, 2, . . . , any x ∈ X, and any independent, zero-mean random vectors, X1 , . . . , Xn , with values in X, n n

p

X X

p kXi kp . Xi ≤ kxk + C E x + i=1

i=1

The behavior of Banach space valued martingales with independent increments, that is sums of independent random vectors, will be discussed at length in later chapters of this book in connection with p-type Banach spaces, that is spaces for which there exists a positive constant C such that, for any n = 1, 2, . . . , and any independent zero-mean random vectors X1 , . . . , Xn with values in X, n n

X

p X

E Xi ≤ C kXi kp . i=1

i=1

Despite the formal similarity of the concepts of p-uniform smoothness and p-type, they do not coincide. Of course, the above discussion shows that every p-uniformly smooth Banach space is also 9

See W.A. Woyczy´ nski (1973,1974).

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55

of p-type. However, the reverse implication is not true (even after renorming). There exists an example10 of a Beck-convex Banach space (thus of p-type for some p > 1) which is non-reflexive, so that it cannot be p-uniformly smooth for any p > 1. Uniform convexity. Now, we turn to the relationship between the concept of uniform convexity of a Banach space and the behavior of martingales with values in such spaces. Theorem 3.2.3.11 For a Banach space X the following four conditions are equivalent: (i) X is q-uniformly convex; (ii) There exists a constant L > 0 such that, for any x, y ∈ X, kx + ykq + kx − ykq ≥ 2kxk + Lkykq , (that is, we are on the other side of the parallelogram equality as compared with p-uniform smoothness); (iii) There exists a constant K > 0 such that, for any X-valued martingale (Mn , n = 0, 1, . . . ) ⊂ Lq , q

q

sup EkMn k ≥ EkM0 k + K n

∞ X n=1

EkMn+1 − Mn kq ;

(iv) There exists a constant K > 0 such that, for any two Xvalued random vectors in Lq , X1 mes Σ1 , and X2 mes Σ2 , with Σ1 ⊂ Σ2 , and such that k2X1 − E(X2|Σ1 )k = kX1 k, we have the inequality, E kX2 kq − kX1 kq |Σ1 ≥ KE kX2 − X1 kq |Σ1 . Proof. We shall prove the following implications : (i) =⇒ (iv) =⇒ (iii) =⇒ (ii) =⇒ (i). 10 11

Due to R.C. James (1972). Due to P. Assuad (1974).

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(i) =⇒ (iv). Assume that X is q-uniformly convex. Hence, it is also reflexive12 , and so is X ∗ , which, additionally, is p-uniformly smooth (with 1/p + 1/q = 1) in view of Corollary 3.2.1. Now, let Y1∗ , and Y2∗ , be two random vectors in p L (Ω, Σ, P; X ∗ ). Then, the random vectors X1∗ = E(Y1∗ |Σ1 ),

and

X2∗ − X1∗ = Y2∗ − E(Y2∗ |Σ1 ),

give rise to a martingale X1∗ , X2∗ , with values in X ∗ . In view of Theorem 3.2.1, E Y1∗ X1 + Y2∗ (X2 − X1 )

h i = E X1∗ X1 +X1∗ (X2 −X1 )+(X2∗ −X1∗ )X1 + Y2∗ −E(Y2∗ |Σ1 ) (X2 −X1 )

h i = E X1∗ X1 + X1∗ (X2 − X1 ) + (X2∗ − X1∗ )X1 + (X2∗ − X1∗ )(X2 − X1 ) 1 1 = EX2∗ X2 ≤ EkX2∗ kkX2 || ≤ EkX2∗ kp + EkX2 kq p q 1 ≤ EkX1∗ kp + p 1 ≤ E kY1∗ kP p

1 1 KEkX2∗ − X1∗ kp + EkX2 kq p q 1 p+1 1 ∗ p q + 2 KkY2 k + kX2 k . p q

Finally, because of the arbitrariness of Y1∗ , and Y2∗ , we get that 1 E kX1∗ kq + kX2 − X1 kq ≤ EkX2 kq , c where the last inequality is motivated by the inequality 1 1 E(Y1∗ X1 ) ≤ EkY1∗ kkX1 k ≤ EkY1∗ kp + EkX1 kq , p q a similar inequality for E(Y2∗ (X2 − X1 )), and the fact that if, for given reals c, b, the inequality, ab ≤ ap /p + c, is satisfied for all real a, then c ≤ bq /q. This completes the proof of the implication(i) =⇒ (iv). 12

See Theorem 1.3 in V.D. Milman (1971).

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(iv) =⇒ (iii). This implication follows directly by the operations of summing up and averaging. (iii) =⇒ (ii). This implication follows by setting M0 = x, M1 = x + εy, and Mn = M1 , for n = 2, 3, . . . . Here, ε is a Bernoulli random variable. (ii) =⇒ (i). Let us apply (ii) to the vectors x = kukv + kvku,

and

y = kukv − kvku,

thus obtaining the inequality,

2kukv q + 2kvku q ≥ 2 kukv + kvku q + L kukv − kvku q ,

wherefrom it follows that,

q

q 2q+1 kukq kvkq ≥ 2 kukv + kvku + L kukv − kvku ,

so that, for any u, v, such that kuk = kvk = 1, and ku − vk = ǫ, we get that

u + v L q

1−

2 ≥ 2q+1 ǫ .

Finally, because, for any real number α, 0 ≤ α ≤ 1, 1 − αq ≤ q(1 − α), we get that

u + v

δX (ǫ) = inf 1 −

2 : kuk = kvk = 1, ku − vk = ǫ

u + v q L 1

: kuk = kvk = 1, ku − vk = ǫ ≥ q+1 ǫq , ≥ inf 1 −

q 2 q2

which implies the q-unifrom convexity of X and completes the proof of the theorem. QED

Paley-Walsh Martingales. The results of the previous subsection can be refined using the concept of the Paley-Walsh martingale: Definition 3.2.2. Paley-Walsh martingale is the dyadic martingale on Ω = {+1, −1}×{+1, 1}×. . . , equipped with the Bernoulli

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product probability, and adapted to the natural σ-field filtration, Σn , spanned by the first n factors of the product space Ω. In this context, the following result characterises p-smooth and q-convex spaces: Theorem 3.2.4.13 (i) A Banach space X is q-uniformly convexifiable if, and only if, there exists a constant C > 0 such that, for any X-valued Paley-Walsh martingale (Mn ), q

q

C sup EkMn k ≥ EkM0 k + n

∞ X n=0

EkMn+1 − Mn kq ,

(3.2.3)

(ii) A Banach space X is p-uniformly smoothable if, and only if, there exists a constant C > 0 such that, for any X-valued Paley-Walsh martingale (Mn ), ∞ h i X p q sup EkMn k ≤ C EkM0 k + EkMn+1 − Mn k , p

n

(3.2.4)

n=0

Proof. The “only if” parts of statements (i), and (ii), automatically follow from the results of the previous subsection (Corollary 3.2.1 and Theorem 3.2.1). So, next, we will prove the “if” part of the statement (i) (The proof of (ii) is completely analogous and we will omit it). Assume that (3.2.3) holds true and define the new norm on X by means of the formula ∞ h i1/q X q q EkMn+1 − Mn k . ♯x♯ := inf C sup EkMn k − n

n=0

where the infimum is taken over all Paley-Walsh martingales (Mn ) with values in X, such that M0 = EMn = x, and satisfying the condition sup EkMn kq < ∞. Evidently, the norm ♯.♯ is positively homogeneous, and additionally it satisfied the inequality, kxk ≤ ♯x♯ ≤ C 1/q kxk, 13

Due to G. Pisier (1974).

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59

so it is equivalent to the original norm k.k. The first inequality follows directly from (3.2.3), and the second from the definition of ♯.♯, if one takes Mn = x, n = 0, 1, 2, . . . . So, in view of the above inequality and Theorem 3.2.3, it is sufficient to show that, for all x, y ∈ X, ♯(x + y)/2)♯q + k(x − y)/2kq ≤ ♯x♯q + ♯y♯q /2,

which also will show the convexity of the set {x : ♯x♯ ≤ 1}, that is the triangle inequality for the norm ♯.♯. Now, let x, y ∈ X. By the definition of the new norm, for any α > 0, there exist Paley-Walsh martingales, (Xn ), and (Yn ), with values in X such that X0 = x,

sup EkXn kq < ∞,

Y0 = y,

sup EkYn kq < ∞,

and C sup EkXn kq − n

q

C sup EkYn k − n

n

n

∞ X

EkXn+1 − Xn kq ≤ ♯x♯q + α,

∞ X

EkYn+1 − Yn kq ≤ ♯y♯q + α.

n=0

n=0

At this point we will construct a new Paley-Walsh martingale (Zn ) via the following formulas: Z0 =

x+y , 2

1 + ε1 1 − ε1 Xn−1 (ε2 , ε3, . . . ) + Yn−1 (ε2 , ε3, . . . ). 2 2 Evidently, supn EkZn kq < ∞, and Zn =

q

q

♯(x + y)/2♯ ≤ C sup EkZn k − n

∞ X n=0

EkZn+1 − Zn kq

EkXn kq + EkYn kq

x − y q ≤ C sup −

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Geometry and Martingales in Banach Spaces −

∞ X EkXn+1 − Xn kq + EkYn+1 − Yn kq

2

n=0

x − y q ♯x♯q + ♯y♯q

≤ +α−

, 2 2 which proves the theorem in view of the arbitrariness of α. QED Convergence of rearranged nonrandom series in puniformly smooth Banach spaces. The classical Steinitz Theorem states that if a subsequence of the partial sums of a real series converges then one can finid a rearrangement of the series which converges.14 The result below shows its validity for general p- uniformly smooth Banach spaces . Theorem 3.2.5.15 Let x be a p-uniformly Banach P smooth p space. If the sequence (xi ) ⊂ X is such that i kxi k < ∞, and nk X i=1

xi → x ∈ X,

as

k → ∞,

for a certain subsequence (nk ) ⊂ N , then there exists a rearrangement σ of positive integers such that ∞ X

xσ(i) = x.

i=1

Proof. Let us start with an observation that if X is p-uniformly smooth then there exists a constant K > 0 such that, for any n ∈ N , and any x1 , . . . , xn ∈ X such that x1 + · · · + xn = 0, one can find a permutation σ of the set (1, 2, . . . , n) such that, for each k, 1 ≤ k ≤ n, k k

X

p X

xσ(i) ≤ K kxσ(i) kp .

i=1

(3.2.5)

i=1

An analogue of this theorem for Lp has been demonstrated by M.I. Kadec (1954). 15 Due to P. Assuad (1974). 14

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Indeed, to prove the above statement it is sufficient to note that given such x1 , . . . , xn ∈ X, and taking as the probability space Ω the set of all permutations σ of the set {1, . . . , n} equipped with the uniform probability distribution, xσ(1) , . . . , xσ(n) , become exchangeable random variables so that Yk (σ) =

Xσ(1) + · · · + Xσ(n−k) , n−k

k = 0, 1, . . . , n − 1,

becomes a martingale with values in X (relative to the natural σalgebras). Since, for any real-valued random variables, ξ1 , . . . , ξn , such that E(ξi+1 |Σi ) ≥ 0, for i = 0, . . . , n − 1, one can find an ω ∈ Ω such that, for every i = 1, . . . , n, ξi (ω) ≥ 0, applying Theorem 3.2.1 to the norm supermartingale, Yk (n − k) = xσ(1) + · · · + xσ(n−k) , we obtain the existence of σ such that (3.2.5) is satisfied. An obvious calculation shows that, for any x1 , . . . , xn ∈ X, with x1 + · · · + xn = y, one can find a permutation σ of {1, . . . , n} such that, for each k, 1 ≤ k ≤ n, k k

p

X X

kxσ(i) kp , xσ(i) ≤ L kykp +

i=1

(3.2.6)

i=1

Thus, the above inequality P implies the statement of the theorem after rearranging the series i xi in blocks (ni , ni+1 − 1) according to the permutation guaranteeing (3.2.6). QED Remark 3.2.2. It is also worth recalling the theorem16 which states that every uniformly smooth (uniformly convex) Banach space may be renormed so that it becomes p-uniformly smooth (q-uniformly convex) for some p > 1 (q < ∞).

3.3

General concept of super-property

Definition 3.3.1. Let P be a property of a Banach space. We say that the Banach space X has the property super P if every Banach space Y that is finitely representable in X has the property P. 16

Due to G. Pisier (1974), see also Pisier’s monograph (2016).

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Recall that Y is said to be finitely representable in X if, for any ǫ > 0, and any finite-dimensional Y f d ⊂ Y , one can find a finite-dimensional subspace X f d ⊂ X such that dist(X f d , Y f d ) := inf kT k kT −1 k : isomorphism T : X f d 7→ Y f d ≤ 1 + ǫ.

The following implications are evident: super P =⇒ P, super (super P) =⇒ super P, (P =⇒ Q ) =⇒ (super P =⇒ super Q). Roughly speaking, X ∈ super-P if any Banach space Y the finite dimensional subspaces thereof are “similar” to those of X has P. Below, we collect some purely geometric characterizations of of the property of super-reflexivity that will be needed in further sections. Let us begin by introducing the concept of the finite tree property.17 Definiton 3.3.2. Banach space X is said to have the finite tree property if there exists an ǫ > 0 such that, for any n ∈ N, one can find a binary tree x(ε1 , . . . , εk ) : 1 ≤ k ≤ n, εi = ±1 , contained in the unit ball of X, and such that x(ε1 , . . . , εk ) =

1 x(ε1 , . . . , εk , 1) + x(ε1 , . . . , εk , −1) 2

and ǫ kx(ε1 , . . . , εk ) − x(ε1 , . . . , εk , εk+1)k ≥ , 2

k = 1, . . . n − 1.

Note that the above tree is essentially a finite Paley-Walsh martingale with uniformly big increments. 17

The concepts of super-reflexivity, and finite tree property, were introduced by R.C. James (1972).

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Definition 3.3.3. A sequence (xn ) ⊂ X is said to be basic with constant δ if, for any scalar sequence (αn ) ⊂ R, and all integers n, m, n

X

n+m

X

δ αi x i ≤ αi x i . i=1

i=1

In the following theorem we cite several characterizations of super-reflexivity without proofs. Theorem 3.3.1.18 The following properties of a Banach space X are equivalent: (i) X is super-reflexive; (ii) There exists an n ∈ N, and ǫ > 0, such that, for all x1 , . . . , xn ∈ X, k n

X X

xi − xi ≤ n(1 − ǫ) sup kxi k; inf

1≤k≤n

i=1

i=k+1

1≤i≤n

(iii) X does not possess the finite tree property; (iv) For each δ > 0, there exists a p > 1, and a constant C > 0, such that, for any finite basic sequence (xn ) ⊂ X with constant δ, X

X 1/p

; kxn kp xn ≤ C

(v) For each δ > 0, there exists a q < ∞, and a constant C > 0, such that, for any finite basic sequence (xn ) ⊂ X with constant δ,

X 1/q 1 X

kxn kq ; xn ≥ C (vi) L2 (Ω, µ; X), µ(Ω) > 0, is super-reflexive.

3.4

Martingales in super-reflexive Banach spaces

The following theorem provides a characterization of superreflexivity of X in terms of convergence properties of x-valued 18

For proofs, see J.J. Sch¨ affer and K. Sundaresan (1970) ((a) ⇔ (b)), R.C. James (1972, 1974) ((a) ⇔(c) ⇔(d) ⇔(e) ) , and G. Pisier (1974)((a) ⇔ (f)).

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martingales. For the definitions of the Radon-Nikodym Property (RNP ), and the Lp Martingale Convergence Theorem (MCTp ), see Chapter 2. Theorem 3.4.1.19 For the Banach space X the following properties are equivalent: (i) X is super-reflexive; (ii) X has the super-MCTp property; (iii) X has the super-RNP property; (iv) There exists a constant C > 0, and p > 1, such that, for each X-valued martingale (Mn ) ⊂ L2 , ∞ h i1/p X p p sup kMn k2 ≤ C kM0 k2 + kMn+1 − Mn k2 , n

(3.4.1)

n=0

where kMk2 := (EkMk2 )1/2 ; (v) There exists a constant C > 0, and p > 1, such that, for each X-valued Paley-Walsh martingale (Mn ) the inequality (3.3.1) holds true; (vi) There exists a constant C > 0, and q < ∞, such that, for each X-valued martingale (Mn ) ⊂ L2 , ∞ h i1/q X q q kM0 k2 + kMn+1 − Mn k2 ≤ C sup kMn k2 ; n=0

(3.4.2)

n

(vii) There exists a constant C > 0, and q < ∞, such that for each X-valued Paley-Walsh martingale (Mn ) the inequality (3.4.2) holds true. Proof. We will prove the following implications: (i) =⇒ (ii) =⇒ (i) , (ii) ⇔ (iii), (i) =⇒ (vi) =⇒ (vii) =⇒ (i), (i) =⇒ (iv) =⇒ (v) =⇒ (vii). (i) =⇒ (ii) This implication follows from the fact that superreflexive spaces are reflexive, and that reflexive spaces satisfy (MCTp ) (see Chapter 2). 19

Due to G. Pisier (1974).

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(ii) =⇒ (i) Assume that X is not super-reflexive, i.e., there exists a Banach space Y which is finitely representable in X, and which has the infinite ǫ-tree in its unit a ball, for some ǫ > 0. But the same tree can be viewed as a bounded Paley-Walsh martingale with values in X which does not converge a.s. This evidently contradicts (ii). (ii) =⇒ (iii) This follows directly from Theorem 2.2.1 in Chapter 2. (i) =⇒ (iv), and (i) =⇒ (vi), follow directly from Theorem 3.3.1 because the increments of a square integrable martingale form a basic sequence in L(Ω, Σ, P ; X) with constant 1. Indeed, for any (αi ) ⊂ R,

2 1/2 h n+m

2 i1/2 n+m

X

X

E αi (Mi −Mi−1 ) = E E αi (Mi −Mi−1 ) Σn i=1

i=1

2 i1/2 h n+m

2 i1/2 h n+m X

X

≥ E E αi (Mi −Mi−1 ) Σn = E αi (Mi −Mi−1 ) i=1

i=1

(iv) =⇒ (v), and (vi) =⇒ (vii) are obvious implications. (vii) =⇒ (i) Assume that X is not super-reflexive. Then, by Theorem 3.1.1, it has the finite tree property, that is there exists an ǫ > 0 such that, for all n ∈ N, there exists a Paley-Walsh martingale (Mn ) of length n, with values in the unit ball of X, such that kMn+1 − Mn k ≥ ǫ/2.

Thus, were the inequality (3.4.2) satisfied, we would have that n

2 1/2

1 ≥ (EkMn k )

i1/q ǫ 1/q 1 hX kMn − Mn−1 kq2 ≥ n , ≥ C k=1 2C

a contradiction. (v) =⇒ (vii) We will prove this implication for X ∗ but that suffices because, by the implication (vii) =⇒ (i), we would have that X ∗ is super-reflexive, and so is X, since super-reflexivity is a self-dual property.20 20

See R.C. James (1972).

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So, let (m∗N ) be a Walsh-Paley martingale with values in X ∗ . Then n 1/q X ∗ q ∗ kM0 k2 + kMk∗ − Mk−1 kq2 k=1

n n X ∗ = sup EM0 X0 + E(MK∗ − Mk−1 )Xk : kX0 kp2 k=1

+

n X k=1

kXk kp2 ≤ 1, Xi ∈ Lp (X)

o

n n X ∗ o ∗ = sup E[Mn E(X0 |Σ0 )] + E Mn (E(Xk |Σk ) − E(Xk |Σk−1 ) k=1

n o n X

≤ kMn∗ k2 sup E(X0 |Σ0 ) + E(Xk |Σk ) − E(Xk |Σk−1 ) 2

k=1

≤

CkMn∗ k2

sup

n

kE(X0|Σ0 )kp2 +

n X k=1

kE(Xk |Σk )−E(Xk |Σk−1 )kp2

o1/p

≤ 2CkMn∗ k2 , which completes the proof of the Theorem. QED Remark 3.4.1. One can show21 that the super-reflexivity is equivalent to uniform convexifiability, and this property, in turn, is equivalent to uniform smoothability. Actually, in this case the space may be equipped with an equivalent norm which is at the same time uniformly smooth and uniformly convex. Using the martingale techniques developed above, one can demonstrate,22 that uniform convexifiability is equivalent to q-uniform convexifiability for some q < ∞.

21 22

See P. Enflo (1972). See G. Pisier (1974).

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Chapter 4 Spaces that do not contain c0 4.1

Boundedness and convergence of random series

Let us begin by discussing the boundedness of the vector-valued Rademacher series, and its relationship to the Banach space c0 of all real sequences convergent to 0.. The Rademacher vector sums PnDefinition 4.1.1. i=1 ri xi , P(ri = ±1) = 1/2, (xi ) ⊂ X, are said to be almost surely bounded if n

X

P sup ri xi = ∞ = 0. n

i=1

P It is clear that ni=1 ri xi are if, for Pn a.s. bounded if, and only 1 each ǫ ∈ (0, 1), the gauges Jǫ ( i=1 ri xi ; P) are bounded.

Theorem 4.1.1.2 The following properties of the Banach space X are equivalent: (i) X does not contain a subspace isomorphic to c0 ; 1 2

See Chapter 1 for the definition of gauges Jǫ . Due to S. Kwapie´ n (1974).

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(ii) For each sequence (xn ) ⊂ X such that bounded, we have xn → 0, as n → ∞.

Pn

i=1 ri xi

is a.s.

Proof. (ii) =⇒ (i). Assume P X ⊃ c0 , and let (ei ) be the canonical basis in c0 . Then, k ni=1 ri ei k = 1, but still en ’s do not converge to 0. (i) =⇒ (ii). Suppose (ii) is not satisfied. Then there exists a sequence (xi ) ⊂ X, with inf i kxi k > 0, and a constant M < ∞, such that n

1 X

P sup ri xi < M > . 2 n i=1 Utilizing the fact that for each set A in the σ-algebra spanned by (ri , i ∈ N), lim P(A ∩ (ri = 1)) = lim P(A ∩ (ri = −1)) = P(A)/2, i

i

we can find by induction a sequence ni ↑ ∞ of integers such that for every sequence of signs, εi = ±1, n

X

P sup ri xi < M, εi rni = 1, i = 1, . . . , k n

(4.1.1)

i=1

n

1 1

X > P sup ri xi < M, εi rni = 1, i = 1, . . . , k − 1 > 2−k , 2 2 n i=1

k ∈ N.

Let us now define ri′ = ri , for i ∈ (ni ), and ri′ = −ri , for i ∈ / (ni ). ′ Since (ri ) and (ri ) are identically distributed, for each εi = ±1, and each k ∈ N, n

X

′ ′ ri xi < M, εi rni = 1, i = 1, . . . , k > 2−(k+1) . (4.1.2) P sup n

i=1

Since P(εi rni = 1, i = 1, . . . , k) = 1/2k , it follows from (4.1.1), and (4.1.2), that for each ε = ±1, and each k ∈ N, there exists an ω ∈ Ω such that k

X

ε x

i ni = i=1

nk nk

X 1

X

′ r (ω)x + r (ω)x

i i i < M. i 2 i=1 i=1

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Spaces that do not contain c0

69

P Therefore, the series i xni is weakly unconditionally convergent with inf kxni k > 0 and, by the Bessaga-Pelczynski Theorem,3 there exists a subsequence (y i ) ⊂ (xni ) such that (y i ) is isomorphic to the canonical basis (ei ) ⊂ c0 . QED The above result can be extended to the case of general series of independent random vectors with values in a Banach space X. Theorem 4.1.2.4 Let 1 ≤ p < ∞. The following properties of a Banach space X are equivalent: (i) X does not contain an isomorphic copy of c0 ; (ii) Lp (Ω, Σ, P; X) does not contain an isomorphic copy of c0 ; Pn(iii) For each sequence (xn ) ⊂ X suchP that the sums i=1 ri xi , n ∈ N, are a.s. bounded, the series i ri xi converges a.s. ; (iv) For each sequence (Xn ) of symmetric, and independent Pn random vectors in X, the a.s. boundedness P of the sums i=1 Xi implies the a.s. convergence of the series i Xi .

Proof. We will prove the following implications: (iii) ≡ (iv), (i) =⇒ (iii) =⇒ (ii) =⇒ (i). (iii) ≡ (iv). The equivalence follows immediately from the following Lemma which itself is a straightforward corollary to the Fubini Theorem, and the fact that if (Xi ), and (ri ) are independent and (Xi ) is sign-invariant, then (Xi ), and (ri Xi ) are identically distributed. Lemma 4.1.1. If (Xn ) is a sign-invariant sequence of random vectorsPin a Banach space X, and (rn ) is independent of (Xn ), then ni=1 Xi is a.s. bounded P [convergent] if, and only if, for almost every ω ∈ Ω, the sums ni=1 ri Xi (ω) are a.s. bounded [ convergent]. (i) =⇒ (iii). Assume P that (iii) is not satisfied, and let (xi ) ⊂ X be such that i ri xi is a.s. bounded, but not a.s. convergent. In view of the Kahane Theorem (see, Chapter 1), Pn i=1 ri xi is not a Cauchy sequence in L1 (X), so that there exists 3

See C. Bessaga and A. Pelczynski (1958). See J. Hoffmann-Jorgensen (1972/73, 1974). A variant of the proof for (i) =⇒ (ii) can also be found in J. Hoffmann-Jorgensen (1973/74). 4

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an ǫ > 0, and a sequence ni ↑ ∞, such that, for each i ∈ N, EkXi k > ǫ, where X Xi := rj xj , i ∈ N. ni ≤j 0, since EkXi k ≥ ǫ. Now, from Theorem 4.1.1, it follows that X contains an isomophic copy of c0 . (iii) =⇒ (ii). Suppose, to the contrary, that Lp (X) contains a copy of c0 . This implies the existence of constants a, b, c > 0, and a sequence (Xn ) ⊂ Lp (X), such that 1/p a ≤ EkXn kp ≤ b, n ∈ N, (4.1.3) and n

p 1/p X

cj X j ≤ c max |cj |, E j=1

1≤j≤n

n ∈ N, (cj ) ⊂ R. (4.1.4)

Define Xj′ = rj Xj , where (rj ) are independent of (Xj ). PnEvidently, ′ (Xj ) is sign-invariant, satisfies (4.1.3-4), and Yn =P j=1 Xj′ are bounded in Lp (X). Thus, by the above Lemma, rj′ Xj′ (ω) are a.s. bounded for almost all ω ∈ Ω (with (rj′ ) being independent P from (Xj′ ) ). Therefore, (iii) cannot hold because, were rj′ Xj (ω) convergent a.s., then also Yn would converge a.s. to, say, Y (again, in view of the above Lemma), and by Fatou Lemma, EkY k ≤ lim inf EkYn kp ≤ cp . n

Since P(sup kYn k ≥ t) ≤ 2P(kY k > t), we would have sup kYn k ∈ Lp . Therefore Yn → Y in Lp (X) by the Lebesgue Dominated

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Convergence Theorem, which would contradict EkXj′ kp > ap (i.e., (4.1.3)). (ii) =⇒ (i). This implication is obvious since X ⊂ Lp (Ω, Σ, P; X). QED Theorem 4.1.3.5 Let X be a Banach space which does not contain an isomorphic copy of c0 . If (Xi ) is a sequence of P independent random i conP vectors in X such that the series i XP verges a.s., and i Xi ∈ L∞ (Ω, Σ, P; X), then the series i Xi converges in L∞ (Ω, Σ, P; X). P Proof. Assume that i Xi ∈ L∞ (X). P By Hoffman-Jorgensen P Theorem (Part (ii)), it follows that E i Xi = i EXi , so that we can assume, without loss of generality, that EXi = 0, i ∈ N. P Suppose, to the contrary, that i Xi does not converge in L∞ (X). So, in particular, there exists a constant a > 0, and a sequence 0 = n0 < n1 < . . . of integers such that, for every j ∈ N we have ess sup kYj k > a, where nj+1

Yj =

X

Xi .

i=nj +1

P

P

Evidently, j Yj = i Xi converges a.s. Now, take a sequence (Zi ) of zero-mean random vectors in X such that, for every j ∈ N, Zj =

∞ X k=1

xjk IAjk ,

and ess sup kZj − Yj k ≤ a2−(j+1) ,

where (Ajk ), k ∈ N, are pairwise disjoint, Ajk ∈ σ(Yj ), and P (xjk ) ⊂ X. Clearly, the P series of independent random vectors j Zj conP verges a.s., and j Zj ∈ L∞ (X) because j ess sup kZj − Yj k ≤ a < ∞. Furthermore, for each j ∈ N, ess sup kZj k ≥ ess sup kYj k − a2−(j+1) > a/2, so that there exists a subsequence (kj ) ⊂ N such that kxjkj k ≥ a/2, and P(Ajkj ) > 0. 5

See J. Hoffmann-Jorgensen (1972/73).

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By the Hoffmann-Jorgensen Theorem there exists a constant K > 0 such that, for each n ∈ N, n

X

ess sup Zj ≤ K. j=1

Then, by a symmetrization argument we get that, for every B ⊂ {0, . . . , n}, n

X

X

Zj ≤ K, Zj ≤ ess sup ess sup j=1

j∈B

so that there exists Ω0 ∈ Σ, with P(Ω0 ) = 1, such that, for all finite B ⊂ N,

X

Zj ≤ K. ess sup j∈B

Since Tn P(Ajkj ) > 0, and the events (Ajkj ) are independent, P j=1 T Ajkj > 0, and, for each n ∈ N, there exists an ωn ∈ Ω0 ∩ nj=1 Ajkj . Because Zj (ωn ) = y j , 0 ≤ j ≤ n, we have that

X

y j ≤ K,

j∈B

for each B ⊂ {1, . .P . , n}, and each n ∈ N. Therefore, for each ∗ ∗ x ∈ X , we have j |x∗ y j | ≤ 2Kkx∗ k, and since ky j k ≥ a/2, the space X contains a copy of c0 in view of Bessaga-Pelczynski Theorem. A contradiction. QED

4.2

Pre-Gaussian random vectors

In this section we will consider the situation when the summands are in the domain of attraction of a Gaussian random vector with values in a Banach space X. Theorem 4.2.1.6 The following two properties of a Banach space X are equivalent: 6

See G. Pisier and J. Zinn (1978).

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(i) X does not contain an isomorphic copy of c0 ; (ii) For each sequence (Xn ) of independent and identically distributed random vectors in X such that supn EkX1 + · · · + Xn k/n1/2 < ∞ there exists a Gaussian random vector Y in X such that E exp[ix∗ Y ] = exp[−E(x∗ X1 )2 ],

x∗ ∈ X ∗ .

(4.2.1)

Proof. (ii) =⇒ (i). It is sufficient to construct an appropriate example let (en ) be the canonical basis in c0 , and let P in c0 . So, −1/2 X = r e log n. Evidently, X is a random vector in c0 n n n and, if X1 , X2 , . . . , are independent copies of X, then, in view of the Khinchine Inequality, (k) EkX1 +· · ·+Xn k/n1/2 = n−1/2 E sup r1 +· · ·+rn(k) log−1/2 k ≤ const. k

On the other hand,

exp[−E(x∗ X)2 ] = exp[−

X

αn2 / log n],

n

x∗ = (αn ) ∈ l1 ,

so thatP the Gaussian random vector Y would have the represen−1/2 tation n, where γn are independent and identin γn en log cally distributed real symmetric Gaussian random variables. HowP ever, the latter series diverges a.s. in c0 because n P(|γn | > 1/2 −1/2 log n) = ∞, so that |γn | log n ≥ 1 infinitely often with probability 1. A contradiction. (i) =⇒ (ii). Assume that X does not contain c0 , and that (Xn ) is such that supn EkX1 + · · · + Xn k/n1/2 < ∞. Let (k) (Xn ), n, k ∈ N, be independent copies of a martingale of simple random vectors in X, with finite range, and such that (k)

lim EkX1 − X1 k = 0.

k→∞

(4.2.2)

Then (k)

sup EkX1 +· · ·+Xn(K) k/n1/2 ≤ sup EkX1+· · ·+Xn k/n1/2 < const., N

n

(4.2.3)

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where the constant is independent of k. By the Central Limit (k) Theorem in the finite-dimensional subspace of X spanned by X1 , (k) there exists a Gaussian random vector Y1 such that (k)

(k)

E(x∗ Y1 )2 = E(x∗ X1 )2 ,

∀k ∈ N,

(4.2.4)

and for which, in view of (4.2.3), (k)

(k)

sup EkY1 k ≤ sup lim sup EkX1 + . . . Xn(k) k/n1/2 < const. k

k

n→∞

(4.2.5) (k) On the other hand, from (4.2.4), and from the fact that (X1 ) (k) (k−1) is a martingale, it follows that (Y1 − Y1 ), k ∈ N, form a sequence of independent Gaussian random vectors in X. Now, P (k) (k−1) (4.2.5) implies that the partial sums of − Y1 ) are k (Y1 bounded a.s. Since, X does not contain a copy of c0 , by Theorem (k) 4.1.2, there exists Y = limk→∞ Y1 (a.s., but also in L2 ) which is a desired Gaussian random vector in X satisfying (4.1.1), because of (4.1.2) and (4.1.4). QED The Banach spaces X in which c0 is not finitely representable can also be characterized by the behavior of sums of X-valued random vectors. These results will be discussed at length in Chapter 5 since the proofs depend on the techniques developed for spaces of cotype q.

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Chapter 5 Cotypes of Banach spaces 5.1

Infracotypes of Banach spaces

Let us begin by introducing a purely geometric concept of infracotype1 of a normed space and define, for q ∈ [1, ∞], and n ∈ N, the numerical constants n n X 1/q q + dn (X) := inf d ∈ R : x1 , . . . , xn ∈ X, kxi kq i=1

n

o

X

εi xi . ≤ d sup εi =±1

i=1

Definiton 5.1.1. A normed space X is said to be of infracotype q if there exists a constant C > 0, such that, for each n ∈ N dqn (X) ≤ C < ∞. In other words, X is of infracotype q if, and only if, for some constant C > 0, and any finite sequence (xi ) ⊂ X, n X i=1

1

kxi kq

n

X

1/q

≤ d sup εi xi . εi =±1

i=1

In the classical terminology the space is of infracotype q if, and only if, the identity operator is (1, q)-absolutelly summing. Results of this section are in the spirit of those obtained by G. Pisier (1973) for spaces of infratype p (see, Chapter 6.1).

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It is also easy to see that X is of infracotype P q if, and only if, for each unconditionally convergent series i xi in X, we have P q i kxi k < ∞. Also, in view of the Dvoretzky Theorem of Chapter 1, the following Proposition is evident: Proposition 5.1.1. (i) The space Rn is of infracotype 1, for each n ∈ N. (ii) If X is infinite-dimensional and of infracotype q, then necessarily q ≥ 1. (iii) If X is of infracotype q, then dqn (X) ≥ n1/q−1 . (iv) If X is of infracotype q, and q1 > q, then X is of infracotype q1 . The case q = 1 plays a special role in the investigation of the infracotypes of normed spaces and here are some properties, such as the monotonicity and submultiplicativity, of the sequence d1n (X): Proposition 5.1.2. (i) If X 6= {0}, then 1 ≤ d1n (X) ≤ n, and if X is infinite-dimensional then n1/2 ≤ d1n (X) ≤ n, (ii) If n ≤ m, then d1n (X) ≤ d1m (X). (iii) For any n, k ∈ N, d1nk (X) ≤ d1n (X) · d1k (X). 1 Proof. (i) The P fact that dn (X) ≤ n follows from the inequality kxi k ≤ sup{k i εi xi k : ε = ±1} which, in turn, is a consequence of the fact that, for any x, y ∈ X either kxk ≤ kx + yk, or kxk ≤ kx − yk. The inequality d1n (X) ≥ n1/2 is an immediate consequence of the Dvoretzky Theorem of Chapter 1. (ii) This statement follows directly from the definition of 1 dn (X).

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(iii) Let x1 , . . . , xnk ∈ X. Choose εij , (i − 1)k < j ≤ ik, i = 1, . . . , n, so that ik

X

j=ik−k+1

and define

yi =

εij xj

ik X

ik

X

= sup εj =±1

εij xj ,

j=ij−k+1

εj xj ,

i = 1, . . . , n.

j=ik−k+1

By the construction itself, ik X

j=ik−k+1

kxi k ≤ d1k (X)ky i k,

so that nk X j=1

kxi k ≤

d1k (X)ky i k

≤

d1k (X)d1n (X)

n

X

sup εi y i

εi =±1

i=1

nk

X

εi y i , ≤ d1k (X)d1n (X) sup εi =±1

i=1

which completes the proof of the Proposition. QED It turns out that the sequence d1n (X) contains complete information about the infracotype of the space X. Indeed, we have the following Proposition 5.1.3. If X 6= {0} is a normed space, and n0 > 1, then there exists q0 , 1 ≤ q0 ≤ ∞ (2 ≤ q0 ≤ ∞, if X is infinite-dimensional) such that 1−1/q0

d1n0 (X) = n0

,

and, for each q > q0 , X is of infracotype q. Proof. The existence of such a q0 follows from Proposition 5.1.2 (i). Using the submultiplicativity property proven in Proposition 5.1.2 (iii) we immediately obtain that k(1−1/q0 )

d1nk (X) ≤ n0 0

,

k ∈ N.

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Put C = n0

, and consider n ∈ [nk0 , nk+1 0 ). Then. (k+1)(1−1/q0 )

d1n (X) ≤ d1nk+1 (X) ≤ n0

≤ Cn1−1/q0 .

0

Now, take x1 , . . . , xn ∈ X, and order them in such a way that kxi+1 k ≤ kxi k. Then, kxk k = inf kxj k ≤ k 1≤j≤k

≤

k −1 d1k (X)

−1

k X j=1

kxj k

k

X

εj xj ≤ Ck −1 k 1−1/q0 sup sup

εj =±1

εj =±1

j=1

so that, if q > q0 , then n X k=1

kxk k

q

1/q

k

X

εj xj ,

≤C

∞ X

k

−q/q0

k=1

1/q

j=1

n

X

εj xj , sup

εj =±1

j=1

which shows that X is of infracotype q. QED

In the above context let us define the quantity, qinf (X) = inf{q : X is of infracotype q}. It turns out that qinf (X) can be explicitely expressed in terms of the quantities d1n (X). Theorem 5.1.1. Let 0 < q < ∞, and X 6= {0}. Then log n . n→∞ log n/d1 (X) n

qinf (X) = lim

Proof. If X is of infracotype q then, by H¨older Inequality, n1/q−1

n X i=1

kxi k ≤

n X i=1

kxi kq

so that d1n (X) ≤ Cn1−1/q , and lim sup n→∞

1/q

n

X

εi xi , ≤ C sup εi =±1

i=1

log n log n ≤ lim sup = q. 1 log n/dn (X) n→∞ (1/q) log n − log C

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Cotypes of Banach spaces Therefore, lim sup n→∞

log n ≤ qinf (X). log n/d1n (X)

If q < qinf (X), then, by Proposition 5.1.3, d1n (X) > n1−1/q for each n ∈ N, and lim inf n→∞

QED

5.2

log n log n ≥ = q ≥ qinf (X). 1 (1/q) log n log n/dn (X)

Spaces of Rademacher cotype

In this section we will adapt the definition of the infracotype to the random environment by replacing the deterministic εi = ±1 signs used in the previous section by the random Rademacher coefficients. So, for a normed space X, q > 0, and an arbitrary n ∈ N, we define the constants, cqn (X)

n

+

:= inf c ∈ R : x1 , . . . , xn ∈ X,

n X i=1

kxi k

q

1/q

n

q 1/q o X

. ri x i ≤ c E i=1

Definition 5.2.1. The normed space X is said to be of Rademacher cotype q (or, simply, of cotype q)2 , if there exists a constant C such that, for each n ∈ N, cqn (X) ≤ C < ∞. 2

The notion of cotype 2 was introduced by D. Mouchtari (1973) who called it the Mazur-Orlicz Property, by E. Dubinsky, A. Pelczynski, and H.P. Rosenthal, who called it the superquadraticity of Rademacher averages, and by B. Maurey (1972/73). For general q, the notion was considered by J. HoffmannJorgensen who called it weak cotype q. Results of this section are due to B. Maurey and G. Pisier (1976).

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Note that, because of Kahane Theorem of Chapter 1, X is of cotype q if, and only if, there exists a constant C < ∞, and an α, 0 < α < ∞, such that for any finite sequence (xn ) ⊂ X, n X i=1

n

α 1/α o X 1/q

kxi k ≤ C E ri x i q

(5.2.1)

i=1

or, alternatively, X is of cotype q if, and only if, for each α, 0 < α < ∞ there exists a constant C = Cα such that, for any finite sequence (xi ) ⊂ X, the inequality (5.2.1) holds true.

Remark 5.2.1. If one replaces in the above definition (with apppripriate modifications) Rademacher random variables (ri ) by stable random variables (θi ) of exponent q, then one obtains a definition of a space of stable-cotype q. However, because of the tail behavior of (θi ), and Borel-Cantelli Lemma, each Banach space is of stable cotype q if q < 2. For q = 2, the stable cotype and Rademacher cotype coincide as we will see later on in this chapter. Remark 5.2.2. In Banach lattices, and spaces with unconditional basis (and in spaces with local unconditional structure), if q > 2, then X is of infracotype q if, and only if, X is of cotype q.3 Proposition 5.2.1. (i) If X is of cotype q, and q1 > q, then X is of cotype q1 . (ii) If X is of cotype q and X = 6 {0}, then q ≥ 2. (iii) If X is of cotype q, then X is of infracotype q.

We omit the obvious proof of (i). The implication (ii) immediately follows from the Khinchine Inequality on the real line, and the implication (iii) follows directly from the definitions. The properties of constants cqn (X), are similar to the properties of constants d1n (X) investigated in the previous section in connection of the infracotype of a Banach space. Proposition 5.2.2. (i) The sequence (cqn (X)) is monotone, that is, if n ≤ m, n, m ∈ N, then cqn (X) ≤ cqm (X). (ii) For any q, 0 < q < ∞, and n, k ∈ N, cqnk (X) ≤ cqn (X) · cqk (X). 3

See B. Maurey (1973/74).

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Proof. The property (i) is obvious, so we will only prove (ii). Take x1 , . . . , xnk ∈ X and, for each i = 1, . . . , n, define the random vectors ik X rj xj , Xi = j=ik−k+1

so that, if (rj′ ) is a Rademacher sequence independent of (rj ) then n X i=1

kXi (ω)k

q

1/q

≤

cqn (X)

n

q 1/q X

′ ′ , E ri Xi (ω) i=1

where E′ denotes integration with respect to (rj′ ). Furthermore, by the symmetry argument, nk X i=1

≤ cqk (X)

n X i=1

kxi k

EkXi kq =

q

1/q

1/q

=

ik X

n X

i=1 j=ik−k+1

kxi kq

1/q

n

q 1//q

X

ri′ Xi (ω) ≤ cqk (X)cqn (X) EE′ i=1

cqk (X)cqn (X)

nk

q 1//q X

. ri x i E

QED

i=1

Knowledge of the sequence (cqn (X)) for any particular q, 0 < q < ∞, provides information about the cotype of X. Proposition 5.2.3. If X 6= {0} is a normed space, and n0 > 1, 0 < q < ∞, then there exists a q0 ≥ q such that 1/q−1/q0

cqn0 (X) = n0

,

and X is of cotype q1 , for each q1 > q0 . Proof. The existence of such a q0 is obvious because cqn ≥ 1. Using the submultiplicativity property we get the inequality k(1/q−1/q0 )

cqnk (X) ≤ n0 0

,

k ∈ N.

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Martingales and Geometry in Banach Spaces 1/q−1/q0

Now, set C = n0 Then,

, and select any integer n ∈ [nk0 , nk+1 0 ). (k+1)(1/q−1/q0 )

cqn (X) ≤ cqnk+1 (X) ≤ n0 0

≤ Cn1/q−1/q0 .

Let x1 , . . . , xn ∈ X, and assume that these vectors are ordered in such a way that kxi+1 k ≤ kxi k. In this case, k 1/q X −1 kxk k = inf kxj k ≤ k kxj kq 1≤j≤k

j=1

n k

q 1/q

q 1/q X X

rj xj ≤ k −1/q cqk (X) E rj xj ≤ k −1/q cqk (X) E j=1

j=1

≤k

−1/q

Ck

1/q−1/q0

j=1

so that, if q1 > q0 , then n X k=1

kxk kq1

1/q1

n

q 1/q X

, rj x j E

≤C

∞ X

k −q1 /q0

n

q 1/q 1/q1 X

, rj x j E j=1

k=1

which, in view of (5.2.1), shows that X is of cotype q1 . QED Now, proceeding in a way analogous to that of Section 5.1.1, let us define the quantity qrad (X) = inf{q : X is of cotype q}. And again, it turns out that qrad (X) can be explicitly expressed in terms of the quantities cpn (X). Theorem 5.2.1. Let 0 < p < ∞, and X = 6 {0}. Then qrad (X) = lim

n→∞

log n log

n1/p /cpn (X)

.

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Proof. If X is of cotype q then, in view of (5.2.1), and H¨older inequality, 1/q−1/p

n

n X i=1

kxi k

p

1/p

≤

n X i=1

kxi k

q

1/q

Therefore, cpn (X) ≤ Cn1/p−1/q , and lim sup n→∞

log n log

n1/p /cpn (X)

so that lim sup n→∞

≤ lim sup n→∞

n

p 1/p X

≤ C E ri x i . i=1

log n =q (1/q) log n − log C

log n ≤ qrad (X). log n1/p /cpn (X)

If q < qrad (X) then, by Proposition 5.2.3, cpn (X) > n1/p−1/q , for all n ∈ N, and lim inf n→∞

log n log

n1/p /cpn (X)

so that lim sup n→∞

log n log

n1/p /cpn (X)

≥

log n = q, (1/q) log n

≥ qrad (X).

QED

As far as subspaces of spaces of cotype q are concerned, the following Proposition is evident. Proposition 5.2.4. If X is of cotype q, and Y is a subspace of X then Y is of cotype q, and cqn (X) ≥ cqn (Y ), for each q > 0, and n ∈ N. Less trivial is the following result which shows circumstances under which the cotype of X is preserved by formation of spaces Lp (X).

Theorem 5.2.2. Let (T, Σ, µ) be a σ-finite measure space. Then, (i) cqn (X) = cqn Lq (T, Σ, µ; X) , and (ii) if p ≤ q then X is of cotype q if, and only if, Lp (T, Σ, µ; X) is of cotype q.

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Proof. (i) Evidently, cqn (X) ≤ cqn Lq (T, Σ, µ; X) . On the other hand, if (Xi ) ⊂ Lq (X), then for each t ∈ T , n X i=1

q

|Xi (t)k ≤

n

X

(cqn (X))q E

i=1

q

ri Xi (t) .

Integrating both sides with respect to µ we get that cqn (X) ≤ cqn Lq (X) , which concludes the proof of (i). (ii) The“if” part being evident, let us assume that X is of cotype q. Then, by (5.2.1), if (Xi ) ⊂ Lp (X) then, for each t ∈ T , n X i=1

kXi (t)k

q

1/q

n

p 1/p X

, ri Xi (t) ≤ C E i=1

so that, because q ≥ p, n Z n X q/p 1/q Z X p/q 1/p p q kXi (t)k µ(dt) ≤ kXi (t)k µ(dt) i=1

T

T

i=1

n

p 1/p Z X

. ri Xi (t) µ(dt) ≤C E T

QED

i=1

Corollary 5.2.1. The following two properties of a normed space X are equivalent: (i) X is of cotype q; (ii) There exists a constant C > 0 such that, for each finite sequence, X1 , . . . , Xn , of independent, zero-mean random vectors in Lq (X), n X i=1

EkXi kq

1/q

n

q 1/q X

Xi . ≤ C E i=1

Proof. The implication (ii) =⇒ (i) is obvious. The reverse implication, (i) =⇒ (ii) follows from the above Theorem 5.2.2 by the standard symmetrization procedure. QED The above results permit us to give a number of concrete examples of Banach spaces which are (or, are not) of cotype q. More examples will be provided at the end of this chapter.

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85

Example 5.2.1. (i) The space c0 is not of cotype q, for any q < ∞. Indeed, take (xi ) = (ei ), where the latter is the canonical basis in c0 . Then the left-hand side ofP(5.2.1) is equal to n1/q , while the right-hand side (for q = 1), Ek i ri ei k = E max |ri | ≤ 1. A contradiction. (ii) As a consequence of (i), neither l∞ , nor L∞ , nor C[0, 1], is of cotype q, for any q < ∞. (iii) By Theorem 5.2.2, and Khinchine Inequality, for any n ∈ N, the space Rn ,and any Hilbert space, are of cotype 2. (iv) The spaces Lp , and lp , are of cotype max(2, p). This fact is a direct consequence of Theorem 5.2.2 and Proposition 5.2.1. (v) More generally, if the modulus of convexity of X (see, Chapter 3) satisfies the asymptotic condition, δX (ǫ) = O(ǫq ), as ǫ → 0, then X is of cotype q. (vi)4 The triple projective tensor product space X = lp1 ⊗π lp2 ⊗π lp3 , with 1 ≤ p1 ≤ p2 ≤ p3 ≤ ∞, is not of cotype q in the following cases: • p1 ≤ 2, 1/p1 + 1/p2 < 1/2, and 1/p1 + 1/p2 + 1/p3 ≥ 1, and for q < 3/(1 + 1/p2 + 1/p3 ); • p1 < 2, 1/p1 + 1/p2 + 1/p3 ≤ 1, and for q < 3/(2 − 1/p1 ); • 1/p1 + 1/p2 + 1/p3 < 1/2, and for q < 3/(1 + 1/p1 + 1/p2 + 1/p3 ).

