Nonconventional Limit Theorems and Random Dynamics 9789813235007, 9813235004

Citation preview

Nonconventional Limit Theorems and Random Dynamics

10849hc_9789813235007_tp.indd 1

29/1/18 6:58 PM

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

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

Nonconventional Limit Theorems and Random Dynamics

Yeor Hafouta

Hebrew University of Jerusalem, Israel

Yuri Kifer

Hebrew University of Jerusalem, Israel

World Scientific NEW JERSEY

•

LONDON

10849hc_9789813235007_tp.indd 2

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TAIPEI

•

CHENNAI

•

TOKYO

29/1/18 6:58 PM

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

Library of Congress Cataloging-in-Publication Data Names: Hafouta, Yeor, author. | Kifer, Yuri, 1948– author. Title: Nonconventional limit theorems and random dynamics / by Yeor Hafouta (Hebrew University of Jerusalem, Israel), Yuri Kifer (Hebrew University of Jerusalem, Israel). Description: New Jersey : World Scientific, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2018000570 | ISBN 9789813235007 (hardcover : alk. paper) Subjects: LCSH: Limit theorems (Probability theory) | Random dynamical systems. Classification: LCC QA273.67 .H335 2018 | DDC 519.2--dc23 LC record available at https://lccn.loc.gov/2018000570

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

For any available supplementary material, please visit http://www.worldscientific.com/worldscibooks/10.1142/10849#t=suppl

Desk Editor: Benny Lim Printed in Singapore

Benny - 10849 - Nonconventional Limit Theorems.indd 1

06-03-18 9:51:58 AM

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Preface

Nonconventional ergodic theorems dealt with the limits of expressions haPN ving the form 1/N n=1 T q1 (n) f1 · · · T q` (n) f` where T is a measure preserving transformation, fi ’s are bounded measurable functions and qi ’s are linear or polynomial functions taking on integer values on integers. These results were used in the ergodic theory proof of Szemer´edi’s theorem on arithmetic progressions (see [21] and [22]) but since then their various extensions became a topic of its own interest. From the probabilistic point of view ergodic theorems can be regarded as laws of large numbers and once they are established it is natural to inquire about other limit theorems of probability such as the central limit theorem, Poisson limit theorem, large deviations etc. During the last decade these questions were studied in a number of papers (see, for instance, [29], [30], [41]-[43] and [45]-[47]). These limit theorems were studied for nonconventional sums of the form N X SN = F (ξq1 (n) , ..., ξq` (n) ) n=1

where {ξn , n ≥ 0} is a sufficiently fast mixing process with some stationarity properties, F is a Borel function with some regularity properties and qj ’s are functions taking on integer values on integers and satisfying certain conditions (for instance, linear or polynomial ones). These results hold true, in particular, when ξ1 , ξ2 , ξ3 , ... form a Markov chain satisfying some form of the Doeblin condition while in the dynamical systems setup these results are applicable to topologically mixing subshifts of finite type considered with appropriate Gibbs measures with implications to systems having corresponding symbolic representations such as C 2 Axiom A diffeomorphism (in particular, Anosov) in a neighborhood of an attractor or an expanding C 2 endomorphism of a Riemannian manifold. v

page v

March 13, 2018 15:46

vi

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Some of number theoretic (combinatorial) applications of such limit theorems can be described in the following way. For each point ω ∈ [0, 1) consider its base m or continued fraction expansions with digits ξk (ω), k = 1, 2, .... Next, count the number SN (ω) of those `-tuples q1 (n), ..., q` (n), n ≤ N for which, say, ξqj (n) (ω) = aj , j = 1, ..., ` for some fixed integers a1 , ..., a` with the most notable qj ’s forming an arithmetic progression qj (n) = jn, j = 1, ..., `. Now we can write SN (ω) =

N Y ` X

δaj ξqj (n) (ω)

n=1 j=1

where δks = 1 if k = s and = 0 otherwise, arriving at the above setup. To make ξk ’s random variable, we supply the unit interval with an appropriate probability measure such as the Lebesgue measure for base m expansions and the Gauss measure for continued fraction expansions. We mention also another application to limit theorems for numbers of certain patterns in random sets. Namely, define a random set Γ in positive integers via a sequence of random variables ξ1 , ξ2 , ... taking on values 0 or 1 by saying that n ∈ Γ if and only if ξn = 1. Then, for instance, SN =