5.3

Local structure of spaces of cotype q

The following Proposition follows directly from the definitions. Proposition 5.3.1. Cotype q is a superproperty, i.e., if X is of cotype q and Y is finitely representable in X, then Y is also of cotype q. In particular, by the local reflexivity principle (see Section 1.3) X is of cotype q if, and only if, the space X ∗∗ is of cotype q. 4

Due to O. Giladi, J. Prochno, C. Sch¨ utt, N. Tomczak-Jaegermann, and E. Werner (2017). Results on cotypes of k-fold projective tensor products of lp spaces are also available in the paper.

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The next result about the local structure is much more subtle and requires a nontrivial proof. Theorem 5.3.1.5 If X is an infinite-dimensional normed space, then the canonical injection lqrad (X) 7→ l∞ , is finitely factorable through X. Proof. If qrad (X) = 2, then the Theorem is a corollary to the Dvoretzky-Rogers Lemma from Section 1.3 which states, in our current terminology, that the canonical injection l2 7→ l∞ is finitely factorable through any infinite-dimensional normed space. If qrad (X) > 2, then it is sufficient to show that, for any q such that qrad (X) > q > 2, the injection lq 7→ l∞ is finitely factorable through X, because the interval of those q for which the injection lq 7→ l∞ is finitely factorable through X is closed (see, Section 1.3). So, let 2 < q < qrad (X). Then, by Theorem 5.2.1, 1 1 q log cn (X) ≥ − log n, n ∈ N. q qrad (X) Now, because, for any sequence (an ) ⊂ R+ , a log an n lim sup ≤ lim sup n −1 , log n an−1 n n (since, limt→∞ [(tα /(t − 1)α ) − 1] = α) there exists an N1 ⊂ N, a constant b > 0, and a sequence (ǫn ) ⊂ (0, 1) such that, for each n ∈ N1 , 1 cq (X) b b 1/q · qn ≥1− ≥ 1+ . (5.3.1) 1 + ǫn cn−1 (X) n n One can assume that b ≤ 1, so that n/(n+ b) ≥ 1/2, for all n ∈ N. By the definition of cqn (X), for n ∈ N, there exists a finite Peach n sequence (xi ) ⊂ X such that i=1 kxi kq = n, and n

q 1/q X (1 + ǫn )n1/q

E ri x i ≤ . cqn (X) i=1 5

See, B. Maurey and G. Pisier (1976).

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Cotypes of Banach spaces Let us set α := inf{kxi k : 1 ≤ i ≤ n}, and let kxi0 k = α. Then

q 1/q X X 1/q

kxi kq = (n − αq )1/q ≤ cqn−1 (X) E ri xi i6=i0

≤

i6=i0

cqn−1 (X)

n

q 1/q X (1 + ǫn )n1/q

. E ri x i ≤ cqn−1 (X) cqn (X) i=1

Now, if n ∈ N1 then, by (5.3.1), 1− Therefore,

b αq ≥1− , q n−α n

αq ≥

and

inf{kxi k : 1 ≤ i ≤ n} ≥ (b/2)1/q ,

nb b ≥ . n+b 2

n ∈ N1 .

(5.3.2)

By the Contraction Principle (see, Chapter 1), for each (ai ) ⊂ R, n

q 1/q X 2n1/q

≤ q ai ri xi E sup |ai |. cn (X) 1≤i≤n i=1

At this point, let us define the operators Un :

l(n) ∞

∋ a = (ai ) 7→

n X i=1

ai ri xi ∈ Lq (Ω, Σ, P; X).

The preceding inequality implies that kUn k ≤ 2n1/q /cqn (X). Because, in view of Theorem 5.2.2, cqn (X) = cqn (Lq (X)), if (ak ) ⊂ (n) l∞ , k = 1, . . . , n, we have the inequalities, n X k=1

≤

k

kUn (a )k

cqn (X)kUn k

q

1/q

n

X

E

k=1

≤

cqn (X)

n

q 1/q X

Un (ak )rk E k=1

n

q 1/q X

q a rk ≤ cn (X)kUn k · sup |aki |. k

1≤i≤n

k=1

By the Nikishin Lemma (see, Chapter 1) there exists a subset A ⊂ {1, . . . , n}, |A| ≥ n/2, such that, for every (ai ) with supp(ai ) ⊂ A, n

q 1/q X

E ai ri x i ≤ 21/q cqn (X)kUn kn−1/q σq ((ai )) ≤ 21+1/q σq ((ai )). i=1

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By (5.3.2), and the discussion that precedes (5.3.2) , there exist a subset N2 ⊂ N, a constant d > 0, an integer n ∈ N2 , and xn1 , . . . , xnn ∈ X, such that inf kxni k ≥ d,

n

q 1/q X

E ai ri xni ≤ σq ((ai )).

and

1≤i≤n

i=1

(5.3.3)

Let k0 be such that

1/2 dK0

−

1/q k0 1/2

δ := k0−1 (dK0

> 0, and define 1/q

− k0 ).

We shall show that if n ∈ N2 , n ≥ k0 , then the sequence, xn1 , . . . , xnn , satisfies the property P (k0, δ) of the Johnson Lemma cited in Section 1.5 of Chapter 1. Indeed, let A ⊂ {1, . . . , n}, |A| = k0 . Choosing xni0 in such a way that kxni0 k = supi∈A kxni k, i0 ∈ A, we have

q 1/q X

n ri xi E i∈A

q 1/q X q 1/q X

n n n ri (xi − xi0 ) − E ri ≥ E xi0 i∈A

i∈A

X q 1/q ri − k0 sup kxni − xnj k ≥ sup kxni k E i∈A

i,j∈A

i∈A

1/2

≥ dk0

− k0 sup kxni − xnj k, i,j∈A

so that, in view of (5.3.3), 1/2

sup kxni − xnj k ≥ k0−1 (dk0

i,j∈A

1/q

− k0 ) = δ.

Now, let m ∈ N, and take n ≥ N(k0 , δ, 1, m) (existence thereof being guaranteed, again, by the Johnson Lemma), n ∈ N2 , and a subsequence (xni1 , . . . , , xni2m ) such that the differences (xni2j − xni2j−1 ), j = 1, . . . , m, form an unconditional sequence with constant ≤ 3, and inf ky i k ≥ δ/2,

1≤j≤m

where

y j := xni2j − xni2j−1 .

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Cotypes of Banach spaces Now, it follows from (5.3.3) that, for each (ak ) ⊂ R, m m

X

q 1/q X

a y ≤ 3 E a r y

k k k k k k=1

k=1

m m

q 1/q X

q 1/q X

n n ≤ 3 E ak rk xi2k +3 E ak rk xi2k−1 ≤ 6σq ((ai )). k=1

k=1

Therefore, since for every p, 1 < p < q < ∞, σq ((ai )) ≤

1/p p∗ 1/p∗ X p |a | , i p∗ − q ∗ i

where 1/p + 1/p∗ = 1, we have, for each (ak ) ⊂ R, that m

X p∗ 1/p∗ X 1/p δ

p ak y k ≤ 6 ∗ sup |ak | ≤ |a | . i ∗ 6 k p − q i k=1

Thus, for each q < qrad (X), the embedding lq 7→ l∞ is crudely finitely factorable through X and, in view of the James theorem (Section 1.3), it is also finitely factorable. QED In the context of the above result it now becomes interesting to define a new parameter of of a normed space. Definition 5.3.1. For a normed space X we define the constant q(X) as the supremum of those q for which the embedding lq 7→ l∞ is finitely factorable through X. The next result shows that the constant q(X) introduced above is closely tied to the other geometric parameters of X we have introduced before. Theorem 5.3.2.6 For any infinite-dimensional normed space X, qrad (X) = qinf (X) = q(X).

6

See B. Maurey and G. Pisier (1976).

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Proof. The inequality qrad (X) ≥ qinf (X) follows from Proposition 5.2.1, and the inequality qrad (X) ≤ q(X). from Theorem 5.3.1. The inequality qinf (X) ≥ q(X) can be verified as follows. Since the interval of those q for which the embedding lq 7→ l∞ is finitely factorable through X is closed, the embedding lq(X) 7→ l∞ is finitely factorable through X. Therefore, for each n ∈ N there exist x1 , . . . , xn ∈ X such that, for every (ai ) ⊂ R

so that

n n

X

X 1/q(X ) 1

q(X ) |ai | , ai xi ≤ sup |ai | ≤ 2 i i=1 i=1 n

X

sup εi xi ≤ n1/q(X ) .

εi =±1

i=1

Hence, if q < q(X), then X cannot be of infratype q. Indeed, were X of infratype q, we would have n n

X 1/q 1 1/q X

q n ≤ kxi k ≤ sup εi xi ≤ n1/q(X ) , 2 εi =±1 i=1 i=1

n ∈ N.

A contradiction. Thus q(X) ≤ qinf (X). QED

Finally, the next theorem shows the importance of parameter q(X) in finite representablitiy of spaces lq in X.

Theorem 5.3.3.7 If X is an infinite-dimensional Banach space, then lq(X) is finitely representable in X. Proof. For the sake of simplicity, put q = q(X) throughout the proof. By Theorem 5.3.1, for each n ∈ N there exist xn1 , . . . , xnn ∈ X such that , for each (ai ) ⊂ R,

X

X 1/q 1

ai xni ≤ |ai |q . (5.3.4) sup |ai | ≤ 2 i i i

Let U be a non-trivial ultrafilter on X (on canPuse the Banach limits instead). The formula, k(ai )k := limn∈U k i ai xni k, defines 7

See B. Maurey and G. Pisier (1976), but the proof strongly depends on the result of J.L. Krivine (1976).

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a norm on S, say k.k1 , such that, if (ei ) is the canonical basis of S then, for each (ai ) ∈ S,

X

X 1/q 1

sup |ai | ≤ ai ei ≤ |ai |q . 2 i 1 i i Utilizing the Brunel-Sucheston procedure from Chapter 1 one can find a subsequence (e1n ) ⊂ (en ) such that the formula,

X X

ak eik , lim ai ei :=

i

i1 0 : lim λ ai = ∞ , n

i=1

1

we get that b = 2−1/q . Now, by Krivine Theorem of Chapter 1, lq is finitely representable in Y , and thus also in X. QED

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5.4

Operators in spaces of cotype q

Recall (see Section 1.5) that an operator U : X 7→ Y is said to be q-absolutely summing if there exists a constant C > 0 such that, for each (xi ) ⊂ X, X i

kUxi kq

1/q

X 1/q ≤ C sup |x∗ xi |q kx∗ ≤1 i

o n X X 1 1

|ai |p ≤ 1, + = 1 . ai xi : ai ∈ R, = C sup p q kx∗ ≤1 i i

The space of q-absolutely summing operators is denoted Πq (X, Y ), and it is a subspace of the space B(X, Y ) of all bounded operators. The minimal constant C in the above inequality will be denoted πq (U), and it serves as a complete norm on the space Πq (X, Y ). Theorem 5.4.1.8 Let 2 < q < ∞. Then Πq (c0 , X) = B(c0 , X) if, and only if, q > q(X), where the constant q(X) was defined in Definition 5.3.2. In particular, q(X) = inf{q : Πq (c0 , X) = B(c0 , X)}.

Proof. We start with the “only if” implication. Assume that Πq = B. We have to prove that the mapping lq 7→ l∞ is not finitely factorable through X (since lq(X) 7→ l∞ is finitely factorable through X). Suppose, to the contrary, that lq 7→ l∞ is finitely factorable through X. Let (θi ) be a sequence of i.i.d. stable random variables with the characteristic function E exp[itθi ] = exp[−|t|p ], and Ω = [0, 1], 1/p + 1/q = 1. Then, in view of the Schwartz Theorem of Chapter 1, there exists a constant C such that C

−1

(log n)

1/p

n 1/p X −1 ≤E n |θi |p , i=1

8

n ∈ N.

See, B. Maurey and G. Pisier (1976).

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Cotypes of Banach spaces For each n ∈ N, define the operator, Zn : C[0, 1] ∋ g 7→ Zn (g) = E(gθi )

n

i=1

∈ l(n) q .

Denote by Jn the canonical embedding l(n) 7→ l(n) q ∞ , and set Un = Jn ◦Zn . Now, it is sufficient to prove that πq (Un ) tends to infinity as n → ∞, whereas kZn k remains bounded. Indeed, because lq 7→ l∞ is finitely factorable through X, for each n ∈ N there exists a (n) factorization Jn = Wn ◦Vn , with Vn ∈ B(l(n) q , X), Wn ∈ B(X, l∞ ), and kVn k, kWn k ≤ 2. Then, since πq (Un ) ≤ 2πq (Vn ◦ Zn ), and kVn ◦ Zn k ≤ 2kZn k, we would have lim πq (Vn ◦ Zn ) = ∞, n

sup kVn ◦ Zn k < ∞,

and

n

and this would contradict the equality, Πq (C[0, 1], X) = B(C[0, 1], X), in view of the Closed Graph Theorem. Now, the boundedness of kZn k follows, from P P by transposition, r 1/r r p 1/p the equality (E| n an θn | ) = C ( n |an | ) , and we can prove that πq (Un ) → ∞ as follows: n 1/p X C −1 (log n)1/p ≤ E n−1 |θi |p i=1

n n X o X = sup E n−1/p gi θi : gi ∈ C[0, 1], |gi (t)|q ≤ 1 i=1

≤ sup ≤ sup

i

n nX

n nX i=1

i=1

o 1/q X q q Egi θi | : |gi (t)| ≤ 1

kUn gkq

i

1/q X o |gi (t)|q ≤ 1 ≤ πq (U). : i

The proof of the “if” part is more straightforward. Indeed, if q > p > q(X) then, by Theorem 5.3.2, X is of infracotype p, which means that the identity mapping, X 7→ X, is (p, 1)summing. Thus, Πp,1(c0 , X) = B(c0 , X) and, because for q > p we have the inclusion Πq (c0 , X) ⊃ Πp,1 (c0 , X) (see Chapter 1), we get the desired equality. QED

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Corollary 5.4.1 (i) Let 2 < q < ∞. The canonical embedding lq 7→ l∞ is finitely factorable through X if, and only if, Πq (c0 , X) 6= B(c0 , X). (ii) Let, again, 2 < q < ∞. If Πq (c0 , X) = B(c0 , X), then there exists a p < q such that Πp (c0 , X) = B(c0 , X). Proof. The Corollary follows immediately from the above Theorem, and the fact that the set of q’s for which the embedding lq 7→ l∞ is finitely factorable through X is closed (see, Chapter 1). Remark 5.4.1. (a) By duality, Πq (c0 , X) = B(c0 , X) if, and only if, for any Banach space Y , Πp (X, Y ) = Π1 (X, Y ), with 1/p + 1/q = 1. (b) In the equality Π2 (c0 , X) = B(c0 , X) on can replace c0 by l∞ , or C[0, 1], and get equivalent results. Theorem 5.4.2.9 The following properties of a Banach space X are equivalent: (i) There exists a q < ∞ such that Πq (c0 , X) = B(c0 , X); (ii) There exists a q < ∞ such that X is of cotype q; (iii) The space c0 is not finitely representable in X. Proof. (i) =⇒ (ii) By Theorem 5.4.1, if Πq (c0 , X) = B(c0 , X) then q > q(X), so that X is of cotype q, by Theorem 5.3.2. (ii) =⇒ (iii) Cotype q is a superproperty, and we know, in view of Example 5.2.1, that c0 is not of cotype q for any q < ∞. Therefore, a space in which c0 is finitely representable cannot be of cotype q for any q < ∞. (iii) =⇒ (i) By Corollary 5.4.1(i), if Πq (c0 , X) = 6 B(c0 , X) then, for each q < ∞, the embedding lq 7→ l∞ is finitely factorable through X. Because the interval of such q’s is closed (see, Chapter 1), also the identity mapping, l∞ 7→ l∞ , is finitely factorable through X, that is, c0 is finitely representable in X. QED Remark 5.4.2. It is easy to see that the following statement is also equivalent to the statements (i) − (iii) of the above Theorem: (iv) There exists a q < ∞ such that X is of infracotype q. 9

See B. Maurey and G. Pisier (1976).

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In the case of spaces of cotype 2 we also have the following result which is only a one-sided implication: Theorem 5.4.3.10 If X is of cotype 2, then Π2 (c0 , X) = B(c0 , X). Proof. Asssume that Π2 (c0 , X) = 6 B(c0 , X). Then, by a du∗ ∗ ality argument, Π1 (X , l1 ) = 6 Π2 (X , l1 ), and there exists an operator W : X ∗ 7→ L1 which is not factorable through any Hilbert space. Now, consider two cases. Case 1. There exists a p ∈ (1, 2) such that W = U ◦ V, where U : X ∗ 7→ Lp , and V : Lp 7→ l1 . Because p < 2, and q > 2, the space Lq is of Rademacher type 2, and X ∗∗ is of cotype 2. Therefore, (see Remark 7.6.1 in Chapter 7) by duality, U admits a factorization through a Hilbert space. A contradiction. Case 2. If W : X ∗ 7→ l1 does not factor through any Lp with 1 < p < 2, then, by Maurey-Rosenthal Theorem (see, Chapter 1), ∗∗ for each p ∈ (1, 2), and each n ∈ N, there exist x∗∗ 1 , . . . , xn ∈ SX ∗∗ such that, for any (ai ) ⊂ R, n n

X X 1/q

∗∗ |aj |q , aj xj ≤ 2

j=1

j=1

1 1 + = 1. p q

Therefore, were X of cotype 2, we would have the contradictory inequalities n

X

1/2 n ∈ N. QED rj x∗∗ ≤ 2n1/q , cn ≤ E j=1

However, in spaces with unconditional basis we also have the reverse implication. Theorem 5.4.4.11 Let X be a Banach space with an unconditional basis (en ). Then X is of cotype 2 if, and only if, Π2 (c0 , X) = B(c0 , X). 10

See E. Dubinsky, A. Pelczynski and H.P. Rosenthal (1972). This result is valid even for X with local unconditional structure, see B. Maurey (1973/74), Exp. XIV-XV. Moreover, one can prove that, if X has a local unconditional structure, then X is of type 2 if, and only if, Π2 (c0 , X ∗ ) = B(c0 , X ∗ ), and l1 is not finitely representable in X ∗ . 11

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Proof. In view of Theorem 5.4.3 it suffices to prove only the “if” part of the above statement. Denote by (e∗n ) the sequence of coefficient functionals of (en ). It is known12 that the equality Π2 = B implies that there exists a constant C such that, for all n ∈ N, and all (xi ) ⊂ X, ∞ X m m

X X 1/2 1/2

∗ 2 |en xj | en ≥ C kxj k2 .

n=1

j=1

j=1

Since the basis (en ) is unconditional, and the real line R is of cotype 2, there exists a constant A such that ∞ X m m

X

X

rj e∗n xj en rj xj = E E n=1 j=1

j=1

m ∞ ∞ m X

X

X

X ∗ ∗ rj en xj en E rj en xj en ≥ A ≥ AE n=1

n=1 j=1

∞ X m

X

2

≥A

n=1

j=1

|e∗n xj |2

1/2

so that X is of cotype 2. QED

j=1

m

X 1/2

2 kxj k2 , en ≥ A C j=1

Remark 5.4.3. By duality, if X is of cotype 2 (and also “only if”, in the case X when has an unconditional basis) then Π1 (X, Y ) = B(X, Y ), for any Banach space Y . The above analysis of absolutely summing operators was conducted in the context of Rademacher random multipliers. Next, we will conduct a similar analysis in the case when the multipliers (γi ) form a sequence of independent and identically distributed Gaussian random variables with the standard N(0, 1) distribution . Definition 5.4.1. The operator U : X 7→ Y is said to be Gabsolutely summing if there exists a constant C > 0 such that, for each n ∈ N, and any (xi ) ⊂ X, n n

2 1/2 X X 1/2

∗ 2 E γi Uxi ≤ C sup (x xi ) . kx∗ k≤1 i=1 i=1 12

See E. Dubinsky, A. Pelczynski and H.P.Rosenthal (1972).

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The space of G-absolutely summing operators will be denoted by ΠG (X, Y ), and the smallest C in the above inequality will be denoted by πG (U). Note that πG (U) is the complete norm on the space ΠG (X, Y ). Theorem 5.4.5.13 The following properties of a Banach space X are equivalent: (i) The space X is of cotype 2; (ii) For any Banach space Y , Π2 (Y , X) = ΠG (Y , X); (iii) If H is a Hilbert space, then Π2 (H, X) = ΠG (H, X). Proof. The implication (i) =⇒ (ii) follows from Corollary 5.2.1, and the implication (ii) =⇒ (iii) is evident. So, let us prove the implication (iii) =⇒ (i). (iii) =⇒ (i) We begin with checking that, for every finite dimensional U : H 7→ X, Z 1/2 πG (U) = kUhk2 mH (dh) , (5.4.1) H

where mH is the canonical Gaussian cylindrical measure on H with the characteristic functional Z exp[i(h∗ , h)]mH (dh) = exp[−(1/2)kh∗ k2 ], h, h∗ ∈ H. H

Pn Indeed, let Uh = i=1 (ei , h)xi , where (ei ) is an orthonormal basis in H, and x1 , . . . , x − n ∈ X. Denote by Pn : H 7→ Rn the operator of projection on the first n coordinates. Then Pn∗ : Rn 7→ H, with kPn k = kPn∗k = 1, U = UP ∗ P , and, by the definition of πG (U), Z 1/2 Z 1/2 2 ∗ 2 −1 kUhk mH (dh) = kUP sk (mH Pn )(ds) Rn

H

=

Z

Rn

n

2

X 1/2

si Uei (mH Pn−1)(ds) ≤ πG (U),

i=1

where s = (s, . . . , sn ) ∈ Rn . 13

See V. Linde and A. Pietsch (1974).

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To prove the converse inequality let us define, for each (hi ) ⊂ H, the operator, k

V : R ∋ t = (t1 , . . . , tk ) 7→ V t = Since kV k = sup we have

Z

Rk

Z

k nX

(hi , h)

2

i=1

1/2

k X i=1

ti hi ∈ H.

o : khk ≤ 1 ,

k

2

X 1/2

−1 ti Uhi (mH Pn )(dt)

i=1

1/2 k UPn∗ Pn V tk2 (mH Pn−1 )(dt) Rk Z 1/2 ∗ 2 −1 ≤ kPn V k kUPn sk (mH Pn )(dt) =

Rk

≤

Z

H

2

kUhk mH (h. )

n nX 1/2 o sup (hi , h)2 : khk ≤ 1 ,

1/2

so that πG (U) ≤

i=1

Z

H

kUhk2 mH (dh)

1/2

,

which proves (5.4.1). Now, assume that Π2 (H, X) = ΠG (H, X). By the Closed Graph Theorem, there exists a constant C ≥ 0, such that π2 (U) ≤ CπG (U),

U ∈ ΠG (H, X).

For each (xi ) ⊂ X, define the operator, U0 : Rn ∋ t 7→

, X i=1

ti xi ∈ X.

By (5.4.1) πG (U0 ) =

Z

Rn

n

X

2 1/2

−1 t x (m P )(dt) .

i i H n i=1

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99

Cotypes of Banach spaces Therefore, n X i=1

=C

Z

Rn

kxi k2

1/2

≤ π2 (U0 Pn ) ≤ CπG (U0 Pn )

n n

X

2

2 1/2 X 1/2

−1 = C E ti xi (mH Pn )(dt) γ i xi ,

i=1

i=1

that is, X is of cotype 2. QED

In the next section we will address the issue of convergence of vector series in Banach spaces with Rademacher and Gaussian coefficients, and its relationship to the geometric structure of those Banach spaces.

5.5

Random series and law of large numbers

The next result provides necessary conditions for the convergence of a series of independent random vectors with values in Banach spaces of cotype q. Theorem 5.5.1. The following properties of a Banach space X are equivalent: (i) The space X is of cotype q; P (ii) If the P series i ri xi , (xi ) ⊂ X, converges a.s. (or, in q Lq (X)), then i kxi k < ∞; (iii) For eachP sequence (Xi ) of independent, zero-mean random P vectors in X, if i Xi converges in Lq (X), then i EkXi kq < ∞.

Proof. (iii) =⇒ (ii) The Lq convergence part is evident, and the almost sure convergence follows from the Ito-Nisio Theorem (see Chapter 1). (ii) =⇒ (i) This implication follows from the Closed Graph Theorem. (i) =⇒ (iii) This implication follows directly from Corollary 5.2.1. QED

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Random series in spaces of cotype q, for some q < ∞, that is spaces in which c0 is not finitely representable (see Theorem 5.4.2), enjoy a number of other remarkable properties. Theorem 5.5.2.14 Let (γi ) be a sequence if i.i.d. N(0, 1) Gaussian random variables. The following properties of a Banach space X are equivalent: (i) c0 is not finitely representable in X; (ii) There exists a constant C > 0 such that, for each n ∈ N, and each finite set (xi ) ⊂ X,

2 1/2

2 1/2 X X

; ri xi ≤ C E γi xi E i

i

P

(iii)PIf the series i ri xi , (xi ) ⊂ X, converges a.s. then the series i γi xi also converges a.s.

Proof. (i) =⇒ (ii) Because c0 is not finitely representable in X, and because q(X) = q(L2 (X)) (see, Theorem 5.2.2), it follows from Theorem 5.4.2 that there exists a q, 2 < q < ∞, such that Πq (c0 , L2 (X)) = B(c0 , L2 (X)). Therefore, by the Closed Graph Theorem, there exist a K > 0 such that πq (U) ≤ KkUk,

U ∈ B(c0 , L2 (X)).

Let x1 , . . . , xn ∈ X, and define the operator, U : c0 ∋ (ci ) 7→ U((ci )) =

n X i=1

ri ci xi ∈ L2 (X).

By the Pietsch Factorization P Theorem (see, Chapter 1), there ex+ ists an (an ) ⊂ R , with n an = 1, such that for any (ci ) ∈ c0 X 1/q X 1/q kU((ci ))k ≤ πq (U) ai |ci |q ≤ KkUk ai |ci |q . i

i

Therefore, after integration,

X

2 1/2 X 2/q

Eγ Er ri γi xi ≤ KkUk Eγ ai |γi |q 1/2 i

14

i

See, B. Maurey and G. Pisier (1976).

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101

Cotypes of Banach spaces X 1/q 1/q ≤ KkUk Eγ ai |γi |q ≤ KkUk E|γ|q . i

On the other hand, (ri ), and (ri γi ), are identically distributed, so that

2 1/2 X

2 1/2 X 1/q

E γi xi = E ri γ i x i ≤ KkUk E|γ|q . i

i

Now, by the Contraction Principle (see Chapter 1), we have

2 1/2

2 1/2 o X n X

, ri x i : |ci | ≤ 1 ≤ E ci ri x i kUk = sup E i

i

which completes the proof of the implication (i) =⇒ (ii). (ii) =⇒ (iii) This implication is evident if one takes into account the Ito-Nisio Theorem (see Chapter 1), the ClosedPGraph Theorem, and the fact that the a.s. convergent series, i ri xi , P and i γi xi , have all the moments finite. (ii) =⇒ (i) If c0 is finitely representable in X then, for each n ∈ N, there exists (xi ) ⊂ X such that, for every (ai ) ⊂ R, n

X 1

ai xi ≤ sup |ai |. sup |ai | ≤ 2 i i i=1

Therefore, for each n ∈ N,

and

n

X

2 1

2 γ i xi , E sup |γi | ≤ E 4 1≤i≤n i=1 n

X

2

ri xi ≤ 1. E i=1

Thus (ii) is violated because, by the Borel-Cantelli Lemma, sup1≤i≤n |γi |2 goes almost surely to +∞, as n → ∞, and in view of the fact that the tail of the distribution of γi behaves like t exp[−t2 ]. QED

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Remark 5.5.1. (a) It follows immediately from Corollary 5.4.1 that the statements (i)−(iii) of the above Theorem are also equivalent to the following statement: (iv) For each sequence (ξn ) of independent, zero-mean real random variables which are uniformly bounded in PLq (Ω, Σ, P), for each q < ∞, and for each (x ) ⊂ X such that n n rn xn converges P a.s., also n ξn xn converges a.s. (b) The above Theorem also gives the following alternative characterization of spaces of cotype q: X is of cotype q if, and only if, for each p, 0 < p < ∞, there exists a constant C > 0 such that, for every n ∈ N, and each sequence (xi ) ⊂ X, X i

kxi kq

1/q

p 1/p X

. γi xi ≤ C E i

Now, we turn to the investigation of Gaussian measures on spaces that are of cotype q, for some q < ∞, and which, additionally, have an unconditional basis (en ) with the coordinate functionals (e∗ ) ⊂ X ∗ . Recall that, for any random vector X (always zero-mean) in X such that E(x∗ X)2 < ∞, x∗ ∈ X ∗ , there exists a symmetric (i.e., x∗ Ry ∗ = y ∗ Rx∗ ), and positive (i.e., x∗ Rx∗ ≥ 0) covariance operator RX : X ∗ 7→ X.

Theorem 5.5.3.15 The following properties of a Banach space X with an unconditional basis (en ) are equivalent: (i) The space c0 is not finitely representable in X; (ii) A positive, symmetric operator R ∈ B(X ∗ , X) is the covariance of a Gaussian measure on X if, and only if, the P operator ∗ ∗ 1/2 series n (en Ren ) en converges in X. P Proof. First, observe that the convergence of n (e∗n Re∗n )1/2 en is a necessary condition in any Banach space with an unconditional basis. Indeed, we can assume (up to an equivalent renorming) that k.k does not change under sign changes of coordinates. Therefore, 15

See D. Mustari (1973), but the proof provided below is due to V. Mandrekar (1977).

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103

Cotypes of Banach spaces if X is a Gaussian random vector in X, then Ek EkXk < ∞, so that E

X n

|e∗n X|en =

2 1/2 X π

P

n

|e∗n X|en k =

(e∗n Re∗n )1/2 en ,

n

has to converge in X. A A∗ Now, let R : X ∗ 7→ H P 7→ X, be the standard factorization of R. The convergence of n (e∗n Re∗n )1/2 en is equivalent to the P ∗ convergence of n kAen ken . First, we shall prove that we have a factorization, A A (5.5.1) A : X ∗ 7→1 l1 7→2 H, such that A∗1 : l∞ 7→ X, and kA2 k ≤ 1 , which, in particular, would show that A ∈ Π1 (X ∗ , H). Indeed, because A∗ H ⊂ X, A is a continuous operator from (X ∗ , σ(X ∗ , X)) into (H, σ(H, H)). Since (en ) is unconditional, for each x∗ ∈ X ∗ , the P ∗ ∗ series x∗ = in the weak n (x en )e converges P unconditionally ∗ ∗ ∗ ∗ topology σ(X , X). So Ax = n (x en )Aen , where the series converges unconditionally in the weak topology, and the metric topology on H. Define, x∗ ∈ X ∗ , A1 x∗ := (x∗ en )kAe∗n k n∈N ∈ l1 , and

A2 (an ) =

∞ X n−1

an hn ,

(an ) ∈ l1 ,

Ae∗n /kAe∗n k,

where hn = if Ae∗nP= 6 0 , and 0, otherwise. Evidently, A∗1 (l∞ ) ⊂ X, and kA2 (an )k ≤ n |an |. This shows the validity of (5.5.1). Since A∗1 ∈ B(l∞ , X), by Theorem 5.4.2, A∗1 ∈ Πq (l∞ , X), for some q, 1 < q < ∞, so that, also, A∗ ∈ Πq (H, X). Therefore mH (A∗ )−1 , where Z exp[i(h∗ , h)]mH (h. ) = exp[(−1/2)kh∗ k2 ], H

is σ-additive, and its extension has covariance R.

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P (ii) =⇒ (i) In view of Theorem 5.5.2 P it suffices to show that i γi xi converges a.s. in X, whenever i ri xi does. P By Ito-Nisio, and Kahane Theorems (see, Chapter 1), if Y = Pi ri xi converges a.s., then it also converges in L2 (X), and Ek i ri xi k2 < ∞. Thus, arguing P as in the proof of the implication (i) =⇒ (ii), we get that n (e∗n RY e∗n )en converges in X. By assumption, there exists a Gaussian random vector with the covariance operator RY . Checking its Pfinite-dimensional distributions we conclude that it has to be i γi xi , the series being convergent in distribution. However, the Ito-Nisio Theorem guarantees the a.s. convergence P of i γi xi , as well. QED

Corollary 5.5.1 The following properties of a Banach space X with unconditional basis (en ) are equivalent: (i) The space c0 is not finitely representable in X; (ii) Let mH be the standard cylindrical measure on the Hilbert space H with the Fourier transform exp[−khk2 ]. The operator U : H 7→ H maps mH into a σ-additive measure on X if, and only if, U ∗ ∈ Π1 (X, H); (iii) The functional exp[−x∗ Rx∗ ] is theP Fourier transform of a Gaussian measure on X if, and only if, n (e∗n Re∗n )1/2 en converges in X. 5.5.2. (a) In general, the convergence of the series P Remark ∗ ∗ 1/2 (e Re ) en means that R1//2 ∈ Π1 (X, H), but if X = H, n n n then the summability condition on R means, simply, that R is nuclear. (b) The cylindrical measure mH in the statement (ii) of the above Corollary may be replaced by any cylindrical measure invariant under unitary transformations, and such that finitedimensional projections thereof are absolutely continuous with respect to the Lebesgue measure. Corollary 5.5.2. Let p > 0, and let X be a Banach space with unconditional basis (en ) , and such that c0 is not finitely in X. Then the norms (EkXkp )1/p , P ∗representable ∗ and k n (en Ren )en k, are equivalent on the subspace of Lp (X) spanned by all Gaussian random vectors X in X. Proof. The Corollary follows from Theorem 5.5.3 by an obvious

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application of the Closed Graph Theorem and Kahane’s Theorem of Chapter 1. The above discussion directly implies the following result for Gaussian random series: Corollary 5.5.3. The following properties of a Banach space X with unconditional basis (en ) are equivalent: (i) The space c0 is not finitely representable in X; (ii) Let (Xn ) be a sequence of independent zero-mean Gaussian P random vectors in X with covariance operators (R ). Then n i Xi P P converges a.s. if, and only if, the series i ( n e∗i Rn e∗i )1/2 ei converges in X; P (iii) For each sequence (x ) ⊂ X, the series i i γi xi converges P P ∗ 2 1/2 a.s. if, and only if, i ( n (ei xn ) ) ei converges inX; Finally, the above Corollary and the standard Kronecker Lemma give the following Strong Law of Large Numbers:

Theorem 5.5.4 (SLLN).16 If c0 is not finitely representable in the Banach space X, and (Xn ) is a sequence of independent zero-mean Gaussian random vectors in X with covariance operators (Rn ) such that 2 1/2 XX −2 ∗ ∗ n ei Rn ei ei , i

n

converges in X, then X1 + · · · + Xn = 0, n→∞ n lim

a.s.

In the remainder of this section we will concentrate on spaces of cotype 2, Gaussian measures on such spaces, and convergence of general series of independent random vectors in such spaces, as well as the Law of Large Numbers for them. Let us begin with the basic result that shows that cotype 2 of X is indispensable for the validity of the vector analogue of the classical Bochner Theorem which characterizes functions which are Fourier transforms of positive measures (on X, in our case). 16

See, S.A. Chobanyan and V.I. Tarieladze (1977).

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Theorem 5.5.5. Let X be a Banach space, and assume that there exists a topology τ on X ∗ such that φ : X ∗ 7→ C is positivedefinite, τ -continuous with φ(0) = 1, if, and only if, there exists a probability measure m on X with Z exp[ix∗ x]m(dx) = φ(x∗ ), X Then X is of cotype 2. Proof. Assume that X is of cotype 2.P Then there exists a sequence (x Pi )n ⊂ X 2such that the series P i ri x2i converges a.s., and an = i=1 kxi k ↑ ∞. Then also k kxk k /ak = ∞. Now, define a sequence, (Yn ), of independent random vectors in X with the distribution laws (µk ) such that P(Yk = ±ak xk /kxk k) = (1/4)kxk k2 /a2k , P(Yk = 0) = 1 − kxk k2 /(2a2k ). Then, by the Borel-Cantelli Lemma, Yk ’s do not converge to 0, P a.s., so that the series k Yk diverges a.s. P Now, the assumption that r x i i i converges a.s. implies that Q∞ ∗ ∗ φ1 (x ) := k=1 cos(x xk ) is the characteristic functional of a measure on X, and thus is τ -continuous, by assumption. let δ > 0 be such that , if |t| ≤ π/2, cos t ≥ 1−δ, then 1−t2 /2 ≤ cos t ≤ 1−t2 /4. If φ1 (λ) > 1 − δ, for |λ| ≤ 1, then φ1 (x∗ ) ≤

∞ Y

k=1

1 − (1/4)(x∗ xk )2

∞ Y ≤ 1 − kxk k2 /2a2 ) cos x∗ ak xk /kxk k =: φ2 (x∗ ). k=1

The functional φ2 is also τ -continuous at 0, and positive-definite as a limit of the characteristic functions of µ1 ∗ · · · ∗ µn . Therefore, again by the assumption, a probability measure m on R there exists ∗ ∗ X such P that φ2 (x ) = exp[ix x]m(dx). That means, however, that k Yk converges in law and, by the Ito-Nisio Theorem (see Chapter 1), also a.s. A contradiction. QED

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107

Now, we are ready to study measures with Gaussian covariance on spaces of cotype 2. Theorem 5.5.6. The following properties of a Banach space X are equivalent: (i) Space X is of cotype-2; (ii) Let the operator U ∈ B(H, X), and let mH be the standard cylindrical measure with the Fourier transform exp[[−khk2 ]. Then mh U −1 is countably additive if, and only if, U ∈ Π2 (H, X); (iii) For each probability R ∗ 2measure m on X such that, for each ∗ ∗ x ∈ X the integral (x x) m(dx) < ∞, and which has a Gaussian covariance operator, there exists U ∈ B(l2 , X), and a meaR 2 sure µ on l2 , with khk µ(dh) < ∞ such that m = µU −1 .

Proof. (i) =⇒ (ii) The “if” part is true in any Banach space X because, by Pietsch Factorization Theorem (see Chapter 1), V we have the decomposition U : H 7→ H 7→ X, where V is a Hilbert-Schmidt operator. Now, assume that U ∈ B(H, X), mH U −1 is countP and that ably additive. Let (g n ) ⊂ H, and n (g n , h)2 < ∞, h ∈ H. Denote by (hn ) an orthonormal basis in PH, and define the bounded operator, V , byP the formula VPh = n (h, hn )g n . We shall show that the series, n γn Ug n = n γn UV hn , converges a.s. Indeed, by our assumption, UU ∗ is a covariance operator of a Gaussian ∗ ∗ measure. P Therefore UV V U is also a Gaussian covariance, so that n γn Ug n converges in law, and thus also a.s. by the ItoNisio Theorem (see, 1). Because X is of cotype 2, by P Chapter 2 Corollary 5.2.1, n kUg n k < ∞. Hence, for each h ∈ H, the condition ((g n , h)) ∈ l2 implies that (Ug n ) ∈ l2 (X), so that, by the Closed Graph Theorem, U ∈ Π2 (H, X). (ii) =⇒ (iii) Step 1: Assume m itself is Gaussian with the covariance operator R : X ∗ 7→ X, R = A∗ A, A ∈ B(X ∗ , H). Evidently, m is the countably additive extension of the standard Gaussian cylindrical measure mH (A∗ )−1 . In view of (ii), we have A∗ ∈ Π2 (H, X). Therefore, by the Pietsch Factorization Theorem V U (see Chapter 1), A∗ : H 7→ l2 7→ X, where V is a Hilbert-Schmidt operator, and U ∈ B(l2 , X). Therefore, R = UV V ∗ U ∗ , where V V ∗ : l2 7→ l2 is nuclear. Thus m is the image, under U, of a

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Gaussian measure on l2 with covariance V V ∗ . Step 2: Let m be an arbitrary measure with a Gaussian covariance, and let X be a random vector in X such that its probability distributions L(X) = m. Define the operator, A : X ∗ ∋ x∗ 7→ Ax∗ := x∗ X ∈ L2 (R). Then A∗ A is a Gaussian covariance operator, and, by (ii), A∗ ∈ Π2 (H, X). Hence, A∗ admits a factorization A∗ = UV , where V ∈ Π2 (L2 , l2 ), and U = B(l2 , X). Therefore, V ∗ : l2 7→ L2 (R) is Hilbert-Schmidt, so that there exists a random vector Y in l2 such that EkY k2 < ∞, and V ∗ h = (h, Y ), h ∈ l2 . Clearly, m = L(Y )U −1 , which completes the proof of the implication (ii) =⇒ (iii). P (iii) =⇒ (i) Assume that n γn xn , (xn ) ⊂ X, coverges a.s. In view of (iii), every measure m of weak second order with a Gaussian covariance has second moments finite. Consider the −1/2 measure m on X concentrated at the points ±pn xn , n ∈ N, with X m({±pn−1/2 xn }) = pn /2, where pn = 1, pn ≥ 0. n

P 2 Evidently, Rm = RPn γn xn . Therefore, = n kxn k R 2 kxk m(x. ) < ∞, and X is of cotype 2 in view of Remark 5.5.1 (b). QED Corollary 5.5.4. The following properties of a Banach space X are equivalent: (i) The space X is of cotype 2; (ii) For each random vector X in X with Gaussian covariance, and E(x∗ X)2 < ∞, for each x∗ ∈ X ∗ , we have EkXk2 < ∞; (iii) A function φ : X ∗ 7→ C is the Fourier transform of a zero-mean Gaussian measure on X if, and only if, φ(x∗ ) = exp[−x∗ UV U ∗ x∗ ),

x∗ ∈ X ∗ ,

where V : l2 7→ l2 is nuclear, and U ∈ B(l2 , X). (iv) There exist constants, C1 , C2 , C3 ≥ 0, such that, for each random vector X in X , with E(x∗ X)2 < ∞, each x∗ ∈ X ∗ ,

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and the covariance operator R of a Gaussian random vector Y , we have the inequalities, EkXk2

1/2

≤ C1 Π2 ((R1/2 )∗ ) ≤ C2 EkY k2

1/2

≤ C3 Π2 ((R1/2 )∗ ).

(v) Under the additional assumption that X has an unconditional basis (en ), there exists a constant C ≥ 0 such that, for each X with E(x∗ X)2 < ∞, each x∗ ∈ X ∗ , and a Gaussian covariance operator R,

X

∗ ∗ 1/2 2 1/2 ≤ C (en Ren ) en . EkXk n

As corollaries to the above Theorem 5.5.6 we obtain the following results on the convergence of random series and the Strong Law of Large Numbers in spaces of cotype 2. Theorem 5.5.7.17 Let X be of cotype 2, and let (en ) be an unconditional basis in X. Then, for any independent zero-mean random vectors (X in X with the covariance operators (Rn ) such Pn )P ∗ ∗ 1/2 that ei converges in X, the series i( n ei Rn ei ) P the series X converges a.s. in X. i i Proof. By Theorem 5.5.3, and in view of the assumption, the P operator, R = n Rn , is the covariance operator of a Gaussian A

A∗

measure. Therefore, in the factorization, R : X ∗ 7→ H 7→ X, we A∗ ∈ Π2 (H, X), so that, for each x∗ ∈ X ∗ , the series P have ∗ a.s. By the Ito-Nisio Theorem (see Chapter n x Xn converges P 1), the series n Xn converges a.s. as well. QED.

Corollary 5.5.5 (SLLN). Let X be of cotype 2, and let (en ) be an unconditional basis in X. If (Xn ) is a sequence of independent zero-mean random vectors with the covariance operators P Pin X −2 ∗ (Rn ) such that the series i ( n n ei Rn e∗i )1/2 ei converges in X, then X1 + · · · + Xn lim = 0, a.s. in X. n→∞ n 17

See S.A. Chobanyan and V.I. Tarieladze (1977).

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5.6

Central limit theorem, law of the iterated logarithm, and infinitely divisible distributions

In this section we will discuss how cotype 2 of a Banach space is related to the validity of the Central Limit Theorem (CLT) for random vectors of weak and strong second order, and to the Law of the Iterated Logarithm (LIL) for them. The section concludes demonstrating that the cotype q of the Banach space is equivalent to the min(1, xq ) integrability of the Levy measure for infinitely divisible probability distributions on a Banach space. Theorem 5.6.1 (CLT).18 The following properties of a Banach space X are equivalent: (i) Space X is of cotype 2; (ii) For any sequence (Xi ) of zero-mean, independent, and identically distributed random vectors in X the probability distribution, X + · · · + X 1 n L , n1/2 converges weakly (to a Gaussian measure) if, and only if, E(x∗ X1 )2 < ∞, for each x∗ ∈ X ∗ , and X1 has a Gaussian covariance operator. Proof. The implication (i) =⇒ (ii) is immediate in view of Theorem 5.5.6 since X1 is a continuous linear image of a Y in l2 with EkY k2 < ∞, and in the Hilbert space the Central Limit Theorem for i.i.d summands of second order holds true. (see also Chapter 7). (ii) =⇒ (i) Assume that if X has a Gaussian covariance operator then it satisfies the CentralPLimit Theorem. Then, necessarily, EkXk < n γn xn that converges a.s., P ∞. Now, take and let pn ≥ 0, n pn = 1. Consider a random vector X with −1/2 the probability distribution concentrated at the points ±pn xn , −1/2 with P({±pn xn }) = pn /2. Clearly, RX = RP γn xn and, by asP 1/2 sumption, = EkXk < ∞. Since this holds true n kxn kpn 18

See S.A. Chobanyan and V.I. Tarieladze (1977), and also N. Jain (1976).

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Cotypes of Banach spaces 1/2

for every sequence (pn ) such that (pn ) ∈ l2 , we also have that P 2 n kxn k < ∞, which implies that X is of cotype 2 in view of Remark 5.5.1 (b). QED As an immediate consequence of the above result we also have the following Corollary: Corollary 5.6.1. A Banach space X is of cotype 2 if, and only if, for any sequence of independent, and identically distributed random vectors in X, for which L((X1 +· · ·+Xn )/n1/2 )) converges weakly, we have EkX1 k2 < ∞.

Theorem 5.6.2 (LIL)19 The following properties of a Banach space X are equivalent: (i) Space X is of cotype 2; (ii) For any sequence (Xi ) of independent, identically distributed random vectors in X with E(x∗ X1 )2 < ∞, and Gaussian covariance operator R, X +···+X 1 n P lim distance , K = 0 = 1, n→∞ (2n log log n)1/2

and n X +···+X o 1 n = K = 1, P cluster points of (2n log log n)1/2 where the set K = {(R1/2 )∗ h : khk < 1}. Proof (i) =⇒ (ii) In view of Theorem 5.5.6 it is sufficient to check that the Law of the Iterated Logarithm holds true in the Hilbert space. And, indeed, it does because Hilbert space is of Rademacher type 2 (see Chapter 7). (ii) =⇒ (i) The Law of the Iterated Logarithm implies that (X1 + · · ·+ Xn )/n → 0, a.s. This, however, even in general Banach spaces, implies that EkX1 k < ∞20 .Thus we obtain that X is of 19

See S.A. Chobanyan and V.I. Tarieladze (1977), but the result depends strongly on earlier work of J. Kuelbs (1977), and G. Pisier (1975/76) and (1976). 20 See W.A. Woyczy´ nski (1974).

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cotype 2 exactly as in the proof of the implication (ii) =⇒ (i) of the preceding theorem. QED The final result of this chapter deals with the L´evy-Khinchine representation of non-Gaussian, infinitely divisible probability distributions on spaces of cotype q. Theorem 5.6.3.21 The following properties of a Banach space X are equivalent: (i) Space X is of cotype q; (ii) If m is a symmetric, infinitely divisible probability distribution on X with the L´evy-Khinchine representation, Z hZ i ∗ exp[ix x]m(dx) = exp cos(x∗ x) − 1 µ(dx) , X X then

Z

X

min(1, kxkq )µ(dx) < ∞.

Proof. (i) =⇒ (ii) By a standard argument22 one can reduce the proof to showing that if µn are symmetric, finite, satisfy the condition µn (x : kxk ≤ 1) = kn ↑ ∞,

as

n → ∞,

and if the measures mn , determined by the formula Z hZ i ∗ ∗ exp[ix x]mn (dx) = exp cos(x x) − 1 µn (dx) , X X are unifomly tight, then Z sup kxkq µn (dx) < ∞. (5.6.1) n∈N X Define νn = µn kn−1 , and ρn = νn∗kn , n ∈ N (convolution power). Then, since X is of cotype q, Z Z Z q q kxk ρn (dx) ≥ Ckn kxk νn (dx) = C kxkq µn (dx), X X X 21 22

See A. Araujo and E. Gine, (1978). See K.R. Parthasarathy (1967).