N Y ` X

ξjn

n=1 j=1

counts the number of arithmetic progressions of length ` in Γ starting at n and having the step n where n is between 1 and N . Most of previous papers dealing with the central limit theorem for the above sums relied on the well known martingale approximation method and in Chapter 1 we return to this question showing how Stein’s method can be successfully applied in the nonconventional situation. The advantage of this method is that we obtain simultaneously the speed of convergence in the corresponding central limit theorem improving estimates from [30], so that the central limit theorem and a version of the Berry-Esseen theorem come together. In Chapter 2 we return to the study of the nonconventional local (central) limit theorem (LLT) which was considered first in [29]. The method of [29] was restricted to Markov chains with certain regularity of their transition probability (a version of the Doeblin condition) which excluded applications to important classes of dynamical systems such as subshifts of finite type. In order to overcome this restriction we had to extend the Ruelle-Perron-Frobenius (RPF) type theorems to random complex operators which is developed in Part 2 of the manuscript. After establishing

page vi

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Preface

LNcov

vii

this LLT, we present in Section 2.10 what seems to be a new approach to certain type of random dynamical systems which stems from the proof of the nonconventional LLT and is based on certain regularity conditions of the system around periodic points. In Chapter 3 we consider nonconventional arrays of the form SN =

N X

F (ξq1 (n,N ) , ..., ξq` (n,N ) )

n=1

where now the summands themselves depend on the number N of summands. Ergodic theorems for such arrays were recently studied in [48]. Here we restrict ourselves to the linear case qj (n, N ) = pj n + qj N and obtain under certain conditions the strong law of large number, the central limit theorem and the Poisson limit theorem for such expressions. Though our motivation for the results in Part 2 came from the need to extend the nonconventional local limit theorem to additional important classes of processes, the RPF type theory for random complex operators developed there is certainly interesting by its own. For instance, in Chapter 7 we apply it in order to obtain, for the first time, a version of the Berry-Esseen and the local limit theorems for processes in random dynamical environment, where the proof of the latter involves ideas from the proof of the nonconventional LLT, as well. Extensions of the nonconventional LLT (and CLT) for such processes will also be discussed, and this theory should find additional applications. The RPF theorem for products of real (random and deterministic) operators is a well-studied topic being the main step in the thermodynamic formalism constructions for many families of operators (see [10], [37], [40] and [53]), but this is the first exposition of the RPF theorem for random complex operators which emerge naturally in the study of the nonconventional LLT. The RPF theorem for a single deterministic complex operator was established, for instance, in [28] and [54], where both works rely on quasi compactness of a fixed real operator and the perturbation theory (see [36]). Recently, a complex version of the Hilbert metric was introduced and the corresponding theory of cones contractions was established yielding the complex deterministic RPF theorem, as well (see [18], [19] and [58]). In Part 2 we adapt this theory to the situation of complex random operators and as an application we produce the random complex thermodynamic formalism type constructions.

page vii

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

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Contents

Preface

v

Nonconventional Limit Theorems

1

1.

3

Stein’s method in the nonconventional setup 1.1 1.2

1.3

1.4

1.5

1.6

Introduction: local (strong) dependence structure . . . . Stein’s method for normal aproximation . . . . . . . . . 1.2.1 A short introduction . . . . . . . . . . . . . . . . 1.2.2 Normal approximation for graphical indexation . 1.2.3 Weak local dependence coefficients . . . . . . . . 1.2.4 Relations with more familiar mixing coefficients Nonconventional CLT with convergence rates . . . . . . 1.3.1 Assumptions and main results . . . . . . . . . . 1.3.2 Asymptotic variance . . . . . . . . . . . . . . . . 1.3.3 CLT with convergence rate . . . . . . . . . . . . 1.3.4 The associated strong dependency graphs . . . . 1.3.5 Expectation estimates . . . . . . . . . . . . . . . 1.3.6 Proof of Theorem 1.3.7 . . . . . . . . . . . . . . 1.3.7 Back to graphical indexation . . . . . . . . . . . General Stein’s estimates: proofs . . . . . . . . . . . . . 1.4.1 Proof of Theorem 1.2.1 . . . . . . . . . . . . . . 1.4.2 Proof of Theorem 1.2.2 . . . . . . . . . . . . . . Stein’s method for diffusion approximations . . . . . . . 1.5.1 A functional CLT via Stein’s method . . . . . . 1.5.2 Finite dimensional convergence rate . . . . . . . A nonconventional functional CLT . . . . . . . . . . . . ix