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Cotypes of Banach spaces so that (5.6.1) follows because Z sup kxkq ρn (dx) < ∞. n∈N X in view of the inequality Z (a + 1)q + aq (1 − 21−q ) , kxkq ρ(dx) ≤ 1 − 2q ρn (kxk ≥ a) X

valid for any a > 0, and which is a corollary to the following inequality (a + c)q + aq (1 − 21−q ) , P kX1 + · · · + Xn k > a) ≥ 2−q 1 − EkX1 + · · · + Xn kq

valid in any Banach space for independent, zero-mean random vectors uniformly bounded by a constant c 23 , and the fact that tightness of (ρn ) = (νn∗kn ) is implied by tightness of (mn )24 . (ii) =⇒ (i) We first prove that (ii) P implies that c0 is not contained in X. If c0 ⊂ X then the series n πn n−1/q en , where (πn ) are i.i.d. real symmetrized Poissonian random variables, converges a.s. The sequence (en ) here is the canonical basis in c0 . Indeed, because the tail of the distribution of π1 satisfies the inequality, P(|π1 | > n) ≤ C/(n!n) we have X X 1 P kπn n−1/q en k > n−1−1/q ≤ C < ∞, n!n n n P −1/q so that, by Borel-Cantelli Lemma, n πn nP en converges a.s. Therefore, the probability distribution L( n πn n−1/q en ) is infinitely divisible with the Fourier transform ∞ i hZ X exp (cos(x∗ x) − 1) δ(en n−1/q ) (dx) , X n=1 so that Z 23 24

q

X

min(1, kxk )

∞ X

δ(en n−1/q ) (dx) =

n=1

X1 = ∞, n n

See A. de Acosta and J. Samur (1979). See L. LeCam (1970), Theorem 3.

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which would contradict (ii). Now, because (ii) is a superproperty (since any measure can be weakly approximated by atomic measures), we conclude that c0 is not finitely representable in X. Therefore, in view of Theorem 5.5.2, Kahane Theorem, and Hoffmann-Jorgensen Theorem (seeP Chapter 1), to show (i) it suffices that if the series P n γn xn converges a.s., then P to prove q n kxn k < ∞. So, assume that n γn xn , where (xn ) ⊂ X, converges a.s. As we observed above, the tail of the distributions of the Poisson random variable π1 is thinner than the tail of the Gaussian random variable γ1 , so P that by the Comparison Theorem P (see Chapter 1), the series n πn xn converges a.s. Evidently, L( n πn xn ) is an infinitely divisible probability disR ∗ tribution with the Fourier transform exp[ (cos(x x) − 1)µ(dx), X P P where µ = n δxn . By assumption (ii),P n min(1, kxn kq ) < ∞ so that, (xn ) is bounded (since n γn xn converges a.s.), P because q also n kxn k < ∞. QED

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Chapter 6 Spaces of Rademacher and stable types 6.1

Infratypes of Banach spaces

For a normed space X we define numerical constants apn (X), 1 ≤ p ≤ ∞, n ∈ N, as follows: apn (X)

n

X

εi xi := inf a ∈ R ; ∀x1 , . . . , xn ∈ X, inf

n

∗

εi =±1

i=1

n X 1/p o p ≤a kxk . i=1

Definition 6.1.1. We shall say that the normed space X is of infratype p1 if there exists a constant C > 0 such that, apn (X) ≤ C < ∞, for all n ∈ N. In other words, X is of infratype p if, and only if, for some constant C, and any finite sequence (xi ) ⊂ X, n n

X

X 1/p

εi xi ≤ C kxkp . inf

εi =±1 1

i=1

i=1

The notion of infratype is due to G. Pisier (1973), and it was also investigated in B. Maurey and G. Pisier (1976).

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Remark 6.1.1. It is easy to see that X isP of infratype p if, and p only if, for each sequence (xi ) ⊂ X with P i kxi k < ∞, there exists a sequence ε = ±1 such that the series i εi xi converges. The following Proposition is evident.

Proposition 6.1.1. (i) Any normed space is of infratype 1. (ii) If X is of infratype p, and 1 ≤ p1 < p, then X is of infratype p1 . 1/p (iii) If X is of infratype p, then a∞ , for some n (X) ≤ Cn C > 0. The properties of the sequence (a∞ n (X)) stated below will be useful in developing the theory of spaces of type p in the remainder of the chapter. Proposition 6.1.2. (i) If X = {0}, then a∞ n (X) = 0, for each n ∈ N, but if X 6= {0} then 1 ≤ a∞ n (X) ≤ n,

n ∈ N,

and if, additionally, X is infinite-dimensional, then n1/2 ≤ a∞ n (X) ≤ n,

n ∈ N;

(ii) Monotonicity: If n ≤ m, n, m ∈ N, then ∞ a∞ n (X) ≤ am (X);

(iii) Subadditivity: For any n, k ∈ N, ∞ ∞ a∞ n+k (X) ≤ an (X) + ak (X);

(iv) Submultiplicativity: For any n, k ∈ N, ∞ ∞ a∞ nk (X) ≤ an (X) · ak (X).

Proof. (i) Only the fact that in infinite-dimensional X we 1/2 have the inequality a∞ is non-trivial, but it follows n (X) ≥ n immediately from the Dvoretzky Theorem of Chapter 1. Indeed,

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Dvoretzky Theorem states that for each n ∈ N, and any ǫ > 0, there exist x1 , . . . , xn ∈ X such that, for any a1 , . . . , an ∈ R, n X i=1

|ai |

2

1/2

n

X

X

≤ ai xi ≤ (1 + ǫ) |ai |, i

i=1

which gives the necessary estimate. (ii), and (iii), have straightforward proofs which we omit. (iv) To prove the submultiplicativity of a∞ n (X) take x1 , . . . , xnk ∈ X. Then, for each i = 1, . . . , n, select εij , (i − 1)k < j ≤ ik, such that ik

X

j=ik−k+1

and define y i = struction,

Pik

εij xj

ik

X

= inf εi =±1

i j=ik−k+1 εj xj , i

ky i k ≤ a∞ k (X)

j=ik−k+1

εj xj ,

= 1, . . . , n. By the very consup

(i−1)k 1, then there exists p0 , 1 ≤ p0 ≤ ∞ (1 ≤ p0 ≤ 2 if the space X is infinite-dimensional), such that 1/p0

a∞ n0 (X) = n0

,

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and X is of infratype p, for each p < p0 . Proof. The existence of such a p0 is implied by Proposition 6.1.2 (i). Now, let p < p0 , and (xn ) be a finite sequence in X. For each k ∈ N, we define the set of integers, Ak , as follows: P P n ( n kxn kp )1/p ( n kxn kp )1/p o Ak := n : < kxn k ≤ . (k+1)/p k/p n0 n0 Evidently, by the triangle inequality, ∞

X

X

X

εn xn , inf εn xn ≤ inf

εn =±1

n

k=0

εn =±1

n∈Ak

from which it follows that

P ∞

X

X ( n kxn kp )1/p

∞ a|Ak | (X) εn xn ≤ inf . k/p εn =±1 n n 0 k=0

(6.1.1)

However, on the other hand, X n

kxn k

p

1/p

≥

X

n∈Ak

kxn k

p

1/p

≥ |Ak |

so that |Ak | ≤ nk+1 (we assume here that 0 by Proposition 6.1.2 (ii) and (iv),

1/p (

P

n

P

n

kxn kp )1/p

(k+1)/p

n0

,

kxn kp = 6 0). Now, (k+1)/p0

∞ ∞ k+1 a∞ = n0 |Ak | (X) ≤ ank+1 (X) ≤ (an0 (X)) 0

,

and, by (6.1.1), P ∞

X

X p 1/p

(k+1)/p0 ( n kxn k ) inf εn xn ≤ n0 k/p εn =±1 n0 n k=0 1/p0

= n0

∞ 1/p X (1/p0 −1/p) k X n0 · kxn kp . k=0

n

This proves that X is of infratype p, because the series on the right converges in view of the assumption, p < p0 . QED

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119

Finally, in view of the above Proposition, it is of interest to calculate the largest value of p such that X is of infratype p. Definition 6.1.2. pitype (X) := sup{p : X is of infratype p}. It turns out that the above parameter can be explicitly expressed in terms of the constants a∞ n (X). Theorem 6.1.1. log n . n→∞ log[a∞ (X)] n

pitype (X) = lim

Proof. If X is of infratype p then, by Proposition 6.1.1(iii), lim inf n→∞

log n log n ≥ lim inf = p, ∞ n→∞ log C + (1/p) log n log[an (X)]

so that lim inf n→∞

log n ≥ pitype (X). log[a∞ n (X)]

If q > pitype (X), in view of Proposition 6.1.3, for each n ∈ N, 1/q a∞ , so that n (X) ≥ n lim sup n→∞

log n log n ≤ lim sup = q, ∞ log[an (X)] n→∞ (1/q) log n

and lim sup n→∞

6.2

log n ≤ pitype (X). log[a∞ n (X)]

QED

Banach spaces of Rademachertype p

For a normed space X we define numerical constants bpn (X), p ≥ 1, n ∈ N, as follows: bpn (X) :=

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n n

p 1/p n X X 1/p o

+ inf b ∈ R : ∀x1 , . . . , xn ∈ X, E ri xi ≤b kxi kp . i=1

i=1

Definition 6.2.1. We shall say that X is of Rademacher-type p if there exists a constant C > 0 such that, bpn (X) ≤ C < ∞, for all n ∈ N.

2

Remark 6.2.1. The Kahane Theorem (see, Chapter 1) implies that a Banach space X is of Rademacher-type p if, and only if, there exists a constant C and an α, 0 < α < ∞, such that, for any finite sequence (xi ) ⊂ X n n

α 1/α X 1/p X

p ri xi ≤C kxi k , E i=1

(6.2.1)

i=1

or, alternatively, X is of Rademacher-type p if, and only if, for each α, 0 < α < ∞, there exists a constant C = Cα such that, for any finite sequence (xi ) ⊂ X, the inequality (6.2.1) holds true. Remark 6.2.2. One can also define the concept of a Banach space of Rademacher-type Φ, where Φ is an Orlicz-type function. Namely, a Banach space X is said to be of Rademacher-type Φ if, for every p ≥ 1, there exists a constant C such that, for any sequence x1 , . . . , xn ∈ X,

p 1/p n o X X

≤ C inf c > 0 : ri x i Φ(kxi k/c) ≤ 1 . E i

i

Many results valid in spaces of Rademacher-type p may be generalized to spaces of Rademacher-type Φ. The following proposition partially explains the relationship between the infratype and the Rademacher-type of a normed space. Proposition 6.2.1. (i) Any normed space X is of Rademacher-type 1, and if the dimension of X is greater than 1, and X is of Rademacher-type p, then p ≤ 2; 2

The concept of Rademacher-type 2 is due to E. Dubinsky, A. Pelczynski and H.P. Rosenthal (1972). For general p the notion was introduced by J. Hoffmann-Jorgensen (1972/73).

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(ii) If X is of Rademacher-type p, and 1 ≤ p1 < p, then X is of Rademacher-type p1 ; (iii) If X is of Rademacher-type p, then it is also of infratype p. We omit the obvious proofs of the first part of (i), and of (ii) and (iii). The second part of (i) is an immediate corollary to the Khinchine Inequality (see Introduction). Now, we turn to the investigation of properties of the sequences of constants bpn (X), 1 ≤ p ≤ ∞.

Proposition 6.2.2.3 (i) If X = {0}, then bpn (X) = 0, n ∈ N. If X 6= {0} then 1 ≤ apn (X) ≤ bpn (X) ≤ n1−1/p ,

n ∈ N,

and, if X is infinite-dimensional, then 1/p p n1/2 ≤ a∞ bn (X) ≤ n, n (X) ≤ n

n ∈ N.

(ii) Monotonicity: If n ≤ m, n, m ∈ N, then bpn (X) ≤ bpm (X); (iii) Subadditivity: For any n, k ∈ N, bpn+k (X) ≤ bpn (X) + bpk (X); (iv) Submultiplicativity: For any n, k ∈ N, and 1 ≤ p < ∞, bpnk (X) ≤ bpn (X) · bpk (X); Proof. Statements (i), (ii), and (iii), have straightforward proofs based on Proposition 6.1.2, and we omit them. (iv) Take x1 , . . . , xn ∈ X, and for each i = 1, . . . , n, define random vectors ik X Xi = rj xj , j=ik−k+1

3

This result is due to G. Pisier (1973).

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so that, if (rj′ ) is a Rademacher sequence independent of (rj ), we have the inequality, n n

p 1/p X X 1/p

E′r ri′ Xi (ω) ≤ bpn (X) kXi (ω)kp , i=1

i=1

∀ω ∈ Ω,

(here, E′r denotes the integration with respect to (rj′ )), from which, by symmetry, we obtain the estimate, n n nk

p 1/p

X

p 1/p X 1/p X

≤ bpn (X) EkXi kp = EE′ ri′ Xi E rj x j i=1

i=1

j=1

≤

bpn (X)

n X i=1

=

bpn (X)

·

ik X

(bpk (X))p

j=ik−k+1

bpk (X)

·

nk X i=1

kxj kp

kxi k

p

1/p

1/p

.

QED

As we saw in the case of infratype, the knowledge of the sequence (bqn (X)), for any particular q, 1 ≤ q < ∞, provides complete information about the Rademacher-type of X. Proposition 6.2.3. If X = 6 {0} is a normed space, and n0 > 1, 1 ≤ q < ∞, then there exists a p0 , 1 ≤ p0 ≤ q (1 ≤ p0 ≤ 2, if X is infinite-dimensional) such that 1/p0 −1/p

bqn0 (X) = n0

,

and X is of Rademacher-type p, for each p < p0 . Proof. The existence of such p0 is an immediate consequence of Proposition 6.2.2 (i). By submultiplicativity, k(1/p0 −1/q)

bqnk (X) ≤ n0 0

1/p0 −1/q

Now, set C = n0

,

k ∈ N.

, and take any n ∈ [nk0 , nk+1 0 ). Then, (k+1)(1/p0 −1/q)

bqnk (X) ≤ bqnk+1 (X) ≤ n0 0

0

≤ Cn1/p0 −1/q .

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Spaces of Rademacher and stable types so that, for any finite subset A ⊂ N,

q 1/q X X 1/q

E rn x n ≤ C|A|1/p0 kxn kq /|A| n∈A

n∈A

≤ C|A|1/p0 sup kxn k.

(6.2.2)

n∈A

At this point, take a finite sequence (xi ) ⊂ X , and p < p0 . For each k, define the set of integers, Ak , as follows: n X 1/p X 1/p o Ak := n ∈ N; kxi kp 2−(k+1)/p < kxn k ≤ kxi kp 2−k/p . i

i

Then, |Ak | ≤ 2k+1 , and in view of (6.2.2), ∞ X

q 1/q

q 1/q X X

ri x i ri x i ≤ E E i

≤ ≤

i∈Ak

k=0

∞ X

∞ X k=0

C2

C|Ak |1/p0 sup kxn k n∈Ak

(k+1)/p0

i

k=0

≤ C2

1/p0

X

∞ X

2

kxi k

k(1/p0 −1/p)

1/p

X i

k=0

p

2−k/p

kxi k

p

1/p

,

so that X is of Rademacher-type p in view of the Remark 6.2.1. QED Like in the case of infratype we can now calculate the supremum of p’s for which the space X is of Rademacher-type p in terms of the constants bqn (X). Definition 6.2.2. pRtype (X) := sup{p : X is of Rademacher type p}. Theorem 6.2.1.4 Let q < ∞, and let X 6= {0}. Then log n . q n→∞ log(n1/q bn (X))

pRtype (X) = lim 4

This result is due to G. Pisier (1973).

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Proof. If X is of Rademacher-type p then, by Remark 6.2.1, and H¨older’s Inequality, there exists a constant C > 0 such that, for any finite (xi ) ⊂ X, n n n

q 1/q X X 1/p X 1/q

E ri x i ≤C kxi kp ≤ Cn1/p−1/q kxi kq , i=1

i=1

i=1

so that bqn (X) ≤ Cn1/p−1//q , and lim inf n→∞

log n log n ≥ lim inf = p. q n→∞ log C + (1/p) log n log(n1/q bn (X))

Therefore, lim inf n→∞

log n ≥ pRtype (X). log(n1/q bqn (X))

If p > pRtype (X), then, by Proposition 6.2.3, bqn (X) ≥ n1/p−1//q , n ∈ N, and lim sup n→∞

Hence, lim inf n→∞

log n ≤ p. log(n1/q bqn (X))

log n ≤ pRtype (X). log(n1/q bqn (X))

QED

In the next step we will verify how the Rademacher-type is preserved under standard operations on normed spaces. These results will enable us to give a number of concrete examples of spaces of Rademacher-type p at the end of this section. Proposition 6.2.4. (i) If X is of Rademacher-type p, and Y is a subspace of X , then Y is of Rademacher-type p as well. (ii) If X is of Rademacher-type p, and Y is a closed subspace of X, then the quotient space X/Y is also of Rademacher-type p. Proof. The statement (i) is obvious, so let us prove (ii). Let ¯ i, . . . , x ¯ n ∈ X/Y , and let π : X 7→ X be of Rademacher-type p, x X/Y be the standard surjection. For any ǫ > 0, one can find

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¯ i , and kxi k ≤ (1 + ǫ)k¯ x1 , . . . , xn ∈ X such that π(xi ) = x xi k, so that

X

p

X

X

p p

¯ i = E π ¯ i ≤ E E ri x ri x ri xi i

i

≤C

X i

i

kxi kp ≤ C(1 + ǫ)p

X i

k¯ x i kp .

QED

Theorem 6.2.2.5 Let (T, Σ, µ) be a σ-finite measure space. Then, (i) The constants bpn (X) = bpn (Lp (T, Σ, µ; X)), 1 ≤ p < ∞; (ii) If p ≤ q < ∞, then X is of Rademacher-type p if, and only if, Lq (T, Σ, µ; X) is of Rademacher-type p. Proof. (i) Since X can be identified with a subspace of Lp (X), we have bpn (X) ≤ bpn (Lp (X)). To prove the converse inequality let X1 , . . . , Xn ∈ Lp (X). Then, for each t ∈ T ,

p

X X

kXi (t)kp . ri Xi (t) ≤ (bpn (X))p E i

i

Integrating this inequality with respect to measure µ we obtain the inequality

p

X X

kXi kp , ri Xi ≤ (bpn (X))p E i

i

which shows that bpn (X) ≥ bpn (Lp (X)). (ii) The “if” part is obvious because X is a subspace of Lq (X). The “only if’ part follows from (i) in the case p = q, and in the case p < q < ∞ it follows from the fact that if X1 , . . . , Xn ∈ Lq (X), then

q 1/q

q 1/q Z X X

ri X i = E ri Xi (t) dµ E q L

i

≤C 5

i

hZ X i

kXi (t)kp

q/p

dµ

ip/q·1/p

See B. Maurey and G. Pisier (1976).

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Geometry and Martingales in Banach Spaces ≤C

hXZ i

kXi (t)kp·q/p

p/q i1/p

≤C

hX i

kXi kpLq

i1/p

.

Above, we utilized Remark 6.2.1, and the triangle inequality for the norm in lq/p . QED The next result deals with the so-called “three-space problem” which provides an additional exploration of the quotient spaces of spaces of Rademacher-type which have been already investigated in Proposition 6.2.4. Theorem 6.2.3.6 Let 1 < p < q ≤ 2, and let Y be a closed subspace of a Banach space X. If Y , and X/Y , are of Rademacher-type q, then X is of Rademacher-type p. The proof of the above theorem will rely on the folllowing Lemma which can be viewed as a generalization of the submultiplicativity property from Proposition 6.2.2 (iv). Lemma 6.2.1. Let X be a Banach space, and Y be a closed subspace of X. (i) If 1 < p ≤ 2, then bpnk (X) ≤ bPn (Y )bPk (X) + bPn (Y )bPk (X/Y ) + bPn (X)bPk (X/Y ),

for all n, k ∈ N; (ii) If Y and X/Y are of Rademamcher-type q, then there exist constants C, and γ, such that bqn (X) ≤ (log n)γ ,

n ≥ 2.

Proof. As above, π stands for the canonical projection of X onto X/Y , with the norm kπ(x)kX/Y = inf{kx + yk : y ∈ Y }. Now, let x1 , . . . , xnk ∈ X, and define Xi =

ik X

rj xj .

j=ik−k+1 6

See G. Pisier (1975), and P. Enflo, J. Lindenstrauss, and G. Pisier (1975).

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Then, for each ω ∈ Ω, i = 1, 2, . . . , n,, and γ > 0, there exists a Yi (ω) ∈ Y such that kXi (ω) + Yi (ω)kX ≤ kπ(Xi (ω))kX/Y + γ. Then, choosing (ri′ ) to be independent of (ri ), by the convexity of the norm, we obtain the inequality n

p 1/p X

′ E ri′ Xi (ω) i=1

n n

p 1/p X X

p 1/p ′ ′ ′ ′ + E ri Xi (ω) + Yi (ω) , ≤ E ri Yi (ω) i=1

i=1

so that, by definition of

bpn (X),

and bpn (Y ),

n

p 1/p X

′ ′ E ri Xi (ω) i=1

≤ bpn (Y )

n X i=1

kYi (ω)kp

1/p

+ bpn (X)

n X i=1

kXi (ω) + Yi (ω)kp

1/p

,

Furthermore, kYi (ω)k ≤ kXi (ω)k + kYi (ω) + Xi (ω)k and, on the other hand, kXi (ω) + Yi (ω)k ≤ kπ(Xi (ω))k + γ, so that we have n n

p 1/p X X 1/p

′ ′ p E ri Xi (ω) ≤ bn (Y ) kXi (ω)kp i=1

+

bpn (Y

i=1

)+

bpn (X)

n hX i=1

kπ(Xi (ω))k

p

1/p

1/p

+ γn

i .

After integration, and employing an obvious symmetry argument, we obtain that nk nk

p 1/p

p 1/p

X X

′ = EE ri′ Xi ri xi E i=1

i=1

≤ bpn (Y )

n X i=1

EkXi kp

1/p

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Geometry and Martingales in Banach Spaces +

bpn (Y

)+

bpn (X)

n hX

Ekπ(Xi )kp

i=1

1/p

+ γn1/p

i

nk h iX 1/p p p p p p ≤ bn (Y )bk (X) + bn (Y ) + bn (X) bk (X/Y ) kxj kp i=1

+

bpn (Y

)+

bpn (X)

1/p

γn

,

which proves (i) in view of the arbitrariness of γ. (ii) Let C1 = supn bqn (X) < ∞, and C2 = supn bqn (X/Y ) < ∞. In view of (i), bqnk X) ≤ C1 bqk (X) + C1 C2 + bqn (X)C2 ,

n ∈ N.

Since 1 ≤ bqn (X), n ∈ N, (see Proposition 6.2.1), bqn2 (X) ≤ (C1 + C1 C2 + C2 )bqn (X). Chose γ so that 2γ = C1 + C1 C2 + C2 , and set cn = bqn (X)log −γ n, n = 2, 3, . . . , Then the above inequality reads Cn2 ≤ Cn ,

n ∈ N.

(6.2.3)

Now, let n be an integer ≥ 2. Choose k ∈ N such that k

k+1

Nk := 22 ≤ n < 22

= Nk2 .

Since bpn are increasing (see Proposition 6.2.2), cn = bpn (X) log−γ n ≤ bpN 2 (X) log−γ n ≤ 2γ bpN 2 (X) log−γ Nk2 = 2γ cNk2 . k

k

However, (6.2.3) implies that, for each k ≥ 0, we have cNk ≤ cN0 = c2 , so that cn ≤ 2γ c2 , which proves (ii). QED Proof of Theorem 6.2.3. By Theorem 6.2.1, and Lemma 6.2.1 (ii), log n pRtype (X) = lim n→∞ log n1/q bqn (X) log n = q, n→∞ q −1 log n + log C + log[γ 1/2 log n]

> lim

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so that, by Proposition 6.2.1(ii), the space X is of Rademachertype p, for each p, 1 ≤ p < q. QED To conclude this section we will provide a number of examples of spaces which are of Rademacher-type p, as well as the spaces that are not of Rademacher-type p. More examples can be found in the later parts of this chapter. Example 6.2.1. (i) The real line R is of Rademacher-type p for any p ∈ [1, 2]. In fact, it is sufficient to show that R is of Rademacher-type 2, and this fact is a direct consequence of the Khinchine Inequality (see Introduction). (ii)7 If X is p-smoothable, i.e., it has an equivalent norm with the modulus of smoothness, p(τ ), satisfying the condition p(τ ) = O(τ p ), τ → 0, then X is of Rademacher-type p (see Chapter 3). (iii) In particular, Lp , and lp , are of Rademacher-type min(p, 2), for 1 < p < ∞, and if 1 ≤ q < p ≤ 2, then Lq , and lq , are not of Rademacher-type p. In other words, pRtype (Lp ) = min(p, 2). This fact follows from Theorem 6.2.2 (i). (iv)8 Let 1 ≤ p < ∞. By C p (H) denote the Banach space of compact operators, A, on the Hilbert space H such that kAkp := tr(A∗ A)p/2

1/p

< ∞.

Note that, if p 6= 2, then C p (H) is not isomorphic with any subspace of Lp . However, pRtype (C p (H)) = min(p, 2), too. This fact follows directly from (ii), and the evaluation of the modulus of smoothness for C p (H). (v) The spaces c0 , l1 , and l∞ , are not of Rademacher-type p, for any p ∈ (1, 2]. Finally, we have a general result relating the Rademacher-type of the Banach space X to the cotype of its dual space, X ∗ . Theorem 6.2.4.9 If Banach space X is of Rademacher-type p, then its dual, X ∗ is of cotype q, with 1/p + 1/q = 1. 7

See W.A. Woyczynski (1973). See N. Tomczak-Jaegerman (1974). 9 See J. Hoffmann-Jorgensen (1972/73), and G. Pisier (1973/74). 8

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Proof. If X is of Rademacher-type p then, by Remark 6.2.1, there exists a constant C such that, for any n ∈ N, and any (xi ) ⊂ X, n n

2 1/2 X X 1/p

E ri xi ≤C kxi kp . i=1

i=1

Let x∗1 , . . . , x∗n ∈ X ∗ . Then, there exists an ǫ > 0 such that, for any x1 , . . . , xn ∈ X, n X

x∗i xi

i=1

≥

X i

kx∗i kq

1/q

− ǫ,

and

n X i=1

kxi kp ≤ 1.

Therefore, X i

kx∗i kq

1/q

≤

n X

x∗i xi + ǫ = E

i=1

n X i=1

ri x∗i

n X i=1

ri xi + ǫ

n n

2 1/2

2 1/2 X X

ri x i E ri x∗i ≤ E i=1

i=1

n

2 1/2 X

∗ + ǫ. ri x i +ǫ ≤ C E i=1

Hence, since ǫ > 0 was arbitrary, X ∗ is of cotype q. QED Remark 6.2.3. (a) Note that the dual space of the space of cotype q does not have to be of Rademacher-type p, 1/p + 1/q = 1. For example l1 is of cotype 2, but l∞ is not of Rademacher-type 2. However, if X is a Banach lattice with p(X) > 1, then X is of Rademacher-type p if, and only if, X ∗ is of cotype q.10 (b) If X 1 , and X 2 , are two Banach spaces contained in a locally convex topological vector space, and 0 ≤ θ ≤ 1, 1 ≤ p ≤ ∞, then, if X 1 is of Rademacher-type p1 , X 2 is of Rademacher-type p2 , and 1/p = (1 − θ)/p1 + θ/p2 , then the Lions-Petree interpolation space Int(X 1 , X 2 )θ,p is of Rademacher-type p.11 10 11

See B. Maurey (1973/74). See B. Beauzamy (1974/75).

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6.3

Local structures of Rademacher-type p

spaces

of

We begin with a statement that follows directly from the definition. Proposition 6.3.1. Rademacher-type p is a super-property, i.e., if the Banach space X is of Rademacher-type p, and space Y is finitely representable in X, then Y is of Rademacher-type p as well. A factor critical for the investigation of the local structure of spaces of Rademacher-type p is the following result which deals with the concept of finite factorability (see Chapter 1). Theorem 6.3.1.12 If X is an infinite-dimensional normed space, then the canonical injection l1 7→ lRtype (X) is finitely factorable through X. Proof. If pRtype (X) = 2, then the theorem is a corollary to the Dvoretzky-Rogers Lemma (see, Chapter 1) which states, in our current terminology, that l1 7→ l2 is finitely factorable through any infinite-dimensional normed space. If pRtype (X) < 2, then it is sufficient to show that, for any q satisfying the inequality, pRtype (X) < q < 2, the mapping l1 7→ lq is finitely factorable through X because the interval of those q’s for which l1 7→ lq is finitely factorable through X is closed (see Chapter 1). So, let pRtype (X) < q < 2. The assertion of the theorem is then an immediate corollary to the following Lemma (Parts (i), and (iii)).13 Lemma 6.3.1. (i) If 1 ≤ q < 2, and the space Y is the completion of the space S 14 under the norm |.| which is invariant under spreading, and such that n q 1/q X −1/q n E ei ri ≥ δ > 0, n ∈ N, (6.3.1) i=1

12

See B. Maurey and G. Pisier (1976). The technical Part (ii) is needed in the proof of Part (iii). 14 The space of real-valued sequences with a finite number of terms different from zero. 13

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then the embedding l1 7→ lq is finitely factorable through Y . (ii) Let pRtype (X) < q < 2. Then, there exist an infinite set N1 ⊂ N, and a sequence ǫn ∈ (0, 1), n = 1, 2, . . . , such that, for each n ∈ N1Pthere exists x1 , . . . xn ∈ X such that, for each (αi ) ∈ Rn , with i |αi | = 1, Cbqn (X)n1/q−1 ≤ (1 − ǫn )bqn (X) − bqn−1 (X) n1/q

q 1/q X

αi r i x i ≤ E

(6.3.2)

i

where

−1 C = (q − pRtype (X) 2q pRtype (X) · bq2 (X) .

(iii) There exists an invariant under spreading norm |.| on S, satisfying (6.3.1), and such that Y is finitely representable in X. Proof. (i) Let (γn ) be a sequence of independent N(0, 1) Gaussian random variables which are also independent of the sequence (rn ). In view of (6.3.1), −1/q

0 0.

The last equality above took advantage of the invariance under spreading of the norm |.|. Now, we shall show that the canonical embedding lq′ 7→ l∞ , 1/q + 1/q ′ = 1, is finitely factorable through the dual space G∗ , where G := span uk = e2k − e2k−1 : k ∈ N ⊂ Y .

Indeed, were it not the case then, by Theorem 5.3.2, there would exist an r ′ > q such that G∗ is of cotype r ′ , i.e., there exists a constant K > 0 such that, for each n ∈ N, and each α1 , . . . , αn ∈ R, X i

where u∗k ( 1,

|αi |r

P

i

′

1/r′

X

X

αi u∗i , αi u∗i ri ≤ 2K ≤ KE i

i

αi ui ) := αk , and, by transposition, with 1/r +1/r ′ = X X 1/r r |αi | . αi ui ≤ 2K i

i

Above, we used the fact that (uk ) is a (monotone) unconditional basic sequence with constant 2. Then, however, n q 1//q h X X q/r i1/q uk γk ≤ 2K E |γk |r E k=1

k

≤ 2KE

n X k=1

|γk |r

1/r

≤ 2K(E|γ1 |r )1/r · n1/r ,

which contradicts (6.3.3). Hence, by the definition of finite factorability, for each n ∈ N, and each ǫ > 0, there exists a sequence x∗1 , . . . , x∗n ∈ G∗ such that, for each α1 , . . . , αn ∈ R, n

X

X ′ ′ 1/q

(1 − ǫ) sup |αi | ≤ αi x∗i ≤ |αi |q . i

i

i=1

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For a certain H ⊂ G, span[x∗1 , . . . , x∗n ] = (G/H)∗ . By transposi¯ i, . . . , x ¯ n ∈ G/H such that, for all α1 , . . . , αn ∈ tion, there exist x R, n X i=1

|αi |

q

1/q

X

¯ i ≤ αi x i

G/H

≤ (1 − ǫ)

−1

n X i=1

|αi |,

¯ i , we can which concludes the proof of part (i) because, for each X pick a representative X i such that |xi | ≤ (1 + ǫ)k¯ xi kG/H . (ii) By Lemma 6.2.3, bqn (X) ≥ n1/pRtype (X)−1/q , n ∈ N, Since, for any (an ) ⊂ R+ , a log an n ≤ lim sup n −1 , lim sup log n an−1 n→∞ n→∞ (because limt→∞ t(tα /(t − 1)α − 1) = α), setting an = bqn (X) we get that bq (X) 1 n − ≤ lim sup n q −1 , pRtype (X) q bn−1 (X) n→∞ 1

so that there exists N1 ⊂ N, and a sequence (ǫn ) ⊂ (0, 1), such that, for all n ∈ N1 , h i 1 1 bqn (X) 1 − ≤ n (1 − ǫn ) q −1 , 2 pRtype (X) q bn−1 (X) which implies that −1 2qpRtype (X) 1 ≥ bqn−1 (X) (1 − ǫn )bqn (X) − bqn−1 (X) . q − pRtype (X) n Finally, in view of the submultiplicativity property, bqn (X) ≤ bq2 (X) · bqn−1 (X), n ≥ 1, and this yields the inequality −1 2qpRtype (X) · bq2 (X) 1 q bn (X) (1 − ǫn )bqn (X) − bqn−1 (X) ≤ = C, n q − pRtype (X) which gives the first inequality in (6.3.2).

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Now, let n ∈ N1 . P By definition of bqn (X), there exist x1 , . . . , xn ∈ X such that i kxi kq = n, and n

q 1/q X

E ri xi ≥ (1 − ǫn )bqn (X)n1/q . i=1

If (αi ) ∈ Rn ,

P

i

(1 −

|αi | = 1, then

ǫn )bqn (X)n1/q

n

q 1/q X

ri |αi |xi ≤ E i=1

n

q 1/q X

ri (1 − |αi |)xi + E i=1

n

X

≤ E

i=1

q 1/q

+ r i αi x i

supP

(ti )∈Rn +,

n

q 1/q X

. ri (1 − ti )xi E

ti =1

i=1

P

Furthermore, the function (ti ) 7→ (Ek i ri (1 − ti )xi kq )1/q P is a convex, continuous function on the convex set {(ti ) ∈ Rn+ : ti = 1}, so that its maximum is attained at an extreme point of this set, i.e., at a point which has all components except one equal to zero. Therefore, (1−ǫn )bqn (X)n1/q ≤

n n

q 1/q

q 1/q X X

ri x i + sup E r i αi x i E i=1

1≤j≤n

i=1,i6=j

n

q 1/q X

≤ E r i αi x i + bqn−1 (X)n1/q , i=1

which proves the second inequality in (6.3.2).

(iii) Take n ∈ N1 , and x1 , . . . , xn , as in part (ii). Denote ˜ n the subspace of Lq (X) spanned by ri xi , i = 1, . . . , n, and by R define the operator ˜n ∋ Un : R

n X i=1

(n)

αi ri xi 7→ (αi ) ∈ l1 .

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˜ n , then If y 1 , . . . , y n ∈ R n X

sup |Un (y k )(i)| ≤

i=1 1≤k≤n

n n X X E rk Un (y k )(i) i=1

k=1

n n

q 1/q

q 1/q X X

≤ E rk Un (y k ) (n) ≤ kUn k E , y k rk ˜n l1 R k=1 k=1

wherefrom it follows that n X i=1

sup |Un (y k )(i)| ≤

1≤k≤n

bqn (X)kUN k

n X k=1

ky k kq

1/q

,

because bqn (X) = bqn (L(X)), according to Theorem 6.2.5 (i). Therefore, by Nikishin’s Lemma (see Chapter 1), there exists a set An ⊂ {1. . . . , n}, such that |An | ≥ n/2, and such that, for each (αi ) with αi = 0 for i ∈ / An , λq ((αi )) ≤

′ 21/q n−1/q bqn (X)kUn k

q 1/q X

, αi ri xi E i

where (see Chapter 1) λp ((ai )) = sup cp |{I : |ai | > c}| c>0

1/p

,

and |A| denotes the cardinality of the set A. If we define Bn = {i : kxi k ≤ 41/q }, have that |Bn | ≥ Pthen we q 3n/4, and |An ∩ Bn | ≥ n/4, because i kxi k = n. For n ∈ N1 , the second inequality in Part (ii) can be written as follows: −1 kUn k ≤ n−1/q (1 − ǫn )bqn (X) − bqn−1 (X) ,

which, in conjunction with the first inequality in Part (ii) yields that, for each n ∈ N1 , and α1 , . . . , αn with the support in An ∩Bn ,

q 1/q X

λq ((αi )) ≤ 21/q C −1 E αi r i x i , i∈An ∩Bn

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n , 4

i ∈ An ∩ Bn =⇒ kxi k ≤ 41/q .

Hence, one can find an infinite subset N2 ⊂ N, and a constant δ > 0 such that, for each n ∈ N2 , there exist xn1 , . . . , xnn ∈ X such that, for all (αi ) ∈ Rn , n n

q 1/q X X

n αi xi ri ≤ |αi |. δ · λq ((αi )) ≤ E i=1

i=1

(n)

(n)

(This means, essentially, that the canonical embedding λq 7→ l1 is factorable through subspaces of Lq (X), for infinitely many n’s.) Now, according to a classical theorem of Banach, there exists a continuous linear functional µ on l∞ (N2 ) such that kµk = 1 and, for every (βn ) ∈ l∞ (N2 ), lim inf βn ≤ µ(βn ) ≤ lim sup βn . n∈N2

n∈N2

Define on S the seminorm, |||

X i

n

h X

αi xni αi ei ||| = µ

n∈N2

i=1

i .

Evidently, for each (αi ) ∈ S,

q 1/q X X |αi |. ≤ αi ei ri δλq ((αi )) ≤ E i

(6.3.4)

i

¯ i be the image of ei in S/|||.|||. Clearly, (6.3.4) holds in Let e S/|||.||| too. Using the Brunel-Sucheston construction (see Chapter 1) we can find a subsequence (¯ e′n ) ⊂ (¯ en ) such that, n X αi ei := i=1

lim

n X αk e′ik

i1 0

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Spaces of Rademacher and stable types and the gauge Jǫ (f, µ) := inf{c > 0; µ(kf k > c) < ǫ},

0 < ǫ < 1.

Definition 6.4.1. (a) We say that a set A ⊂ L0 (T, Σ, µ) is almost bounded in Λp if, for every ǫ > 0, there exists a measurable set Tǫ ⊂ T such that µ(T − Tǫ ) ≤ ǫ, and such that the set {f ITǫ : X ∈ A} is bounded in Λp (T, Σ, µ; X). (b) We say that the linear operator U : X 7→ L0 (T, Σ, µ; Y ) is almost continuous into Λq (T, Σ, µ; Y ) if the image by U of the unit ball in X is almost bounded in Λq (T, Σ, µ; Y ). Theorem 6.4.1.15 If X is a Banach space of Rademachertype p, 1 ≤ p ≤ 2, and Y is a Banach space, then each linear continuous operator from X into L0 (T, Σ, µ; Y ), where µ is a finite measure, is almost continuous from X into Λp (T, Σ, µ; Y ). The proof of Theorem 6.4.1 will rely on the following two Lemmas which provide alternative characterizations of the concepts of almost boundedness and almost continuity defined above. Lemma 6.4.1. A set A ⊂ L0 (T, Σ, µ; R) is almost bounded in Λ1 (R) if, and only if, for each ǫ ∈ (0, 1), there exists a constant Cǫ such that, for all (cn ) ∈ l1 , and all (fn ) ⊂ A, X Jǫ (sup |cn fn |, µ) ≤ Cǫ |cn |. (6.4.1) n

n

Proof. “If”. By assumption, in view of the inequality (6.4.1), for each ǫ ∈ (0, 1), there exists a constant C > 0 such that, for all (fn ) ⊂ A, and (cn ) ∈ l1 , X |cn | ≤ 1 =⇒ µ{sup |cn fn | > C} ≤ ǫ. (6.4.2) n

For the sake of brevity let’s introduce the concept of the N-set: we shall say that a measurable set B ⊂ T is an N-set if µ(b) > 0, and if there exists a function f ∈ A such that, for µ-almost all t ∈ B, 15

This result is due to B. Maurey (1973/74), Exp. IV and V.

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µ(B)|f (t)| > C. Denote by F the set of families (Bi ) of pairwise a.s. disjoint N-sets. Assume that F is non-empty, and order F by the relation : (Bi ) < (Ci ) if (Bi ) is a subfamily of (Ci ). Note that F is inductive in this order. Let (Bi ) be a maximal element of F . Because µ is finite, and Bi ’s are disjoint, we can assume that i ∈ N. By the definition of an N-set, for each i ∈ N, there exist fi ∈ A such S that µ(Bi )|fi | > C on Bi , so that P supi µ(Bi )|fi | > C on B = i Bi . Put ci = µ(Bi ). We have i |ci | ≤ 1, and by (6.4.2), µ(B) ≤ ǫ. Define Tǫ := T \ B, and let c > 0, and f ∈ A. If we define D := {t ∈ Tǫ : |f (t)| > Cc}, then µ(D) ≤ 1/c. Otherwise D would be an N-set (on D, one would have |f | > Cc ≥ C/µ(D)) disjoint with B, which would contradict the maximality of (Bi ). Hence, for each c > 0, and each f ∈ A, µ{|f χTǫ | > Cc} ≤ 1/c,

(6.4.3)

which proves that A is almost bounded in Λ1 in the case of nonempty F . If F is empty, then there is no N-set. Let c > 0, f ∈ A, and put D = {t ∈ T : |f (t)| > Cc}. Then µ(D) ≤ 1/c since, in the opposite case D would be an N-set. Therefore, one can take Tǫ = T , and the proof of the implication “If” is over. “Only if”. In view of the assumption, for each ǫ > 0, there exists a measurable Tǫ ⊂ T with µ(T \ Tǫ ) ≤ ǫ, such that for each c > 0, Pand each f ∈ A the inequality (6..4.3) holds true. let (cn ) ⊂ R, n |cn | ≤ 1, and (fn ) ⊂ A. Put Bn = {t ∈ Tǫ : |cn fn (t)| > C/c}.

In view of (6.4.3), µ(Bn ) ≤ ǫ|cn |, and µ(

S

n

Bn ) ≤ ǫ. But,

{t : sup |cn fn (t)| > C/c} ⊂ (T \ Tǫ ) ∪ ( n

[

Bn ),

n

so that µ{t : sup |cn fn (t)| > C/c} ≤ 2ǫ, n

which implies (6.4.1). QED

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Lemma 6.4.2. If X, and Y , are Banach spaces, (T, Σ, µ) is a probability space, and 0 < q < ∞, then the linear continuous operator U from X into L0 (T, Σ, µ; Y ) is almost continuous from X into Λq (T, Σ, µ; Y ) if, and only if, for each ǫ > 0, there exists a constant Cǫ such that, for each (xn ) ⊂ X, Jǫ (sup kU(xn )k, µ) ≤ Cǫ n

X n

kxn kq

1/q

.

(6.4.4)

Proof. First of all, notice that a set A is almost bounded in Λq (Y ) if, and only if, the set {kf kq : f ∈ A} is almost bounded in Λ1 (R). To complete the proof with help of Lemma 6.4.1 it is sufficient to observe that the condition (6.4.4) is equivalent to the condition (6.4.1) for fn = kU(xn )kq , with kxn k ≤ 1, because Jǫ (sup |cn fn |, µ) = Jǫ (sup |cn |kU(xn )kq , µ) n

n

= Jǫ sup kU(|cn | n

1/q

xn )k, µ

q

≤ Cq

X n

|cn |.

QED

Now, we are ready to turn to the proof of Theorem 6.4.1. Proof of Theorem 6.4.1. Let us begin by restating the Fubinitype inequality: If γ + δ ≤ αβ, then Jα Jβ (f (t, s), µ(dt)), ν(ds) ≤ Jγ Jδ (f (t, s), ν(ds)), µ(dt) .

Now, let ǫ > 0 be given and choose γ, and δ, so that γ+δ ≤ ǫ/3. By continuity of the operator U, Jδ (U(x), µ) ≤ Kδ kxk,

x ∈ X,

and, because the space X is of Rademacher-type p, Jγ

X i

X 1/p x i r i , P ≤ Kγ x i kp , i

for each finite (xi ) ⊂ X.

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X,

Also, observe that, for any Banach space X, and x1 , . . . , xn ∈ sup kxi k ≤ Jα i

X i

x i ri , P ,

(6.4.5)

whenever α < 1/2. Indeed, let β = 2α < 1. Then 2kxi k = Jβ (2xi ri , P) = Jβ ≤ Jβ xi ri +

X j6=i

h

xi ri +

X j6=i

X i x j rj , P x j rj + x i ri − j6=i

X X xj rj , P = 2Jα x j rj , P . xj rj , P +Jα xi ri − j

j6=i

Using (6.4.5), and Fubini’s Inequality, we get that X Jǫ (sup kU(xi )k, µ) ≤ Jǫ J1/3 U(xi )ri , P , µ i

i

X X

≤ Jγ Jδ U x i r i , µ , P ≤ Kδ J γ x i ri , P i

≤ Kδ Kγ

X i

kxi kp

1/p ,

which, in view of Lemma 6.4.2, concludes the proof of Theorem 6.4.1. QED

6.5

Banach spaces of stable-type p and their local structures

In analogy to the Rademacher-type spaces we will now introduce the concept of a Banach space of stable-type p in which the sequence (ri ) Rademacher random variables is replaced by the sequence (ξi ) of independent, identically distributed, p-stable random variables with the characteristic function p

Eeitξi = e−|t| ,

t ∈ R,

0 ≤ p ≤ 2.

(6.5.1)

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Again, we will take advantage of the numerical constants, called spn (X) in this case, related to the concept of stable-type p, spn

n

X

p/2 2/p +

:= inf s ∈ R : ∀x1 , . . . , xn ∈ X, E ξ i xi i=1

≤s

n X i=1

kxi kp

1/p .

Definition 6.5.1.16 We shall say that a normed space X is of stable-type p if there exists a constant C > 0 such that, for all n ∈ N, spn (X) ≤ C < ∞.

Remark 6.5.1. In view of Hoffmann-Jorgensen Theorem (see Chapter 1) X is of stable-type p if, and only if, there exists a constant C > 0 such that, for each α, 0 ≤ α < p∗ (p∗ = p, if p < 2, and p∗ = ∞, if p = 2), and for each finite (xi ) ⊂ X,

α 1/α X 1/p X

ξi xi ≤C kxi kp , E i

(6.5.2)

i

or, alternatively, X is of stable-type p if, and only if, for each α, 0 ≤ α < p∗ , there exists a constant C > 0 such that, for each finite (xi ) ⊂ X, the inequality (6.5.2) holds true. In the case α = 0, the inequality (6.5.2) has the usual interpretation: X is of stable-type p if, and only if, for each ǫ ∈ (0, 1) there exists a constant C > 0 such that, for every finite (xi ) ⊂ X,

1/p X X

Jǫ ξi xi , P ≤ C kxi kp . i

(6.5.3)

i

The relationship between Rademacher and stable types is explained in the following Proposition. In the case p = 2 both notions coincide and this case will be discussed separately in detail in Chapter 7. 16

Investigation of spaces of stable-type was initiated by J. HoffmanJorgensen (1972/73), B. Maurey (1972/73), Exp, VII, X, XI, and B. Maurey and G, Pisier (1973).

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Proposition 6.5.1. (i) If X is of stable-type p, 1 ≤ p ≤ 2, then it is also of Rademacher-type p. (ii) If X is of Rademacher-type p, 1 ≤ p ≤ 2, and q < p, then X is of stable-type q. Proof. (i) Because each normed space is of Rademacher-type 1, we can assume that p > 1. Let ξ1 , . . . , ξn be independent and identically distributed p-stable random variables with the characteristic function (6.5.1), and the Rademacher sequence (ri ) be independent of (ξi ). If X is of stable-type p, then

X

X

−1 ri Eξ |ξi |xi ri xi = (Eξ |ξi |) E E i

i

X

ri |ξi |xi ≤ (Eξ |ξi |) E −1

i

X X 1/p

−1 p −1 kxi k , ξi xi ≤ (Eξ |ξi |) C ≤ (Eξ |ξi |) E i

i

because (ξi ), and (ri |ξi |) have identical distributions. This proves that X is of Rademacher-type p in view of Remark 6.2.1. (ii) Recall that if (ξi ) are independent α-stable random variables and β < α, then X i

|ri |α

1/α

E|ξi |β

1/β

β 1/β X ri ξ i . = E

(6.5.4)

i

Now, let 0 < r < q < p, (ξi ) be a sequence of independent q-stable random variables, and (ri ) be independent of (ξi ). Since X is of Rademacher-type p, by Remark 6.2.1,

r 1/r X X 1/p

ri x i ≤C kxi kp , E i

i

and, for a fixed ω ∈ Ω,

X

r X r/p

Er ri ξi (ω)xi ≤ C r kxi kp |ξi (ω)|p , i

i

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Spaces of Rademacher and stable types from which, by integration, and (6.5.4) (used twice),

r 1/r X h X r/p i1/r

E ri ξi xi ≤C E kxi kp |ξi |p i

i

r 1/r X r −1/r = C Ekη1 k E kxi kξi ηi i

= C Ekηi k

r −1/r

r/q 1/r 1/r X Ekξ1 kr kxi kq |ηi |q , E i

where (ηi ) are independent, identically distributed p-stable random variables independent of (ξi ). Because (ξi ), and (ri ξi ), are identically distributed, and in view of the Jensen’s Inequality (r/q < 1 !),

r 1/r X r/q 1/r X

q q ≤ C1 E kxi k |ηi | ξi xi E i

i

X 1/q 1/q 1/q X ≤ C1 E kxi kq |ηi |q = C1 Ekηi kq kxi kq , i

i

again by (6.5.4), so that X is of stable-type q. QED The following Corollary is a straightforward consequence of the Proposition 6.5.1, and the results contained in Sections 6.2 and 6.3. Corollary 6.5.1. (i) Each normed space is of stable-type p, whenever 0 < p < 1. (ii) If X is of stable-type p, and p1 < p, then X is also of stable-type p1 . (iii) The spaces Lp , and lp , are of stable-type q, for each q such that 0 < q < p ≤ 2. On the other hand we have the following examples of spaces that are not of stable-type p even though they are of Rademachertype p.

Example 6.5.1. If 0 < p < 2, then the spaces Lp , and lp are not of stable-type p. Indeed, were (say) lp of stable-type p, for each each α ∈ (0, 1) there would exist a constant K > 0 such that

X X 1/p

Jα αi ei ξi , P ≤ K kαi ei kp , i

i

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for each (αi ) ⊂ R, with (ei ) being, as usual, the canonical basis iin lp . Thus, Jα

X i

|αi ξi |p

1/p

X 1/p ,P ≤ K |αi |p , i

P whichPwould imply that the series i |αi ξI |p converges a.s. whenp ever i |αi | < ∞. But this implication is not true (see, e.g., Schwartz Theorem, Chapter 1). For p ≥ 2 the situation is different.

Example 6.5.2. If 2 ≤ p < ∞, then the spaces Lp and lp , are of stable-type 2. To check this fact, let (γi ) be a sequence of independent N(0, 1) Gaussian (i.e., 2-stable) random variables, and (αi ) ⊂ R. In view of the elementary probability result, there exists a constant C > 0, such that p 1/p X 1/2 X 2 ≤C |αi | (6.5.5) αi γ i E i

i

If (xi ) ⊂ Lp (T, Σ, µ), p ≥ 2, then

p 1/p Z X

γi xi = E E

T

i

p X 1/p γi xi (t) µ(dt) i

p (1/p)p 1/p Z X µ(dt) γi xi (t) = E T

≤C ≤C

hXZ i

T

i

Z X T

i

|xi (t)|2

|xi (t)|p µ(dt)

p/2

2/p i1/2

µ(dt)

=C

1/p

X i

kxi k2

1/2

,

with the next to the last inequality being implied by the fact that p/2 ≥ 1. QED

Actually, Example 6.5.1 can be generalized to produce the following important geometric characterization of spaces of stabletype p, for 1 ≤ p < 2.

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Theorem 6.5.1.17 A normed space X is of stable-type p, 1 ≤ p < 2, if, and only if, p < p(X), that is if, and only if, the canonical embedding l1 7→ lp is not finitely factorable through X.

Proof. If p < p(X) then Proposition 6.5.1 (ii), and Corollary 6.3.1, immediately imply that X is of stable-type p. Conversely, if p ≥ p(X) then, by Theorem 6.3.1, and Corollary 6.3.1, the embedding l1 7→ lp is finitely factorable through X, so that for each n ∈ N there exist x1 , . . . , xn ∈ X such that, for all (αi ) ∈ Rn , X i

|αi |p

1/p

X X

|αi |. αi xi ≤ 2 ≤ i

i

In particular, 1 ≤ kxi k ≤ 2. Now, let (ξI ) be standard independent p-stable random variables, and r < p. Then, were X of stable-type p we would have that n n

r 1/r X r/p 1/r X

p p αi ξ i x i E |αi | |ξi | ≤ E i=1

i=1

≤C

n X i=1

|αi |p kxi kp

1/p

≤ 2C

X i

|αi |p

1/p

,

P so that, again, σi |αi ξi | would converge a.s. whenever i |αi |p < ∞, the implication that is not true in view of the Schwartz Theorem (see Chapter 1). QED p

Corollary 6.5.2. (i) A normed space X is of stable-type p, 1 ≤ p < 2, if, and only if, lp is not finitely representable in X. (ii) The interval of those p’s for which X is of stable-type p is open whenever p(X) < 2, that is, if X is of stable-type p, p < 2, then X is of stable-type q, for some q > p. Proof. Part (i) follows directly from the above Theorem 6.5.1, and Theorem 6.3.2. Part (ii) is also a direct consequence of the above Theorem, and the fact that the set of those p’s for which the embedding l1 7→ lp is finitely factorable through X is closed (see Chapter 1). QED 17

This result is due to B. Maurey and G. Pisier (1976).

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In the next Theorem we provide a characterization of subspaces of the space Lq which are of stable-type p. Theorem 6.5.2.18 Let 1 ≤ p < 2, and let X be a closed subspace of the space Lq (T, Σ, µ), where µ is a finite measure. Then, the following four conditions are equivalent: (i) X is of stable-type p; (ii) X does not contain a subspace isomorphic to lp ; (iii) X does not contain a complemented subspace isomorphic to lp ; (iv) For some r, 0 ≤ r < p, the topologies of Lp , and Lr , coincide on X. Proof. (i) =⇒ (ii) follows directly from Example 6.5.1. (ii) =⇒ (iii) is obvious. (iii) =⇒ (iv) This implication is an immediate corollary to the well known Kadec-Pelczynski Theorem which states that if the sequence (xn ) ⊂ Lp , p ≥ 1, is such that, for each ǫ > 0, (xn ) ⊂ x ∈ Lp : µ{x(t) ≥ ǫkxk} ≥ ǫ , then, for each δ > 0, there exists a subsequence (y k ) = (xnk ) such that (y k /ky k k) is a basic sequence, equivalent to the canonical basis in lp with constant (1 + δ), and such that the span of (xn ) is (1 + δ)-complemented in Lp . (iv) =⇒ (i) Assume, first, that 0 < r < p, and that the topologies of Lr , and Lp , coincide on X, that is, there exists a constant C > 0 such that, for all x ∈ X, Z 1/r kxk ≤ C |x(t)|r µ(dt) . T

If (ξi ) is a sequence of independent, identically distributed p-stable random variables, and (xi ) ⊂ X, then r

r 1/r Z X 1/r X

E ξ i xi ≤C E ξi xi (t) µ(dt) i

18

T

i

This result is due to B. Maurey (1972/73), Exp. X, and XI.

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Z

T

=C

Z

T

≤ CE|ξi |

r

Z X T

i

X r 1/r E ξi xi (t) µ(dt) i

E|ξi |r

X i

|xi (t)|p

p

|xi (t)| µ(dt)

1/p

r/p

µ(dt)

= CE|ξi |

r

1//r

X i

kxi k

p

1/p

,

in view of (6.5.4) , and since r/p < 1, so that X is of stable-type p. In the case r = 0 we proceed in a similar fashion. By assumption, for each β ∈ (0, 1), there exists a constant C > 0 such that, for each x ∈ X, kxk ≤ CJβ (|x(t)|, µ). Utilizing the Fubini Inequality (see Chapter 1), we get that for each α ∈ (0, 1/4), i

h X X

ξi xi (t) , µ , P ξi xi , P ≤ CJ4α J1/2 J4α i

i

≤ CJα

h

i X ξi xi (t) , µ , P Jα i

h X 1/p i X 1/p = CJα Jα (ξ1 ) |xi (t)|p , µ ≤ C1 kxi kp , i

i

so that, again, X is of stable-type p. QED Remark 6.5.2. Note that the above result evidently fails for p = 2. To conclude this section let us verify how the stable-type property behaves under standard operations on normed linear spaces. Proposition 6.5.2. (i) If X is of stable-type p and Y is a subspace of X, then Y is of stable-type p, as well. (ii) If X is of stable-type p and Y is a closed subspace of X, then X/Y is of stable-type p.

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Proof. Part ((i) is evident. The staighforward proof of (ii) is analogous to the proof of Proposition 6.2.4(ii) and will be omitted. QED Theorem 6.5.3. If 1 ≤ p < 2, and X is of stable-type p, the space Lq (X) is of stable-type p whenever q > p. Proof. The fact that X is of stable-type p, p < 2, implies the existence of an r, p < r < q, such that X is of stable-type r (see Corollary 6.5.2 (ii)). Now, Proposition 6.5.1(i)implies that X is of Rademacher-type r, so that, by Theorem 6.2.5 (ii), Lq (X) is of Rademacher-type r as well, and therefore, by Proposition 6.5.1 (ii), Lq (X) is of stable-type p. QED Remark 6.5.3. If X is of stable-type 2, then Lq (X) is of stabletype 2, whenever q ≥ 2. We shall prove this fact in Chapter 7.

Finally, here is yet another procedure which produces new spaces of stable-type p Proposition 6.5.3. Denote by P [rn ] ⊗ X the subspace of L2 (Ω, P; X) spanned by the sums n rn xn , (xn ) ⊂ X. Then X is of stable-type p if, and only if, [rn ] ⊗ X is of stable-type p.

Proof. The “if” part is evident. So, let’s prove the “only if” part. Consider the sequence (ξi ) of independent and identically distributed p-stable random variables independent of the sequence (Xi ) ⊂ [rn ]⊗X. Then, if r < p, by Kahane Theorem (see Chapter 1),

r

r X 1/r X 1/r

Eξ Xn ξ n ≤ C Eξ Xn ξ n L2 (X)

n

n

Lr (X)

r 1/r X r/p 1/r X

p ≤ C E Xn ξ n ≤ C1 E r kXn k n

≤ C1

X n

kXn kpLp (X)

n

1/p

≤ C1

X n

kXn kpL2 (X)

1/p

.

QED

The “three space problem” for spaces of stable-type has a solution even neater than the analogous problem for spaces of Rademacher-type.

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Theorem 6.5.4. Let 1 ≤ p < 2, and Y be a closed subspace of X. If both spaces, Y , and X/Y , are of stable-type p, then X is of stable-type p as well. Proof. By Corollary 6..5.2 (ii), there exists a q, p < q < 2, such that Y , and X/Y , are of stable-type q, and thus also of Rademacher-type q. By Theorem 6.2.3, X is of Rademacher-type q1 , for each q1 < q. In particular, if p < q, this implies that X is of stable-type p in view of Proposition 6.5.1(ii). QED

6.6

Operators on spaces of stable-type p

We begin with the result on factorization of operators from a space of stable-type p into the Lebesgue space Lq . Theorem 6.6.1.19 Let X be a normed space of stable-type p, p ≥ 1, and let 0 < q ≤ p ≤ 2. Then each linear continuous operator U : X 7→ Lq (T, µ), where µ is a finite measure, can be factored as follows: Tg

V

X 7→ Lp (T, µ) 7→ Lq (T, µ), where V is linear and continuous, and Tg is the operator of multiplication by a function g ∈ Lr (T, µ), with 1/q = 1/p + 1/r. Proof. We can assume that q < p. In view of Maurey Theorem (see Chapter 1) it is sufficient to prove that Z X q/p X p kxn k < ∞ =⇒ |U(xn )(t)|p µ(dt) < ∞. T

n

n

Indeed, as in the proof of Proposition 6.5.1(ii), if (ξi ) is a sequence of independent and identically distributed p-stable random variables, and X is of stable-type p, then Z X q/p 1/q |U(xi )(t)|p µ(dt) T

19

n

Most of the results in this Section are due to B. Maurey (1974).

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Geometry and Martingales in Banach Spaces = E|ξ1|

q −1/q

Z

T

X q 1/q E U(xi )(t)ξi µ(dt) i

q 1/q X

q −1/q = E|ξ1| E U(xi )ξi i

q 1/q X −1/q

≤ E|ξ1|q kUk E xi ξ i i

≤ C E|ξ1 |

q −1/q

kUk

X i

kxi kq

1/q

,

which concludes the proof of the theorem. QED In the next step we will discuss the relationship between spaces of stable-type and absolutely summing operators which were introduced in Section 1.5. The next result is a corollary to Theorem 6.6.1. Proposition 6.6.1 Let the dual space X ∗ be of stable-type p, p ≥ 1, and let 0 < q ≤ p ≤ 2. Then there exists a constant C > 0 such that, for each operators from X into a Banach space Y, πq (U) ≤ Cπp (U), where πq , and πp , denote the norms in the spaces of q-, and p-, absolutely summing operators, respectively, defined in Section 1.5.

Proof. By Theorem 6.6.1, for any U : X ∗ 7→ Lq , we have the factorization U = tg ◦ V , where V : X ∗ 7→ Lp is bounded, Tg is a multiplication by a function g ∈ Lr , 1/q + 1/p + 1/r, and kg||Lr ≤ 1, and kV k ≤ CkUk. Therefore, if x1 , . . . , xn ∈ X, one can find P α1 , .r . . , αn ∈ R, and y 1 , . . . , y n ∈ X, such that xi = αi y i , i |αi | ≤ 1, and X 1/p 1/q X X ∗ q ∗ p |x xi | . (6.6.1) sup |x y i | ≤C kx∗ k≤1 i i kx∗ k≤1 Indeed, assuming that not all of xi ’s are equal to 0, in view of Theorem 6.6.1, one can factor the operator X ∗ ∋ x∗ 7→ (x∗ xi ) ∈ l(n) q ,

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as follows: U P = α◦V , where α is the diagonal operator of multiplication, with i |αi |r ≤ 1, and V : X ∗ 7→ l(n) p , with kV k ≤ CkUk. Taking y i = xi /αi , we get sup

kx∗ k≤1

X i

1/p X 1/q ∗ q |x y i | ≤ kV k ≤ CkUk = C sup |x xi | , kx∗ k≤1 i ∗

p

which establishes (6.6.1). Now, if W : X 7→ Y , and x1 , . . . xn ∈ X, using (6.6.1), and the H¨older Inequality, we get that X i

≤

X i

|αi |

r

kW (xi )kq

1/r X

1/q

X i

kαi W (y i )kq

1/p

1/q X

≤ πp (W ) sup kx∗ k≤1 X 1/q ≤ Cπp (W ) sup |x∗ xi |q . kx∗ k≤1 i i

kW (y i )k

p

=

i

∗

|x y i |

p

1/p

QED

Theorem 6.6.2. Let 1 ≤ p < 2. The space X is of stable type p if, and only if, there exists a constant C > 0 such that, for every quotient space X ∗ /Z, and each linear operator U acting from X ∗ /Z into a Banach space Y , π1/2 (U) ≤ Cπp (U), i.e., in particular, for any Banach space Y , the operator space Πp (X ∗ , Y ) = Π0 (X ∗ , Y ). In the proof of the theorem we shall have need of the following lemma (also, note that the choice of 1/2 above is arbitrary because, for 0 < α < 1, all the norms πα are equivalent). Lemma 6.6.1. Let 1 ≤ p < 2, and r < p. Denote by Wn the operator from l(n) into l(n) q p (1/p + 1/q = 1) defined by the formula, Wn ((αi )) = (n−1/p αi ). Then πr (Wn ) ≥ K(log n)1/p , where the constant K does not depend on n.

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Proof. By Pietsch Theorem (see Chapter 1), there exists a probability measure µ on the unit ball of l(n) such that, for all p (n) (αi ) ∈ lq , hZ i1/r 1/p X −1 [ n . |αi | ≤ πr (Wn ) |(αi )(βi )|r µ(d(βi )) i

If (ξi ) are independent and identically distributed p-stable random variables then, utilizing the above inequality with αi = ξi (ω), we get that n r i1/r r/p 1/r hZ X X −1 p , βi ξi µ(d(βi )) E n |ξi | ≤ πr (Wn ) E i

i=1

so that, in view of the properties of p-stable laws, r i1/r hZ X −1 1/p βi ξi µ(d(βi )) c1 (log n) ≤ πr (Wn ) E i

= Cπr (Wn )

hX i

|βi |p

r/p

µ(d(βi ))

i1/r

≤ Cπr (Wn ).

QED

Proof of Theorem 6.6.2. If X is of stable-type p then, by Theorem 6.5.2, p > p(X), and the canonical embedding l1 7→ lp is finitely factorable through X, i.e., for each n ∈ N there exist x1 , . . . , xn ∈ X such that, for all αa , . . . , αn ∈ R, X i

|αi |p

1/p

n

X X

|αi |. αi xi ≤ 2 ≤ i=1

i

Put Z = {x∗ ∈ X ∗ : x∗ xi = 0, i = 1, . . . , n}. In the space X ∗ /Z (which is the dual of the span of {x1 , . . . , xn } ), one can find y ∗1 , . . . , y ∗n such that, for every α1 , . . . , αn ∈ R,

X

X 1/q

−1 ∗ q 2 sup |αi | ≤ αi y i ≤ |αi | , 1/p + 1/q = 1. i

i

This means that there exist operators Un , and Vn , U

V

n n (n) ∗ l(n) q 7→ X /Z 7→ l∞ ,

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157

such that kUn k ≤ 1, kVn k ≤ 2, and which factor the embedding (n) lq(n) 7→ l∞ . ˜ n : l(n) 7→ l(n) , defined by the equality, Denote by W ∞ p ˜ n ((αi )) = (n−1/p αi ). W In the notation of the preceding Lemma, ˜ n ◦ Vn ◦ Un , Wn = W

˜ n ) ≤ 1, πp (W

˜ n ◦ Vn : X ∗ /Z 7→ l(n) , one has and, hence, for the operator V = W p the inequalities, K −1 (log n)1/p ≤ π1/2 (Wn ) ≤ π1/2 (V )kUn k ≤ π1/2 (V ), where πp (V ) ≤ kVn k ≤ 2. Therefore, the inequality π1/2 (U) ≤ Xπp (U) would be impossible for the operator U from the quotient of the space X ∗ . To prove the converse we use the Proposition 6.6.1 which states that, if X ∗ is of stable-type p, then π1/2 (U) ≤ Cπp (U), for each U acting on X. If X is of stable-type p then the dual of the dual space, X ∗∗ , is also of stable-type p because X ∗∗ is finitely representable in X (see, Chapter 1), so that the subspaces of X ∗∗ are of stable-type p with the same constant as X. Now, to complete the proof it is sufficient to notice that (X ∗ /Z)∗ is a subspace of X ∗∗ . QED The next result is an analogue of Theorem 6.4.1 for spaces of stable-type p. For the definition of an almost continuous operator, also see Section 6.4. Theorem 6.6.3. If a normed space X is of stable-type p, 1 ≤ p ≤ 2, then each linear continuous operator U from X into L0 (T, µ), where µ is a finite measure, is almost continuous from X into Lp (T, µ). Proof. We shall show that, for each ǫ ∈ (0, 1/8), there exist a measurable set Tǫ ⊂ T , and a constant C > 0, such that µ(T \Tǫ ) ≤ 8ǫ, and Z 1/p |U(x)|p dµ ≤ ckxk, x ∈ X. (6.6.2) Tǫ

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For simplicity’s sake, and with no restriction on generality, take µ(T ) = 1. Let g1 = U(xi ), . . . , gn = U(xn ) ∈ U(BX ), with x1 , . . . , xn , in the unit ball, BX , of the space X, and let λ1 , . . . , λn ∈ R. Then, since for each β ∈ (0, 1), and each c1 , . . . , cn ∈ R, X X 1/p |ci |p = Jβ−1 (ξ1 , P)Jβ ci ξ i , P , i

i

with (ξi ) being, as usual, the sequence of independent and identically distributed p-stable random variables, we get, in view of the Fubini Inequality (see Chapter 1), and the assumption of stabletype p for X, that X 1/p p J4ǫ |λi gi | ,µ (6.6.3) i

−1 (ξ1 , P)J4ǫ = J1//2

X λi gi (t)ξi (ω) , P(dω) , µ(dt) J1/2 i

X −1 λi gi (t)ξi (ω) , µ(dt) , P(dω) (ξ1 , P)Jǫ Jǫ ≤ J1//2 i

X

−1 λi xi ξi , P (ξ1 , P)Jǫ (U(BX ), µ)Jα = J1//2 i

≤

−1 (ξ1 , P)Jǫ(U(BX ), µ)C1 J1//2

X i

|λi |

p

1/p

.

P p Now, consider the set D of functions of the form, i |λi gi (t)| , P where g1 , . . . , gn ∈ U(BX ), and |λi |p ≤ 1. The set D is a convex subset of L0 (T, µ) consisting of non-negative functions. In view of (6.6.3), J4ǫ (D, µ) = sup{J4ǫ (g, µ) : g ∈ D} p −1 ≤ C1 J1/2 (ξi , P)Jǫ (4(BX ), µ) =: C p /2.

Therefore, by the Nikishin Theorem (see Chapter 1), there exists a set Tǫ ⊂ T such that µ(T \ Tǫ ) ≤ 8ǫ, and such that Z g dµ ≤ 2J4ǫ (D, µ) ≤ C p , g ∈ D, Tǫ

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Tǫ

|g|pdµ

1/p

≤ C,

g ∈ U(BX ),

which proves (6.6.2), QED

6.7

Extented basic inequalities and series of random vectors in spaces of type p

We begin this section by extending the basic inequalities defining the Rademacher-type p to a wider class of random vectors in spaces Lp (X). Proposition 6.7.1. The following properties of a normed space are equivalent: (i) The space X is of Rademacher-type p; (ii) There exists a constant C > 0 such that, for all n ∈ N, and all independent, zero-mean X1 , . . . , Xn ∈ Lp (X), X 1/p X p 1/p

p ≤C EkXi k ; Xi E i

i

(iii) There exists an α ∈ (0, p] (or, for each α ∈ (0, p]), and a constant C > 0 such that, for all n ∈ N, and any independent, zero-mean X1 , . . . , Xn ∈ Lp (X), 1/p X α 1/α X

E Xi ≤C EkXi kp . i

i

Proof. (i) =⇒ (ii) At the beginning let us assume that Xi ’s are symmetric, and that the sequence (ri ) is independent of (Xi ). By Theorem 6.2.2, the space Lp (X) is of Rademacher-type p, so that

p 1/p X X 1/p

E ri X i ≤C EkXi kp . i

i

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However, the symmetry of Xi ’s implies that (ri Xi ), and (Xi ) are identically distributed which give (ii) in the symmetric case. If (Xi ) are not symmetric then we proceed by symmetrization as follows: Let (Xi′ ) be independent copies of (Xi ). Then Xi − Xi′ are symmetric,

p 1/p X p 1/p X

E Xi ≤ E (Xi − Xi′ ) i

≤C

X i

i

EkXi −

Xi′ kp

1/p

≤ 2C

X i

EkXi k

p

1/p

.

The implication (i) =⇒ (iii) can be proved exactly as the implication (i) =⇒ (ii). The only additional information that is needed is contained in Remark 6.2.1. The implications (ii) =⇒ (i), and (iii) =⇒ (i), are evident. QED One can further strengthen Proposition 6.7.1 by dropping the assumption of independence, and replacing it by the assumption of sign-invariance which has been defined in Section 1.1. Theorem 6.7.1.20 The following properties of a normed space X are equivalent: (i) The space X is of Rademacher-type p; (ii) There exists a sign-invariant sequence (φn ) of real random variables with inf n E|φn | > 0, and a constant C > 0 such that, for all (xi ) ⊂ X,

X X 1/p

kxn kp ; (6.7.1) E φn xn ≤ C n

n

(iii) For each sign-invariant sequence (φn ) of real random variables with supn E|φn |p < ∞, there exists a constant C > 0 such that, for each (xi ) ⊂ X, the inequality (6.7.1) holds true. Proof. The implication (iii) =⇒ (ii) is trivial. So, next, let’s prove (i) =⇒ (iii). By Remark 6.2.1, there exists a C > 0 such that for any finite (xi ) ⊂ X,

X

X 1/p

[ E ri x i ≤ C kxi k , i

20

i

This results is due to G. Pisier (1973/74), Exp. 3.

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so that, for a fixed ω ∈ Ω (we assume (ri ), and (φi ), to be independent)

X

X 1/p

p p Er ri φi (ω)xi ≤ C kxi k |φi (ω)| . i

i

Integrating both sides we get the inequalities,

X

1/p X

. kxi kp |φi |p E ri φi xi ≤ CE i

i

≤C

X i

kxi kp E|φi |p

1/p

≤ C sup E|φi |p i

X n

kxn kP

1/p

,

which give (iii) because, for sign-invariant (φn ), the sequences (φn ) and (rn φn ) are identically distributed. (ii) =⇒ (i) Notice that if the sequence (φn ) is sign-invariant then, for each (xn ) ⊂ X, and each p ∈ [1.∞),

[ 1/p

[ 1/p X X

. (6.7.2) φn xn ≤ E rn xn inf E|φn | E n

n

n

Indeed, for a fixed ω ∈ Ω,

X

X

rn (ω)|φn |xn rn (ω)|φn |xn ≤ Eφ

Eφ n

n

from which it follows that

p

X

p

X

rn (ω)xn |φn | rn (ω)xn E|φn | ≤ Eφ

n

n

so that, by integration, we get that

p 1/p

p 1/p X X

≤ E rn xn |φn | . E rn xn E|φn | n

n

By the Contraction Principle (see Chapter 1),

p 1/p X

p 1/p X

inf E|φn | E rn x n ≤ E rn xn E|φn | n

n

n

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which implies (6.7.2) because (φn ), and (rn |φn |) are identically distributed. Finally, from (6.7.2), and the assumption, we get that, for each (xn ) ⊂ X,

X

p 1/p X

E rn xn ≤ (inf E|φn |)−1 E φn xn n

n

n

X p 1/p

≤ (inf E|φn |) C E xn , −1

n

n

which proves that X is of Rademacher-type p. QED Further extensions of basic inequalities can also be obtained for weakly exchangeable and exchangeable random vectors (for definitions, see Section 1.1). Let us start with the weakly exchangeable case. Theorem 6.7.2.21 If X is of Rademacher-type p, then there exists a constant C > 0 such that, for each n ∈ N, and arbitrary weakly exchangeable X1 , . . . , Xn ∈ Lp (X), with X1 + · · · + Xn = 0, E sup

n k

p 1/p

X X 1/p

≤C EkXi kp . Xi

1≤k≤n

i=1

(6.7.3)

i=1

Proof. In view of the Maximal Inequality (see Chapter 1), there exists a constant C1 > 0 such that

k n

X

p 1/p

p 1/p X

E sup Xi ≤ C1 E ρi X i , 1≤k≤n

i=1

i=1

where ρi = (1 + ri )/2 , and the sequence (ri ) is independent of (Xi ). However, ρi , i = 1, . . . , n, are independent themselves, and by Proposition 6.7.1, there exists a constant C2 , such that, for each ω ∈ Ω, n n

X

p X

Eρ ρi Xi (ω) ≤ C2 E|ρi |p kXi (ω)kp. i=1

21

i=1

This, and the next result are due to B. Maurey and G. Pisier (1974/75), Annexe I.

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Integrating both sides we get the inequality, n n

X

p 1 X

E EkXi kp , ρi Xi ≤ C2 2 i=1 i=1

because E|ρi | = 1/2. This implies (6.7.3) in view of the definition of weak exchangeability. QED For exchangeable random vectors we get an even stronger result. Theorem 6.7.3. If the normed space X is of Rademachertype p, p > 1, then there exists a constant C > 0 such that, for each n ∈ N, and arbitrary exchangeable X1 , . . . , Xn ∈ Lp (X) with X1 + · · · + Xn = 0, and any α1 , . . . , αn ∈ R, k n n

p 1/p

X 1 X 1/p X 1/p

≤C αi X i |αi |p EkXi kp . E sup n i=1 1≤k≤n i=1 i=1 (6.7.4)

Proof. Let k < n. For the sake of convenience let us introduce the notation, k

p 1/p X

, αi X i φk (α1 , . . . , αk ) = E i=1

φ∗k (α1 , . . . , αk )

= E sup

1≤j≤k

j

X

p 1/p

αi X i .

i=1

Let ε1 , . . . , εk = ±1, and let σ be a permutation of {1, . . . , k} such that in {εσ(1) , . . . εσ(k) } the plus signs precede the minus signs. If we denote by Ai the σ-algebra in F spanned by X1 , . . . , Xi , then, for each j ≤ k, j X i=1

αi X i = E

k X i=1

αi X i A j +

j k X 1 X αi Xi . n − j i=j+1 i=1

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By the Doob’s Martingale Inequality, k

X p ≤ φk (α1 , . . . , αk )+(n−k) |αi |φ∗k (1, . . . , 1). p−1 i=1 (6.7.5) However, the exchangeability of (Xi ) implies that

φ∗k (α1 , . . . , αk )

φk (α1 , . . . , αk ) = φk (ασ(1) , . . . , ασ(k) ) and, by the triangle inequality, we get φk (ασ(1) , . . . , ασ(k) ) ≤ 3φ∗k (εσ(1) ασ(1) , . . . , εσ(k) ασ(k) ), so that (6.7.5) and the exchangeability yield again the inequalities, φk (α1 , . . . , αk ) k 3 X 3p ∗ φk (εσ(1) ασ(1) , . . . , εσ(k) ασ(k) ) + |αi |φ∗k (1, . . . , 1) ≤ p−1 n − k i=1 k

3p ∗ 3 X ≤ |αi |φ∗k (1, . . . , 1). φk (ε1 α1 , . . . , εk αk ) + p−1 n − k i=1 Substituting the above inequality in (6.7.5) we obtain that φ∗k (α1 , . . . , αk ) ≤ 3

p 2 φk (ε1 α1 , . . . , εk αk ) p−1

k 3p 1 X + +1 |αi |φ∗k (1, . . . , 1). p−1 n − k i=1

Averaging over possible choices of ε1 , . . . , εk = ±1 (with respect to the Rademacher measure), we get φ∗k (α1 , . . . , αk ) ≤C

k X i=1

E|αi |p kXi kp

1/p

+

k 3p 1 X +1 |αi |φ∗k (1, . . . , 1). p−1 n − k i=1

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165

Now, it is easy to conclude the proof of (6.7.4) by choosing k = IntegerPart[(n + 1)/2], using Theorem 6.7.2 to estimate φ∗k (1, . . . , 1), and dealing with the interval {k + 1, . . . , n} in a similar fashion. QED The inequalities discussed above arePuseful in studying the problem of convergence of random series, i Xi , in Banach spaces. In the remainder of this section we concentrate on the series of independent random vectors but it is easy to see that Theorems 6.7.2 and 6.7.3 make it possible, using analogous methodology, to extend the results for random series of independent random vectors to the case of weakly exchangeable and exchangeable random series. Theorem 6.7.4.22 The following properties of a Banach space X are equivalent, (i) The space X is of Rademacher-type p; (ii) If (Xi ) is a sequence of independent, zero-mean random vectors in X, then the condition ∞ X i=1

Eφp (kXi k) < ∞,

φp (t) := min{tp , t}, t ≥ 0,

P implies the almost sure convergence of the random series i Xi ; (iii) If (Xi ) is a sequence of independent, P zero-mean random vectors in X, then the convergence of the series iP EkXi kp implies the almost sure convergence of the series i Xi ; P random p P (iv) For any (xn ) ⊂ X with i kxi k < ∞ , the random series i ri xi converges almost surely, and in Lp (X). Proof. (i) =⇒ (ii) Denote Xi′ = Xi I[kXi k≤] ,

Xi′′ = Xi I[kXi k≥] .

Clearly, Xi = Xi′ + Xi′′ , and both (Xi′ ), and (Xi′′ ) are sequences of independent random vectors in X. Notice that m m m

X

X X

E Xi′′ ≤ EkXi′′ k ≤ Eφp (kXi′′ k), i=n

22

i=n

i=n

This result is due to W.A. Woyczynski (1973).

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P ′′ so that, by Cauchy’s argument, the series i Xi converges in L1 (X) and, in view of the Ito-Nisio Theorem (see Chapter 1), also almost surely. Now, because X is of Rademacher-type p, by Proposition 6.7.1, and Jensen’s Inequality, m m m

p 1/p h X

p X

p i1/p X

′ ′ ′ ′ E Xi ≤ E (Xi − EXi ) + EXi i=n

I=n

i=n

m m

p 1/p

X X

′ ′ Xi′ + E ≤ E (Xi − EXi ) i=n

i=n

≤C

m X

EkXi′

i=n

−

EXi′ kp

1/p

+

m X i=n

EkXi′k

m m X 1/p X ′ p ≤ 2C 2 EkXi k + EkXi′ k i=n

≤2

1+1/p

C

m X i=n

i=n

Eφp (kXi′ k)

1/p

+

m X i=n

Eφp (kXi′ k).

P

′ i Xi

Thus converges in Lp (X) and, again by the Ito-Nisio Theorem, also almost surely. This proves (ii). (ii) =⇒ (iii) This implication is immediate because φp (t) ≤ p t , t > 0. P (iii) =⇒ (iv) The almost sure convergence of the series i ri xi is an immediate consequence of (iii), and the Lp (X) convergence follows from the Kahane Theorem (see Chapter 1). (iv) =⇒ (i) Assume (iv) and define two Banach spaces, n X 1/p o kxn kp . 3 2 Because the canonical embedding l1 7→ lp is finitely factorable through X one can find xNk +1 , . . . , xNk+1 such that, for any (αi ) ⊂ R, Nk+1 Nk+1 k+1

1/p NX X 2 X

p |αi |, αi x i ≤ |αi | ≤ 3 i=N +1 i=N +1 i=N +1 k

k

k

so that, for any εi = ±1, one would have k+1

NX −1/p Nk+1 I=1

Nk k+1

i

X

h NX

−1/p εi xi εi xi − εi xi ≥ Nk+1 i=Nk +1

i=1

h2

i 1 (Nk+1 − Nk )1/p − Nk > , 3 2 which completes the construction and the proof of the Theorem. QED −1/p

≥ Nk+1

6.8

Strong laws of large numbers and asymptotic behavior of random sums in spaces of Rademachertype p

We begin with an analogue of the classical Kolmogorov-Chung Strong Law of Large Numbers. Its validity in a Banach space characterizes the space’s Rademacher-type. Theorem 6.8.1.24 Let p ∈ (1, 2]. The following properties of a Banach space X are equivalent: (i) The space X is of Rademacher-type p; 24

This result is due to W.A. Woyczynski (1973), and J. Hoffmann-Jorgensen (1975).

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(ii) For each sequence (Xn ) of independent, random P zero-mean −p p vectors in X, the convergence of the series n n EkXn k implies that X1 + · · · + Xn lim = 0, a.s.; n→∞ n (iii) There exists a constant C > 0 such that, for each finite sequence (xi ) ⊂ X, n n

X kxi kp 1/p 1

X ri x i ≤ C ; E p n i i=1 i=1

(iv) There exists a constant C > 0 such that, for each finite sequence (Xn ) of independent, zero-mean random vectors in X, n n

X EkXi kp 1/p 1

X Xi ≤ C E . p n i i=1 i=1

Proof. (i) =⇒ (ii) follows immediately from Corollary 6.7.1, with φn (t) = tp , and αn = n, and from the Kronecker’s Lemma. (ii) P =⇒ (iii) Using (ii) in the case Xi = ri xi we get that n−1 ni=1 ri xi → 0, a.s., and also in LP 1 (X) (by Kahane’s Lemma– see Chapter 1) as n → ∞, whenever i i−p kxkp < ∞. Thus ∞ n X o kxi kp 1/p (xi ) ⊂ X; < ∞ ip i=1

n

n o 1

X

⊂ (xi ) ⊂ X : sup E ri x i < ∞ , n n i=1

and a standard application of the Closed Graph Theorem yields the desired inequality. (iii) =⇒ (i) By the assumption, for each (xi ) ⊂ X, n n

X

1/p X

E in−1 ri xi ≤ C kxi kp , i=1

i=1

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Spaces of Rademacher and stable types so that

n 2n n

X

X 1/p X

−1 −1 1/p E i(2n) ri xi + ri xi−n i(2n) ≤ C2 kxi kp , i=1

i=n+1

i=1

(6.8.1)

and, by the Contraction Principle (see Chapter 1), n 2n

1 X

1

X

E ri x i = E ri xi−n 2 2 i=1 i=n+1

(6.8.2)

n 2n

X X

−1 i(2n) ri xi + ri xi−n i(2n)−1 . ≤ E i=1

i=n+1

Now, (6.8.1) and (6.8.2) imply that X is of Rademacher-type p. To complete the proof of the Theorem it is sufficient to notice that (iv) follows for (ii) by an application of the Closed Graph Theorem technique used above, and that (iv) trivially implies (iii). QED Utilizing the full version of the Corollary 6.7.1 one can immediately strengthen the implication (i) =⇒ (ii) of the above Theorem. Proposition 6.8.1 If φ : R+ 7→ R+ is continuous and such that φ(t)/t, and tp /φ(t), are non-decreasing, and if (Xi ) is a sequence of independent, zero-mean random vectors in a Banach space X of Rademacher-type p, 1 < p ≤ 2, then the convergence P of the series n Eφ(kXn k)/φ(n) implies that n

1X lim Xi = 0 n→∞ n i=1

a.s.

It is possible to obtain further corollaries to the results of Section 6.7 on random series in spaces X of Rademacher -type p, and get a precise description of the asymptotic behavior of sums of independent random vectors with values in X. To accomplish this task let us introduce a new classes of functions ψ.

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Definition 6.8.1. The set of all functions ψ : R+ 7→ R+ which do not decrease for t > t0 , for some t0 = t0 (ψ), and for which P the series n n−1 ψ −1 (n) converges, will be denoted Ψc (or Ψd , P by −1 −1 in the case of functions ψ for which the series n n ψ (n) diverges). The function inverse to φ will be denoted by φ(−1) . Theorem 6.8.2.25 Let φ : R+ 7→ R+ be a continuous function such that φ(t)/t, and tp /φ(t), are non-decreasing, and let (Xi ) be a sequence of independent, zero-mean random vectors in a Banach space X of Rademacher-type p. Then, if Eφ(kXn k) < ∞, and the P sequence An := nk=1 Eφ(kXn k) ↑ ∞, then n

X

Xk = O φ(−1) (An ψ(An )) ,

k=1

n → ∞,

almost surely, for each function ψ ∈ Ψc .

Proof. Denote bn = φ(−1) (An ψ(An )). Then, of course, bn ↑ ∞, and, furthermore, ∞ X Eφ(kXn k) < ∞. (6.8.3) A ψ(A ) n n n=1 Indeed,Ptake n0 such that An0 > 0, and ψ(An0 ) > 0. Because the series n n−1 ψ −1 (n) converges, the integral Z ∞ dx I := An0 xψ(x) converges as well. Now, the Mean Value Theorem implies that Z An dx = (An − An−1 )cn , An−1 xψ(x) −1 for n > n0 , and some cn such that A−1 n ψ (An ) ≤ cn ≤ −1 −1 An−1 ψ (An−1 ). Remembering that An − An−1 = Eφ(kXn k), and that Z An ∞ X dx I= , xψ(x) n=n +1 An−1 0

25

This result is due to W.A. Woyczynski (1973).

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we getP (6.8.3). Now, φ(bn ) = An ψ(An ) and, by Corollary 6.7.1, the series n b−1 n Xn converges almost surely, so that the Kronecker’s Lemma gives the desired asymptotics of the partial sums. QED Remark 6.8.1. The above result is, in a sense, best possible. Indeed, let φ : R+ 7→ R+ be continuous and strictly increasing with φ(0) = 0, and φ(t) → ∞, as t → ∞. Then, for every function ψ ∈ Ψd (e.g., ψ(t) = log t, or log t log log t) there exists a sequence of independent Pn real random variables (Xi ) with Eφ(|Xi |) < ∞, and An = k=1 Eφ(|Xk |) ↑ ∞, and such that Pn i=1 Xi lim sup (−1) > 0, a.s. φ (An ψ(An )) n Even without any restrictions on the moments of the random vectors (Xi ) in a space of Rademacher-type p it is still possible to obtain some sort of the Strong Law of Large Numbers. Theorem 6.8.3.26 Let (Xi ) be a sequence of independent random vectors taking values in a Banach space X of Rademachertype p, and let (φi ) be a sequence of convex functions, φi : R+ 7→ R+ , such that φi (t)/t, and tp /φi (t), are not decreasiing. If 0 < tn ↑ ∞, then the convergence of the series ∞ X

E

n=1

φn (kXn k) φn (kXn k) + φn (tn )

implies that n 1 X (Xk − EZk ) = 0, n→∞ tn

lim

.a.s.

k=1

where Zn := Xn I[kXn k < tn ]. Proof. The obvious inequalities Eφn (kZn k) 1 φn (kXn k) + P kXn k ≥ tn ≤ E , 2φn (tn ) 2 φn (kXn k) + φn (tn ) 26

n ∈ N,

This result is due to W.A. Woyczynski (1974).

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imply that ∞ X n=1

and

P(Xn = 6 Zn ) < ∞,

∞ X Eφn (kZn k)

φn (tn )

n=1

(6.8.4)

< ∞.

(6.8.5)

Define, again, φp (t) := min(t, tp ), t ≥ 0. Then, φp (t + s) ≤ K φp (t) + φp (s) ,

t, s, ≥ 0,

(6.8.6)

for some constant K which, in general, may depend on p. Indeed, if t+s ≤ 1, then (6.8.6) follows from the boundedness of the function (1 + t)p /(1 + tp ) on R+ . If t + s ≥ 1, and t, s ≤ 1, then (6.8.6) follows from the fact that (t + s)/(tp + sp ) ≤ 2/(tp − (1 − t)p ) < 2. Finally, if t + s ≥ 1, and (say) t > 1, s < 1, then (6.8.6) follows from the inequality (t + s)/(t + sp ) ≤ (t + 1)/t ≤ 2. In view of (6.8.6), the convexity of φn , and the fact that φn (tn t)/φn (tn ) ≥ φp (t), t ≥ 0, we get that

≤ K Eφp Zn kt−1 Eφp (Zn − EZn )t−1 + φp EkZn kt−1 n n N ≤ 2Kφ−1 n (tn )Eφn (kZn k), P so that (6.8.5) implies that n Eφn (kZn − EZn )t−1 n k) < ∞. The random vectors (Zn −EZn ) are independent P and zero-mean, so that Theorem 6.7.4 implies that the series n (Zn − EZn )t−1 n converges almost surely. Now, the Kronecker’s Lemma gives us that lim

n→∞

t−1 n

n X k=1

(Zk − EZk ) = 0,

a.s.

Furthermore, by (6.8.4), and the Borel-Cantelli Lemma, the probability that infinitely often Zk = 6 Xk is equal to 0, so that also lim t−1 n

n→∞

n X k=1

(Xk − EZk ) = 0,

a.s.

QED

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To complete this Section we now turn to the case when the sequence (Xi ) consists of independent, and identically distributed random vectors. Theorem 6.8.4.27 Let (Xi ) be a sequence of symmetric, independent, and identically distributed random vectors in a Banach space X of Rademacher-type p, and let φ : R+ 7→ R+ be a convex function such that φ(t)/t, and tp /φ(t), are non-decreasing. If the sequence (tn ), 0 < tn ↑ ∞ satisfies the condition, ∞ X

n 1 , =O φ(t φ(t k) n) k=n

(6.8.7)

then the condition ∞ X n=1

P kX1 k ≥ tn < ∞,

(6.8.8)

is necessary, and sufficient, for n 1 X Xi = 0. n→∞ tn i=1

lim

a.s.

Proof. Sufficiency: Let Zn := Xn I kXn k < tn as in Theorem 6.8.3. Then, in view of (6.8.7), ∞ X Eφ(kZnk) n=1

φ(tn ) =

∞ X k=1

27

∞ X k=1

n=1

n 1 X Eφ kX1 kI tk−1 ≤ kX1 k < tk φ(tn ) k=1

∞ X Eφ kX1 kI tk−1 ≤ kX1 k < tk

≤ const ≤ const

=

∞ X

n=k

∞ X k=1

1 φ(tn )

k Eφ kX1 kI tk−1 ≤ kX1 k < tk φ(tk )

∞ X kP tk−1 ≤ kX1 k < tk = const P kX1 k ≥ tk . k=0

This result is due to W.A. Woyczynski (1974).

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This inequality, together with the reasoning exactly as in the proof of Theorem 6.8.3 implies that n 1 X lim (Xk − EZk ) = 0, n→∞ tn k=1

a.s.,

which completes the proof of sufficiency because EZk = 0 in view of the symmetry assumption on Xn ’s. Necessity: It follows from the fact that n n−1 Xn 1 X tn−1 1 X = Xi − · Xi → 0, a.s., tn tn i=1 tn tn−1 i=1 P as n → ∞, so that, should the series n P(kX1 k ≥ tn ) diverge, by the Borel-Cantelli Lemma, the probability that kX1 k ≥ tn happens infinitely often is equal to 1, which contradicts (6.8.8). QED

Remark 6.8.2. It is easy to check that the condition (6.8.7) is fulfilled whenever lim inf k φ(t2k )/φ(tk ) > 2. This gives a handy criterion for a sequence (tk ) to satisfy (6.8.7). For instance, if X is of Rademacher-type p, 1 < p ≤ 2, and φ(t) = tq , 1 < q ≤ p, then the sequence tk = k fulfills the condition (6.8.7), but if q = 1, then it does not. In the case of non-symmetric random vectors we need more restrictions on the sequence (tk ). Theorem 6.8.5.28 Let (Xi ) be a sequence of independent, identically distributed random vectors in a Banach space X of Rademacher-type p, and φ and (tn ) be as in Theorem 6.8.4. If, additionally, EX1 = 0, and there exists a constant C such that tk k ≤C , tn n

k ≥ n,

(6.8.9)

then the condition, ∞ X n=1

28

P kXi k ≥ tn < ∞,

(6.8.10)

This result is due to B. Maurey and G. Pisier(1976).

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n 1 X lim Xi = 0 n→∞ tn i=1

a.s.

Proof. The proof follows the lines of the proof of Theorem 6.8.4 but to complete it (with the help of Theorem 6.8.3) we need to show that n 1 X lim EZk = 0. n→∞ tn k=1 This can be accomplished as follows: In view of (6.8.9), and the assumption EX1 = 0, we get that n n n

X X 1 1 X 1

EZk = E(Xk − Zk ) ≤ EkXk − Zk k tn k=1 tn k=1 tn k=1 n ∞ 1 XX ≤ E kX1 k · I tm ≤ kX1 k < tm+1 tn k=1 m=k

n m ∞ n X X 1 X X ≤ + E kX1 k · I tm ≤ kX1 k < tm+1 tn m=1 k=1 m=n+1 k=1 n 1 X ≤ mtm+1 P tm ≤ kX1 k < tm+1 tn m=1 ∞ 1 X ntm+1 P tm ≤ kX1 k < tm+1 . + tn m=n+1

Now, the first term in the above sum tends to zero because of the Kronecker’s Lemma, and because of the fact that that ∞ X

m=0

∞ X (m + 1)P tm ≤ kX1 k < tm+1 = P kX1 k ≥ tm < ∞, m=0

and the second term converges to zero in view of the above inequality, and because in view of (6.8.9) we have ntm+1 /tn ≤ C(m + 1). QED

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6.9

Weak and strong laws of large numbers in spaces of stable-type p

The spaces of stable-type p are characterized by an infinitedimensional analogue of the classical Kolmogorov’s Weak law of Large Numbers which can be also interpreted as a Central Limit Theorem with degenerate stable limit distribution. Theorem 6.9.1.29 Let 1 ≤ p < 2. The following properties of a Banach space X are equivalent: (i) The space X is of stable-type p; (ii) For each sequence (Xn ) of symmetric, independent, and identically distributed random vectors in X, lim

n→∞

n 1 X

n1/p

Xn = 0,

(6.9.1)

i=1

in probability, if , and only if, lim nP kX1 k > n1/p = 0.

n→∞

(6.9.2)

Proof. (i) =⇒ (ii) First of all notice that (6.9.1) implies (6.9.2) in any normed space. Indeed, because of the symmetry assumption,

1 1

X

X

X + X , and X − X

i j i j , n1/p i6=j n1/p i6=j are identically distributed. Thus it follows that n

1 X 1

−1/p P 1/p Xi > ǫ ≥ P sup kn Xj k > ǫ . n 2 1≤j≤n i=1

Hence, (6.9.1) implies the convergence P(sup1≤j≤n kXj k ǫ n1/p ) → 0, and this is equivalent to the condition (6.9.2). 29

>

This result is due to M.B. Marcus and W.A. Woyczynski (1979).