. . . . . . . . . . . . . . . . . . . . .

3 5 5 6 8 10 11 11 13 15 16 18 24 26 32 32 36 43 43 48 51

page ix

March 13, 2018 15:46

x

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

1.7

2.

ws-book9x6

Extensions to nonlinear indexes . . . . . . . . . . . . . . . 1.7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . 1.7.2 Another example with nonlinear indexes . . . . .

56 56 58

Local limit theorem

65

2.1 2.2 2.3

65 67

2.4

2.5

2.6 2.7

2.8

2.9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Local central limit theorem via Fourier analysis . . . . . . Nonconventional LLT for Markov chains by reduction to random dynamics . . . . . . . . . . . . . . . . . . . . . . . Markov chains with densities . . . . . . . . . . . . . . . . 2.4.1 Basic assumptions and CLT . . . . . . . . . . . . 2.4.2 Characteristic functions estimates . . . . . . . . . Markov chains related to dynamical systems . . . . . . . . 2.5.1 Locally distance expanding maps and transfer operators . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Inverse branches, the pairing property and periodic points . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Thermodynamic formalism constructions and the associated Markov chains . . . . . . . . . . . . . . 2.5.4 Relations between dynamical systems and Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Mixing and approximation assumptions . . . . . . 2.5.6 Asymptotic variance and the CLT . . . . . . . . . Statement of the local limit theorem . . . . . . . . . . . . The associated random transfer operators . . . . . . . . . 2.7.1 Random complex RPF theorem . . . . . . . . . . 2.7.2 Distortion properties . . . . . . . . . . . . . . . . 2.7.3 Reduction to random dynamics . . . . . . . . . . . Decay of characteristic functions for small t’s . . . . . . . 2.8.1 The random pressure function . . . . . . . . . . . 2.8.2 The derivatives of the pressure . . . . . . . . . . . 2.8.3 Pressure near 0 . . . . . . . . . . . . . . . . . . . . 2.8.4 Norms estimates: employing the pressure . . . . . 2.8.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . Decay of characteristic functions for large t’s . . . . . . . . 2.9.1 Basic estimates and strategy of the proof . . . . . 2.9.2 Probabilities of large number of visits to open sets 2.9.3 Segments of periodic orbits . . . . . . . . . . . . . 2.9.4 Parametric continuity of transfer operators . . . .

71 74 74 76 81 81 82 84 86 88 89 90 92 93 95 97 99 99 101 103 104 111 113 114 115 118 121

page x

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Contents

xi

2.9.5

3.

Quasi compactness of Rit , lattice and non-lattice cases . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.6 Norms estimates . . . . . . . . . . . . . . . . . . . 2.10 The periodic point approach to quenched and annealed dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Extensions to dynamical systems . . . . . . . . . . . . . . 2.11.1 Subshifts of finite type . . . . . . . . . . . . . . . 2.11.2 The strategy of the proof . . . . . . . . . . . . . . 2.11.3 Local limit theorem: one sided case . . . . . . . . 2.11.4 Two sided case . . . . . . . . . . . . . . . . . . . . 2.11.5 Reduction to one sided case . . . . . . . . . . . . .