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Spaces of Rademacher and stable types We now show that (6.9.2) implies (6.9.1). Let Yk := Xk I[kXk k ≤ n1/p ],

Zn := Y1 + · · · + Yn ,

and Bn := {X1 + · · · + Xn = Zn }. By assumption, and by Corollary 6.5.2(ii), there exists a q, p < q < 2, such that X is of Rademacher-type q. Now, for each ǫ > 0, P kX1 + · · · + Xn kn−1/p ≥ ǫ ≤ P(Bn ) · P kX1 + · · · + Xn k ≥ ǫn1/p Bn +P(Bnc ) · P kX1 + · · · + Xn k ≥ ǫn1/p Bnc 1/p

≤ P kZn k ≥ ǫn ≤ ǫ−q n−q/p

n X k=1

)+P(Bnc ) ≤ ǫ−q Ekn−1/p Zn kq +

n X k=1

P kXk k > n1/p

n X EkXk kq I kXk k ≤ n1/p + P kXk k > n1/p , k=1

where, in the last step we employed the Proposition 6.7.1(ii). Thus, we now have the inequality P kX1 + · · · + Xn kn−1/p ≥ ǫ Z −q 1−q/p ≤ǫ n kXk kq dP + nP kX1 k > n1/p , kXk k≤n1/p

and we only need to show that the first term above, call it In , converges to zero as n → ∞. Indeed, −q 1−q/p

In = ǫ n

n Z X k=1

−q 1−q/p

≤ǫ n

≤ ǫ−q n1−q/p

n X k=1

kX1 kq dP

k q/p P(k − 1 ≤ kX1 kp ≤ k)

n X k X k=1

k−1≤kXk

kp ≤k

i=1

iq/p−1 P(k − 1 ≤ kX1 kp ≤ k)

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Geometry and Martingales in Banach Spaces −q 1−q/p

≤ǫ n

n X i=1

iq/p−1 P(kX1 kp > i − 1).

In view of (6.9.2), for each δ > 0, there exists an i0 ∈ N such that, for all i > i0 , we have the inequality iP(kX1 kp ≥ i − 1) < δ. Therefore, In ≤ ǫ−q n1−q/p

i0 X i=1

iq/p−1 P(kX1 kp > i − 1) + ǫq δn1−q/p

n X

iq/p−2 .

i=i0 +1

The last term is less than δC, where the constant C depends only on p, and q. Since δ is arbitrary, we get that, for all ǫ > 0, P(kX1 + · · · + Xn kn−1/p > ǫ) → 0, as n → ∞. This completes the proof of the implication (i) =⇒ (ii). (ii) =⇒ (i) Because of Corollary 6.5.2 (i) the space X is of stable-type p if, and only if, lp is not finitely representable in X. Thus it is sufficient to construct a counterexample in lp . Consider the random vector, X X(ω) = r(ω) ej N 2 (ω)≤j n ∼ Cn−p , P but n−1/p ni=1 Xi is not bounded in probability, which implies the existence of a random vector Y (ω) in L0 (lp ) such that n−p P(kY k > P n) → 0, but for which n−1/p ni=1 Yi does not converge to 0 in probability. QED In the case of sequences of special p-stable random vectors of the form (ξi xi ), we also have an analogue of the classical Kolmogorov-Chung’s Theorem. Theorem 6.9.2. Let 1 ≤ p ≤ 2. The following properties of a Banach space X are equivalent:

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(i) The space X is of stable-type p; P (ii) For any sequence (xi ) ⊂ X such that i i−p kxi kp < ∞, and each sequence (ξi ) of independent and identically distributed p-stable random variables, n

1X ξi xi = 0, n→∞ n i=1 lim

a.s., and also in Lq , q < p;

(iii) There exists a constant C such that, for any (xi ) ⊂ X, and (ξi ), as above , n n

p/2 2/p X 1/p 1

X ≤C i−p kxi kp . ξ i xi E n i=1 i=1

Proof. (i) =⇒ (ii) This implication follows directly from Theorem 6.7.5, and the Kronecker’s Lemma. (ii) =⇒ (iii) Assuming the almost sure convergence of the averages one gets their Lq , q < p, convergence by the HoffmannJorgensen Theorem (see Chapter 1). Then (iii) follows by a standard application of the Closed Graph Theorem (as in Theorem 6.7.4 (iv) =⇒ (i) ). (iii) =⇒ (i) Let x1 , . . . , xn ⊂ X, and define 0, for 1 ≤ j ≤ N, yj = xj−N for N < j ≤ N + n, for some integer N. Then, by our assumption, the inequality n +n

p/2 2/p NX

p/2 2/p X

E ξ i xi = E ξi y i i=1

≤ C(N + n)

+n NX j=1

j=1

j

−p

kxj−N k

p

1/p

n

1/p N + n X ≤C kxj kp N + 1 j=1

holds for any N ≥ 1, so that n n

p/2 2/p X X 1/p

E ξ i xi ≤C kxj kp . i=1

QED

j=1

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Also, perhaps a little surprisingly, a Strong Law of Large Numbers for Rademacher sequences (rn ) also characterizes spaces of stable-type p. Theorem 6.9.3. Let 1 ≤ p < 2. The following properties of a Banach space X are equivalent: (i) The space X is of stable-type p; (ii) For each bounded sequence (xn ) ⊂ X, lim

n→∞

n 1 X

n1/p

rk xk = 0,

a.s.;

k=1

(iii) For each bounded sequence (xn ) ⊂ X there exists a choice of εi = ±1 such that lim

n→∞

n 1 X

n1/p

εk xk = 0.

k=1

Proof. The implication (i) =⇒ (ii) follows directly from Theorem 6.7.5 (ii) and the Kronecker’s Lemma. The implication (ii) =⇒ (iii) is obvious, and the implication (iii) =⇒ (i) has already been proven in the course of the proof of implication (iv) =⇒ (i) of Theorem 6.7.5. QED

6.10

Random integrals, convergence of infinitely divisible measures and the central limit theorem

We R begin with the construction of random integrals of the form f dM, where f is a deterministic function with values in a Banach space X, and M is a p-stable, real-valued random measure30 . Random integrals of this Ptype are a natural generalization of random series of the form i ξi xi , xi ∈ X. 30

For a complete theory of real-valued random integrals see S. Kwapie´ n and W.A. Woyczy´ nski (1992).

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Definition 6.10.1. Let (T, Σ) be a measurable space. A mapping M : Σ 7→ L0 (Ω, F , P ) is said to be an (independently scattered) random measure if, for any pair-wise disjoint sets A1 , A2 , . . . , ∈ Σ, the random variables M(A S P 1 ), M(A2 ), . . . , are stochastically independent, and M( i Ai ) = i M(Ai ), where the series on the right-hand side converges in probability (or, almost surely). Definition 6.10.2. A random measure M on Σ is said to be p-stable with a finite, non-negative control measure m on (T, Σ), if E exp[itM(A)] = exp[−m(A)|t|p ], A ∈ Σ. It follows from the Kolmogorov’s Consistency Theorem that given such a control measure m one can construct the related random measure M for any value of the parameter p, 0 < p ≤ 2. P If f : T 7→ X is a simple function, i.e., f = i xi IAi , where Ai ∈ Σ are pairwise disjoint, and xi ∈ X, then we define the random integral by the obvious formula, Z X f (t) M(dt) := xi M(Ai ). (6.10.1) T

i

R

In this case, the integral f dM is a p-stable X-valued random vector. For a simple function f in Lp (T, Σ, M; X), and for q < p, the map Z f 7→ f (t) M(dt), T

is a linear operator with values in Lq (Ω, F , P; X). By (6.5.2), if X is of stable-type p, and the random measure M is p-stable, we have the inequality

Z

q 1/q X

M(Ai ) (6.10.2)

f dM = E Lp

T

≤C

X i

m(Ai )kxi k

i

p

1/p

=C

Z

T

kf (tkp m(dt)

1/p

.

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Since the simple functions are dense in Lp (T, Σ, M; X), there exists a unique extension of this operator to the whole of Lp . This R extension will be also denoted by T f dM, f ∈ Lp , and it also satisfies (6.10.2). Summarizing the above discussion we have the following Theorem 6.10.1.31 If X is a Banach space of stable-type p, and M is a p-stable random measure on (T, Σ), with control measure m, then, for each q, 0 < q < p ≤ 2, there exists a linear map, Z Lp (T, Σ, m; X) ∋ f 7→ f dM ∈ Lq (Ω, F , P; X) (6.10.3) T

satisfying (6.10.1), with its values being p-stable random vectors on X. The mapping, which is called the random integral of function f with respect to random measure M, satisfies the inequality

q 1/q Z 1/p Z

≤C kf kp dm , (6.10.4) E f dM T

T

for some constant C which is independent of f .

Remark 6.10.1. (a) The inequality (6.10.4) is also valid if the Lq Rnorm on the left hand side is replaced by the Lorentz norm Λp ( f dM) (see Chapter 1). (b) Of course, the fact the space X is of stable-type p is also necessary in the above Theorem. It is quite easy R to compute the characteristic functional of the random integral Indeed, take a sequence of simple funcP n f dM. n tions fn = i xi I(Ai ) converging Rto f in Lp (m; X). Then, in view R of (6.10.4), the distribution L( fn dM) converges weakly to L( f dM). Therefore, for each x∗ ∈ X ∗ , Z h i h i X ∗ ∗ n n E exp ix f dM = lim E exp ix xi M(Ai ) n→∞

i

31

Most of the results of this Section are due to M.B. Marcus and W.A. Woyczynski (1979).

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n→∞

Y i

exp

−m(Ani )|x∗ xni |p

185

h Z i = lim exp − |x∗ fn (t)|p m(dt) n→∞

h Z i = exp − |x∗ f (t)|p m(dt) .

Theorem 6.10.2. Let 0 < p ≤ 2. The following properties of a Banach space X are equivalent: (i) The space X is of stable-type p; (ii) For any finite measure space (T, Σ, m), and any function f ∈ Lp (T, Σ, m; X), the function, h Z i ∗ ∗ ∗ p X ∋ x 7→ exp − |x f (t)| m(dt) ∈ R, T

is the characteristic functional of a p-stable probability measure R µ = L( f dM) on X.

Proof. Theorem 6.10.1, R and the above computation of the characteristicfunctions of L( f dM), show that (i) =⇒ (ii). We will now show that if (ii) is not satisfied then (i) is not satisfied either. Indeed, if X is not of stable-type pP then, by Theorem 6.7.5, p there exists a sequence P (xn ) ⊂ X, with i kxi k < ∞, such that the random series i ξi xi does not converge almost surely. Now, take T = X, with Σ being the family of Borel sets of X. Let m be concentrated on the set {x1 , x2 , . . . , }, with m({xi }) = kxi kp . Finally, take f (x) = x/kxk. For (ii) to be true, the functional h Z i h X i exp − |x∗ f (x)|p m(dx) = exp − |x∗ xi |p , x∗ ∈ X ∗ , X

i

∗ would have to be a characteristic functional on PX . That is, the random vector defined by the random series i ξi xi would have to exist, but it does not. QED

Remark 6.10.2. As far as representing all stable measures µ on a Banach space X of stable-type p is concerned, we can restrict our attention to measures of the form Z µ=L xM(dx) , SX

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where the control measure σ of M is concentrated on the unit sphere SX of the space X. This fact follows by a simple change of variables. Then, the characteristic functional of µ is of the form, Z h Z i ∗ ∗ p exp[ix x]µ(dx) = exp − |x x| σ(dx) SX

and σ will be called the spectral measure of the p-stable measure µ. In spaces of stable-type p the weak convergence of the spectral measures on the unit sphere SX implies the weak convergence of the corresponding p-stable probability distributions on X. This fact will have important applications in the proof of the Central Limit Theorem to be discussed later on in this Section. Theorem 6.10.3. Let 0 < p ≤ 2. The following properties of a Banach space X are equivalent: (i) The space X is of stable-type p; (ii) If σ1 , σ2 , . . . , are spectral measures on the unit sphere of X such that σi → σ∞ , weakly on SX , and if µ1 , µ2, . . . , are the corresponding p-stable probability measures on X, then there exists a stable measure µ∞ with spectral measure σ∞ , and µi ’s converge weakly to µ∞ on X. Proof. (i) =⇒ (ii) Let Mn , n = 1, 2, . . . , ∞, be p-stable random measures on X, with control measures σn , n = 1, 2, . . . , ∞. By Prokhorov’s Theorem, for any ǫ > 0, one can find a compact set K ⊂ P SX such that, for all n = 1, 2, . . . , ∞, σn (SX \ K) < ǫ. Let f = i xi I(Ai ) be a simple function with a finite range such that Ai ’s are continuity sets of the limit spectral measure σ∞ , kf (x) − xk ≤ ǫ on K, and kf (x)k ≤ 1, elsewhere. Then, Z Z p kx − f (x)k σn (dx) = kx − f (x)kp σn (dx) SX

K

+

Z

SX \K

kx − f (x)kp σn (dx)

≤ ǫp + 2p ǫ,

n = 1, 2, . . . , ∞.

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By Theorem 6.10.1, we have that if q < p, then

q 1/q

p 1/p Z Z

E (x−f (x))Mn (dx) ≤ C E (x−f (x))σn (dx) SX

SX

≤ C(ǫp + 2p ǫ)1/p . R Therefore, because µn = L( xMn (dx)), and Z Z xMn (dx) − xM∞ (dx) Z

Z

(x − f (x))Mn (dx) + (f (x) − x)M∞ (dx) Z Z + f (x)Mn (dx) − f (x)M∞ (dx),

it is sufficient to show that Z Z L f (x) Mn (x. ) −→ L SX

SX

f (x) M∞ (x. )

R

weakly, as n → ∞. However, f (x)Mn (dx) are p-stable random vectors taking values in a fixed finite-dimensional subspace of X spanned by the values of f . Therefore, to prove the weak convergence it is sufficient to prove the convergence of the characteristic functionals. And, for each x∗ ∈ X ∗ , as n → ∞, Z h i h Z i ∗ ∗ p E exp ix f (x)Mn (dx) = exp − |x f (x)| σn (dx) SX

h X i h X i = exp − |x∗ xj |p σn (Aj ) −→ exp − |x∗ xj |p σ∞ (Aj ) j

j

h = E exp ix∗

Z

i f (x)M∞ (dx) ,

because Aj ’s are continuity sets for σ∞ , and R σn → σ∞ weakly. So, clearly, the probability measure µ∞ = L( xM∞ (x. )). (ii) =⇒ (i) As in the proof of Theorem 6.10.1, we will show that if (i) is not satisfied then (ii) is not satisfied either. So, if X is not of stable-type p, then there exists a sequence

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P p (x ) ⊂ X with < ∞, for which the random series i i kxi k P i ξi xi does not converge almost surely. We take for σ∞ the measure concentrated on the set {xi /kx1 k, x2 /kx2 k, . . . } with values σ∞ ({xi /kxi k}) = kxi kp , i = 1, 2, . . . ,, and for σn we take the measure concentrated on the set {xi /kx1 k, . . . , x2 /kxn k}, with values σn ({xi /kxi k}) = kxi kp , i = 1, . . . , n. Clearly, as n → ∞, we have the weak convergence σn → σ∞ on SX . Now, consider the corresponding p-stable measures µn , n = 1, 2, . . . . We have P µn = L( ni=1 ξi xi ). If µn convergedP weakly then, by Ito-Nisio Theorem (see Chapter 1) , the series i ξi xi would converge almost surely, a contradiction. QED Now, we turn to R a study of Poisson random measures M, and random integrals f dM, where the function f takes values in a Banach space X of Rademacher-type p. This will permit us to introduce a representation of a class of infinitely divisible measures on X, and will also serve as a tool in the proof of the Central Limit Theorem. Definition 6.10.3. A random measure M on (T, Σ) is said to be (symmetric) Poissonian, with a σ-finite control measure m, if its characteristic function E exp[itM(A)] = exp m(A)(cos t − 1) ,

t ∈ R,

(6.10.5)

for each set A ∈ Σ , with m(A) < ∞. Note that if p ≤ 2, then d2 E|M(A)| ≤ E(M(A)) = − 2 exp m(A)(cos(t−1) = m(A), dt t=0 (6.10.6) because M(A) is integer-valued. Now, assume that f ∈ Lp (T, Σ, m; X), and X is a BanachPspace of Rademacher-type p. If f is a simple function, f = i xi IAi , Ai ∈ Σ, we, obviously define p

2

Z

f dM =

X

xi M(Ai ).

(6.10.7)

i

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Then, because X is of Rademacher-type p (see, Theorem 6.7.1),

Z

p

X

p X

E f dM = E xi M(Ai ) ≤ C kxi kp E|M(Ai )|p i

≤C

X i

i

kxi kp m(Ai ) =

Z

(6.10.8)

kf kp dm,

in view of (6.10.6). Thus, as has been done above for theRp-stable random integrals, we can extend the integral operator to the whole of Lp while preserving the inequality (6.10.8). The characteristic functional, for a simple f defined above, is of the form Z i h i h X ∗ xi M(Ai ) E exp ix f dM = E exp ix∗ i

=

Y i

Y E exp ix∗ xi M(Ai ) = exp m(Ai )(cos x∗ xi − 1) i

= exp

hZ

i cos(x∗ f ) − 1 dm ,

T

and, because of (6.10.8), the same formula extends to all f ∈ Lp . Summarizing the above analysis we obtain the following, Theorem 6.10.4. If X is a Banach space of Rademachertype p, and M is a Poissonian random measure on (T, Σ) with the control measure m, then there exists a linear map, called the random integral, Z Lp (T, Σ, m; X) ∋ f 7→ f dM ∈ Lp (Ω, F , P; X), (6.10.9) T

satisfying (6.10.7). Its values are infinitely divisible X-valued random vectors with the characteristic functional h Z i hZ ∗ E exp ix f dM = exp cos(x∗f )−1 dm, x∗ ∈ X ∗ , f ∈ Lp , T

(6.10.10)

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satisfying the inequality Z

Z

p

E f dM ≤ C kf kp dm.

(6.10.11)

Remark 6.10.3. Rademacher-type p of X is easily seen to be also the necessary condition in the above Theorem. Remark 6.10.4. A straightforward computationR shows that if R T = X, and X kxkp dm < ∞, then the law of x dM is the symmetrization, call it es (m), of the Poissonization e(m) of m. Recall that ∞ X m∗n −m(X ) e(m) := e n! n=0 where the inequality (6.10.11) gives the above formula its precise interpretation. In this particular case the meaning of (6.10.11) is as follows: If X is of Rademacher-type p, then Z Z p kxk des (m) ≤ C kxkp dm. One can easily extend the above definition of the random integral to all Poissonian random measures M on a Banach space of Rademacher-type p for which the control measure satisfies only the condition, Z min(1, kxkp ) dm < ∞. (6.10.12) X

Thus for all such m, the formula (6.10.10) also represents an infinitely divisible law on X. Here is how the extension can be accomplished. GivenR our previous result, we only have to be concerned about the case kxk>1 x dM. So, let us notice that Z

kxk>1

x dM =

∞ Z X k=1

k 1}) < ∞, P k 0. By (6.10.15), and (6.10.11), there exists an ǫ > 0, and an integer N0 , such that, for each n ≥ N0 , Z

Z

kxkp dmn < δ. (6.10.17) E x dMn ≤ C Bǫ

Bǫ

On the other hand

Z =

Z

Bǫc

Bǫc

x dMn −

Z

x dM∞

(6.10.18)

Bǫc

Z Z (x−f (x)) dMn − (x−f (x)) dM∞ + Bǫc

Bǫc

Z f dMn −

f dM∞ ,

Bǫc

for any f ∈ Lp (m; X). Because of (6.10.14), by Prokhorov’s Theorem, there exists a compact set K ⊂ BǫcP⊂ X such that mn (Bǫc \K) < δ, for all n = 1, 2, . . . , ∞. Let f = i xi IAi , xi ∈ X, be a simple function on (Bǫc , BX ) with a finite range, and such that Ai ’s are continuity sets of m∞ , kf (x)−xk ≤ δ, and kf (x)k ≤ kxk, elsewhere. Then, by (6.10.11), Z

p

Z

kx − f (x)kp mn (dx) E (x − f (x))Mn (dx) ≤ C K

K

p

p

≤ Cδ sup mn (K) ≤ C1 δ , n

n = 1, 2, . . . , ∞,

where C1 is independent of n because, by (6.10.14), supn mn (Bǫc ) < ∞. Outside K we have, for each α > 0 (as in the reasoning following Remark 6.10.4),

Z X 1

P x dMn > α2 k2 Bǫc \K k ∞ n Z [

≤P

k=1

α2

x dMn > 2 k (Bǫc \K)∩{ǫk 2 k (Bǫc \K)∩{ǫk t, x/kxk ∈ A = t−p σ(A),

(6.10.19)

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for any Borel set A ⊂ SX , with σ(∂A) = 0, and t > 0, and if, for some q > p, lim lim sup ǫ→0

n→∞

kn Z X

kxk≤ǫ

k=1

then, as n → ∞, L

kn X k=1

kxkq µnk (dx) = 0,

Xnk −→ µ,

(6.10.20)

(6.10.21)

weakly, where µ is the p-stable measure with the characteristic functional, Z hZ dρ ∗ exp[ix x] µ(dx) = exp cos(x∗ x) − 1 σ(ds) 1+p , ρ X SX ×R+ (6.10.22) ∗ ∗ + where x ∈ X , x = s · ρ, s ∈ SX , and ρ ∈ R .

Proof. (ii) =⇒ (i) follows immediately from the counterexample to the Weak Law of Large Numbers in the proof of Theorem 6.9.1. (i) =⇒ (ii) By Theorem 6.10.2, for any finite measure σ on the unit sphere SX , (6.10.22) is the characteristic functional of a pstable probability measure on the Banach space X of stable-type p. Now, in R view of LeCam’s Theorem, it is sufficient to show that if Ynk = x dMnk , where µnk are control measures of Poissonian Mnk , then the probability distributions of the array Ynk converge weakly to µ, as n → ∞. However, L

kn Z X k=1

x Mnk (dx) = L

Z

X

x Mn (dx) ,

where Mn is Poissonian with control measure µn =

kn X

µnk

k=1

which is easy to check by verifying the characteristic functionals. Because X is of stable-type p, there exists a q > p such

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195

that X is of Rademacher-type q (see Section 6.5). Now, both µn and σ(ds)dρ/ρ1+p integrate min(1, kxkq ), and hence it is sufficient to check the conditions of Theorem 6.10.5 with p replaced by q. Indeed, (6.10.19) implies evidently that dµn converge to σ(ds)dρ/ρ1+p weakly outside each neighborhood of 0 because the limiting measure is a product measure. QED Corollary 6.10.1. Let p < 2. The following properties of a Banach space X are equivalent: (i) The space X is of stable-type p; (ii) For each finite, non-zero Borel measure on SX , and for any sequence (Xi ) if independent, identically distributed random vectors in X with symmetric probability distributiom µ = L(X1 ) such that µ(kxk > st) lim , (6.10.23) s→∞ µ(kxk > s) exists for each t, and such that for a nonnegative sequence bn ↑ ∞, lim nµ {kxk > tbn , x/kxk ∈ A} = t−p σ(A), (6.10.24) n→∞

for each Borel set A ⊂ SX , with σ(∂A) = 0, and any t > 0, we have that, as n → ∞, X + · · · + X 1 n L −→ µ bn weakly, where µ is the p-stable measure determined by the characteristic functional (6.10.22). Proof. In view of (6.10.24), the limit in (6.10.23) is exactly t . Therefore, by the standard Karamata procedure for regularly varying functions, for each q > p one can find a constant K such that Z q Kt µ(kxk > t) ≥ kxkq µ(dx). (6.10.25) kxk ǫn1/p ) ǫ→0

n→∞

≤ K lim ǫq−p = 0, ǫ→0

whenever q > p, so that (6.10.20) is also satisfied, which gives us our Corollary. QED Remark 6.10.5. To conclude this Chapter we would like to mention a non-probabilistic application32 of the above results concerning rearrangement of series in Banach spaces of Rademachertype p. If X is of Rademacher type p, and (xi ) ⊂ X is such that P p i kxi k < ∞, and nk X i=1

xi −→ x ∈ X,

as

k → ∞,

for a certain sequence (nk ) ⊂ N, then there exists a rearrangement γ of positive integers such that ∞ X

xγ(i) = x.

i=1

32

Due to P. Assuad (1974), Exp. XVI, and B. Maurey and G. Pisier (1974/75), Annexe I. For real series it is due to Steinitz, and for Lp spaces, to Kadec.

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Chapter 7 Spaces of type 2 7.1

Additional properties of spaces of type 2

For the parameter value p = 2, the concepts of the Rademachertype p, and the stable-type p spaces, coincide. Proposition 7.1.1. A normed space X is of Rademacher-type 2 if, and only if, it is of stable-type 2. Proof. If X is of stable-type 2 then, by Proposition 6.5.1, it is also of Rademacher-type 2. Conversely, if X is of Rademachertype 2, then, by Proposition 6.7.1, there exists a constant C > 0 such that, for any n ∈ N, and any sequence (xi ) ⊂ X, n n

2 1/2 X X 1/2

E γ i xi ≤C kxi k2 , i=1

i=1

where (γi ) are independent, identically distributed, zero-mean real 2-stable (that is, Gaussian) random variables. Thus X is of stabletype 2. QED So from this point forward we will not use the names “stabletype 2”, and “Rademacher-type 2”, but we will simply talk about spaces of type 2. Of course, results proven in Chapter 6 for spaces of general type p (stable, or Rademacher) apply to spaces of type 2, as well. We already know that if 2 ≤ p < ∞ then Lp is of type 2. 197 ✐

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Here we give an elementary proof of this fact. Let, r1 , . . . , rn , be, as usual, a Rademacher sequence on (Ω, F , P), and let x1 , . . . , xn , ∈ Lp (T, Σ, µ). By Khinchine’s Inequality (see Introduction), for each p ≥ 2, there exists a constant C such that, for each t ∈ T , n X p/2 p p/2 2 E r1 x(t) + · · · + rn x(t) ≤ C |xj (t)| . j=1

Integrating both sides with respect to µ, and applying Fubini’s Theoorem to the left-hand side, we get that n n

p 2/p X X

≤C kxj k2 . rj xj E j=1

j=1

Furthermore, by Jensen’s Inequality, n n

2

X

p 2/p X

rj xj , ≥ E rj x j E j=1

so that

j=1

n n

2

X X

rj x j ≤ C kxj k2 . E j=1

QED

j=1

Many additional probabilistic properties of spaces of type 2 follow from the next Corollary and the results contained in earlier chapters. Corollary 7.1.1.1 If X is of type 2 then there exists a q < ∞ such that X is of cotype q. Proof. Because of Theorem 5.4.2, and because type 2 is a superproperty, it suffices to show that if X contains a copy of c0 then it is not of type 2. Let (e in c0 ⊂ X. Pn ) be the standard basis −1 Consider the random series i γP i xi , where xi = i ei . Evidently, P 2 kx k < ∞, but the series i i i γi xi diverges almost surely in view of the Borel-Cantelli Lemma. Thus the above Proposition implies that X is not of type 2. QED 1

Due to B.Maurey and G. Pisier (1976).

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The next two results2 provide sufficient conditions for a Banach space to be of type 2, and they use orthogonal rather than independent random variables. Theorem 7.1.1. Let (ξi ) be a complete orthonormal system in L2 (Ω, F , P), and let X be a Banach space. If there exists a constant C > 0 such that, for all n ∈ N, and any x1 , . . . , xn ∈ X, n n

1 1/2 X X 1/2

ξ i xi E ≤C , kxi k2 i=1

i=1

then X is of type 2.

Proof. The standard “gliding hump” procedure implies that, for each ǫ > 0, there exist increasing sequences of integers (kj ), and (mj ), and an orthonormal sequence (ηj ) such that kj+1 −1

ηj =

X

E|ηj − rmj |

0 such that, for all n ∈ N, and any sequence x1 , . . . , xn ∈ X, n n

2 1/2 X 1/2 X

ξ i xi ≥C kxi k2 , E i=1

i=1

then the dual space X ∗ is of type 2. Proof. In view of the above Theorem it is sufficient to show that, and any sequence x∗1 , . . . , x∗n ∈ X ∗ ,

P for each

2 n ∈ N, P

E ni=1 ξi x∗i ≤ C −1 ni=1 kx∗i k2 . The completeness of the sequence (ξi ) implies that the linear combinations of ξi ’s are dense in L2P . Hence, the set V of all X-valued functions of the form φ = ni=1 ξi xi , xi ∈ X, i, n ∈ N, is dense in L2 (Ω, F , P; X). Now, it follows the standard duality argument that, for Pn from ∗ ∗ any i=1 ξi xi , and an arbitrary ǫ > 0, there exists φ = Pm φ = ξ x ∈ V , with Ekφk2 = 1, such that i=1 i i min(n,m) n

2 1/2 X X

E ξi x∗i ≤ E|φ∗φ| + ǫ = |x∗i xi | + ǫ i=1

i=1

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201

Spaces of type 2 ≤

n X i=1

kx∗i k2

m m 1/2 X 1/2 X 1/2 2 −1/2 · kxi k +ǫ≤C kxi k2 + ǫ. i=1

i=1

QED

The sufficient conditions appearing in the following Theorem involve independence but random variables are not necessarily identically distributed. Theorem 7.1.2. Let X be a Banach space, and let (ξi ) be a sequence of real independent random variables satisfying the Lindeberg-type conditions, E|ξi |2 = 1,

i = 1, 2, . . . ,

n √ 1X 2 lim E kxi k I(|ξi | > ǫ n) = 0, n→∞ n i=1

(7.1.2) ∀ǫ > 0.

(7.1.3)

If there exists a C > 0 such that, for any n ∈ N, and x1 , . . . , xn ∈ X, n n

X

X 1/2

2 ξ x ≤ C kx k ,

i i i i=1

i=1

then X is of type 2.

Proof. Let x1 , . . . , xn ∈ X. By the Central Limit Theorem the joint probability distribution of the random variables m 1 X ξn(i−1)+j , i = 1, . . . , n, ξi,m = 1/2 m j=1 converges, as m → ∞, to the joint distribution of independent Gaussian variables γ1 , . . . , γn . Hence, lim Eφ(ξ1,m , . . . , ξn,m ) (7.1.3) Z ∞ Z ∞ 2 2 ··· φ(s1 , . . . , sn )e−(s1 +···+sn )/2 ds1 . . . dsn , m→∞

1 (2π)n/2 −∞ −∞ for any bounded and continuous function φ : Rn 7→ R. Consider the Banach space Y of all continuous functions φ : Rn 7→ R such that φ(s1 , . . . , sn ) Pn = 0, Pn lim 2 i=1 |si |→∞ i=1 |si | =

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with the norm kφkY = max P sup

φ(s1 , . . . , sn ) Pn |φ(s1 , . . . , sn )|, P sup . 2 n n |s |≤1 |s |≥1 i i i=1 |si | i=1 i=1

Let us set Φm (φ) = Eφ(ξ1,m , . . . ξn,m ). One can easily check that

|Φm (φ)| ≤ kφkY 1+

n X i=1

Eξi,m ≤ (n+1)kφkY ,

φ ∈ Y , m ∈ N.

Hence, (7.1.3) also holds true for any φ ∈ Y because, by (7.1.3), the limit exists on a dense subset of Y . In particular, we have n n

X

X

γ i xi , ξi,m xi = E lim E

m→∞

i=1

i=1

because the function φ defined by the formula φ(s1 , . . . , sn ) = Pn k i=1 si xi k belongs to the space Y . On the other hand, by assumption, n X m n

X

X xi

ξm(i−1)+j 1/2 ξi,m xi = E E m i=1 j=1 i=1

≤C

n X m X kx k2 1/2

Therefore,

i

i=1 j=1

m

=C

n X i=1

kxi k2

1/2

,

m ∈ N.

n n

X

X 1/2

2 γ i xi ≤ C kxi k , E i=1

i=1

which proves that X is of type 2. QED

7.2

Gaussian random vectors

Spaces of type 2 are characterized by the fact that for any second order random vector in them one can find a Gaussian random vector with the same covariance functional. The following result is a special case of Theorem 6.10.2.

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203

Theorem 7.2.1.3 The following properties of a Banach space X are equivalent: (i) The space X is of type 2; (ii) For any random vector Y ∈ L2 (Ω, F P; X), with EY = 0, there exists a Gaussian random vector X in X such that its characteristic functional h 1 i ∗ ∗ 2 E exp[ix X] = exp − E(x Y ) , x∗ ∈ X ∗ . 2 Given the above result it is of interest to discuss in detail the structure of the covariance operators of Gaussian measures on spaces of type 2. Definition 7.2.1. A linear operator R : X ∗ 7→ X is called symmetric if, x∗ Ry ∗ = y ∗ Rx∗ for any x∗ , y ∗ ∈ X ∗ . The operator R is said to be positive if x∗ Rx∗ ≥ 0 for any x∗ ∈ X ∗ . For every symmetric positive operator R : X ∗ 7→ X there exists a Hilbert space H, and a continuous linear operator A : X ∗ 7→ H such that R = A∗ A. The operator A is called the square root of R and will be denoted R1/2 . Any zero-mean Gaussian measure µ on X has the characteristic functional of the form, Z h 1 i ∗ ∗ ∗ exp[ix x]µ(dx) = exp − x Rx , x∗ ∈ X ∗ , 2 X where R is symmetric and positive. R will be called the covariance operator of µ. Definition 7.2.2. If (γi ) is a sequence of independent and identically distributed Gaussian, zero-mean random variables then the closed linear span in L2 (X) of random vectors of the form P i γi xi , (xi ) ⊂ X, will be denoted [γi ] ⊗ X.

Proposition 7.2.1. If X is of type 2 then [γi ] ⊗ X is a complemented subspace of L2 (X) and L2 (X) = ([γi ] ⊗ X) ⊗ ([γi ] ⊗ X ∗ ). 3

Due to J. Hoffman-Jorgensen (1975). In his paper measures with Gaussian covariances are called pregaussian.

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In particular, the dual space of [γi ] ⊗ X may be identified with [γi ] ⊗ X ∗ . Proof. Let f ∈ L2 (X). Define the operator

Af : X ∗ ∋ x∗ 7→ Af x∗ = x∗ f ∈ L2 (R). By Theorem 7.2.1, the operator Rf = A∗f Af : X ∗ 7→ X is the covariance operator of a Gaussian measure µf on X. Let π : L2 (R) 7→ [γi ] ⊗ R be the orthogonal projection, and consider operators A1 = π ◦ Af , A2 = Af − A1 . Put R1 = A∗1 A : X ∗ 7→ X. Then x∗ R1 x∗ ≤ x∗ Rf x∗ . Thus R1 is also P the covariance operator of a Gaussian measure. Hence, the series i A∗1 γi γi = ξ1 convergences almost surely, and f = ξ1 + ξ2 , where ξ1 ∈ [γi ] ⊗ X, and ξ2 ∈ ([γi ] ⊗ X ∗ )⊥ . QED The next theorem displays the crucial role played by absolutely summing operators. Recall that Π2 (X, Y ) denotes the space of 2-absolutely summing operators from X to Y (see Chapter 1).

Theorem 7.2.2.4 The following properties of a Banach space X are equivalent: (i) The space X is of type 2; (ii) A symmetric positive operator R : X ∗ 7→ X is a covariance operator of a Gaussian measure in X if, and only if, R1/2 ∈ Π2 (X ∗ , H), where H is a Hilbert space.

Proof. (i) =⇒ (ii) Let R = A∗ A with A ∈ B(X ∗ , H), and (ek ), k ∈ N, be an orthonormal basis in H. In order to proveP(i) =⇒ (ii) it suffices to show the convergence of the series k A∗ ek γk . Its summands may be considered as elements of [γi ] ⊗ X, and for our purposes it is sufficient to show that for any linear functional F : [γi ] ⊗ X 7→ R the series P continuous ∗ |F (A e γ )| converges (since [γi ] ⊗ X does not contain a subk k k space isomorphic to c0 ). According to the above Proposition this series can be written in the form X |Eη ∗(A∗ ek γk )|, η ∗ ∈ [γi ] ⊗ X ∗ . k

4

Due to S.A. Chobanyan and V.I. Tarieladze (1977).

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Spaces of type 2

Let R1 : X ∗∗ 7→ X ∗ be the covariance operator of η ∗ , and let R1 = A∗1 A1 be a factorization of R1 through H. Therefore η ∗ ∈ [γi ] ⊗ X ∗ is a Gaussian random vector in the space of cotype 2, and by Theorem 5.5.6 the operator A∗1 is 2-absolutely summing. Hence, X X |F (A∗ ek γk )| = |Eη ∗(A∗ ek γk )| k

=

X k

k

|(A∗ ek )(A∗1 ek )| =

X k

|(AA∗1 ek )ek | < ∞,

AA∗1

since the operator : H 7→ H is nuclear as a superposition of two 2-summing operators. P (ii) =⇒ (i) Let (xk )P⊂ X, with k kxk k2 < ∞. We have to prove that the series k γk xk converges almost surely. The convergence of this series will be established as soon as we show that operator R : X ∗ 7→ X defined by the equality RX ∗ = P the ∗ k (x xk )xk is the covariance operator P of a Gaussian measure, i.e., the operator A : X ∗ 7→ H, Ax∗ = k (x∗ ek )ek isP 2- summing. ∗ ∗ Consider an arbitrary sequence (xn ) ⊂ X such that n (x∗n x)2 < ∞ by the Uniform Boundedness Principle, P for ∗all 2x ∈ X. Then, 2 n (xn x) ≤ Ckxk , for some constant C > 0, and all x ∈ X. Thus we have

2 X X X X X

X ∗

kAx∗n k2 = (x∗n xk )2 ≤ C kxk k2 < ∞,

(xn xk )ek = n

n

k

n

k

k

that is, A is 2-absolutely summing. QED

The next result is an immediate corollary to Theorems 7.2.1 and 7.2.2. Corollary 7.2.1. The following properties of a Banach space X are equivalent: (i) The space X is of type 2; (ii) A symmetric positive operator R : X ∗ 7→ X is a Gaussian covariance if, and only if, for each B ∈ B(X, H), the operator BRB ∗ : H 7→ H is nuclear (trace class); (iii) Let T ∈ B(H, X) and µH be a standard cylindrical Gaussian measure on H with the characteristic functional exp[−kxk2 /2]. A cylindrical Gaussian measure µH ◦ T −1 on X is σ-additive if and only if T ∗ ∈ Π2 (X ∗ , H);

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(iv) A functional Φ : X ∗ 7→ C is the characteristic functional of a zero-mean Gaussian measure on X if, and only if, Φ(x∗ ) = exp[−kT x∗ k2H ], where T : X ∗ 7→ H is 2-absolutely summing. (v) There exist constants C1 , C2 , C3 > 0 such that, for each zero-mean measure νR on X with finite second moments, Z 1/2 Z 1/2 2 1/2 2 kxk νR (dx) ≥ C1 π2 (R ) ≥ C2 kxk µR (dx) X

X

≥ C3 π2 (R1/2 ), where µR is a zero-mean Gaussian measure on X with the covariance operator R. (vi) (Under the extra assumption that X has an unconditional basis (ek )). There exists a constant C > 0 such that for each masure νR on X with zero mean, and covariance operator R, Z

2

X

∗ ∗ 1/2 2 kxk νR (dx) ≥ C (ek Rek ) ek . X

7.3

k

Kolmogorov’s inequality three-series theorem

and

Two classical results of probability theory, the Kolmogorov’s inequality for the maximum of sums of independent random variables, and the three-series theorem, giving a sufficient condition for the almost sure convergence of series of independent random variables also have extensions to Banach spaces of type 2. Theorem 7.3.1.5 The Banach space X is of type 2 if, and only if, there exists a constant C > 0 such that, for any n ∈ N, 5

Due to N. Jain (1976).

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and arbitrary independent random vectors X1 , . . . , Xn ∈ L2 (X) with EXi = 0, n = 1, 2, . . . , n, we have the inequality n X

−2

P max X1 + · · · + Xj > λ ≤ Cλ EkXj k2 ,

1≤j≤n

j=1

∀λ > 0.

Proof. If we assume the validity of the Kolmogorov’s inequalP ity, a routine argument shows that the series i Xi of zero-mean independent random vectors in X converges almost surely whenP ever i EkXi k2 < ∞. This and Theorem 6.7.4 ensure that X is of type 2. Conversely, let X be of type 2, and, as usual, define the set Λ = {max1≤j≤n kSj k > λ}, where Sj = X1 + · · · + Xj , and, for 1 ≤ j ≤ n, define the sets n o Bj = kS1 k ≤ λ, . . . , kSj−1k ≤ λ, kSj k > λ . By convention, we set S0 = 0. Then, Z

2 kSn k2 dP = E Sj IBj + (Sn − Sj )IBj . Bj

Using Jensen’s Inequality, and the independence assumption, we get that

2

2 E′′ Sj IBj + (Sn − Sj )IBj ≥ Sj IBj ,

where E′′ denotes integration on variables Xj+1 , . . . , Xn . Therefore, because E = E′ E′′ (with E′ standing for integration on X1 , . . . , Xj ) we have Z Z 2 kSn k dP ≥ kSj k2 dP ≥ λ2 P(Bj ), Bj

Bj

and a simple addition gives the inequality Z kSn k2 dP ≥ λ2 P(Λ), S since Bj ’s are disjoint and j Bj = Λ. Now, the Kolmogorov’s Inequality follows in view of the inequality defining type 2. QED

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Employing the usual real-variable procedure we can deduce from the Kolmogorov’e inequality “one half” of the Three Series Theorem for random vectors taking values in Banach spaces of type 2. Corollary 7.2.2. If X is a Banach space of type 2, and (Xj ) is a sequence of independent random vectors in X, then the convergence following three series, for some c > 0, Pof the (i) j P kxj k > c , P (ii) j E Xj I[kXj k ≤ c] , and

2 P (iii) j E Xj I[kXj k ≤ c] − EXj I[kXj k ≤ c] P, implies the almost sure convergence of the series j Xj .

7.4

Central limit theorem

The following simplest form of the Central Limit Theorem also characterizes Banach spaces of type 2. Theorem 7.4.1.6 The following properties of a Banach space X are equivalent: (i) The space X is of type 2; (ii) For any independent, identically distributed, zero-mean random vectors (Xn ) ⊂ L2 (X) there exists a Gaussian measure γ on X such that X + · · · + X 1 n L −→ γ, 1/2 n weakly, as n → ∞. Moreover, the Fourier transform (characteristic functional) of γ is exp[−E(x∗ X1 )2 /2]. Proof. (i) =⇒ (ii) By Theorems 6.10.1, 6.10.2, and 7.2.1, there exists a Gaussian random vector X in X of the form R xM(dx), where M is a Gaussian random measure on (X, BX ) X with control measure µ = L(X1 ). Moreover, i h 1 E exp[ix∗ X] = exp − E(x∗ X1 )2 , 2 6

x∗ ∈ X ∗ .

Due to J. Hoffmann-Jorgensen and G. Pisier (1976).

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We shall show that we can take γ = L(X1 ). Since the topology induced by Lipschitzian functions on X coincides with the norm topology, it is sufficient to show that for any Lipschtzian (say, with constant K) bounded function φ : X 7→ R, Z −1/2 lim Eφ(n Sn ) = Eφ xM(dx) . n→∞

X

Choose a sequence f (d) , d ∈ RN, of simple (finite range) functions from L2 (X, BX , µ; X) with f (d) dµ = 0, and such that Z

X

kx − f (d) (x)kp µ(dx) → 0,

p = 1, 2,

d → ∞.

(7.4.1)

Then, for each d ∈ N, (Xn − f (d) (Xn )) are again independent random vectors in X, and we have that E φ (X1 + · · · + Xn )n−1/2 − φ (f (d) (X1 ) + · · · + f (d) (Xn ))n−1/2 n

X

Xi − f (d) (Xi ) ≤ KE n−1/2 i=1

n

X

2 1/2 (d) ≤ K n E Xi − f (Xi )

−1

i=1

n 1/2 X ≤ KC 1/2 n−1 EkXi − f (d) (Xi )k2 i=1

≤ KC 1/2

Z

X

kx − f (d) (x)k2 µ(dx)

1/2

.

The next to the last inequality used the fact that X is of type 2. Therefore, by (7.4.1), −1/2 −1/2 (d) (d) E φ (X1 +· · ·+Xn )n −φ f (X1 )+· · ·+f (Xn ) n → 0, (7.4.2) as d → ∞, for each n ∈ N, and uniformly in n.

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Now, since for each particular d, (f (d) (Xn )) is a sequence of finite dimensional i.i.d. random vectors, by the finite-dimensional Central Limit Theorem, weakly, n Z X −1/2 (d) (d) L n f (Xi ) −→ L f (x)M(dx) , X

i=1

as n → ∞,

and, in particular, for each fixed d ∈ N, n Z X −1/2 (d) f (Xi ) −φ f (d) (x)M(dx) −→ 0, as n → ∞. E φ n X

i=1

(7.4.3) In view of (7.4.2), and (7.4.3), and the definition of f (d) , we get that n n X X Xi − φ n−1/2 f (d) (Xi ) E φ n−1/2 i=1

i−1

n Z X −1/2 (d) f (Xi ) − φ f (d) (x)M(dx) +E φ n X

i−1

Z Z (d) f (x)M(dx) − φ xM(dx) +E φ X

n X −1/2 Xi − φ ≥ E φ n i=1

X

Z

X

f

(d)

(x)M(dx) −→ 0,

as n → ∞,

which gives the desired weak convergence. Now, the shape of the Fourier transform of γ is an immediate consequence of the formula, h Z i h 1Z ∗ ∗ 2 E exp ix f (x)M(dx) = exp − (x f (x)) µ(dx) , x∗ ∈ X ∗ , 2 X X valid for any f ∈ L2 (X, BX , µ; X) (see Theorem 6.10.2).

(ii) =⇒ L2 (X)

(i) If for any i.i.d. zero-mean sequence (Xn ) ⊂

L (X1 + · · · + Xn )n−1/2 → L(X),

as n → ∞,

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for some Gaussian random vector X in X, then, in particular, E(x∗ X1 )2 = E(x∗ X)2 . Therefore, for any zero-mean X1 ∈ L2 (X), there exists a Gaussian X such that h 1 i ∗ ∗ 2 E exp[ix X] = exp − E(x X1 ) . 2 Thus, by Theorem 7.2.1, X is of type 2. QED Having completed a discussion of the extension of the analogue of the classical Central Limit Theorem in Banach spaces of type 2, we now turn to a presentation of a different approach to the Central Limit Theorem. The next theorem works in general Banach spaces but, perhaps, gives a deeper insight into the problem. For a sequence (Xi ) of independent copies of a random vector X in X, let us introduce the notation CL(X) = sup En−1/2 kX1 + · · · + Xn k. n∈N

Then, we define the space CL∞ := {X ∈ L0 (X) : CL(X) < ∞} which is a Banach space under the norm CL. By CL(X) we shall denote the closed subspace of CL∞ (X) spanned by the zero-mean simple (finite range) random vectors. Theorem 7.4.2.7 Let X be a Banach space, and let (Xn ) be a sequence of independent, identically distributed random vectors in X. Then the laws L((X1 + · · · + Xn )n−1/2 ) converge weakly if, and only if, X1 ∈ CL(X). Proof. To prove the “if” part of the theorem let us begin by recalling the concept of the L´evy distance on the set of positive measures on X: d(λ, µ) := inf a > 0 : λ(F ) ≤ µ(F a ) + a, µ(F ) ≤ λ(F a ) + a, ∀ bdd F ⊂ X , where F a = {x ∈ X : dist(x, F ) < a}. The metric space created with the help of this distance has the topology equivalent to the 7

Due to G. Pisier (1975/76).

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topology of weak convergence. It is easy to see that the mapping X 7→ L(X) from L0 (X) into the space of measures equipped with the L´evy distance is uniformly continuous. Now, if X1 ∈ CL(X), then for each ǫ > 0 there exists a simple random vector Y such that CL(X1 − Y ) < ǫ. The finite dimensional Central Limit Theorem implies that if (Yi ) are independent copies of Y then the laws L(n−1/2 (Y1 + · · · + Yn )) converge weakly. Hence, the sequence L(n−1/2 (X1 + · · · + Xn )) is arbitrarily uniformly close in the L´evy metric to a convergent sequence of measures. Thus it converges itself in view of the completeness of the space of measures equipped with the L´evy metric. This concludes the proof of the “if” part of the Theorem. In the proof of the “only if” part of the Theorem we shall have need of the following result which is also of independent interest. Lemma 7.4.1.8 Let X be a Banach space, and let (Xn ) be a sequence of independent, identically distributed random vectors in X. If the laws L(n−1/2 (X1 + · · · + Xn )) converge weakly, then (i) supn∈N supc>0 c2 P(n−1/2 kX1 + · · · + Xn k > c) < ∞, and (ii) EkX1 kp < ∞, and supn∈N Ek(X1 + · · · + Xn )n−1/2 kp < ∞, for each p < 2. Proof. If X1 satisfies the Central Limit Theorem then so does ˜ 1 . In particular, its symmetrization X ˜1 + · · · + X ˜ n ) → L(Y ), L n−1/2 (X as n → ∞,

for some Gaussian Y . Also, for each a > 0, ˜1 + · · · + X ˜ n k ≥ a ≤ P(kY k ≥ a). limn→∞ P n−1/2 kX

Therefore, for each ǫ < 1/2, there exists an a > 0 such that ˜1 + · · · + X ˜ n k ≥ a ≤ ǫ, P n−1/2 kX n ∈ N, (7.4.4) and by the L´evy Inequality ˜ i k > 2an1/2 ≤ 2ǫ. P sup kX 1≤i≤n

8

Due to N. Jain (1976).

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Spaces of type 2 Assume that 1/2 ˜ P kX1 k > 2an ≤ 1 − (1 − 2ǫ)1/n , which means that, for a certain constant A(ǫ) ˜ 1 k > 2an1/2 ≤ A(ǫ)n−1 . P kX This leads to the inequality ˜ 1 k > c) ≤ B(ǫ), sup c2 P(kX c>0

where B(ǫ) is another constant. To finish the proof of the Lemma ˜1 + . . . X ˜ N )N −1/2 is substituted for it suffices to notice that if (X X1 in (7.4.4) then we get that ˜1 + · · · + X ˜ N k > c ≤ B(ǫ), sup c2 P N −1/2 kX N ∈ N. c>0

Then the triangle inequality yields that P n−1/2 kX1 + · · · + Xn k > c + α . −1/2 −1/2 ≤P n kX1 + · · ·+ Xn k > c P n kX1 + · · ·+ Xn k ≤ α .

Since X1 satisfies the Central Limit Theorem there exists an α > 0 such that inf P n−1/2 kX1 + · · · + Xn k ≤ α > 0, n

from which statement (i) follows. To obtain (ii) it is enough to use (i) and the obvious formula, Z ∞ p EY = pcp−1 P(Y > c)dc, Y ≥ 0. 0

This concludes the proof of the Lemma. QED Proof of Theorem 7.4.2, continued. Now we are ready to prove the “only if” part of the Theorem. Let X ∈ L0 (X) and assume

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that X satisfies the Central Limit Theorem. By the above Lemma, EkXk < ∞, and EX = 0. One can thus consider X as a L1 -limit of a martingale (X N ), N ∈ N, such that, for each N, X N takes on only finitely many values. We shall show that limN →∞ CL(X − X N ) = 0, which would complete the proof. Notice that, by compactness argument, for each N ∈ N, X − N X also satisfies the Central Limit Theorem. Now, for the sake of simplicity, assume that X

N

=

N X

φi xi ,

i=1

where (xi ) ⊂ X, and (φi ) isPan orthonormal sequence of martinN gale differences. Put Y N = N converges almost i=1 γi xi . Then Y surely to a Gaussian random vector Y . Indeed, for each x∗ ∈ X ∗ , the sequence x∗ Y N converges to a random variable with the law ∗−1 −1/2 γX x , where γX = limn→∞ L n (X1 + · · · + Xn ) . Thus, the Ito-Nisio Theorem (see Chapter 1) guarantees the existence of Y . By Landau-Shepp-Fernique Theorem (see Chapter 1) we also have that EkY − Y N k → 0, as N → ∞. Checking the covariances we get that, as n → ∞, L (X1N + · · · + XnN )n−1/2 → L(Y N ), L (X1 + · · · + Xn )n−1/2 → L(Y ), L ((X1 − X1N ) + · · · + (Xn − XnN ))n−1/2 → L(Y − Y N ).

However, if, say (Zn ) ⊂ L0 (X), and f : X 7→ R is continuous and such that f (Zn ) are equi-integrable, then the weak convergence of Zn to Z implies that limn→∞ Ef (Zn ) = Ef (Z). Hence, in our situation, by virtue of Lemma 7.4.1, we have that n

−1/2 X N (Xi − Xi ) = EkY − Y N k. (7.4.5) lim E n n→∞

i=1

Let now ǫ > 0, and choose N0 such that EkY − Y N0 k < ǫ. By (7.4.5), there exists an m ≥ N0 such that n

X

sup E n−1/2 (Xi − XiN ) < ǫ. (7.4.6) n≥m

i=1

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215

Spaces of type 2 Given m, there exists an M ≥ m such that n

X

sup sup E n−1/2 (Xi − XiN ) < ǫ.

N ≥M 1≤n≤m

(7.4.7)

i=1

Furthermore, for each N ≥ N0 ,

n

−1/2 X N E n (Xi − Xi ) i=1

n n

−1/2 X N

−1/2 X N0 (Xi − XiN0 ) (Xi − Xi ) + E n ≤ E n i=1

i=1

n

X

(Xi − XiN0 ) , ≤ 2E n−1/2 i=1

Pn

N0 N because i=1 (XP i − Xi ) can be obtained by taking a conditional n expectation of i=1 (Xi − XiN0 ). Hence, for each N ≥ N0 , and n ≥ m, by (7.4.6) n

X

(Xi − XiN ) ≤ 2ǫ, E n−1/2 i=1

so that, by (7.4.7), for N ≥ N0 , we have CL(X − X N ) ≤ 2ǫ. QED The Central Limit Theorem for triangular arrays can also be proven for random vectors taking values in spaces of type 2. Below, we discuss two sample results in this area which, however, also rely on the explicit compactness assumptions. Theorem 7.4.3.9 Suppose that (Ynj ), n ∈ N, j = 1, . . . , jn , is an array of row-wise independent random vectors in a Banach space Y of type 2, and satisfying the following three conditions: (i) EYnj = 0; Pn 2 2 2 (ii) 0 ≤ σnj := EkYnj k2 < vnj , with jj=1 vnj = 1; 9

Due to D.J.H. Garling (1976). That paper also contains a functional Central Limit Theorem (invariance principle) written in a similar spirit.

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216

Geometry and Martingales in Banach Spaces (iii) For each η > 0 jn Z X

|y ∗ Ynj |>η

j=1

(y ∗ Ynj )2 dP −→ 0,

as n → ∞,

for each y ∗ ∈ Y ∗ . Suppose that T is a compact linear operator from Y into a Banach space X, and let Xnj = T Ynj . Then, weakly, L

jn X j=1

Xnj −→ γ,

as n → ∞,

(7.4.8)

where γ is a Gaussian measure on X, provided jn X ∗ Xnj exists lim E x

n→∞

j=1

∀x∗ ∈ X ∗ .

Proof. As usual, it is sufficient to show that the sequence in (7.4.8) is uniformly tight. By (i), and the fact that Y is of type 2, jn jn

2

X X

EkYnj k2 ≤ C, Ynj ≤ C E j=1

j=1

so that

n ∈ N,

jn

2 X

P Ynj > C/ǫ ≤ ǫ, j=1

n ∈ N.

Therefore, since K = T BY is a compact set, we have P

jn X j=1

Ynj ∈ / (C/ǫ)

1/2

K ≤ ǫ,

n ∈ N,

which gives the desired uniform tightness. QED The next result also deals with arrays of random vectors but has the advantage of involving no compact operators.

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Theorem 7.4.4.10 Suppose that (Xnj ) is a row-wise independent array of random vectors satisfying conditions (i), (ii), and (iii), of Theorem 7.4.3, and taking values in a Banach space X of type 2. If jn X 2 ∗ lim E x Xnj n→∞

j=1

exists for each x∗ ∈ X ∗ , and for each ǫ > 0 there exists a finitedimensional subspace Y ⊂ X such that jn X 2 E dist(Xnj , Y ) < ǫ, j=1

n ∈ N,

(7.4.9)

then, there exists a Gaussian measure γ on X such that, as n → ∞, weakly, jn X Xnj −→ γ. L j=1

Proof. By Proposition 6.2.4, the quotient space X/Y is of type 2 whenever X is of type 2, and the constant C is the same in both cases. Given ǫ > 0, let η = ǫ3 /2C 2 , and let Y be a finite dimensional subspace corresponding to η in view of (7.4.9). Then, jn i2 h X Xnj , Y ≤ C 2 η, E dist j=1

so that

n ∈ N,

jn h X i P dist Xnj , Y > ǫ ≤ C 2 η/ǫ2 = ǫ/2. j=1

Also, jn

X

P Xnj > 2C/ǫ1/2 ≤ ǫ/4, j=1

10

Due to D.J.H. Garling (1976).

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and, furthermore, if K = {x ∈ Y : kxk ≤ 2C/ǫ1/2 }, jn h X i P dist Xnj , K > ǫ ≤ ǫ/2,

n ∈ N,

j=1

so that, by a standard argument11 the sequence L uniformly tight. QED

7.5

P jn

j=1 Xnj

is

Law of iterated logarithm

Definition 7.5.1. We shall say that X ∈ L0 (X) satisfies the Law of the Iterated Logarithm if, for a sequence of independent copies (Xn ) of X, the sequence n X −1/2 (2n log log n) Xi ,

n = 3, 4, . . . ,

i=1

is almost surely a conditionally compact set in X. For other, but equivalent formulations of the Law of Iterated Logarithm see Section 5.6. In what follows we shall use the following notation: By IL∞ (X) we will denote that set of random vectors in X for which IL(X) := E sup(2n log log n) n

n

X

Xi < ∞.

−1/2

i=1

Note that IL∞ (X) is a Banach space equipped with the norm IL. Denote by IL(X) the closure in IL∞ (X) of the set of zero-mean, simple random vectors in IL∞ (X). In view of the real-valued Law of the Iterated Logarithm, the latter set is contained in IL∞ (X). Theorem 7.5.1.12 Let X be a Banach space, and let X be a random vector in X. Then X satisfies the Law of the Iterated Logarithm if, and only if, X ∈ IL(X). 11 12

See, e..g., K.R. Parthasarathy (1967), p. 49. Due to G. Pisier (1975/76).

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Proof. “If” If X ∈ IL(X) then, for every ǫ > 0, there exists a finite range random vector Y such that IL(X − Y ) < ǫ.

(7.5.1)

Now, consider the random variable αX defined by the formula, n

X

−1/2 αX := lim sup inf (2n log log n) Xi N →∞ n∈N 1≤j≤N

−(2j log log j)

i=1

−1/2

j X i=1

Xi .

Clearly, it suffices to show that αX = 0 almost surely. So, notice that n

X

−1/2 αX ≤ αY + 2 sup(2n log log n)

(Xi − Yi ) . n

i=1

Since Y satisfies the Law of the Iterated Logarithm, αY = 0 almost surely, so that it follows from (7.5.1) that EαX ≤ 2ǫ, for any ǫ > 0. Therefore αX = 0 almost surely. “Only if” Assume that X satisfies the Law of the Iterated Logarithm. First, we shall prove that X ∈ IL∞ (X). By assumption, sup (2n log log n) n∈N

n

X

Xi < ∞,

−1/2

a.s.,

(7.5.2)

i=1

so that, in view of the Hoffmann-Jorgensen Theorem (see Chapter 1), it is sufficient to prove that sup (2n log log n)−1/2 kXn k ∈ L1 .

n∈N

Actually, we shall show that it even belongs to Lp , for every p < 2. It follows from (7.5.2) that supn∈N (2n log log n)−1/2 kXn k < ∞ almost surely, and by Borel-Cantelli Lemma, there exists a d > 0 such that ∞ X Λ= P kXk > d(2n log log n)1/2 < ∞. n=1

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Therefore, for each k ∈ N, ∞ X n=1

1/2 ≤ Λ. kP kXk > d(2nk log log nk)

Furthermore, for some constant B > 0, (2nk log log nk)1/2 ≤ B(2n log log n)1/2 (2k log log k)1/2 , so that ∞ X n=1

1/2 1/2 kP kXk > Bd(2n log log n) (2k log log k) , ≤ Λ.

Hence, for each c > 0 −1/2 Φ(c) := P sup(2n log log n) kXn k > c n

X 1/2 ≤ P kXk > c(2n log log n) , n

so that p Z Z −1/2 E sup(2n log log n) kXn k = n

∞

pcp−1Φ(c)dc

0

p

≤ (Bd) +

Z

∞

pc

p−1

Bd

X 1/2 dc P kXk > c(2n log log n) n

p

≤ (Bd) +

XZ k

Bd(2(k+1) log log(k+1))1/2

Bd(2k

pcp−1 log log k)1/2

X × P kXk > Bd(2n log log n)1/2 (2k log log k)1/2 dc n

p

≤ (Bd) +

XZ k

Bd(2(k+1) log log(k+1))1/2

pcp−1 Λk −1 dc Bd(2k log log k)1/2

≤ (Bd)p + (Bd)p Λ

∞ X

k −1

k=1

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Spaces of type 2 p/2 p/2 × (2(k + 1) log log(k + 1)) − (2k log log k) ,

and, if p < 2, the last series converges because, asymptotically, the numerator behaves like (log log k)p/2 k p/2−1 . Now, as in the proof of Theorem 7.4.2, one can define a martingale X N of simple random vectors such that EkX − X N k −→ 0,

as N → ∞.

Evidently, for each N, the random vector X −X N satisfies the Law of the Iterated Logarithm. We shall show that IL(X − X N ) → 0. By Kuelbs’ Theorem13 limn→∞ (2n log log n)

n

X

(Xi −XiN ) ≤ sup{kxk : x ∈ KX−X N },

−1/2

i=1

(7.5.3) where KX−X N is the compact set for X − X as defined in Definition 7.5.1. Denote by λN the right-hand side of (7.5.3). By the method of the gliding hump, in view of the compactness of KX , one can deduce immediately that λN → 0 as N → ∞. Now, let N0 be such that λN0 < ǫ. By (7.5.3), there exists an m ≥ N0 such that n

X

−1/2 E sup (2n log log n)

(Xi − XiN0 ) < ǫ. N

n≥m

i=1

Then, for each N ≥ N0 ,

n

X

E sup (2n log log n)−1/2 (Xi − XiN ) n≥m

≤ 2E sup (2n log log n) n≥m

i=1

n

X

(Xi − XiN0 ) < 2ǫ.

−1/2

i=1

Now, for a fixed m, there exists an M ≥ m such that sup E sup (2n log log n) n≥M 13

1≤n≤m

See, J. Kuelbs (1977).

n

X

(Xi − XiN ) < ǫ.

−1/2

i=1

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Finally, N

IL(X − X ) ≤ E sup (2n log log n) n≤m

+E sup (2n log log n) n≥m

n

X

N

(Xi − Xi )

−1/2

i=1

n

X

(Xi − XiN ) ,

−1/2

i=1

so that if N ≥ M, we have IL(X − X N ) < ǫ + 2ǫ. QED

As a corollary to the above theorem we have the following analogue of the Law of the Iterated Logarithm in spaces of type 2.

Theorem 7.5.2.14 Let (Xi ) be a sequence of independent and identically distributed random vectors in a Banach space X of type 2. If EX1 = 0, and EkX1 k2 < ∞, then there exists a constant C > 0 such that 1/2 , IL(X1 ) ≤ C EkX1k2 and, in particular, (Xi ) satisfies the Law of the Iterated Logarithm.

Proof. Take the Rademacher sequence (rn ) independent of (Xn ) and assume (without loss of generality) that the random vectors (Xn ) are symmetric. Then, (rn Xn ) and (Xn ) are identically distributed and n

X

2 i1/2 h

IL(X1 ) ≤ E sup(2n log log n)−1 ri X i . n

i=1

By Kwapie´ n’s Theorem (see Chapter 1) the random variable P P 2 exp k i ri xi k is integrable if i ri xi converges almost surely, where (xn ) ⊂ X. Then, by a standard real-line argument based on L´evy’s Inequality, we get that, for a constant D, n

2 1/2 X

IL(X1 ) ≤ D sup n−1/2 Er ri X i . n

14

i=1

Due to G. Pisier (1975/76).

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Spaces of type 2 Since X is of type 2, n 1/2 X IL(X1 ) ≤ CD E sup n−1 kXi k2 , n

i=1

so that, by the maximal inequality associated with the Strong Law of Large Numbers for real-valued random variables (kXi k2 ), for a constant M, IL(X1 ) ≤ CDM(EkX1 k2 )1/2 . QED

Remark 7.5.1. G. Pisier (1975/76) has also demonstrated that if X is a Banach space then each zero-mean random vector X ∈ L2 (X) satisfies the Law of the Iterated Logarithm if, and only if, there exists a constant C such that for all n ∈ N, and every sequence (xi ) ⊂ X, n n

X X 1/2

1/2 2 kxi k . ri xi ≤ C(log log n) E i=1

i=1

This theorem depends on a result of J. Kuelbs (1977) which says that if EkXk2 < ∞ then X ∈ IL(X) if, and only if, (2n log log n)−1/2

n X i=1

Xi −→ 0

in probability, as n → ∞.

7.6

Spaces of type 2 and cotype 2

Theorem 7.6.1.15 If a Banach space is of both type 2 and cotype 2, then it is isomorphic to a Hilbert space. Proof. Step 1. We first prove that if X is of type 2, and cotype 2, then there exists a constant C such that, for each n ∈ N, each sequence (xi ) ⊂ X, and each matrix (aij ) ⊂ R, n X n n

2 X X

aij xj ≤ C 2 kak2 kxi k2 , (7.6.1)

i=1

j=1

i=1

15

Due to S. Kwapie´ n (1972/73). The paper also contains other characterizations of Banach spaces isomorphic to a Hilbert space H. In particular, X is isomorphic to H iff the Fourier transform is a bounded operator from L2 (R; X) into itself.

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(n)

where kak denotes the norm of the operator a : l2 7→ l2 defined by the matrix (aij ). From Remark 5.5.1 (b), and Proposition 7.1.1, it follows that, in view of our assumptions, there exists a constant C such that, for each n ∈ N, and each sequence (xi ) ⊂ X, C

−1

n X i=1

n n

X

2 X

kxi k ≤ E γ i xi ≤ C kxi k2 , 2

i=1

(7.6.2)

i=1

where (γi ) is the sequence of standard independent Gaussian random variables. Now, it is known that the extreme points of the (n) (n) unit ball of the bounded operator space B(l2 , l2 ) are exactly linear isometries. Hence, by the Krein-Milman Theorem, any (n) (n) a ∈ B(l2 , l2 ) with kak ≤ 1 is a convex combination of matrices of isometries. Thus it is clear that it suffices to prove (7.6.1) in the case when the matrix a is an isometry. However, by (7.6.2), in this situation n n n n X

X

2 X 2 X

γ a x ≤ CE a x

i ij j ij j i=1

i=1

j=1

j=1

n n n

2

X

X X 2

xj γ j xj γi aij = CE = CE j=1

j=1

i=1

≤ C2

n X j=1

kxj k2 ,

because the isometry does not change the distribution of a Gaus(n) sian symmetric measure on l2 . Step 2. Let u : l1 (I; R) 7→ X be a bounded linear operator onto X (which can be achieved if |I| is sufficiently large). We shall show first that u ∈ Π2 (l1 (I); X). (y i ) ∈ l1 (I) be suchPthat, for each y ∗ ∈ l∞ (I), the series P Let ∗ 2 i (y y i ) < ∞. Then also i ky i k < ∞, and there exists a linear bounded operator, v : l2 (I) 7→ l1 (I), such that y i = v(e2i ), where (ei ) is the canonical basis in l2 (I). By Grothendieck’s Factorization Theorem, d

a

v : l2 (I) 7→ l2 (I) 7→ l1 (I),

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225

Spaces of type 2

where d = (di ) is diagonal and Hilbert-Schmidt, and a = (aij ) is bounded. Therefore, by Step 1, X X X kuy i k2 = kuv(e2i )k2 ≤ kuk2 kad(e2i )k2 i∈I

i∈I

≤

X i∈I

i∈I

kuk2kak2 |di |2 < ∞,

so that u ∈ Π2 . Now, by the Pietsch Factorization Theorem (see Chapter 1) u

u

u : l1 (I) 7→1 H 7→2 X, where H is a Hilbert space, and u1 , and u2 , are bounded. Since u was an operator onto X, the operator u2 maps H onto X as well, so that X is isomorphic to a Hilbert space. QED Remark 7.6.1. Almost the same proof as above shows that if u : X 7→ Y , and X is of type 2, and Y is of cotype 2, then u can be factorized through a Hilbert space. Remark 7.6.2. Tail probabilities of sums of random vectors in Banach spaces of type 2 can also be controlled. Here are two sample results16 : (a) Let (Xn ) be independent, identically distributed, zero-mean random vectors in a Banach space X of type 2 with EkX1 k < ∞. If t > 0, and r > max(t, 2)/2 then EkX1 k2 < ∞ if, and only if, for each ǫ > 0, X j r−2 P kX1 + · · · + Xj k > ǫj r/t < ∞. j

(b) Let (Xn ) be independent, identically distributed, random vectors in a Banach space X of type 2. Then EkX1 k < ∞, and EX1 = 0, if, and only if, for each ǫ > 0, X j −1 P kX1 + · · · + Xj k > ǫj < ∞. j

16

Due to N. Jain (1975).

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Geometry and Martingales in Banach Spaces

Remark 7.6.3. If X is of type 2 then it is also possible to prove the following extension of the Salem and Zygmund Theorem on exponential moments for lacunary trigonometric series with coefficients in X 17 : If G is a compact Abelian group and ∆ = (χj ) is a Sidon set P in the dual group of G then, if j kxj k2 < ∞ then, for all β > 0, and any bounded sequence (αj ) ⊂ C, Z

G

n

2 i

X h

exp β sup αj χj (g)xj dg < ∞, n

j=1

where dg denotes the integration with respect to the Haar measure on G. Remark 7.6.4. Also, X is of type 2 if, and only if, the class of Gaussian covariance operators coincides with the class of positive, symmetric and nuclear operators.18

17 18

Due to J. Kuelbs and W.A. Woyczy´ nski (1978). Due to S. Chevet, S.A. Chobanyan, W. Linde and V.I. Tarieladze ( 1977).

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Chapter 8 Beck convexity 8.1

General definitions and properties and their relationship to types of Banach spaces

We begin this chapter with the basic definition of the Beckconvexity1 , which for the sake of brevity will be called here Bconvexity. We shall also briefly discuss its relationship to other geometric properties of Banach spaces which were introduced and discussed in depth in the previous chapters. Definition 8.1.1. Let k ∈ N+ , and ǫ ∈ (0, 1). A real, normed space X is said to be (k, ǫ)-convex if, for each sequence x1 , . . . , xk ∈ SX , where SX denotes the unit sphere in X, inf kǫ1 x1 + · · · + ǫk xk k ≤ k(1 − ǫ).

ǫi =±1

The space X is said to be B-convex if it is (k, ǫ)-convex for some k ∈ N+ , and some ǫ ∈ (0, 1). The following simple Proposition shows certain relationship between (k, ǫ)- and (j, δ)- convexity. 1

Introduced by A. Beck (1962), and (1963).

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Geometry and Martingales in Banach Spaces

Proposition 8.1.1.2 (i) If X is (k, ǫ)-convex, 2 ≤ j < k, and kǫ > k −j, then X is also (j, δ)-convex with δ = (kǫ−k −j)/j. (ii) If X is (k, ǫ)-convex and Y ⊂ X then Y is also (k, ǫ)convex. (iii) If X is (k, ǫ)-convex and δ < ǫ, then X is (k, δ)-convex. (iv) If a normed space X is (k, ǫ)-convex then its completion is also (k, ǫ)-convex. Proof. (i) Take an arbitrary sequence x1 , . . . , xj ∈ SX , and put xj+1 = · · · = xk = 0. Since X is (k, ǫ)-convex, there exist ε1 , . . . , εk = ±1 such that kε1 x1 + · · · + εj xj k = kε1 x1 + · · · + εk xk k ≤ k(1 − ǫ) = j(1 − δ),

so that x is (j, δ)-convex. The proofs of (ii) − (iv) are obvious and we omit them. QED In infinite-dimensional normed space X there exists also a less trivial relation between k, and ǫ.

Theorem 8.1.1. If X is an infinite-dimensional (k, ǫ)-convex Banach space then 1 − ǫ ≥ k −1/2 .

Proof. Dvoretzky’s Theorem (see Chapter 1) assures the existence of finite-dimensional spaces of arbitrarily high dimension, say (k) k, approximating Hilbert spaces l2 with any prescribed accuracy. (k) Therefore, it is sufficient to check the inequality in l2 . However, (k) the inequality is evident in l2 because, for an orthonormal basis (k) e1 , . . . , ek ∈ l2 we have the equality kεe ei + · · · + εk ek k = k 1/2 . QED

Remark 8.1.1. The above estimate cannot be improved in general because in the Hilbert space H, for any x1 , . . . , xk ∈ SH , there are ε1 , . . . , εk = ±1 such that k −1 kε1 x1 + · · · + εk xk k ≤ k −1/2 ,

in view of the generalized parallelepiped equality X

ε1 ,...,εk =±1

kε1 x1 + · · · + εk xk k2 = 2k

k X i=1

kxi k2 .

2

This Proposition, Theorem 8.1.1, and Propositions 8.1.3, and 8.1.5 are due to D.P. Giesy (1966).

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229

Beck convexity

The concept of (k, ǫ)-convexity is a property of normed spaces X that imposes restrictions only on the k-dimensional subspaces and, similarly, B-convexity is a local property of normed spaces in the sense that it imposes restrictions only on the structure of the finite-dimensional subspaces of X. In particular, we have the following statement. Proposition 8.1.2. If a normed space Y is finitely representable in a B-convex space X, then Y is also B-convex. In other words, B-convexity is a super-property of a normed space. Also, the B-convexity of the following classes of normed spaces is not difficult to check. Theorem 8.1.2. (i) If X is less than k-dimensional, then it is (k, k −1 )-convex, and thus B-convex. (ii) If X is uniformly convex then it is (2, ǫ)-convex for some ǫ > 0, and thus B-convex. Proof. (i) For any x1 , . . . , xk ∈ SX there exist α1 , . . . , αk ∈ R, with maxi |αi | = 1, such that α1 x1 + · · · + αk xk = 0. Define λi = αi /|αi |, if αi 6= 0, and λi = 1, otherwise. Then, |λi − αi | ≤ 1, for i = 1, . . . , k, and for some i, λi = αi . Then k k

X

X

k kλ1 x1 + · · · + λk xk k = k αi xi + (λi − αi )xi −1

−1

≤k

−1

k X i=1

i=1

i=1

|λi − αi | ≤ k −1 (k − 1) = 1 − k −1 .

(ii) The uniform convexity implies existence of a δ > 0 such that, for any x, y ∈ SX with kx − yk > 1, we have kx + yk ≤ 2(1 + δ). Now, the space X is (2, min((2−1 , δ)-convex because , for any x, y ∈ SX , either kx − yk ≤ 1 = 2(1 − 1/2) ≤ 2(1 − min(2−1 , δ)), or kx + yk ≤ 2(1 − δ) ≤ 2(1 − min(2−1 , δ)).

QED

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Geometry and Martingales in Banach Spaces

Below we list examples of space that are B-convex and that are not B-convex. Example 8.1.1. (a) By Part (ii) of the above Theorem the spaces Lp (T, Σ, µ), 1 < p < ∞, over any measure space (T, Σ, µ) are B-convex. (b) The space l1 is not B-convex. Indeed, if (ei ) is the standard basis in l1 then, for each n ∈ N, n ≥ 2,, and arbitrary εi = ±1, we have the equality kε1 ei + · · · + εn en k = n. (c) The space c0 , and thus also l∞ , are not B-convex. Indeed, take an arbitrary n ∈ N, n ≥ 2, and define x1 = (+1, −1, +1, −1, . . . , +1, −1, 0, 0, 0, . . . ), x2 = (+1, +1, −1, −1, . . . , −1, −1, 0, 0, 0, . . . ), .......................................... xn = (+1, +1, +1, +1, . . . , −1, −1, 0, 0, 0, . . . ),

where in each vector there are 2n non-zero terms, and the non-zero terms in xi consist of alternating blocks of +1’s and −1’s, each block of length 2i−1 . By the very construction, for any sequence ε1 , . . . , εn = ±1, there exists a j, 1 ≤ j ≤ 2n , such that the j’s coordinates of x1 , . . . , xn , are exactly ε1 , . . . , εn , so that kε1 x1 + · · · + εn xn k = n. (d) If X, and Y are infinite-dimensional normed spaces, and if X is a dual space, then the space B(X, Y ) of bounded operators from X into Y is not B-convex. To verify this fact let n ∈ N, n ≥ 2, and m = 2n . Then, for (k) each k, let (βi ) be the periodic sequence with period 2k which starts with 2k−1 terms equal to +1 followed by 2k−1 terms equal to −1. By Dvoretzky’s Theorem (see Chapter 1), for each ǫ > 0 there exists x1 , . . . , xm ∈ SX such that, for all α1 , . . . , αm ∈ R, m m

X

X 1/2

(1 − ǫ) αi xi ≤ αi2 . i=1

i=1

Now, for each j = 1, . . . , n, let us define a linear, continuous operator Tj : l2 7→ X, determining its values on elements of the

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231

Beck convexity standard basis (ek ) ⊂ l2 as follows: (j)

Tj (ei ) = (1 − ǫ)βi xi ,

if

1 ≤ i ≤ m,

and 0 otherwise. Then Tj ∈ B(l2 , X), and kTj k ≤ 1. Again, as in the preceding example (B(X, Y ) has the sup-type norm !), for arbitrary ε1 , . . . , εn = ±1, kε1 T1 + · · · + εn Tn k ≥ n(1 − ǫ), so that B(l2 , X) is not (n, ǫ)-convex. Since n ≥ 2, and ǫ > 0, were arbitrary, we conclude that B(l2 , X) is not B-convex. The adjoint mapping of B(l2 , X) into B(X ∗ , l2 ) is an isometry. Hence, the latter space is not B-convex either, so that, for any n ∈ N, n ≥ 2, and ǫ > 0, one can find T1 , . . . , Tn ∈ SB(X ∗ ,l2 ) such that, for arbitrary ε1 , . . . , εn = ±1, we have the inequality kε1 T1 + · · · + εn Tn k ≥ n(1 − ǫ). By considering the image of points where these 2n linear combinations of Ti ’s nearly achieve their norms, we find a projection P of l2 onto a finite-dimensional subspace such that, for arbitrary ε1 , . . . , εn = ±1, kε1 P T1 + · · · + εn P Tn k ≥ n(1 − ǫ). Utilizing again Dvoretzky’s Theorem we find a linear map S; P (l2 ) 7→ Y , of norm 1, which is so nearly an isometry that, for all ε1 , . . . , εn = ±1, kε1 SP T1 + · · · + εn SP Tn k ≥ n(1 − 3ǫ). Since SP Tj is an element of B(X ∗ , Y ) of norm at most 1, we see that B(X ∗ , Y ) is not (n, 3ǫ)-convex, for any n ≥ 2, and ǫ > 0. This proves our original assertion. In the next step we will indicate the relationship between the B-convexity and the infratype of a Banach space that was studied in Section 6.1. Let us begin by recalling the definition of the numerical constant a∞ k (X) that were essential in investigation of the cotype of a Banach space X: a∞ n (X) = inf

n

n

X

o

εi xi ≤ a max kxi k . a ∈ R : ∀x1 , . . . , xn ∈ X, inf +

εi =±1

i=1

1≤i≤n

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Geometry and Martingales in Banach Spaces

It is easy to see that a∞ n (X)

n

X

n o

+ = inf a ∈ R : ∀x1 , . . . , xn ∈ SX , inf εi xi ≤ a . εi =±1

i=1

The following Proposition provides a characterization of Bconvexity in terms of the constants a∞ n . Proposition 8.1.3. The following properties of a normed space X are equivalent: (i) The space X is B-convex; (ii) The constants a∞ n (X) < n, for some n ≥ 2; (iii) Asymptotically, n−1 a∞ n (X) → 0, as n → ∞; ∞ (iv) The constants an (X) = O(nγ ), for some γ ∈ [1/2, 1). Proof. Implications (iv) =⇒ (iii) =⇒ (ii) =⇒ (i) being obvious it is sufficient to prove that (i) =⇒ (iv). The latter implication is an immediate corollary to the Lemma 6.1.3 and Proposition 6.1.1. QED The above Proposition permits us to prove the following general statement about the relationship between B-convexity and the infratype of a normed space. Theorem 8.1.3. A normed space X is B-convex if, and only if, it is of infratype p for some p ∈ (1, 2]. Proof. If X is B-convex then, by Proposition 8.1.3 (iv), we γ can find a γ ∈ [1/2, 1) such that a∞ n (X) = O(n ). Therefore, by Theorem 6.1.4, −1 pinf = lim [log n/ log a∞ > 1, n (X)] = (1 − γ) n→∞

so that X is of infratype p for some p > 1. Conversely, if X is of infratype p for some p ∈ (1, 2] then Proposition 6.1.1(iii) gives the statement (iv) of the preceding Proposition and, hence, also B-convexity of X. QED In a similar fashion one can relate the concepts of Rademacher and stable type to B-convexity. Again, let us recall the definitions of the constants that played important roles in the study of

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233

Beck convexity Rademacher and stable types in Chapter 6: b2n (X) = inf

n

2 1/2 n X

+ b ∈ R : ∀x1 , . . . , xn ∈ X, E ri x i i=1

n X 1/2 o ≤b kxi k2 . i=1

Proposition 8.1.5. The following properties of a normed space X are equivalent: (i) The space X is B-convex; (ii) b2n (X) < n1/2 , for some n ≥ 2; (iii) n−1/2 b2n (X) → 0, as n → ∞; (iv) b2n (X) = O(nγ ), for some γ ∈ [0, 1/2).

Proof. Implications (iv) =⇒ (iii) =⇒ (ii) are evident. The implication (ii) =⇒ (i) follows from the fact that a∞ n (X) ≤ 1/2 2 n bm (X) in view of Proposition 6.2.2(i), and from Proposition 8.1.4. So, we only need to prove the implication (i) =⇒ (iv). Assume to the contrary that b2n (X) = n1/2 , for all n. Then, for P each 2n ∈ N, and each ǫ > 0, there exist x1 , . . . xn ∈ X, with i kxi k = n,, such that n n

2 X

X 2

ri x i ≤ kxi k , (1 − ǫ)n2 ≤ E i=1

so that

i=1

n n X 2 X 2 1 X 2 kxi k ≤ ǫk 2 . kxi k − kxj k = n kxi k − 2 1≤i,j≤n i=1 i=1

In particular, if i0 , 1 ≤ i0 ≤ n, is such that kxi0 k = sup kxi k,

then

1≤i≤n

n X j=1

kxi0 k − kxj k

2

≤ 2ǫn2 ,

so that 1/2

n

=

n X i=1

kxi k

2

1/2

1/2

≥n

kxi0 k −

n X j=1

kxi0 k − kxj k

2 1/2

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234

Geometry and Martingales in Banach Spaces ≥ n1/2 kxi0 k − n(2ǫ)1/2 ,

and kxi0 k ≤ 1 + (2ǫn)1/2 . Finally, we have that n

2 1/2 X

E ri xi ≥ i=1

1−ǫ n sup kxi k, 1 + (2ǫn)1/2 1≤i≤n

from which we get that

n

2 1/2 X 1−ǫ

ri x i E n≤ sup 1/2 1 + (2ǫn) x1 +···+xn ∈SX i=1

≤

2−n/2

sup x1 +···+xn ∈SX

n

X 1/2

εi xi + n2 (2n − ‘1) inf

εi =±1

i=1

a∞ (X) + n2 (2n − 1) 1/2 n

≤ n. 2n Therefore, for each n ∈ N, we have a∞ n (X) = n, and by Proposition 8.1.4, X is not B-convex, a contradiction. QED =

Finally, the next result shows the connection between spaces that are B-convex and spaces of Rademacher and stable type. Theorem 8.1.4.3 The following properties of a normed space X are equivalent: (i) The space X is B-convex; (ii) X is of Rademacher type p, for some p ∈ (1, 2]; (iii) X is of stable type p, for some p ∈ (1, 2]; (iv) X is of stable type 1. Proof. We prove the implications (i) =⇒ (ii) =⇒ (iii) =⇒ (i). (ii) =⇒ (ii) If X is B-convex then, by Proposition 8.1.5 (iv), there exists a γ ∈ [0, 1/2) such that b2n (X) = O(nγ ). Therefore, by Theorem 6.2.1, the supremum of those p for which X is of Rademacher-type p, log n 1 ≥ ≥ 1, 1/2 2 n→∞ log n bn (X) γ + 1/2

pRtype (X) = lim 3

Due to G. Pisier (1973/73).

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235

Beck convexity

and (ii) is satisfied. (ii) =⇒ (iii) This implication follows directly from Proposition 6.5.1 (ii) which says that if X is of Rademacher-type p then it is of stable-type q for each q < p. (iii) =⇒ (iv) This implication is obvious. (iv) =⇒ (i) Assume that X is not B-convex. We first prove that in this case the space l1 is not finitely representable in X. Indeed, for each n ∈ N, and each ǫ > 0, there exist x1 , . . . , xn ∈ SX such that, for any ε1 , . . . , εn = ±1 kε1 x1 + · · · + εn xn k > n(1 − ǫ). This implies that, for all real sequences (αi ), if i0 is chosen in such a way that |αi0 | = max{|αi | : 1 ≤ i ≤ n} then n n n

X

X X

[sgn(αi0 )|αi0 | − αi ]xi αi xi = [sgn(αi )|αi |]xi −

i=1

i=1

i=1

n n

X

X

sgn(αi )xi − ≥ |αn | |αi0 | − |αi | kxi k i=1

i=1

n X

n X

≥ n|αi0 |(1−ǫ)+

i=1

(|αi |−|αi0 |) =

i=1

(8.1.1)

|αi |−nǫ|αi0 | ≥ (1−nǫ)

n X i=1

|αi |,

so that l1 is finitely representable in X. Now, if X were of stable-type 1 there would exist a constant C > 0 such that, for all n ∈ N, and all sequences x1 , . . . , xn ∈ X, n n

1/2 2 X X

E θi xi ≤C kxi k, i=1

i=1

where (θi ) are i.i.d. stable random variables of exponent 1. Then (8.1.1) would imply that, for any n ∈ N, (αi ) ⊂ R, (xi ) ⊂ SX , n n n

1/2 2 X 1/2 2 X X

E |αi ||θi | ≤ C1 E αi θi xi ≤ C1 C |αi |. i=1

i=1

i=1

However, this inequality may not beP true because, by Schwartz’ Theorem (see Chapter 1), thePseries i αi θi converges absolutely almost surely if, and only if, i |αi |(1 + log(1/|αi |)) < ∞. QED ✐

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8.2

Local structure of B-convex spaces and preservation of B-convexity under standard operations

Proposition 8.2.1.4 A normed linear space X is (k, ǫ)-convex [B-convex] if, and only if, each k-dimensional [separable] subspace of X is (k, ǫ)-convex [B-convex]. Proof. The “only if” part is obvious because every subspace of a (k, ǫ)-convex space X is (k, ǫ)-convex. Conversely, if X is not (k, ǫ)-convex then one can find x1 , . . . , xk ∈ SX such that, for any ε1 , . . . , εk = ±1, kεx1 + · · · + εk xk k > k(1 − ǫ), so that span[x1 , . . . , xk ] ⊂ X is an – at most k-dimensional – subspace of X which is not (k, ǫ)-convex. Similarly, if X is not B-convex, then X is not (k, n−1 )-convex for any k ≥ 2, n ≥ 1, and, as above, one can find a sequence X n,k ⊂ X of finite-dimensional subspaces that are not (k, n−1 )convex. The span of X n.k , k ≥ 2, n ≥ 1, in X is a separable subspace of X which is not B-convex. QED Geometrically speaking, the following important theorem states that a normed space which is not B-convex must contain (k) arbitrarily good approximations of l1 , for any k ∈ N. It is an immediate corollary of Theorem 8.1.6, and the nontrivial Corollary 6.5.2(i). However, we provide here an independent elementary proof. Theorem 8.2.1.5 The following properties of a normed space X are equivalent: (i) The space X is B-convex; (ii) The space l1 is not crudely finitely representable in X; (iii) The space l1 is not finitely representable in X. 4 5

Due to D.P. Giesy (1966). Due to D.P. Giesy and R.C. James (1973).

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Beck convexity

Proof. (i) =⇒ (ii) Assume to the contrary that l1 is crudely finitely representable in X. Then, in particular, there would exist a constant λ ∈ (0, 1] such that, for each k ∈ N, there exist x1 , . . . , xk ∈ SX such that, for any sequence ε1 , . . . , εk = ±1, kλ ≤ kε1 x1 + · · · + εk xk k. This implies that k −1 a∞ k (X) ≥ λ > 0, for all k ≥ 1. Hence, by Proposition 8.1.5(iii), the space X is not B-convex. A contradiction. (ii) =⇒ (iii) This implication is obvious. (iii) =⇒ (i) Proof of this implication is contained in the proof of the implication (iv) =⇒ (i) of Theorem 8.1.6. QED Corollary 8.2.1. If a normed space X is B-convex and Y is isomorphic to X in the sense of normed spaces then Y is B-convex as well. Now we are turning to the problem of preservation of Bconvexity under standard operations on normed spaces such as selection of a subspace, the closure, completion, mapping via linear bounded operators, and taking duals and preduals. The following Proposition follows directly from the definitions so we omit the proofs. Proposition 8.2.2. (i) If a normed space Y ⊂ X, and X is (k, ǫ)-convex, then Y is also (k, ǫ)-convex; (ii) If a normed space Y ⊂ X, and Y is (k, ǫ)-convex, then the closure of Y in X is also (k, ǫ)-convex; (iii) The completion of a B-convex normed space is B-convex. The following result shows that B-convexity is preserved under some linear mappings. Theorem 8.2.2.6 If X is a B-convex normed space and T : X 7→ Y is a continuous, linear and open mapping into a normed space Y , then the image T X is B-convex.

6

Proof. Since T is open, one can find a δ > 0 such that y ∈ T X : kyk < δ ⊂ T (SX ). Due to D.P. Giesy (1966).

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Geometry and Martingales in Banach Spaces

Now, let y 1 , . . . , y k ∈ ST X . Because δy i ∈ T SX , there exist xi ∈ SX such that T xi = δy i . In view of B-convexity of X, and Proposition 8.1.5, one can find a k ∈ N, and ε1 , . . . , εk = ±1, such that kε1 x1 + · · · + εk xk k ≤ kδkT k/2. Therefore, k k k

1 X

1

X

X

1

εi y i = εi xi ≤ , εi T xi ≤ kT k

δ i=1 δ 2 i=1 i=1

so that T X is (k, 1/2)-convex. QED

Remark 8.2.1. In the above Theorem the openness of the operator T is essential. Indeed, define α2 αn T : l1 ∋ (α1 , α2 , . . . ) 7→ α1 , , . . . , , . . . ∈ l1 . 2 n The operator T is linear, continuous and T l2 is dense in l1 . But l2 is B-convex, and T l2 is not . If it were B-convex then its completion, l1 , would be B-convex and it is not. From Proposition 6.5.4, Theorem 6.5.6, and Theorem 8.1.6(iv) we immediately get a result concerning the factoring operation. Theorem 8.2.3.7 Let X be a normed space and Y be its closed subspace. Then X is B-convex if, and only if, both Y and X/Y are B-convex. Corollary 8.2.2. Let X, and Y , be normed spaces, and T : X 7→ Y be linear, continuous and open map. The X is B-convex if, and only if, Ker T , and Im T are B-convex. Proof. If X is B-convex then Ker T is its linear subspace and also B-convex by Proposition 8.2.2. At the same time Im T is B-convex by Theorem 8.2.2. Conversely, if Ker T , and Im T , are both B-convex then X is also B-convex by the above Theorem because X/Ker T is also B-convex as an image of Im T = T X by a continuous linear open map U(T (x)) = x + Ker T . 7

Theorems 8.2.3 and 8.2.4 are due to D.P. Giesy (1966).

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Beck convexity

Theorem 8.2.4. A normed space X is B-convex if, and only if, its dual X ∗ is B-convex. Proof. Assume that X is not B-convex. Then, by Theorem 8.2.2, the space l1 is crudely finitely representable in X, i.e., there exists a λ > 1, and subspaces X n ⊂ X, such that, for every (n) n ∈ N, the distance d(X n , l1 ) < λ. By duality, for each n ∈ N, we have the distance d(X ∗n , l(n) ∞ ) < λ. ∗ On the other hand, X n ∼ X ∗ /X 0n , where En0 = {X ∗ ∈ X ∗ : ∗ x x = 0, ∀x ∈ X n }. In particular, for every n ∈ N, n) d X ∗ /X 02n , l(2 < λ. ∞ n

(n)

) However, l1 may be embedded isometrically into l(2 ∞ so that, for (n) each n ∈ N, each sequence (¯ xi )1≤i≤n in X 2n , and each (αi ) ∈ Rn , n X i=1

n

X

(n) ¯ i , αi x |αi | ≤ i=1

and

(n)

xi k < λ. sup k¯

1≤i≤n

(n)

(n)

Now, let (xi )1≤i≤n be in X n , and such that xi represents (n) (n) ¯ i mod X 02n with kxi k < λ. Then, for every (αi ) ∈ Rn , x n X i=1

n n

X X

(n) |αi |, αi x i ≤ λ |αi | ≤ i=1

i=1

and this means that l1 is crudely finitely representable in X ∗ , so that, again by Theorem 8.2.2, X ∗ is not B-convex. Conversely, if X ∗ is not B-convex then l1 is crudely finitely (n) representable in X ∗ . However, l1 ’s are necessarily duals of quotients of X which are close to l(n) ∞ ’s, so that one can finish the proof proceeding as above. QED The (k, ǫ)-convexity is also inherited by the biduals, and the result also offers an alternative proof of the “only if” part of the above Theorem 8.2.4. Theorem 8.2.5.8 A Banach space X is (k, ǫ)-convex if, and only if, the bidual space X ∗∗ is (k, ǫ)-convex. 8

Due to D.P. Giesy (1966).

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Proof. Because X embeds isometrically into X ∗∗ , we immediately get that the (k, ǫ)-convexity of X ∗∗ implies (k, ǫ)-convexity of X. Conversely, assume that X is (k, ǫ)-convex and choose ∗ ∗∗ ∗ x1 , . . . , x∗∗ k ∈ SX ∗∗ , and ε1 , . . . , εk = ±1. Choose xε1 ,...,εk ∈ X , with kx∗ε1 ,...,εk k = 1 in such a way that k k

X

X

∗∗ ∗∗ ∗ ε x − δ < ε x (x )

i i i i ε1 ,...,εk . i=1

i=1

For each i consider the X ∗ neighborhood N of x∗∗ i determined by δ, and the 2k functionals x∗ε1 ,...,εk , ε, . . . , εk = ±1. Since the canonical mapping U : X 7→ X ∗∗ maps SX into a dense set of SX ∗∗ , one can find an xi ∈ SX such that Uxi ∈ N. This means that ∗∗ ∗ ∗ ∗ (x )−x (x ) < δ, xi (xε1 ,...,εk )−(Uxi )(x∗ε1 ,...,εk ) = x∗∗ i i ε1 ,...,εk ε1 ,...,εk for each (ε1 , . . . , εk ). In view of the (k, ǫ)-convexity of X there exist δ1 , . . . , δk = ±1 such that k

X

δi xi < k(1 − ǫ + δ).

i=1

Then

k k

X

X

∗∗ ∗ ∗∗ δi xi (xδ1 ,...,δk ) + kδ δi xi ≤

i=1

i=1

k k X X ∗ ∗ ∗ δ x (x ) δi x∗∗ x − x x + ≤ i δ1 ,...,δk i + kδ i δ1 ,...,δk δ1 ,...,δk i i=1

i=1

≤

k X i=1

k X ∗ ∗∗ ∗ ∗ |δi | xi xδ1 ,...,δk − xδ1 ,...,δk xi + xδ1 ,...,δk δi xi + kδ i=1

k

X

δi xi + kδ ≤ k(1 − ǫ + δ), < kδ + kx∗δ, ...,δK k · i=1

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241

which implies the (k, ǫ)-convexity of X ∗∗ in view of arbitrariness of δ > 0, and the above sharp inequality. QED It is also elementary to check that B-convexity is preserved under finite direct sums and span operations. Proposition 8.2.3. (i) Let X 1 , . . . , X n be normed spaces and X = X 1 ⊕ · · · ⊕ X n be the direct sum under component-wise (n) arithmetic and (say) l1 -norm. Then X is B-convex if, and only if, all of X 1 , . . . , X n are B-convex. (ii) Let X be a Banach space and X 1 , . . . , X n be its linear subspaces such that X = span(X 1 , . . . , X n ). Then X is B-convex if, and only if, all of X 1 , . . . , X n are B-convex. Proof. (i) Because X i may be identified with a subspace of X, Proposition 8.2.2 implies that if X is B-convex then, for each i = 1, . . . , n, the space X i is B-convex. The converse may be proven by induction on n. For n = 1 the result is trivial. Suppose it is true for n − 1, and that X 1 , . . . , X n are B-convex. The projection T : X 7→ X n is continuous, linear, and open, with both Im T = X n , and the Ker T = X 1 ⊕ · · · ⊕ X n−1 ⊕ 0 being B-convex. Thus, by Corollary 8.2.2, X is B-convex. (ii) Again, by Proposition 8.2.2, if X is B-convex then for each i = 1, . . . , n the space X i is B-convex. To prove the converse, by Proposition 8.2.2, we may assume that X i are closed and Bconvex. Then, by (i), the space Y = X = X 1 ⊕ · · · ⊕ X n is B-convex, and of course complete, and the linear operator, T : Y ∋ (x1 , . . . , xn ) 7→ x1 + · · · + xn ∈ X, is onto, and continuous, because kT (x1 , . . . , xn )kX = kx1 + · · · + xn k ≤ kx1 k + · · · + xn k = k(x1 , . . . , xn )kY . Therefore, T is open by the Banach Open Mapping Theorem, and an application of Theorem 8.2.2 completes the proof. QED Finally, we shall verify that some spaces of functions with values in B-convex spaces are also B-convex.