126 128 128 129 130 131 132

Nonconventional arrays

137

3.1 3.2

3.3

3.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Strong law of large numbers for nonconventional arrays . 3.2.1 Setup and the main result . . . . . . . . . . . . 3.2.2 Auxiliary lemmas . . . . . . . . . . . . . . . . . 3.2.3 Ordering and decompositions . . . . . . . . . . . 3.2.4 Proof of Theorem 3.2.2 . . . . . . . . . . . . . . Central limit theorem . . . . . . . . . . . . . . . . . . . 3.3.1 I.i.d. case . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Convergence of covariances . . . . . . . . . . . . 3.3.3 The number of solutions . . . . . . . . . . . . . 3.3.4 Martingale approximation . . . . . . . . . . . . . Poisson limit theorems for nonconventional arrays . . . . 3.4.1 Preliminaries and main results . . . . . . . . . . 3.4.2 Stationary sequences . . . . . . . . . . . . . . . 3.4.3 Poisson limits for subshifts . . . . . . . . . . . .

. . . . . . . . . . . . . . .

123 124

137 139 139 141 144 146 149 149 152 156 157 159 159 162 166

Thermodynamic Formalism for Random Complex Operators and applications 171 4.

Random complex Ruelle-Perron-Frobenius theorem via cones contractions 4.1 4.2 4.3

Preliminaries . . . . . . . . . . . Main results . . . . . . . . . . . . Block partitions and RPF triplets 4.3.1 Reverse block partitions .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

173 . . . .

. . . .

. . . .

. . . .

. . . .

173 175 177 177

page xi

March 13, 2018 15:46

xii

4.5 4.6

LNcov

4.3.2 Forward block partition and dual operators . . . . 4.3.3 RPF triplets . . . . . . . . . . . . . . . . . . . . . Exponential convergences . . . . . . . . . . . . . . . . . . 4.4.1 Taylor reminders and important bounds . . . . . . 4.4.2 Exponential convergences . . . . . . . . . . . . . . 4.4.3 Additional types of exponential convergences . . . Uniqueness of RPF triplets . . . . . . . . . . . . . . . . . The largest characteristic exponents . . . . . . . . . . . . 4.6.1 Analyticity of the largest characteristic exponent around 0 . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 The pressure function . . . . . . . . . . . . . . . . 4.6.3 Proof of Theorem 4.6.3 . . . . . . . . . . . . . . .

Application to random locally distance expanding covering maps 5.1 5.2 5.3 5.4 5.5 5.6

5.7

5.8 5.9 5.10 5.11 5.12 5.13 6.

Nonconventional Limit Theorems and Random Dynamics

Nonconventional Limit Theorems and Random Dynamics

4.4

5.

ws-book9x6

Random locally expanding covering maps . . . . . . . . . Transfer operators . . . . . . . . . . . . . . . . . . . . . . Real and complex cones . . . . . . . . . . . . . . . . . . . RPF triplets . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of cones: proof of Theorem 5.3.1 . . . . . . . . Properties of transfer operators: proof of Theorem 5.4.1 (i) 5.6.1 Continuity and analyticity . . . . . . . . . . . . . 5.6.2 Analyticity in z . . . . . . . . . . . . . . . . . . . Real Hilbert metric estimates . . . . . . . . . . . . . . . . 5.7.1 General estimates . . . . . . . . . . . . . . . . . . 5.7.2 Real cones invariances and diameter of image estimates . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of real and complex operators . . . . . . . . . Complex image diameter estimates . . . . . . . . . . . . . Real RPF triplets and Gibbs measures . . . . . . . . . . . Complex Gibbs functionals . . . . . . . . . . . . . . . . . The largest characteristic exponents . . . . . . . . . . . . Extension to the unbounded case for minimal systems . .

Application to random complex integral operators 6.1 6.2 6.3

180 183 184 185 186 188 189 191 192 193 194

197 197 201 202 203 205 210 210 211 213 213 216 218 221 223 226 228 230 231

Integral operators . . . . . . . . . . . . . . . . . . . . . . . 231 Real and complex cones . . . . . . . . . . . . . . . . . . . 233 The RPF theorem for integral operators . . . . . . . . . . 234

page xii

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Contents

6.4 6.5 6.6 6.7 7.

xiii

Properties of cones: proof of Theorem 6.2.1 . . . . . . . Real cones: invariance and diameter of image estimates Comparison of real and complex operators . . . . . . . . Complex image diameter estimates . . . . . . . . . . . .

. . . .