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Theorem 8.2.6.9 Let 1 < p < ∞, and let (T, Σ, µ) be a measure space. Then, a Banach space X is B-convex if, and only if, the space Lp (T, Σ, µ; X) is B-convex. Proof. If Lp (X) is B-convex then X is also B-convex as a subspace of Lp (X). Conversely, if X is B-convex then, by Theorem 8.1.4, X is of stable-type 1 and, by Theorem 6.5.3, Lp (X) is also of stable type 1, i.e. B-convex. QED

8.3

Banach lattices and reflexivity of B-convex spaces

We begin by characterizing subspaces of the space of integrable functions which are B-convex. Theorem 8.3.1.10 Let (T, Σ, µ) be a measure space, and let X be a closed subspace of L1 (T, Σ, µ). Then the following properties of X are equivalent: (i) The space X is B-convex; (ii) X is reflexive; (iii) X does not contain an isomorphic copy of l1 ; (iv) X does not contain an isomorphic copy of l1 complemented in L1 . Proof. Because X is B-convex if, and only if, X is of stabletype 1 (see Theorem 8.1.4), Theorem 6.5.2 gives the equivalence of (i), (iii), and (iv). The implication (ii) =⇒ (iii) is evident. Therefore it is sufficient to show that if X is a close non-reflexive subspace of L1 then it contains a basic sequence which is equivalent to the standard basis of l1 . In the course of the proof we shall have need of the following Lemma: Lemma 8.3.1. Let en , n ∈ N be the standard basis in l1 , and let kj , j ∈ N, and nk , k ∈ N, be two sequences of increasing 9

Due to G. Pisier (1973/74). An alternative proof can be found in H.P. Rosenthal (1976). 10 Due to G. Pisier (1972/73), Exp. XVIII and XIX.

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243

Beck convexity integers. Denote kj+1

zj =

X

(j)

ai

i=kj +1

en2i − en2i+1 ,

P (j) (j) with ai ≥ 0, for kj < i ≤ kj+1, and i ai = 1. Then, z j , j ∈ N, is a basic sequence in l1 which is equivalent to en , n ∈ N, and for P ∗ which the coordinate functional z j ( i αi z i ) = αj , has the norm 1/2, for each j. Proof. The proof of the Lemma is obvious in view of the equality

m m

X X

|tj |. tj z j = 2

l1

j=1

j=1

Proof of Theorem 8.3.1, continued. Now, assume that X is non-reflexive. By the Dunford-Pettis compactness criterion, the unit sphere SX is not equiintegrable in L1 , so that Z |x|dµ = δ > 0. lim sup a→∞ x∈SX

|x|>a

Hence, one can find an , n ∈ N, an ↑ ∞, such that Z δ δ δ− |x|dµ < δ + , < sup n ∈ N, 2n x∈SX |x|>an 2n

(8.3.1)

from which it follows that there exists a sequence (xn ) ⊂ SX , such that, for each n ∈ N, Z δ δ δ− < |xn |dµ < δ + , n ∈ N, (8.3.2) 2n 2n |xn |>an Let us define x′n = xn I[|xn | > an ],

and

x′′n = xn − x′n .

Then, for each ǫ > 0, µ |x′n | ≥ ǫkx′n k ≤ µ{|x′n | ≥ 0} ≤ µ{|xn | > an } ≤ 1/an , ✐

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and, because 1/an → 0, as n → ∞, for each ǫ > 0, {x′n : n ∈ N} ⊂ Mpǫ := x ∈ Lp : µ{|x| ≥ ǫkxkLp } ≥ ǫ .

Therefore, by Kadec-Pelczynski’s Theorem11 , one can find a basic subsequence (x′ni ) ⊂ (x′n ) which is equivalent to the standard basis of l1 because, in view of (2), we have δ/2 ≤ kxni k ≤ 3δ/2. On the other hand, the sequence (x′′n ) is equi-integrable because Z Z sup |x|dµ = sup |x′′p |dµ p>n |x′′ x∈(x′′i ) |x|>a p |>an = sup p>n

Z

|xp |>an

|xp |dµ −

Z

|xp |>ap

|xp |dµ

δ δ δ ≤ → 0, ≤ sup δ − − δ− 2n 2p n p>n as n → ∞. Hence, one can find a subsequence (x′′nk ) ⊂ (x′′ni ) which converges weakly so that xn2k − xn2k+1 → 0, weakly. Zero is a strong accumulation point of the convex envelope of the latter sequence. So, there exists an increasing sequence (kj ), (j) and ai > 0, with kj+1 X (j) ai = 1, i=kj +1

such that, if kj+1

zj =

X

i=kj +1

(j)

ai

xn2i − xn2i+1

then lim kz j − z ′j k = lim kz ′′j k = 0,

j→∞ 11

j→∞

See Studia Mathematica 21(1962), 161-176.

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245

Beck convexity where kj+1

z ′j

=

X

(j)

ai

i=kj +1

x′n2i − x′n2i+1 ,

and z ′′j = z j − z ′j .

Now, by the above Lemma, (z ′j ) is a basic sequence equivalent to the standard basis of l1 , and one can choose (z j ) such that ∞ X j=1

kz ∗j kkz ′′j k =

∞ X j=1

kz ∗j kkz j − z ′j k < 1.

By Bessaga-Pelczynski’s Theorem12 , (z j ) is equivalent as a basic sequence to (z ′j ). Thus we have obtained a basic sequence in X which is equivalent to the standard basis of l1 . QED In the next Theorem we turn to a characterization of B-convex Banach lattices and Banach spaces with unconditional bases. Theorem 8.3.2. Let X be either a Banach lattice, or a Banach space with unconditional basis. Then the following properties of X are equivalent: (i) The space X is B-convex; (ii) X is reflexive; (iii) X is superreflexive; (iv) X does not have subspaces isomorphic to either c0 , or to l1 . Proof. It is sufficient to prove the above theorem for Banach lattices because all isomorphically invariant properties of a Banach lattice are shared by Banach spaces with an unconditional basis. This follows from the fact that for every such space one can find an equivalent norm which makes the space isometrically isomorphic to a Banach lattice13 . Now, the implication (i) =⇒ (iv) is trivial because neither c0 , not l1 are B-convex, and because B-convexity is preserved by subspaces. The implications (iv) =⇒ (ii), and (iv) =⇒ (iii) follow from James’ Theorem (see Chapter 1). If X is superreflexive, 12 13

See Studia Mathematica 17 (1958), 151-164. See, e.g., M.M. Day (1973), p. 73, Theorem 1.

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then, by Enflo’s Theorem (see Chapter 1), X admits an equivalent uniformly convex norm. Therefore, by Theorem 8.1.2, X is B-convex. QED Not all B-convex spaces are reflexive. In particular, there exists a (3, ǫ)-convex space which is not reflexive whenever ǫ < 1 − (3−1/2 + 2−1 (2/3)1/2 )14 . However, we have a positive result for (2, ǫ)-convex Banach spaces. Theorem 8.3.3.15 If X is a (2, ǫ)-convex Banach space for some ǫ > 0, then X is reflexive (and even superreflexive). Proof. Suppose that X is non-reflexive. For each sequence (x∗j ) ⊂ SX ∗ , and each p1 , . . . , p2n ∈ N, define n o 3 S(p1 , . . . , p2n , (x∗k )) := x : ∀k, i ∈ N, ≤ (−1)i−1 x∗k (x) ≤ 1 , 4 if p2i−1 ≤ k ≤ p2i , and 1 ≤ i ≤ n. Let

K(n, (x∗j )) := lim inf . . . lim inf inf kzk : z ∈ S(p1 , . . . , p2n , (x∗k )) . . . , p1 →∞

and

p2n →∞

Kn := inf K(n, (x∗j )); kx∗j k = 1, ∀j .

To show that Kn < ∞ let us suppose that {p1 , . . . , p2n } is an increasing sequence of integers. It is known16 that for each r < 1 there exist (z i ) ⊂ BX , and (x∗j ) ⊂ SX ∗ , such that x∗n (z i ) > r if n ≤ i, and x∗n (z i ) = 0, if n > i. Let w=

n X j=1

(−1)

j−1

−z p2j−1 −1 + z p2j .

Then x∗k (w)

=

n X j=1

(−1)

j−1

−x∗k (z p2j−1 −1 )+x∗k (z p2j )

=

n X

(−1)j−1 Akj ,

j=1

14

See R.C. James (1974). Other versions can be found in R.C. James and J. Lindenstrauss (1975), and J. Farahat (1974/75). 15 Due to R.C. James (1974). For another proof see A. Brunel and L. Sucheston (1974). 16 See R. C. James, Studia Mathematica 23(1964), 205-216.

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247

Beck convexity where Akj = 0, if k > p2j , Akj = x∗k (z p2j ),

if

p2j−1 ≤ k ≤ p2j ,

and |Akj | < 1 − r, if k < p2j−1 . Therefore, if p2i−1 ≤ k ≤ p2i , then (−1)i−1 x∗k (w) = x∗k (z p2i ) + Ak ,

with |Ak | < n(1 − r).

Therefore, to have w ∈ S(p1 , . . . , p2n , (x∗i )) it is sufficient to have x∗k (z p2i ) > 7/8, and n(1 − r) < 1/8. To accomplish this we can choose r as the larger of the 7/8, and 1 − 1/(8n). Since kwk ≤ 2n, it follows that K(n, (x∗i )) ≤ 2n, and, hence, Kn ≤ 2n. Now, we shall show that X is not (2, ǫ)-convex for any ǫ > 0. Since Kn is positive and monotonically increasing, for any number r satisfying 1 > r > 1 − δ, there is a δ > 0, and an N, such that (Kn − δ)/(Kn + 2ǫ) > r > 1 − ǫ,

if

n > N.

(8.3.3)

Since Kn − Kn−1 → 0 if lim Kn is finite, and the conditions Kn ≤ 2n, and lim Kn = +∞, imply that lim inf Kn /Kn−1 = 1, it follows from (8.3.3) that there is an m > N such that (Km−1 − δ)/(Km + 2ǫ) > 1 − ǫ.

(8.3.4)

In view of the definition of the sequence K −N, there is a sequence (x∗i ) ⊂ SX ∗ such that K(m, (x∗i )) < Km + ǫ.

(8.3.5)

Also, by choosing p1 , . . . , p2m , and q1 , . . . , q2m successively in the followiing order p1 , q1 , p2 , p3 , q2 , q3 , p4 , p5 , . . . , q2m−2 , q2m−1 , p2m , q2m , and in such a way that the above sequence is increasing, we can get increasing sequences (p1 , . . . , p2m ), and (q1 , . . . , q2m ), with the following three properties: (a) S(p1 , . . . , p2m ; (x∗i )), and S(q1 , . . . , q2m ; (x∗i )), have elements u, and v, respectively, such that kuk ≤ K(m, (x∗i )) + δ,

and kvk ≤ K(m, (x∗i )) + δ;

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Geometry and Martingales in Banach Spaces (b) K(m, (x∗i )) − δ ≤ kzk, if z ∈ S(q1 , p2 , q3 , p4 , q5 , p6 , . . . , q2m−1 , p2m ; (x∗i )); (c) K(m − 1, (x∗i )) − δ ≤ kzk, if z ∈ S(p3 , q2 , p5 , q4 , p7 , q6 . . . , p2m−1 , q2m−2 ; (x∗i )). Then (u + v)/2 ∈ S(q1 , p2 , q3 , p4 , q5 , p6 , . . . , q2m−1 , p2m ; (x∗i )),

and from (b) we have ku + vk/2 > K(m, (x∗i )) − δ.

(8.3.6)

Also, (u − v)//2 ∈ S(p3 , q2 , p5 , q4 , p7 , q6 . . . , p2m−1 , q2m−2 ; (x∗i )), and ku − vk/2 > K(m − 1, (x∗i ) − δ.

(8.3.7)

Finally, let x = u/(Km + 2δ), and y = v/(Km + 2δ). Then, from (a) and (8.3.5), we have kxk, kyk ≤ 1. Since K(m, (x∗i )) ≥ Km , it follows from (8.3.6) and (8.3.3) (and (8.3.7), and (8.3.4), respectively) that kx + y|/2 > (Km − δ)/(Km + 2δ) > 1 − ǫ, and kx − y|/2 > (Km−1 − δ)/(Km + 2δ) > 1 − ǫ, so that X is not (2, ǫ)-convex, for any ǫ > 0. QED Corollary 8.3.1. If X is a (3, ǫ)-convex Banach space for some ǫ > 1/3, then it is reflexive. Proof. The result is an immediate consequence of the above Theorem, Proposition 8.1.1, and Theorem 8.1.1. QED On the other hand, there are reflexive (even locally uniformly convex) Banach spaces that are not B-convex.

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i) Example 8.3.1. Let ni ↑ ∞, pi ↓ 1, and let X i = l(n p1 . Then X i are uniformly convex, and hence locally uniformly convex. The space X i is finite-dimensional and thus reflexive. Define X = (X i ⊕ X 2 ⊕ . . . )l2 . Note that the l2 sum of reflexive spaces is reflexive, and the l2 sum of locally uniformly convex spaces is such as well17 . Thus X is reflexive and locally uniformly convex. Pick k ≥ 2, and ǫ > 0. As p ↓ 1, k (1/p−1) ↑ 1. Obviously, X i is embedded isometrically in X. In X i take xj = (0, . . . , 0, 1, 0, . . . , 0), with 1 in the j-th position. Then, let y j = (0, . . . , 0, xj , 0, . . . , 0) ∈ X. Then ky j k = kxj k = 1, and for εj = ±1,

k k k

X

X

X 1/pi

pi = k 1/pi > k(1 − ǫ), = |ε | ε x = ε y

j j j j j j=1

j=1

j=1

so that X is not (k, ǫ)-convex for any k ∈ N, ǫ > 0, and, by definition, it is not B-convex. Remark 8.3.1. Other characterizations of Banach spaces which are not B-convex are also possible18 : (a) The Banach space X is not B-convex if, and only if, there exists an ǫ ∈ (0, 1), such that for each k ≥ 2 there exist x∗1 , . . . , x∗k ∈ BX ∗ such that, for each ε1 , . . . , εk = ±1 there exists an x ∈ BX such that for each j = 1, . . . , k, x∗j (εj x) > ǫ. (b) The Banach space X is not B-convex if, and only if, there exists an ǫ ∈ (0, 1), such that for each k ≥ 2 there exist x1 , . . . , xk ∈ BX such that, for each ε1 , . . . , εk = ±1 there exists an x∗ ∈ BX ∗ such that for each j = 1, . . . , k, x∗ (εj xj ) > ǫ. (c) If X is not B-convex then it contains a subspace with basis that is not B-convex either.

8.4

Classical weak and strong laws of large numbers in B-convex spaces

In view of Theorems 8.1.4 and 6.9.1, the validity of the classical Kolmogorov’s Weak Law of Large Numbers characterizes B-convex 17 18

See M.M. Day (1973), p. 31. See D.R. Brown (1974).

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Banach spaces. Theorem 8.4.1.19 A Banach space is B-convex if, and only if, for each sequence (Xn ) of symmetric, independent, and identically distributed random vectors in X, X1 + · · · + Xn = 0, n→∞ n lim

in probability,

if, and only if, lim nP(kX1 k > n) = 0.

N →∞

A study of the following Strong Law of Large Numbers initiated investigation of B-convex spaces. Theorem 8.4.2.20 The following properties of a Banach space X are equivalent: (i) The space X is B-convex; (ii) For any sequence (Xn ) of independent, zero-mean random vectors in X with supn kXn k2 < ∞, lim

n→∞

X1 + · · · + Xn = 0, n

with probability 1,

(iii) For any bounded sequence (xn ) ⊂ X, r1 x 1 + · · · + rn x n = 0, n→∞ n lim

in

L1 (Ω, F , P; X).

Proof. (i) =⇒ (ii) By Theorem 8.1.4, the space X is of Rademacher-type p, for some p ∈ (1, 2]. The boundedness of (Xn ) in its boundedness in Lp (X). Therefore, the series P L2 (X) implies p p EkX k /n converges, and by Theorem 6.8.1, we obtain (ii). n n (ii) =⇒ (iii) This implication is obvious in view of the Lebesgue Dominated Convergence Theorem. 19 20

Due to M.B. Marcus and W.A. Woyczy´ nski (1977), and (1979). Due to A. Beck (1962), and (1963).

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Beck convexity

(iii) =⇒ (i) Suppose that X is not B-convex. Then, for each n ∈ N, there exist xn1 , . . . , xnnn ∈ BX such that −n

n

nn

X

E ri xni ≥ 1/2. i=1

Now, let us construct a sequence (xn ) ⊂ BX , as follows: xj = xni , when j = kn + i, where i ∈ 1, 2, . . . , (n + 1)n+1 , and kn = 1 + 22 + 33 + · · · + nn .

Then,

kn

1

X 1

ri xi ≥ nn − kn−1 ≥ nn − (n − 1)n , E 2 2 i=1

from which it follows that

kn kn

1 1

X

X ri x i lim sup E ri xi ≥ lim sup n E kn 2n n n i=1 i=1

1h1 1 n i ≥ lim − 1− > 0. n 2 2 n A contradiction. QED

It is possible to prove the equivalence (i) ⇐⇒ (ii) using only basic definitions21 Here we include only the proof of the implication (ii) =⇒ (i) because it gives an additional insight into the structure of B-convex spaces. Alternative proof of (ii) =⇒ (i). Assume, a contrario, that X is not B-convex, i.e., for each k ∈ N, and each ǫ > 0, there exist x1 , . . . xk ∈ SX such that, for all ε1 , . . . , εk = ±1, kε1 x1 + · · · + εk xk k > k(1 − ǫ). 21

Due to A. Beck (1963).

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Pick arbitrary sequences (βi ), and (δi ), of real numbers converging to zero and proceed as follows: Choose k1 ∈ N with k1 > (1 − (1() (1) δ1 )/δ1 , and xi , . . . , xk1 ∈ SX such that, for each ε1 , . . . , εk1 = ±1, (1) (1) kε1 x1 + · · · + εk1 xk1 k ≥ k1 (1 − β1 ). Then, for each n ∈ N, set mn = k1 + · · · + kn−1 , and choose (n) (n) kn > mn (1 − δn )/δn , and x1 , . . . xkn ∈ SX such that, for every ε1 , . . . , εkn = ±1, (n)

kε1 xi

(n)

+ · · · + εkn xkn k ≥ kn (1 − βn ).

This gives us the inequalities, kn > 1 − δn , mn+1

and

mn < δn . mn+1

For any integer i, we have the bounds mj < i ≤ mj+1 , for some (j) value of j, i.e., i = mj + r, where 1 ≤ r ≤ kj . Define y i = xr . This gives us a sequence (y i ) ⊂ X which is uniformly bounded. On the other hand, mj+1 j +1 j +1

1 mX

1

1 mX

X

ri xi ≥ ri (ω)xi − ri (ω)xi

mj+1 i=1 mj+1 i=m +1 mj i=1 j

≥

mj 1 (j)

ε1 x(j) − 1 + · · · + ε kj mj+1 mj+1

≥ kj (1 − βj )/mj+1 − mj /mj+1 > (1 − δj )(1 − βj ) − δj . Thus, n

X

Xi (ω) n−1 = 1, lim sup n

i=1

∀ω ∈ Ω.

A contradiction. QED

The following result demonstrates that there is a uniformity in the Law of Large Numbers in B-convex Banach spaces.

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Beck convexity

Theorem 8.4.3.22 If X is a B-convex Banach space, then there exists a sequence (pn ), 0 < pn ↑ 1, such that, for any ǫ ∈ (0, 1), and any sequence (xn ) ⊂ BX ,

r x + · · · + r x

1 1 m m P sup ∀n ∈ N.

≤ ǫ > pn , m m>n Proof. We proceed by contradiction and will show that if this result is false then the Strong Law of Large Numbers of Theorem 8.4.2 fails in X as well. Suppose that there exist ǫ, η > 0 such that, for each n ∈ N, (n) there exist independent random vectors (Xi ), i, n ∈ N such that

X (n) + · · · + X (n) m

P sup 1

≤ ǫ < 1 − η. m m>n

We will construct inductively a pair of sequences of integers, (ni ), and (mj ), as follows: Choose n1 = 0. Since

(n ) (n1 ) P sup X1 1 + · · · + Xm m ≤ ǫ < 1 − η,

m>n

we can find m1 > n1 for which

(n1 ) (n1 ) P sup X1 + · · · + Xm m ≤ ǫ < 1 − η, n1 nj so that P

m

X

(n ) sup Xi j m ≤ ǫ < 1 − η.

nj 0. n n i=1

This will contradict (8.4.1) Since lim supn bn > 2β, for each n ∈ N, there exists m(n) ≥ n, the (k, ǫ)-convex spaces X n , probability spaces (Ωn , Fn , Pn ), (n) (n) and zero-mean, independent random vectors Y1 , . . . , Ym(n) on Ωn , (n)

with EkYi kq ≤ 1, such that

1 m(n) X (n)

Y ≥ 2β.

m(n) i=1 i

Q Let X = (X 1 ⊕ X 2 ⊕ . . . )l2 , and (Ω, F , P) = (Ωn , Fn , Pn ). Then, by Theorem 8.2.6, X is B-convex. For all n ∈ N, and (n) 1 ≤ i ≤ m(n), define random vectors Xi : Ω 7→ X, by the formula (n) Xi (ω1 , ω2 , . . . ) = (x1 , x2 , . . . ), where xn = Yi (ωn ), and xi = 0, for j 6= n. Then, for all n ∈ N, (n) (n) and 1 ≤ i ≤ m(n), we have E(Xi ) = 0, and EkXi kq ≤ 1, and

1 m(n)

1 m(n) X (n) X (n)

p

p E Xi = E Yi ≥ (2β)p . m(n) i=1 m(n) i=1 ✐

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Beck convexity (n)

The random vectors {Xi : 1 ≤ i ≤ m(n), n ≥ 1} form a set of independent X-valued random vectors. (n ) Let n1 = 1, p1 = m(n1 ), and Xi = Xi i , for 1 ≤ i ≤ m(n1 ). Then p1

1 X

p

E Xi ≥ (2β)p > β p . p1 i=1 Suppose we have chosen p1 < p2 < · · · < ps , and a sequence Xi , 1 ≤ i ≤ ps , such that

Let

pj

p

1 X

Xi ≥ β p , E pj i=1

1 ≤ j ≤ s.

ps

1 X

p

γs = E Xi ps i=1

Choose ns=1 so large that if we let ms+1 = m(ns=1 ) then 2ms+1 − ps (γs /β) (ms+1 + ps ) ≥ 1. (n

(8.4.2)

)

For 1 ≤ i ≤ m(ns+1 ) define Xps +i = Xi s+1 . Let ps+1 = m(ns+1 )+ ps . Then, from (8.4.2), we have 2ms+1 − ps (γs /β) ps+1 ≥ 1. (8.4.3) Finally, in view of (8.4.3), we obtain the inequality,

ps+1 s+1

p 1/p

p 1/p 1 pX 1

X

E Xi ≥ E Xi ps+1 i=1 ps+1 i=p +1 s

ps

1

X p 1/p − E Xi ps+1 i=1

ms+1 ps

p 1/p X ms+1 1

X (ns+1 ) p 1/p ps 1

(ns+1 ) E Xi − E Xi =

ps+1 ms+1 ps+1 ps i=1 i=1 ms+1 ps ≥ (2β) − γs = β 2ms+1 − ps (γs /β) /ps+1 ≥ β. ps+1 ps+1

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Hence, by induction, we have chosen from the independent family (n) Xi : 1 ≤ i ≤ m(n), n ≥ 1} of random vectors in X, with zeromeans, and q-moments bounded by 1, a sequence (Xn ) such that lim supn n−1 (EkX1 + · · · + Xn kp )1/p ≥ β. A contradiction. Hence limn→∞ bn = 0. To complete the proof for an arbitrary bound M it is sufficient to use the homegeneity of the q-norm. QED

8.5

Laws of large numbers for weighted sums and not necessarily independent summands

We begin by proving a Law of Large Numbers for nonstandard weighted averages. Theorem 8.5.1.24 Let 1 ≤ p < 2, p < q, and let X be a (k, ǫ)-convex Banach space with k(1 − ǫ) < k 1/p . If (Xn ) is a sequence of independent random vectors in X, bounded in Lq (X), then n

1

X q 1/q lim 1/p E = 0, (8.5.1) Xk n→∞ n k=1 and

lim

n→∞

n 1 X

n1/p

Xi = 0,

a.s

(8.5.2)

i=1

Proof. By our assumption, for each x1 , . . . , , xk ∈ BX , k

n X o

inf εi xi : ε = ±1 ≤ k(1 − ǫ) < k 1/p , i=1

so that the embedding l1 7→ lp is not finitely representable in X. Therefore X is of stable-type p by Theorem 6.5.1, so that Lq (X) 24

Due to G. Pisier and B. Maurey (1976).

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259

Beck convexity

is of stable-type p by Theorem 6.5.3. Without loss of generality we can assume that X i ’s are symmetric. Then, by Theorem 6.9.3, n n

q 1/q 1 1

X q 1/q

X

lim E Xi = lim 1/p E ri X i = 0, n→∞ n1/p n→∞ n k=1 k=1

(ri )-almost surely. This proves (8.5.1). Statement (8.5.2) can be proven in a similar fashion. QED

Below we discuss a Strong Law of Large Numbers for general weighted sums of independent random vectors in B-convex spaces. However, its validity is restricted by rather stringent condition on weights. Theorem 8.5.2.25 Let X be a B-convex Banach space and let (ank ) ⊂ R, n, k ∈ N, be an array such that lim ank = 0,

n→∞ ∞ X k=1

ank ≥ 0,

and

|ank | ≤ 1,

lim

n hX

n→∞

k=1

∀k ∈ N,

(8.5.3)

∀n ∈ N,

(8.5.4)

i ank − n max ank = 0, 1≤k≤n

max ank = O(n−α ),

1≤k≤n

(8.5.5) (8.5.6)

where 0 < 1/α < p − 1, for some p > 1. Then, if (Xn ) is a sequence of zero-mean, independent random vectors in X, with supn EkXn kp = M < ∞, then, almost surely, lim

n→∞

n X

ank Xk = 0.

k=1

Proof. Step 1. Suppose additionally that Xn ’s are symmetric and bounded by 1. Denote, βn = min1≤k≤n ank . Then, we have n

X

0 ≤ ess sup limn ank Xk k=1

25

Due to W.J. Padgett and R.L. Taylor (1975).

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260

Geometry and Martingales in Banach Spaces n n

X

X

≤ ess sup limn (ank − βn )Xk + βn Xk k=1

≤ ess sup limn

k=1

n X

n

1 X

(ank − βn )kXk k + ess sup limn Xk , n k=1 k=1

because the inequality, nβn ≤ an1 + · · · + ann ≤ 1, implies that βn ≤ 1/n. On the other hand, the Strong Law of Large Numbers (Theorem 8.4.1) implies that the last term is zero. Since kXk k ≤ 1, by (8.5.5) we also see that the next to the last term vanishes as well. Therefore, n

X

ank Xk = 0. ess sup limn k=1

Step 2. Now, we drop the assumption of uniform boundedness of (Xn ) while preserving the symmetry assumption. Suppose that EkXn kp ≤ M = 1. For a positive integer q, define Xn′ = Xn , Xn′ = 0,

and Xn′′ = 0,

if

kXn k ≤ q,

and Xn′′ = Xn ,

if

kXn k > q.

The random sequences (Xn′ ), and (Xn′′ ), are independent, and n n

X 1 X′

X

′ ank n −→ 0 ank Xn =

q k=1 q k=1

as n → ∞, in view of Step 1. Also, for each n ∈ N, EkXn′′ k ≤ q 1−p EkXn kp ≤ q 1−p .

(8.5.7)

Define a probability density function of a random variable ξ as follows: fξ (s) = 0, for s ≤ 1, and = p/s1+p , for s > 1. Then, for each α > 0, P(|ξ| ≥ s) = 1, if s ≤ 1, and = 1/αp , if s > 1. Also Z ∞ 1+1/α E|ξ| = t1+1/α tp−1 pdt < ∞, (8.5.8) 1

since 1/α < p − 1. For s ≥ 1, in view of the above tail estimate, P kXk′′ k − EkXk′′k ≥ s ≤ P kXk′′ k ≥ s ✐

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Beck convexity ≤ EkXk′′ k/sp ≤ EkXk k/sp ≤ s−p = P(|ξ| ≥ s),

261 (8.5.9)

since the inequality (EkXk′′k − kXk′′ k) ≥ s) is impossible if a ≥ 1. Now the symmetry assumption can be removed by the standard symmetrization procedure.26 QED To a certain extent the assumption of independence can also be removed from the Strong Law of Large Numbers in B-convex Banach spaces. Definition 8.5.1. A family of random vectors (Xα ), α ∈ A, is said to be mutually symmetric if, for every α ∈ A, and every set S ∈ σ(Xβ , β ∈ A, β = 6 α), IS Xα is symmetric. Equivalently, if for every (n + 1)-tuple X0 , . . . , Xn ⊂ (Xα )), and every B0 , . . . , Bn ∈ B(X) P X0−1 (B0 ) ∩ X1−1 (B1 ) ∩ · · · ∩ Xn−1 (Bn ) = P X0−1 (−B0 ) ∩ X1−1 (B1 ) ∩ · · · ∩ Xn−1 (Bn ) ,

then the family (Xα ) is mutually symmetric. Evidently, an independent family of symmetric random vectors is mutually symmetric. Theorem 8.5.3. 27 If X is a B-convex Banach space, and (Xi ) is a sequence of mutually symmetric random vectors in X such that kXi (ω)k ≤ 1, for each i ∈ N, and ω ∈ Ω, then, almost surely, X1 + · · · + Xn = 0. lim n→∞ n Proof. Consider first the case when each Xi takes on only countably many values. Let ǫ > 0. We shall show that the set of points for which the lim sup of Cesaro averages is greater than ǫ has probability 0. Let (pn ) be the sequence defined in Theorem 8.4.3. For each 0 ≤ n ≤ m consider every sequence (y 1 , . . . , y m ) of possible values of X1 , . . . , Xm , respectively. We define F (y 1 , . . . , y m ) as the 26 27

See, e.g., A. Beck (1963). Due to A. Beck (1976).

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set of all sample points where Xi = ±y i , for each i = 1, . . . , m. Note that there are at most countably many sets F (y 1 , . . . , y m ). In each such set there are exactly 2m different m-tuples which Xi can take as values, corresponding to 2m possible ways of assigning +, and − signs to the y i ’s. By the condition of mutual symmetry, each of these 2m sets has the same probability, 2−m P(F (y1 , . . . , y m )). Thus, if we restrict our attention to the set F (y 1 , . . . , y m ), and the conditional probability on that set, the random vectors X1 , . . . , Xn , defined there satisfy the hypotheses of Theorem 8.4.3. Therefore, k

1 X

P sup Xi ≤ ǫ F (y 1 , . . . , y m ) ≥ pn . n 0, and all p, q ≥ N, the exists an integer M ∈ N such that, for all sequences (ni ) ⊂ N, M ≤ n1 < n2 < · · · < np < np+1 < · · · < np+q , we have p p+q

1 X 1 X

e ni − eni < ǫ.

p i=1 q i=p+1

(8.6.2)

Now, let ǫ = 2−n . Given N, choose a sequence (Pn ) ⊂ N satisfying the conditions, Pn ≥ N, Pn > nPn−1 . Finally, take p = Pn , q = Pn+1 , and take the corresponding M, which we will call νn , and define n X vn = (νj + Pj ). j=1

Consider a sequence an =

1 X ev +j ∈ X. Pn j≤P N n

The inequality (8.6.2) implies that kan − an+1 k < 2−n , so that the ¯ ∈ X. Consider the terms, ei − a, sequence (an ) converges to an a with the indices appearing in the sequence v1 +1, v1 +2, . . . , v1 +P1 , v1 +1, . . . , v2 +P2 , . . . , , vn +1, . . . , vn +Pn , . . . and call them ei − a, y 1 , y 2 , . . . , taking them in that order. The inequality (8.6.2) shows that Pn

1 X

¯ ≤ 2−n+3 , emi − a

pn i=1

(8.6.3)

whenever vn < m1 < m2 < · · · < mPn . Now, let (in ) ⊂ N be strictly increasing, and consider the sequence y i1 , y i2 , . . . , y in . Put z l := y il , for l − 1, 2, . . . , n. Define k ∈ N by the inequality P1 + · · · + Pk ≤ n < P1 + · · · + Pk+1 ,

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and m = n − (P1 + · · · + Pk ). The Euclidean sieve, m = dk Pk + Qk , Qk = d′k Pk−1 + Q′k ,

Qk < Pk , Qk < Pk−1,

defines the integers, dk , d′k , Qk .Q′k . Now, it follows from (8.6.3) that n X k=1

z k = (z 1 + · · · + zP1 ) + (zP1 +1 + · · · + z P1 +P2 )

+ · · · + (zP1 +···+Pk−1 + · · · + zP1 +···+Pk ) + · · · + (· · · + z n ),

is bounded in the norm k.k by

(P1 23−1 + . . . Pk 23−k ) + dk Pk 23−k + d′k Pk−1 23−(k−1) + Q′k . Hence, one obtains the inequality k k n

1 X

X X

−j Pj 2 Pj +23−k +23−(k−1) +Pk−1 /Pk , y ij ≤ 8

n j=1 j=1 j=1

which shows the stability of (y n ) in X. QED

Before formulating the next three lemmas we need to introduce the new seminorm |||.|||. put

Definition 8.6.2. Let a = (a1 , a2 , . . . ) and, for any n1 mn2 , ∈ N, n1 n1 +n2 a X a2 X 1 M(n1 , n2 ; a) = ei + ei . n1 i=1 n2 i=n +1 1

By the convexity argument (see Section 1.2) for the definition of L(a)), L(a) ≥ M(n1 , n2 , ; a) ≥ M(n1 n′1 , n2 n′2 ; a),

n1 , n′1 , n2 , N2′ ∈ N.

In an analogous manner one defines M(n1 , n2 , . . . , nk ; a) (for an arbitrary sequence a ∈ s with finitely many non-zero terms) which has similar properties to M(n1 , n2 ; a), so that |||Φ(a)||| :=

lim

n1 0. However, the former assumption implies that n X lim (−1)k ek = +∞.

n→∞

(8.6.6)

k=1

Indeed, put u1 = e1 − e2 , u2 = e3 − e4 , . . . , and consider the subspace U = span[ui ] ⊂ G. Let a, a” ∈ s, and supp(a′ ) ⊂ supp(a). It follows from Lemma 8.6.3 that X X ′ ai ui , ai ui ≥ i

so that

i

X X ai ui , εi ai ui ≤ 2 i

i

forPall choices of εi = ±1, from which we deduce that were ||| ni=1 ui ||| bounded, then U would be isomorphic to c0 . Thus we obtain (8.6.6). Now, let n ∈ N, and u = αn (e1 − e3 + e5 + · · · + e4n−3 − e4n−1 ) ∈ Φ(s), v = αn (e2 − e4 + e6 + · · · + e4n−2 − e4n ) ∈ Φ(s), where αn are chosen so that |||u||| = |||v||| = 1. Furthermore, for each ǫ ∈ (0, 1), choosing n sufficiently large, one can find u, v ∈ SX such that inf ku + εvk > 2(1 − ǫ), ε=±1

so that G is not (2, ǫ)-convex for any ǫ > 0. QED Definition 8.6.3. A Banach space X is said to have the alternate signs Banach-Saks property if from every bounded sequence (xn ) ⊂ X one can choose a subsequence (y n ) such that 1 y 1 − y 2 + y 3 − · · · + (−1)n+1 y n = 0. n→∞ n lim

(8.6.7)

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Theorem 8.6.2.32 If X is a B-convex Banach space then it has the alternate signs Banach-Saks property. Proof. We may assume that (xn ) is not stable since, otherwise, (y n ) satisfying (8.6.7) may be obtained as a union of two stable subsequences of (xn ). Let F 1 be a subspace of the space F (defined earlier in this section) generated by u1 = e1 − e2 , u2 = e3 − e4 , . . . . If X is B-convex then F 1 is also B-convex. Therefore, by the Strong Law of Large Numbers proven earlier in this chapter there exist εi = ±1 such that n 1 X lim εi ui = 0. n→∞ n i=1

(8.6.8)

Since the norm |.| is invariant under spreading (see Chapter 1), n n 1 X 1 X lim ui ≤ 2 lim εi ui = 0. n→∞ n n→∞ n i=1 i=1

Repeating the proof of Lemma 8.6.1, with (en ) replaced by (un ), one obtains a stable subsequence of (un ) wich proves (8.6.7). QED Remark 8.6.2. There exist Banach spaces which are not Bconvex and still have the alternate signs Banach-Saks property. (i) One such example is the space c0 . Indeed, if xn = (xn ), i = 1, 2, . . . , n = 1, 2, . . . , with kxn k ≤ 1, then, for each ǫ > 0, there exists a subsequence (y n ) ⊂ (xn ) such that, for each n ∈ N, m m

X

X

j+1 j+1 (i)

(−1) y j = sup (−1) y j ≤ 2 + ǫ. j=1

i

(8.6.9)

j=1

To see this (up to taking subsequences, and applying the diagonal (i) (i) procedure) put ai = limn→∞ xn , where also |xn − ai | < 2−n ǫ, (i) if |xk > 2−k ǫ, for some k < n. Then, for a subsequence (y n ), (i) (8.6.9) is satisfied since, for each i we can replace each xn by ai , 32

Due to A. Brunel (1973/74), and A. Brunel and L.Sucheston (1974), and (1975).

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and there exists a k < n such that |xk | > 2−k ǫ, thus implying the inequality, m ∞ X X (i) j+1 (i) 2−n + |xk | + |ai | ≤ 2 + ǫ. (−1) y j < ǫ n=1

j=1

8.7

Trees in B-convex spaces

Definition 8.7.1. Let X be a Banach space. We say that x1 , x2 ∈ X form a (1, ǫ)-symmetric branch if kx1 − x2 k ≥ ǫ,

and

kx1 + x2 k ≥ ǫ.

Proceed by induction and suppose that we have defined an (n − 1, ǫ)-symmetric branch. Then we say that the 2n -tuple x1 , . . . , x2n ∈ X forms an (n, ǫ)- symmetric branch if, for any choice of ε = ±1, i = 1, . . . , 2n−1 , we have the inequality kx2i−1 + εx2i k ≥ ǫ,, and if the 2n−1 -tuple, (x2i−1 + εi x2i )/2, i = 1, . . . , 2n−1 , forms an (n − 1, ǫ)-symmetric branch. We say that a Banach space has the finite symmetric tree property if there exists an ǫ > 0 such that, for each n ∈ N one can find an (n, ǫ)-symmetric branch in its unit ball. Theorem 8.7.1.33 A Banach space X is B-convex if, and only if, it does not have the finite symmetric tree property. Proof. If X is not B-convex then, by Theorem 8.2.1, l1 is finitely representable in X, and, evidently, it has the nfinite sym(2 ) metric tree property because the canonical basis in l1 forms an (n, ǫ)-symmetric branch, for each n ∈ N. Conversely, if X has the finite symmetric tree property then we shall show that X is of Rademacher-type p, for no p > 1, that is, by Theorem 8.1.4, it is not B-convex. Indeed, for every ǫ > 0, and n ∈ N, and any (n, ǫ)-symmetric branch x1 , . . . , x2n ∈ X, one has the inequality 2n

X

E ri xi ≥ 2n−1ǫ.

(8.7.1)

i=1

33

Due to B. Beauzamy (1973/74).

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We will prove (8.7.1) by induction. For n = 1, in view of the definition of a symmetric branch, Ekr1x1 + r2 x2 k = 2−1 kx1 + x2 k + 2−1 kx1 − x2 k ≥ ǫ. Assume that (8.7.1) is true for n − 1. Putting rik = ri (t),

t ∈ (k/2n , (k + 1)/2n ),

for

one has n

22 X 2n 2n

X X

−2n −2n I. r x ri xi = 2 E

ik i =: 2 i=1

k=1

i=1

Decompose I as follows,

2n−1

X X

X ′ I= εi (x2i−1 + ε2i x2i ) .

ε2i =±1 ε′i =±1

i=1

By the inductive hypothesis, fixing ε2i , one obtains the inequality, 2n−1

X n−1

X ′ εi (x2i−1 + ε2i x2i ) ≥ 2n−1 22 ǫ,

ε′i =±1

so that

i=1

2n

X n

ri xi = 2−2 I ≥ 2n−1 ǫ. E i=1

Now, in view of (8.7.1), the proof is immediate. Suppose that X has the finite symmetric tree property, and X is of Rademachertype p, for some p > 1. Then 2n 2n

X

X 1/p

E ri x i ≤ C kxi kp , i=1

i=1

∈∈ N,

and since kxi k ≤ 1, by (8.7.1), we have the inequality, ǫ · 2n−1 ≤ C · 2n/p , for all n ∈ N. A contradiction, QED

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Chapter 9 Marcinkiewicz-Zygmund Theorem in Banach spaces 9.1

Preliminaries

In this chapter we study1 the asymptotics of almost sure and tail behavior of sums, (Sn /n1/p ) = (X1 + · · · + Xn )/n1/p ), 1 ≤ p < 2, for independent, centered random vectors Xn , n = 1, 2, . . . , and of martingales, (Mn ), taking values in a Banach space X. The obtained results are in the spirit of classical theorems of Marcinkiewicz-Zugmund, Hsu-Robbins-Erd¨os-Spitzer, and Brunk, for real-valued random variables, and show the essential role played by the geometry of X in the infinite-dimensional case. In particular, we will show that for independent (Xi ) with uniformly bounded tail probabilities the implication EkXi kp < ∞,

EXi = 0 =⇒ Sn /n1/p = 0,

depends in an essential way on lp not being finitely representable in X. We also prove that a Banach space analogue of the Brunk’s strong law of large numbers depends on the Rademacher-type of X. Recall that the Brunk’s law is particularly useful in cases where one has information about existence of moments of Xi ’s 1

The results of this chapter are based on the papers by W.A. Woyczy´ nski (1980, 1982).

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of order higher than 2. Such information may not be utilized in the framework of the more classical Kolmogorov-Chung’s law. Extensions of these types of results to the case of Banach space valued martingales are also provided. As far as the rates of convergence are concerned a number of simple observations is in order. Directly from definitions and from Chebyshev;s Inequality one obtains the following “trivial” rate: Proposition 9.1.1. Let 1 ≤ p ≤ 2, and let X be of Rademacher-type p. If (Xi ) are independent and identically distributed random vectors in X with EkXi kp < ∞, and EX1 = 0, then, for every ǫ > 0, P(kSn /nk ≥ ǫ) = O(n1−p ). Also, some exponential rates can be immediately obtained without any restrictions on the geometric structure of the space X. Proposition 9.1.2. If (Xi ) are independent and identically distributed random vectors in X with EX1 = 0, and such that, for every ǫ > 0, there exist constants Cǫ , and βǫ , such that, for every β ≤ βǫ , E exp[βkX1 k] ≤ Cǫ exp[βǫ], then, for every ǫ > 0, there exists an α < 1 such that P(kSn /nk > ǫ) = O(αn ). Proof. By Chebyshev’s Inequality, and for δ < ǫ, we get P(kSn /nk > ǫ) ≤ exp[−βδ nǫ] E exp[βδ kSn k] ≤ exp[−βδ nǫ](E exp[βδ kX1 k])n ≤ Cδ (exp[(δ − ǫ)βδ ])n .

QED

It is also interesting to notice that a sufficiently rapid rate of convergence to zero of the tail probabilities, P(kSn /nk > ǫ), implies similar rates of convergence in the Strong Law.

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275

Proposition 9.1.3. Let X be a Banach space, and let (Xi ) be a sequence of independent, symmetric random vectors in X. Let (ai ), (bi ), (ci ) ⊂ R be such that 0 < ai ↑ ∞, and let

bi , ci ↓ 0, ∞ X n=1

and

j X

2i b2i = O(2j c2j ),

i=1

cn P(kSn /an k > ǫ) < ∞,

for every ǫ > 0. Then, for every ǫ > 0, ∞ X n=1

bn P(sup kSk /ak k > ǫ) < ∞. k≥n

Proof. Grouping the terms in exponential blocks, (n : 2j < n ≤ 2j+1 ) we get A≡

∞ X n=1

bn P(sup kSk /ak k > ǫ) ≤ k≥n

≤

∞ ∞ X X i=1 j=i

∞ X i=1

b2i · 2i P( max

b2i · 2i P(sup kSk /ak k > ǫ)

2i ǫ),

and, by L´evy’s Inequality, A≤2

=2

∞ X ∞ X i=1 j=i

∞ X ∞ X j=1

≤ 2C

i=j

∞ X j=1

b2i · 2i P kS2j+1 /a2j+1 k > ǫ

b2i · 2i P kS2j+1 /a2j+1 k > ǫ

c2j 2j P kS2j+1 /a2j+1 k > ǫ .

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Now, by the symmetry assumptions, grouping the terms again as follows, Sn = S2j+1 − X2j+1 − X2j+1 −1 − · · · − Xn+1 ,

2j−1 ≤ n < 2j ,

we obtain the inequality A ≤ 8C

∞ X n=1

cn P(kSn /an k > 2ǫ).

QED

Two special cases of the above Proposition will be of interest later on. Corollary 9.1.1. Let X be a Banach space, and let (Xi ) be a sequence of independent symmetric random vectors in X. Then: (i) For every q > 1, there exists a constant C > 0 such that ∞ X n=1

−q

n P(sup kSk /ak k > ǫ) ≤ C k≥n

∞ X n=1

n−q P(kSk /ak k > ǫ);

(ii) There exists a constant C > 0 such that ∞ X n=1

9.2

−1

n P(sup kSk /ak k > ǫ) ≤ C k≥n

∞ X n=1

n−1 (log n)P(kSk /ak k > ǫ).

Brunk-Prokhorov’s type strong law and related rates of convergence

In Proposition 9.1.1 we could have used only moments of order p, 1 ≤ p ≤ 2, and in Proposition 9.1.2 exponential moments were needed. The following analogue of the classical MarcinkiewiczZygmund inequality permits us to use the information about moments of arbitrary order. Proposition 9.2.1. Let 1 ≤ p ≤ 2, and q ≥ 1. The following properties of a Banach space X are equivalent:

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277

(i) The space X is of Rademacher-type p; (ii) There exists a constant C > 0 such that, for every n ∈ N, and for any sequence (Xi ) of independent random vectors in X with EXi = 0, n n

X

q X q/p

E Xi ≤ CE kXi kp . i=1

i=1

˜ i ) = (Xi − X ′ ) be a symmetrization Proof. (i) =⇒ (ii) Let (X i of the sequence (Xi ), and let the Rademacher sequence (ri ) be independent of (Xi ), and (Xi′ ). Then n n n

X

q

X

q

X

q

˜ ˜ E Xi ≤ E Xi = E ri X i i=1

i=1

i=1

n n

p q/p X X q/p

q p ˜i ≤ C · 2 E ≤ CE X kX k ,

i i=1

i=1

where the first inequality follows from the condition EXi = 0, and because (Xi′ ) are independent of (Xi ), the equality is a consequence ˜ i ), the second inequality is a consequence of of the symmetry of (X the space X being of Rademacher-type p and the Fubini’s Theorem, and the third simply, by the triangle inequality. (ii) =⇒ (i) This implications is obvious. QED Corollary 9.2.1. Let X be a Banach space of Rademachertype p, and q ≥ p. If (Xn ) are independent, identically distributed random vectors in X with EkX1 kq < ∞, and EX1 = 0, then EkSn kq = O(nq/p ). Proof. If p = q then the estimate follows directly from the definition of Rademacher-type p. If q > p, then by H¨older’s Inequality with exponents q/p, and q/(q − p), and by proposition 9.2.1, we obtain the estimate, n n

q

X X q/p

Xi ≤ CE kXi kp E i=1

i=1

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Geometry and Martingales in Banach Spaces ≤ CE

n X i=1

kXi kq n(q−p)/p = Cnq/p EkX1 kq .

QED

Hence, by Chebyshev’s Inequality we immediately obtain the following result. Corollary 9.2.2. Let X be a Banach space of Rademachertype p, and q ≥ p. If (Xn ) are independent, identically distributed random vectors in X with EkX1kq < ∞, and EX1 = 0, then, for every ǫ > 0, P kSn /nk > ǫ = O(nq(1/p−1) ). The following result provides an extension to Banach spaces of the Brunk-Prokhorov’s Law of Large Numbers2 that is an extension of the classical Kolmogorov-Chung’s Strong Law. Theorem 9.2.1. Let 1 ≤ p ≤ 2, X be of Rademacher-type p, and q ≥ 1. If (Xn ) is a sequence of independent, zero-mean random vectors in X such that ∞ X EkXn kpq n=1

then, almost surely,

npq+1−q

< ∞,

(9.2.1)

S

n lim = 0. n→∞ n

Proof. For q = 1 the theorem boils down to the classical Kolmogorov-Chung’s Strong Law mentioned above. So, assume that q > 1. Then kSn kpq is a real submartingale and, by the well known Hajek-Renyi-Chow type inequality, we get that for every ǫ > 0, ǫpq P sup kSj /jk > ǫ = ǫpq lim P sup kSj /jkpq > ǫpq m→∞

j≥n

≤ n−pq EkSn kpq + 2

∞ X

j=n+1

m≥j≥n

(9.2.2)

j −pq E kSj kpq − kSj−1kpq .

See H.D. Brunk (1948), and Yu.V. Prokhorov (1950).

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279

Marcinkiewicz-Zygmund theorem in Banach spaces By Proposition 9.2.1, and H¨older’s Inequality, EkSj k

pq

≤ CE

j X i=1

kXi k

p

q

≤ Cj

q−1

j X i=1

EkXi kpq ,

so that, by (9.2.1), and Kronecker’s Lemma, we obtain that lim j −pq EkSj kpq = 0.

j→∞

Also, the series on the right-hand side of (9.2.2) converges because of Proposition 9.2.1. Hence, summing by parts, we obtain the estimate, n X j=1

−pq

(j − 1)

≤C

+j

−pq

n X j=1

pq

EkSj k

pq

j n X X −pq −pq q−1 ≤ (j − 1) +j j EkXi kpq j=1

EkXj k /j

pq+1−q

i=1

+

n X i=1

EkXi kpq /npq+1−q .

Therefore, for every ǫ > 0, lim P sup kSj /jk > ǫ = 0.

n→∞

9.3

QED

j≥n

Marcinkiewicz-Zygmund type strong law and related rates of convergence

In this section we will prove a result that extends to some Banach spaces the classical theorem of Marcinkiewicz and Zygmund which was developed for real random variables.3 Theorem 9.3.1. Let 1 < p < 2. Then the following properties of a Banach space X are equivalent: (i) The space lp is not finitely representable in X; 3

See, J. Marcinkiewicz and A. Zygmund (1937).

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(ii) For any sequence (Xn ) of zero-mean, independent random vectors in X, with tail probabilities uniformly bounded by tail probabilities of a random vector X0 ∈ Lp , the series ∞ X Xn n1/p n=1

converges almost surely in norm. (iii) For any sequence (Xn ) satisfying the assumptions of statement (ii), Sn /n1/p → 0, almost surely.

The proof of the above Theorem will be based on the following Lemma.

Lemma 9.3.1. Assume that 1 ≤ p < 2, lp is not finitely representable in X, and the assumptions of Theorem 9.3.1 (iii) are satisfied. Then the series ∞ X n=1

(Xn − EYn )/n1/p ,

Yn = Xn I(kXn k ≤ n1/p ),

where

converges almost surely. Proof. Since ∞ X n=1

≤C

P(Xn 6= Yn ) =

∞ X n=1

∞ X n−1

P(kXn k > n1/p )

P(|X0 | > n1/p ) ≤ C1 E|X0 |p < ∞,

in viewPof the Borel-Cantelli Lemma, it suffices to show that the 1/p series ∞ converges almost surely. n=1 (Yn − EYn )/n Let r > p. Then ∞ X n=1

r

r/p

EkYn − EYn k /n =2

r+1

∞ X n=1

−r/p

n

≤2

Z

r+1

∞ X n=1

kXn k≤n1/p

EkYn kr /nr/p

kXn kr dP

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Marcinkiewicz-Zygmund theorem in Banach spaces =2

r+1

∞ X

Z

−r/p

n

n=1

=2

r+1

∞ X

−r/p

n

n=1

≤ C1

n=1

n1/p

tr kXn kr dP(kXn k ≤ t)

0

Z r/p 1/p n P(kXn k ≤ n ) − r

∞ X

−r/p

1 − rn

∞ X

= C1

n1/p

Z

t

0

= C1

rn

Z

−r/p

rn

0

≤ C2 E|X0 |

Z

t

0

n=1 p

n1/p r−1

Z

−r/p

n1/p 0

P(kXn k < t)dt

1 − P(|X0| > t) dt

r−1

n=1

∞ X

281

P(|X0 | > t)dt

1

P(|X0s−1/r | > n1/p )ds

1

s−p/r ds = C2

0

r E|X) |p < ∞. r−p

By Theorem 6.5.1 and Corollary 6.5.2, and in view of our assumptions, there exists an r > p such that X is of Rademachertype r. Therefore, the above estimate, give P∞and Theorem 6.7.4, 1/p the desired convergence of the series n=1 (Yn − EYn )/n . QED

Proof of Theorem 9.3.1. (i) =⇒ (ii) In view of the above LemmaPit is sufficient to prove the absolute convergence of the −1/p series . Since EXn = 0, and p > 1, we have the n EYn n estimate, ∞ X n=1

=

∞ X n=1

−1/p

kEYnkn

1/p

P(|X0| > n

≤

)+

∞ X n=1

Z

1

−1/p

n

Z

∞

n1/p

t dP(kXn k ≤ t)

∞

P(|X0/s| > n1/p )ds ≤ C E|X0 |p .

which give (i) =⇒ (ii). (ii) =⇒ (iii) This implication follows by a straightforward application of Kronecker’s Lemma.

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(iii) =⇒ (i)4 In view of Kronecker’s Lemma it suffices to construct, in any Banach space X such that lp is finitely representable in X, a sequence (xn ) ⊂ X, kx − nk ≤ 1, such that, for a sequence (nk ) ⊂ N, Nk → ∞, for all choices εn = ±1, and for all k ∈ N, Nk

X

1

−1/p Nk (9.3.1) εi xi > . 2 i=1

Put N1 = 1, and choose an arbitrary x1 ∈ X with kx1 k = 1. Suppose that N1 , . . . , Nk , and x1 , . . . , xNk , have been chosen so that kxi k ≤ 1, i = 1, . . . , Nk , and for all choices of εi = ±1 the inequality (9.3.1) is satisfied. Next, choose Nk+1 ∈ N large enough so that h i 1 −1/p 2 Nk+1 (Nk+1 − Nk )1/p − Nk > . 3 2 Since lp is finitely representable in X, we can find xNk +1 , . . . , xNk+1 such that, for all (αk ) ⊂ R, Nk+1 k+1 k+1

NX 1/p 1/p NX 2 X

p |αi |p . αi x i ≤ |αi | ≤ 3 i=N +1 i=N +1 i=N +1 k

k

k

Therefore, for all ε = ±1, k+1

NX −1/p Nk+1 i=1

Nk k+1

X

NX

−1/p εi xi εi xi − εi xi ≥ Nk+1 i=Nk +1

−1/p

> Nk+1

h2

i=1

i 1 (Nk+1 − Nk )1/p − Nk > . 3 2

QED

For Banach spaces X in which l1 is not finitely representable, that is for B-convex Banach spaces, Lemma 9.3.1 permits us to prove the following result: Theorem 9.3.2. The following properties of a Banach space X are equivalent: (i) l1 is not finitely representable in X; 4

This implication is essentially due to B. Maurey and G. Pisier (1976).

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283

(ii) For any sequence (Xn ) of zero-mean, independent random vectors in X with tail probabilities uniformlyPbounded by tail prob∞ abilities of an X0 ∈ L log+ L, the series n=1 Xn /n converges almost surely; (iii) For any sequence (Xi ) satisfying assumptions of statement (ii) Sn /n → 0 almost surely, as n → ∞. Proof. to prove P (i) =⇒ (ii) In view of Lemma 9.3.1 it suffices + that n kEYn k/n converges, whenever X0 ∈ L log L. Since, EXn = 0 , integrating by parts we obtain the following estimates, Z ∞ ∞ ∞ X X −1 kEYn k/n ≤ n t P(kXn k ≤ t)dt n=1

n=1

N

Z ∞ h X −1 = P(kXn k > n) + n

∞

n

n=1

i P(kXn k ≤ t)dt

∞ ∞ h i X X −1 ≤ C1 E|X0| + n P(|X0 | > k) n=1

k=n

∞ X k h i X = C1 E|X0 | + n−1 P(|X0 | > k) k=1 n=1

∞ h i X = C1 E|X0| + (log k)P(|X0| > k) k=1

h i + ≤ C1 E|X0| + E|X0 | log |X0 | < ∞.

The implication (ii) =⇒ (iii) follows directly from the Kronecker’s Lemma, and (iii) =⇒ (i) can be proven exactly the same way the same implication in Theorem 9.3.1 was demonstrated. QED Theorem 9.3.3. (i) Let X be a Banach space, 1 < p < 2, and let α ≥ 1/p. Then lp is not finitely representable in X if, and only if, for each sequence (Xi ) of zero-mean, independent random vectors in X with tail probabilities uniformly bounded by tail probabilities of an X0 ∈ Lp , we have, for every ǫ > 0, ∞ X nαp−2 P max kSi k > nα ǫ < ∞. n=1

1≤i≤n

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(ii) Let X be a Banach space, 1 ≤ p < 2. Then lp is not finitely representable in X if, and only if, for each sequence (Xi ) of zeromean, independent random vectors in X, with tail probabilities uniformly bounded by tail probabilities of an X0 ∈ Lp log+ L, we have, for every ǫ, ∞ X n=1

n−1 (log n)P kSi k > n1/p ǫ < ∞.

Proof. (i) First, we shall prove the sufficiency of the condition that lp is not being finitely representable in X. By Theorem 9.3.1, Sn /n1p → 0, almost surely and, as is easy to see, also Mn /n1/p → 0 a.s., where

Mu := max kSi k, u ∈ R, 1≤i≤[u]

with [u] denoting the integer part (entier) of u. Hence, if we introduce the Chow’s delayed sums X Su,v = X[u]+j , u, v ∈ R, 1≤j≤v

we get Mn,n n−1/p ≤ (Mn + M2n )n−1/p → 0,

a.s., as

n → ∞.

Now, in the case α = 1/p, since M2n ,2n , n = 1, 2, . . . , are independent, the Borel-Cantelli Lemma implies that, for every ǫ > 0, ∞>

∞ X

P(M2n ,2n > 2

n=1

n/p

ǫ) =

∞ X

P(M2n > 2n/p ǫ)

n=1

≥ > (log 2)

Z

∞

P(M2t > 2(t+1)/p ǫ) dt !

−1

Z

∞

u−1 P(Mu > 21/p ǫu1/p ) du.

1

So,

P

n

n−1 P(mn > n1/p ǫ) < ∞, for every ǫ > 0.

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285

In the case α > 1/p, for m ≥ 1, we have (m + 1)αp/(αp−1) ≥ mαp/(αp−1) +

αp m1/(αp−1) αp − 1

≥ mαp/(αp−1) + m1/(αp−1) ,

so that the random variables Mmαp/(αp−1 ,m1/(αp−1) , m = 1, 2, . . . , are independent. Moreover, by Theorem 9.3.1, m−α/(αp−1) Mmαp/(αp−1) ,m1/(αp−1) ≤ m−α/(αp−1) Mmαp/(αp−1) ,mαp/(αp−1) → 0,

almost surely as m → ∞. Therefore, again using the BorelCantelli lemma, we obtain that ∞>

∞ X

m=1

= Z

P(Mmαp/(αp−1) ,m1/(αp−1) ≥ mα/(αp−1) ǫ)

∞ X

m=1 ∞

P(Mm1/(αp−1) ≥ mα/(αp−1) ǫ)

P(Mt1/(αp−1) ≥ (t + 1)α/(αp−1) ǫ) dt 1 Z ∞ ≥ (αp − 1) uαp−1P(Mu ≥ 2α/(αp−1) uα ǫ) du, ≥

1

which gives the desired rate of convergence. The necessity of the condition of lp not being representable in X follows directly from the example developed in the proof of the implication (iii) =⇒ (i) in Theorem 9.3.1. (ii) First, let us prove the sufficiency of the condition that lp is not finitely representable in X. Without loss of generality we can assume that Xn ’s are symmetric. The case of zero expectations can be handled by adapting, in a standard way, the method presented below. Put Ykn = Xk I(kXk k < n1/p ). Then ∞ X log n n=1

n

P(kSn k > n1/p ǫ)

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∞ n ∞ n

X X log n [ log n

X 1/p ≤ P P Ykn > n1/p ǫ . (kXk k > n ǫ) + n n n=1 n=1 k=1 k=1

The series on the right-hand side can be estimated from above by the quantity, C

∞ X n=1

(log n)P(|X0 | > n1/p ǫ) ≤ C1 E|X0 |p log+ |X0 | < ∞,

and the convergence of the second series can be verified as follows: Since lp is not finitely representable in X, by the Maurey-Pisier Theorem mentioned earlier, there exists a δ > 0 such that X is of Rademacher-type (p + δ). Hence, making use of Chebyshev’s Inequality, and integrating by parts, we obtain the estimates, n ∞

X log n

X Ykn > n1/p ǫ P n n=1 k=1

≤ C1 ≤ C2

∞ X

−1−(p+δ)/p

n

(log n)

n−1

∞ X n=1

n X k=1

log n n1+(p+δ)/p

n Z X k=1

EkYknkp+δ

n1/p

tp+δ dP(kXk k ≤ t)

0

Z n1/p ∞ X log n ≤ C2 tp+δ−1 P(|X0| > t)dt (p+δ)/p n 0 n=1 = C2

Z

1

s

δ/p

0

≤ C2

∞ X

(log n)P(|X0 s−1/p > n1/p )ds

n=1

Z

0

1

sδ/p E|X0s−1/p |p log+ |X0 s−1/p |ds p

+

≤ C3 E(|X0| log |X0 |)

Z

0

1

s−1+δ/p ds < ∞.

This completes the proof of the sufficiency. The necessity can be obtained exactly as in part (i). QED

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287

Corollary 9.3.1. If lp is not finitely representable in X, 1 < p < 2, and (Xn ) are zero-mean, independent and identically distributed random vectors in X with EkX1 kp < ∞, then, for every ǫ > 0, P(kSn /nk > ǫ) = o(n1−p ). Corollary 9.3.2. Let X be of Rademacher-type p, 1 < p ≤ 2, and let (Xn ) be a sequence of independent, zero-mean vectors in X such that, uniformly in k, P(kXk k > n) = o(n−p ),

(9.3.2)

then, for every δ, ǫ > 0, P(kSn /nk > ǫ) = o(n1−p+δ ). Proof. Since X is of Rademacher-type p, for every δ > 0, the space lp−δ is not finitely representable in X. Now, (9.3.2) implies that Xk ’s have tail probabilities uniformly bounded by tail probabilities of an X0 ∈ Lp−δ . Therefore, by Theorem 9.3.3, ∞ X n=1

np−δ−2 P(kSn /nk > ǫ) < ∞,

so that np−δ−2 P(kSn /nk > ǫ) = o(n−1 ).

QED

From Corollary 9.1.1, and Theorem 9.3.3, we immediately obtain the final result: Corollary 9.3.3. If 1 ≤ p < 2, and lp is not finitely representable in X, then, for any sequence (Xi ) of zero-mean, independent random vectors in X, with tail probabilities uniformly bounded by tail probabilities of an X0 ∈ Lp , if 1 < p < 2, and of an X0 ∈ L log+ L, in the case p = 1, ∞ X n=1

np−2 P(sup kSk /kk > ǫ) < ∞, k≥n

∀ǫ > 0.

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9.4

Brunk and MarcinkiewiczZygmund type strong laws for martingales

For an X-valued martingale (Mn , Fn ), on a probability space (Ω, F , P), consider their difference sequence Dn = Mn −Mn−1 , n = 1, 2, . . . ,. In what follows we shall assume that M0 = 0, a.s. Proposition 9.4.15 The Banach space X is p-smoothable if, and only if, for any q ≥ 1, there exists a constant C > 0 such that, for each |XX-valued martingale (Mn ) q

EkMn k ≤ CE

n X i=1

kDi kp

q/p

.

(9.4.1)

Remark 9.4.1. A Banach space X is said to be ζ-convex if there exists a symmetric, biconvex function ζ on X × X satisfying the following two conditions: ζ(0, 0) > 0, and ζ(x, y) ≤ kx + yk, if kxk ≤ 1 ≤ kyk. It turns out6 that X is ζ-convex if, and only if, for any p, 1 < p < ∞, there exists a constant Cp such that Ek ± D1 ± · · · ± Dn kp ≤ Cpp EkMn kp ,

n = 1, 2, . . . , (9.4.2)

for all X-valued martingales (Mn ), and all sequences ±1. In connection with ζ-convexity, and the MarcinkiewiczZygmund type inequality (9.4.1) it is worthwhile to observe that if X is of Rademacher-type p, and is also ζ-convex, then the inequality (9.4.1) is satisfied. Indeed, let (ri ) be a Rademacher sequence independent of (Mn ). Then, by ζ-convexity and Fubini’s Theorem, n n

q

q 1/q·q

X X

r i Di EkMn k ≤ CEr E ri Di ≤ CE Er q

i=1

n n

p 1/p·q X X q/p

kDi kp , ≤ CE Er r i Di ≤ CE i=1

5

i=1

i=1

Due to P. Assuad (1975). This result is due to D.L. Burkholder (1981), who also introduced the concept of ζ-convexity. 6

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289

where the last two inequalities follow, respectively, from Kahane’s Theorem (see Chapter 1), and the definition of Rademacher-type p. The above fact immediately implies that ζ-convex Banach spaces of Rademacher-type p are p smoothable. On the other hand, although p-smooth spaces are necessarily of Rademachertype p, they need not be ζ-convex. The next result is an extension of the Brunk’s type Strong Law proven for independent summands in the previous section. The theorem, as well as the following later extension of the Marcinkiewicz-Zygmund Law, permit use of moments of order greater than 2 in establishing the asymptotics of Banach space valued martingales. Theorem 9.4.1.7 Let 1 ≤ p ≤ 2, q ≥ 1, and assume that the Banach space X is p-smoothable. Then: (i) If (Mn ) is an X-valued martingale such that ∞ X EkDn kpq n=1

npq+1−q

< ∞,

(9.4.3)

then kMn k = o(n), almost surely. (ii) For every ǫ > 0 there exists a positive constant C such that, for any X-valued martingale (Mn ) ∞ X n=1

−1

n P(kMn /nk > ǫ) ≤ C

∞ X EkDn kpq n=1

npq+1−q

.

(9.4.4)

Proof. (i) The case q = 1 is covered in Section 3.2 of Chapter 3. So, assume that q > 1. Then kMn kpq is a real-valued submartingale, and by the well-known Hajek-Renyi-Chow’s type inequality we get that, for every ǫ > 0 ǫpq P sup kMj /jk > ǫ = ǫpq lim P sup kMj /jkpq > ǫpq j≥n

m→∞

n≤j≤m

(9.4.5)

7

Due to W.A. Woyczy´ nski (1982).

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290

Geometry and Martingales in Banach Spaces −pq

≤n

∞ X

pq

EkMn k +

j=n+1

j −pq E kMj kpq − kMj−1 kpq .

In view of Proposition 9.4.1, and H¨older’s Inequality, EkMj k

pq

≤ CE

n X

kDi k

i=1

p

q

≤ Cj

q−1

j X i=1

EkDi kpq ,

so that, by Kronecker’s Lemma, j p q EkMj kpq → 0, as j → ∞. Also, the series on the right-hand side of (9.4.5) converges because of Proposition 9.4.1. Hence, summing by parts, we obtain the inequalities, n X (j − 1)−pq + j −pq EkMj kpq j=1

≤

n X j=1

j X EkDi kpq (j − 1)−pq + j −pq j q−1 i=1

n n X X 1 EkDj kpq pq + EkD k , ≤ const j j pq+1−q npq+1−q i=1 j=1

so that, for every ǫ > 0, P(supj≥n kMj /jk > ǫ) → 0, as n → ∞. (ii) By Chebyshev’s and H¨older’s Inequalities, in view of Proposition 9.4.1, ∞ X n=1

−1

n P(kMn k > ǫn) ≤ −pq

≤ǫ

C

∞ X

∞ X n=1

n−1 n−pq ǫ−pq EkMn kpq

−1+(q−1)−pq

n

n=1

≤ Cǫ−pq

∞ X k=1

≤ Cǫ−pq

n X k=1

EkDk kpq

∞ X k=1

∞ X

EkDk kpq

n−pq+q−2

n=k

E kDk kpq /k pq+1−q .

QED

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Marcinkiewicz-Zygmund theorem in Banach spaces

In the next step we will prove the analogue of the Marcinkiewicz-Zygmund Strong Law of Large Numbers for vectorvalued martingales with uniformly bounded tail probabilities. Recall that a sequence (Xi ) of random vectors is said to have uniformly bounded tail probabilities by tail probabilities of a positive real random variable X0 , if there exists a positive constant C such that, for all t > 0, and all i = 1, 2, . . . , P(kXi k > t) ≤ CP(X0 > t). Theorem 9.4.28 Let (Mn ) be a martingale with values in a Banach space X, with the difference sequence (Dn ). (i) If the difference sequence (Dn ) has uniformly bounded tail probabilities by an X0 ∈ L log L, and X is superreflexive, then Mn = o(n),

a.s.

(9.4.6)

(ii) If the difference sequence (Dn ) has uniformly bounded tail probabilities by an X0 ∈ Lp , 1 < p < 2, and X is r-smoothable for an r > p, then Mn = o(n1/p ), a.s. (9.4.7) The proof of the above theorem depends on the following two Lemmas: Lemma 9.4.1. Let 1 ≤ p < 2, and let (Xn ) be a sequence of real-valued random variables with tail probabilities uniformly bounded by tail probabilities of X0 ∈ Lp . Then, if Xn′ := Xn I[|Xn | ≤ n1/p ], and r > p, then ∞ X E|X ′ |r n

n=1

< ∞.

nr/p

Proof. The proof involves a straightforward calculation: ∞ X E|X ′ |r n=1

8

n nr/p

=

∞ X

−r/p

n

n=1

Z

0

n1/p

tr dP(|Xn | ≤ t)

Due to W.A. Woyczy´ nski (1982).

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Geometry and Martingales in Banach Spaces

=

∞ X n=1

−r/p

n

Z h r/p 1/p n P(|Xn | ≤ n ) − r

n1/p

0

Z ∞ h X −r/p ≤C 1 − rn n=1

=C

∞ X

rn

t 0

Z

0

n=1

Z

1

i 1 − P(|Xn | > t) dt

n

0

∞ Z X

≤ CEX0

r−1

−r/p

n=1

=C

n1/p

i tr−1 P(|Xn | ≤ t) dt

tr−1 P(|Xn | > t) dt

P X0 s−1/r > n1/p ds

1

s−p/r ds = C

0

r EX0p < ∞. r−p

QED

Lemma 9.4.2. Let (Xn ) be a sequence of real-valued random variables with tail probabilities uniformly bounded by tail probabilities of X0 . (i) If X0 ∈ L log L, and Xn′′ := Xn I[|Xn | > n], then ∞ X E|X ′′ | n

n

n=1

< ∞.

(ii) If X0 ∈ Lp , p > q ≥ 1, and Xn′′ := Xn I[|Xn | > n1/p ], then ∞ X E|X ′′ |q n

n=1

nq/p

< ∞.

Proof. Again, the proof depends on straightforward calculations which we are including for the sake of completeness. (i) Z ∞ ∞ X E|Xn′′ | X −1 ∞ tdP(|Xn | ≤ t) = n n n n=1 n=1 Z ∞ ∞ h i X −1 = P(|Xn | > n) + n P(|Xn | > t) dt n

n=1

∞ ∞ i h X X n−1 P(X0 > k) ≤ C EX0 + n=1

k=n

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293

Marcinkiewicz-Zygmund theorem in Banach spaces ∞ X k h i X −1 = C EX0 + n P(X0 > k) k=1 n=1

∞ h i X = C EX0 + log kP(X0 > k) ≤ C EX0 + EX0 log+ X0 < ∞. k=1

(ii) Similarly, ∞ X E|X ′′ |q n=1

n q/p n

=

∞ X

n

Z

∞

n1/p

n=1

Z ∞ h X 1/p = P(X0 > n ) + q n=1

−q/p

tq dP(|Xn | ≤ t)

∞

s

q−1

1/p

P(X0/s > n

1

i

) ds ≤ CEX0p . QED

Proof of Theorem 9.4.2. For n = 1, 2, . . . , denote Dn′ = Dn I[kDn k ≤ n1/p ],

Dn′′ = Dn I[kDn k > n1/p ],

∆′n = Dn′ − E(Dn′ |Fn−1),

∆′′n = Dn′′ − E(Dn′′ |Fn−1 ).

Then (∆′n ), and ∆′′n ), are martingale difference sequences and, since E(Dn′ + Dn′′ |Fn−1) = 0, Dn = ∆′n + ∆′′N .

(9.4.8)

To prove (i) set p = 1 in the above notation, and observe that since X is superreflexive it is r-smoothable for an r > 1. Thus, by (9.4.8), the triable inequality, and the Proposition 9.4.1, n n n

X X ∆′k Dk

r 1/r X Ek∆′′k k + E

≤ E

k k k k=1 k=1 k=1 n X Ek∆′ kr 1/r

+

n X EkD ′ kr 1/r

+

≤C ≤C

k=1

k=1

k r k

k kr

(9.4.9)

n X Ek∆′′ k k

k=1

k

n X EkD ′′k k

k=1

k

.

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The first series converges by Lemma 9.4.1, andPthe second, by n Lemma 9.4.2(i). Thus, the martingale Nn := k=1 Dk /k, n = 1, 2, . . . , is L1 -bounded, and since the superreflexive space is dentable, Nn converges almost surely. Now, an application of Kronecker’s lemma completes the proof of (i). To show (ii) we proceed exactly as in (9.4.9), except that now p > 1. Thus we get that n n n

X X X Dk Ek∆′k kr 1/r EkDk′′k

E +2 ,

≤C k 1/p k r/p k 1/p k=1 k=1 k=1

and an application of Lemma 9.4.1, and Lemma 9.4.2(ii), together with of X, give us the almost sure convergence of Pn the dentability 1/p , thus assuring, again via Kronecker’s Lemma, that k=1 Dk /k Mn = o(n1/p ). QED Finally, we shall prove two results about integrability of the maximal function M (p) (ω) := sup n

kMn (ω)k , n1/p

for Banach space-valued martingales. Theorem 9.4.3. Let (Mn ) be a martingale with values in a Banach space X. (i) If the difference sequence (Dn ) has uniformly bounded tail probabilities by an X0 ∈ L log L, and X is superreflexive, then M (1) ∈ L1 . (ii) If the difference sequence (Dn ) has uniformly bounded tail probabilities by an X0 ∈ Lp , 1 ≤ q < p < 2, and X is r-smoothable for an r > p, then M (p) ∈ Lq . Proof. By Kronecker’s lemma, and the Closed graph Theorem, there exists a constant C > 0 such that, for any (xn ) ⊂ X, sup n

n n

X xk 1

X

x ≤ C sup

. k 1/p n1/p k n k=1

(9.4.10)

k=1

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295

Therefore, using the notation from the proof of Theorem 9.4.1, we get that M

(p)

n n

X

X ∆′k ∆′′k

≤ C sup

+ sup

. k 1/p k 1/p n n

k=1

k=1

(i) Here, the assumption is that X is superreflexive and hence r-smoothable for an r > 1. So, by the Davis type inequality9 , ∞ n

X X Ek∆′k kr ∆′k

. E sup

≤C r k k n k=1 k=1

The latter series is finite by Lemma 9.4.1. On the other hand, n ∞

X X ∆′′k EkDk′′ kr

E sup ,

≤C r/p k k n k=1 k=1

and the last series is also finite by Lemma 9.4.2(i). Thus EM (1) < ∞. (ii) In this case, with 1 ≤ q < p < r < 2, and X being rsmoothable, using the cited above Davis type inequality, we get the inequality

n ∞

X X ∆′k Ek∆′k kr

q r/q E sup ≤ C ,

k 1/p k r/p n n=1 k=1

which is finite by Lemma 9.4.1. Then again, applying the above Davis type inequality, and the fact that r-smoothablility implies q-smoothability for q < r, we get that n ∞ ∞

X X X ∆′′k Ek∆′′n kq EkDn′′ kq

q E sup ≤ C ≤ C ,

k 1/p nq/p nq/p n n=1 n=1 k=1

and the last series converges in view of Lemma 9.4.2(ii). Thus E(M (p) )q < ∞. QED 9

See W.A. Woyczy´ nski (1976), Theorem 6.

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Bibliography Araujo, A, and Gine, E., (1978) Type, cotype, and L´evy measures in Banach spaces, Annals of Probability 6, 637-643. (1979) On tails and domains of attraction of stable measures in Banach spaces, Transactions Amer. Math.Soc. 248, 105-119. Asplund, E., (1967) Averaged norms, Israel J. Math 5, 227-233 Asplund, E., and Namioka, I., (1967) A geometric proof of Ryll-Nardzewski’s fixed point theorem, Bull. Amer. Math.Soc. 73, 443-445. Assuad, P., (1974) Martingales et rearrangement dans les espaces uniformement lisses, preprint, Orsay, Julliet 1974. (1974/75) Espaces p-lisses, rearrangements, Seminaire Maurey-Schwartz, Exp. XVI. (1975) Espaces p-lisses at q-convexes. Inegalites de Burkholder. Seminaire Maurey-Schwartz, Exp. XV. Azlarov, T.A., and Volodin, N.A., (1981) The law of large numbers for identically distributed Banach space valued random variables, Theorya veroyatnostey i prim 26, 584-590. Badrikian, A., (1976) Prolegomenes au calcul des probbilites dans les Banach, Springer Lecture Notes in Math. 539,1-167. Beauzamy, B., (1973/74) Espaces de Banach uniformement convexifiables, Seminaire Maurey-Schwartz, XIII, 1-18. 297 ✐

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(1974) Operateurs uniformement convexifiables, Preprint, Centre de Mathematiques, Ecole Polytechnique, Paris, Octobre 1974. (1974/75) Proprietes geometriques des espaces d’interpolation, Seminaire Maurey-Schwartz, Exp. XIV, 1-17. Beck, A., (1962) A convexity condition in Banach spaces and the strong law of large numbers, Proc. Amer. Math. Soc. 13, 329-334. (1963) On the strong law of large numbers, Ergodic Theory, Proc. Intern. Symp., Academic Press, New York, 21-53. (1976) Conditional independence, Z. Warsch.verw. Geb. 33, 253-268. Bessaga, C., and Pelczynski, A., (1958) On basis and unconditional convergence of series in Banach spaces, Studia Math. 17, 151-164. Brown, D.R., (1974) B-convexity and reflexivity in Banach spaces, Transactions Amer. Math. Soc. 187, 69-76. (1974) P-convexity and B-convexity in Banach spaces, Transactions Amer. Math. Soc. 187, 77-81. Brunel, A., (1973/74) Espaces associes `a une suite born´e dans un espace de Banach, Seminaire Maurey-Schwartz, Exp. XV. et XVIII. Brunel, A., and Sucheston, L., (1974) On B-convex Banach spaces, Math Systems Theory 7, 294-299. (1975) Sur quelques conditions equivalentes a la superreflexivit´e dans les espaces de Banach, Comptes Rendus Acad. Sci., Paris A275, 993-994. (1975) On J-convexity and some ergodic super-properties of Banach spaces, Transactions Amer. Math. Soc. 204, 21-33. (1976) On sequences invariant under spreading in Banach spaces, Springer Lecture Notes in Math. 204, 21-33. Brunk, H.D., (1948) The strong law of large numbers, Duke Math. J. 15, 181-195. Buldygin, V.V.,

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Day, M.M., (1944) Uniform convexity in factor and conjugate spaces, Ann. Math. 45, 375–385. (1973) Normed Linear Spaces, Third Edition, Springer-Verlag, New York. de Acosta, A., (1981) Inequalities of B-valued random vectors with applications to the strong law of large numbers, Annals of Probability 9, 157-161. de Acosta, A., and Samur, J, (1979) Infinitely divisible probability measures and the converse Kolmogorov inequality in Banach spaces. Studia Math. 66, 143-160. Diestel, J., ( 1975) Geometry of Banach Spaces, Lecture Notes in Mathematics 485, Springer-Verlag, New York. Diestel, J., and Uhl, Jt, J.J., (1976) The Radon-Nikodym Theorem for Banach space valued measures, Rocky Mountains J. 6, 1-46. Dubinsky, E., Pelczynski, A., and Rosenthal, H.P., (1972) On Banach spaces X for which Π2 (L∞ , X) = B(L∞ , X), Studia Math. 44, 617-648. Dunford, N., and Schwartz, J.T., (1958) Linear Operators, Part I: General Theory, Interscience, New York. Dvoretzky, A., (1961) Some results on convex bodies in Banach spacers, Proc. International Symposium on Linear Spaces, New York, 123-160. Elton, J., (1981) A law of large numbers for identically distributed martingale differences, Annals of Probability 9, 405-412. Enflo, P., (1972) Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13, 281-288. Enflo, P., Lindenstrauss, J., and Pisier, G.,

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Kottman, C.A., (1970) Packing and reflexivity in Banach spaces, Trans. Amer. Math. Soc. 150, 565-576. Krivine, J.L., (1973/74) Th´eor`emes de factorization dans les espaces r´eticul´es, Sem. Maurey-Schwartz, Exp. XXII et XXIII. (1976) Sous-espaces de dimension finie des espaces de Banach r´eticul´es, Ann. Math. 104, 1-29. Kuelbs, J., (1976) A counterexample for Banach space valued random variables, Ann. Prob. 4, 690-694. (1977) Kolmogorov’s law of the iterated logarithm for Banach space valued random variables, Illinois J. Math.21, 784-800. Kuelbs, J., abd Woyczy´ nski, W.A., (1978) Lacunary series and exponential moments, Proc. Amer. Math. Soc. 68, 281-291. Kvaracheliya, V.A., and Tien, N.Z., (1976) The central limit theorem and the strong law of large numbers in lp (X)-spaces, 1 ≤ p < ∞, Teorya Veroyat. i Prim., 21, 802-812. Kwapie´ n, S., (1972/73) Isomorphic characterization of Hilbert spaces by orthogonal series with vector valued coefficients, Sem. MaureySchwartz, Exp. VIII. (1974) On Banach spaces containing c0 , Studia Math. 52, 187188. (1976) A theorem on the Rademacher series with vector valued coefficients, Springer Lecture Notes in Math. 526, 157-158. Kwapie´ n, S., and Woyczy´ nski, W.A., (1992) Random Series and Stochastic Integrals: Single and Multiple, Birkh¨auser. Boston. Lai T.L., (1974) Convergence rates in the strong law of large numbers for random variables taking values in Banach spaces, Bull. Inst. Math. Academia Sinica 2, 67-85. Landau, H.J., and Shepp, L.A.,

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(1970) On the supremum of a Gaussian process, Sankhya 32, 369-378. LeCam, L., (1970) Remarque sur le theoreme limite central dans les espaces localement convexes, Les Probabilities sur les Structures Algebriques, Colloq. CNRS Paris, 233-249. Lewis, D.R., (1972) A vector measure with no derivative, Proc. Amer. math. Soc. 32, 535-536. Linde, V., and Pietsch, A., (1974) Mappings of Gaussian cylindrical measures in Banach spaces, Theorya Veroyat. i Prim. 19, 472-487. Lindenstrauss, J., (1963) On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J. 10, 241-252. (1975/76) The dimension of almost spherical sections of convex bodies, Seminaire Mayrey-Schwartz, Exp. XIX. (1976) Type and superreflexivity, Bull. London Math. Soc. 8, 15. Lindenstrauss, J., and Tazfriri, L., (1973) Classical Banach Spaces, Lecture Notes in Math. 338, Springer. Mandrekar, V., (1977) Characterization of Banach spaces through validity of Bochner theorem, Proc. Conf. Dublin, Lecture Notes in Math. Springer, Berlin. Marcinkiewicz, J. and Zygmund, A., (1937) Sur les fonctions independantes, Fundamenta. Math. 29, 60-90. Marcus, M.B., and Woyczy´ nski, W.A., (1977) Domaines d’attraction normale dans l´espace de tupe p-stable, Comptes Rendus Aced. Sci. Paris, 285, 915-917. (1978) A necessary condition for the central limit theorem on spaces of stable type, Proc. Conf. Dublin, Lecture Notes in Math. 644, 327-339.

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(1979) Stable measures and central limit theorems in spaces of stable type, Transactions Amer. math.Soc 251, 71-102. Maurey, B., (1972/73) Espaces de cotype p, 0 < p ≤ 2, Sem. MaureySchwartz, Exp. VII. (1972/73) Theoremes de Nikishin: Theoremes de factorization pour les applications lineaires a valeus dans un espace L0 , Sem. Maurey-Schwartz, Exp. X, XI. (1972/73) Une nouvelle demonstration d’un theoreme de Grothendiesk, Sem. Maurey-Schwartz, Exp. XXII. (1974) Theoremes de factorization pour les operateurs lineaires `a valeurs dans les espaces Lp , Asterisque 11. (1973/74) Nuveaux theoremes de Nikishin, Sem. MaureySchwartz, Exp. IV, V. (1973/74) Type et cotype dans les espaces munis de structures locales unconditionelles, Sem. Maurey-Schwartz, Exp. XXIV, XXV. Maurey, B., and Pisier, G., (1973 Characterization d’une classe d’espaces de Banach par desproprietes de s´esries al´eatoires vectorielles, Comptes Rendus Acad. Sci, Paris. 277), 687-690. (1974/75) Remarques sur l´expose d’Assuad, Sem. MaureySchwartz, Annexe 1. (1976) Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de Banach, Studia Math. 58, 45-90. Maynard, H.B., (1973) A geometrical characterization of Banach spaces with the Radon-Nikodym Property, Trans. Amer. Math. Soc. 185, 493-500. Metivier, M., (1963) Limites projectives de mesures; martingales; applications, Ann. Math. Pura Appl. 63, 225-352. Milman, V.D., (1971) Geometric theory of Banach spaces, Part II; Geometry of the unit sphere, Uspekhi Mat. Nauk 26, 73-149.

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Mouchtari, D., (1975/76) Sur l’existence d’une topolgie du type de Sazonov sur un espace de Banach, Sem. Maurey-Schwartz, Exp. XVII. (1973) Some general questions of the theory of probability measures in linear spaces, Teorya Veroyat. i Prim. 18, 66-77. Musial, K., Ryll-Nardzewski, C., and Woyczy´ nski, W.A., (1974) Convergence presque sure des series aleatoires vectorielles a multiplicateur bornes, Comptes Rendus Acad. Sci Paris 279, 225-228. Neveu, J., (1972) Martingales a temps discret, Masson et Cie, Paris. Nielsen, N.J., (1973) On Banach ideals determined by Banach lattices and their applications, Dissertationes Math. 109, 1-66. Nishiura, T. and Waterman, D. (1963) Reflexivity and summability, Studia Math. 23, 53-57. Nordlander, G. (1961) On sign-independent and almost sign-independent convergence in normed linear spaces, Arkiv Mat. 21, 287-296. Padgett, W.J., and Taylor, R.L., (1975) Stochastic convergence of weighted sums in normed linear spaces, J. Multivariate Analysis, 5, 434-450. (1973) Laws of Large Numbers for Normed Linear Spaces and Certain Frechet Spaces, Springer Lecture Notes in Math. 360. (1973) Weak laws of large numbers in Banach spaces and their extensions, Springer Lecture Notes in Math. 360, 66-83. (1976) Almost sure convergence of weighted sums of random elements in Banach spaces, Springer Lecture Notes in Math. 526, 187-202. Parthasarathy, K. R., (1967) Probability Measures on Metric Spaces, Academic Press, New York. Paulauskas V., (1976) Infinitely divisible and stable probability measures on separable Banach spaces, University of G¨oteborg Report-15.

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Pelczynski, A., (1975) On unconditional bases and Rademacher averages, Springer Lecture Notes in Math. 472, 119-130. Petrov, V.V. (1972) Sums of independent random variables, Moscow. Pietsch, A., (1967) Absolut p-summierende Abbildungen in normierten Raumen, Studia Math. 28, 333-353. Pisier, G., (1972/73) Bases, suites lacunaires dans les espaces Lp Sem. Maurey-Schwartz, Exp. XVIII, XIX. (1972/73) Sur les espaces qui ne contiennent pas de ∞n uniformement, Sem. Maurey-Schwartz, Annexe. (1973/74) Sur les espaces qui ne contiennent pas de ln1 uniformement, Sem. Maurey-Schwartz, Exp. VII., and Comptes Rendus Acad. Sci. Paris 277(1973), 991-994. (1973/74) Type des espaces norm´ees, Sem. Maurey-Schwartz, Exp. III, and Comptes Rendus Acad. Sci. Paris 276(1973), 16731676. (1973/74) Une propriete du type p-stable, Sem. MaureySchwartz, Exp. VIII. (1975) B-convexity, superreflexivity and the 3-space problem, Proc. Sem. ”Random Series, Convex Sets and Geometry of Banach Spaces, Aarhus, 151-175. (1975) Martingales with values in uniformly convex spaces, Israel J. Math. 20, 326-350. (1975/76) Le theoreme limite centrale et la loi du logarithme iteree dans les espaces de Banach, Sem. Maurey-Schwartz , Exp. III, IV. (1976) Sur la loi du logarithme iteree dans les espaces de Banach, Springer Lecture Notes in Math. 526, 203-210. (2016) Martingales in Banach Spaces, Cambridge University Press. Pisier, G., and Zinn, J., (1978) On the limit theorems for random variables with values in the spaces Lp (2 ≤ p < ∞), Z. Wahr. verw. Geb. 41, 289-304.

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Prokhorov, Yu,V., (1950) On a strong law of large numbers (in Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 14, 523-536. Retherford, J.R., (1975) Applications of Banach ideals of operators, Bull. Amer. Math. Soc. 81, 978-1012. Revesz, P., (1968) Laws of Large Numbers , Academic Press, New York. Phelps, R.R., (1974) Dentability and extreme points in Banach spaces, J. Functional Anal. 16, 78-90. Rieffel, M.A., 1967 Dentable subsets of Banach spaces with application to a Radon-Nikodym Theorem, Functional Analysis, Thompson Book Co. (1968) The Radon-Nikodym Theorem for the Bochner integral, Trans. Amer. Math. Soc. 131, 466-487. Rosenthal, H.P., (1970) On the subspaces of Lp (p > 2) spanned by sequences of independent random variables, Israel J. Math. 9, 272-303. (1973) On subspaces of Lp , Annals of Math. 97, 344-373. (1976) Some applications of p-summing operators to Banach space theory, Studia Math. 58, 21-43. Ryll-Nardzewski, C., and Woyczy´ nski, W.A., (1975) Bounded muliplier convergence in measure of random vector series, Proceedings of the American Mathematical Society 53, 96-98. Scalora, F.S., (1961) Abstract martingale convergence theorem, Pacific J. math. 11, 347-374. Schaffer, J.J., and Sundaresan, K., (1970) Reflexivity and the girth of spheres, Math. Ann. 184, 163-168. Schwartz, L., (1969/70) La theoreme de dualite pour les applications radonifiantes, Sem. Schwartz, Exp. XXIV.

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(1969/70) Les applications o-radonifiantes dans les espaces de suites, Sem. Schwartz , Exp. XXVI. (1974) Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Tata Institute Monographs on Mathematics and Physics, Bombay. (1974) Les espaces de type et cotype 2 et leurs applications, d’apres Bernard Maurey, Ann. Inst. Fourier 24, 179-188. Sundaresan, K, and Woyczy´ nski, W.A., (1980) Laws of large numbers and Beck convexity in metric linear spaces, Journal of Multivariate Analysis 10, 442-459. Szankowski, A., (1974) On Dvoretzky’s theorem on almost spherical sections of convex bodies, Isreal J. mamth. 17, 325-338. Szarek, S.J., (1976) On the best constant in the Khinchin inequality, Studia Math. 58, 197-208. Szulga, J., (1977) On the Lr -convergence, r > 0, for n−1/r Sn in banach spaces, Bull. Polon. Acad. Sci. 25, 1011-1013. Szulga , J., and Woyczy´ nski, W.A., (1976) Convergence of submartingales in Banach lattices, Annals of Probability 4, 464-469. Tortrat, A., (1976) Sur les lois e(λ) dans les espaces vectorielles, Applications aux lois stables, Z. Wahr. verw. Geb. 27, 175-182. Tulcea, A.I., and Tulcea, C.I., (1962) Abstract ergodic theorems, Proc. Nat. Acad. Sci 48, 204-206. Tzafriri, L., (1972) Reflexivity in Banach lattices and their subspaces, J. Functional Anal. 10, 1-18. Uhl, Jr, J.J., (1972) A note on the Radon-Nikodym Property for Banach spaces, Rev. Roum. Mat. 17, 113-115. Vakhania, N.N.,

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(1971) Probability Distributions in Linear Spaces, Tibilisi. Warren, P., and Howell, J., (1976) A strong law of large numbers for orthogonal Banach spaces valued random variables, Springer Lecture Notes in Math. 526, 253-262. Woyczy´ nski, W.A., (1973) Random series and laws of large numbers in some Banach spaces, Teorya Veroyatnostej i Prim. 18, 371-377. (1974) Strong laws of large numbers in certain linear spaces, Ann. Institute Fourier 24, 205-223. (1975) Geometry and martingales in Banach spaces, Springer Lecture Notes in Math. 472, 229-276. (1975) A few remarks on the results of Rosinski and Suchanecki concerning unconditional convergence and C-spaces, S´eminaire Maurey-Schwartz (Paris), XXVII, 1-9. (1975) Laws of large numbers for vector valued martingales, Bulletin de l’Academie Polonaise des Sciences 23, 1199-1201. (1975) Asymptotic behavior of martingales in Banach spaces, in Probability in Banach Spaces, Oberwolfach, Springer’s Lecture Notes in Mathematics 526(1976), 273-284. (1975) A central limit theorem for martingales in Banach spaces, Bulletin de l’Academie Polonaise des Sciences 23, 917-920. (1977) Weak convergence to a Gaussian measure of martingales in Banach spaces, Symposia Math. 21, 319-331. (1978) Geometry and martingales in Banach spaces, Part II: Independent increments, Probability on Banach Spaces, J. Kuelbs, Editor, Marcel Dekker, New York, 267 -518. (1980) Tail probabilities of sums of random vectors in Banach spaces and related mixed norms, Conference on Measure Theory, Oberwolfach 1979, Springer’s Lecture Notes in Mathematics 794, 455-469. (1980) On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence, Probability and Mathematical Statistics 1, 117-131. (1983) Survey of asymptotic behavior of independent random vectors and general martingales in Banach spaces, Fourth Conference on Probability Theory in Banach Spaces, Springer’s Lecture

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Notes in Mathematics 990, 215-220. Zinn, J., (1977) A note on the central limit theorem in Banach spaces, Ann. Prob. 5, 283-286.

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Index

315

strongly exposed point, 21 sub-martingales, 31 super-property of Banach space, 51 super-reflexive Banach space, 52 Three Series Theorem, 157 Tortrat Theorem, 4 trees in B-convex spaces, 204 uniform convexity, 41 uniform smoothness, 41 Weak Laws of Large Numbers, 137, 189 weighted random sums, 194

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Index

313 ✐

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314

Geometry and Martingales in Banach Spaces Kwapie´ n Theorem, 4

absolutely summing operators, 11 asymptotics of random sums, 131 Banach-Mazur distance, 6 Beck convexity, 173 Brunel-Sucheston Theorem, 9 Brunk-Prokhorov Strong Law, 209 Central Limit Theorem, 88, 159 Closed Interval Lemma, 8 Comparison Theorem, 3 Contraction Principle, 3 cotype of Banach space, 63 dentable set, 21 dentable Banach space, 21 Dvoretzky Theorem, 6 Dvoetzky-Rodgers Lemma, 9

Landau-Shepp-Fernique Theorem, 5 lattice bounded operator, 31 Law of the Iterated Logerithm, 88, 166 Laws of Large Numbers, 81 Lorentz space, 1 Marcinkiewicz-Zygmund Theorem, 207 martingale convergence property, 7 Maurey Theorem, 14 Maurey-Rosenthal Theorem, 15 modulus of convexity,

ergodic properties of B-convex 9 spaces, 198 exchangeable random vectors, 2

Nikishin Lemma, 13 Nikishin Theorem, 14 norm invariant under spreading,

order bounded set, 14 finite representability, 6 finite tree property, 52 Fubini inequality, 1

Paley-Walsh martingale, 32 Pietsch Theorem, 12 pre-Gaussian random vectors, 61

Gaussian random vectors, 155 Rademacher cotype, 66 Rademacher type, 93 Radon-Nikodym Property, 25 Ramsey Lemma, 9 random integrals, 141 random series, 124 reflexivity of B-convex spaces,

Hoffman-Jorgensen Theorem, 3, 5 infinitely divisible distributions, 88 infracotype of Banach space, 63 infratype of Banach space, 93 Ito-Nisio Theorem, 2 James Theorem, 7 Johnson Lemma, 15 Kahane Theorem, 4 Khinchine Inequality i Kolmogorov’s Inequality, 157 Krivine Theorem, 7

183 Schwartz Theorem, 5 sign invariance, 2 spaces of type 2, 151 stable type, 114 Strong Law of Large Numbers, 45, 131, 137, 189 strong laws for martingales, 217

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