235 237 239 240

Limit theorems for processes in random environment

243

7.1

243 244 244 246 251 253 253

7.2 7.3 7.4

7.5

The “conventional” case: preliminaries and main results . 7.1.1 Self normalized Berry-Esseen theorem . . . . . . . 7.1.2 Local limit theorem . . . . . . . . . . . . . . . . . Pressure near 0 . . . . . . . . . . . . . . . . . . . . . . . . A fiberwise (self normalized) Berry-Esseen theorem: proof A fiberwise local (central) limit theorem: proof . . . . . . 7.4.1 Characteristic functions estimates for small t’s . . 7.4.2 Characteristic functions estimates for large t’s: covering maps . . . . . . . . . . . . . . . . . . . . . 7.4.3 Characteristic functions estimates for large t’s: Markov chains . . . . . . . . . . . . . . . . . . . . 7.4.4 An extension . . . . . . . . . . . . . . . . . . . . . Nonconventional limit theorems for processes in random environment . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Central limit theorem . . . . . . . . . . . . . . . . 7.5.2 Local limit theorem . . . . . . . . . . . . . . . . .

Appendix Appendix A A.1 A.2

254 255 257 257 257 263

269 Real and complex cones

Real cones and real Hilbert metrics . . . . . . . . . . Complex cones and complex Hilbert metrics . . . . . A.2.1 Basic notions . . . . . . . . . . . . . . . . . . A.2.2 The canonical complexification of a real cone A.2.3 Apertures and contraction properties . . . . A.2.4 Comparison of real and complex operators . A.2.5 Further properties of complex dual cones . .

271 . . . . . . .

. . . . . . .

. . . . . . .

271 272 272 273 274 275 276

Bibliography

279

Index

283

page xiii

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

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

PART 1

Nonconventional Limit Theorems

1

LNcov

page 1

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

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Chapter 1

Stein’s method in the nonconventional setup

1.1

Introduction: local (strong) dependence structure

Let ξ = {ξn : n ≥ 0} be a sequence of random variables and let ` ∈ N. ConPN sider the sums SN = n=1 F (Ξn ), N ∈ N, where Ξn = (ξn , ξ2n , ..., ξ`n ) and F is a function which satisfies some regularity conditions. The summands in SN are long range dependent even when ξ is a sequence of independent random variables. For instance, Ξn and Ξkn are strongly dependent for any n and 1 ≤ k ≤ `. Still, each Ξn can depend only on the random vectors Ξm for m’s which are members of { in j : 1 ≤ i, j ≤ `}. When this type of dependence structure occurs, it is convenient to think about the indexes 1, 2, ..., N as vertices of a graph, and from this point of view the random variables {F (Ξn ) : 1 ≤ n ≤ N } are “locally dependent”. More precisely, we will say that n and m are “connected” if in = jm for some 1 ≤ i, j ≤ `. Then each n can be connected to at most `2 integers m, and the random vectors Ξn ˜ n = {Ξm : m and n are not connected} are independent. When ξn ’s and Ξ are not independent but instead satisfy some mixing (weak dependence) conditions then we fix some l ≥ 1, which is small relatively to N , and say that n and m are “connected” if |in − jm| < l for some 1 ≤ i, j ≤ `. Then each n is connected to at most `2 (2l+1) integers m, and the random vectors ˜ n defined in a similar to the above way are “weakly dependent”. Ξn and Ξ We see then that the summands of SN are “locally (strongly) dependent” in the sense that each summand F (Ξn ) can strongly depend only on the summands indexed by members from the neighborhood of n which is the set of all m’s connected with n. This local dependence structure is one of the main situations in which Stein’s method is effective, and, in fact, is not restricted to the above specific arithmetic progression structure of the indexes n, 2n, ..., `n. For instance, such local dependence structure occurs 3

page 3

March 13, 2018 15:46

4

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

PN also for sums of the form n=1 F (ξq1 (n) , ξq2 (n) , ..., ξq` (n) ) where qi ’s are polynomials with integer coefficients whose leading coefficients are positive, and, more generally, when the qi ’s satisfy some growth conditions. In this situation we will say that n and m are connected if |qi (n) − qj (m)| < l for some 1 ≤ i, j ≤ `. In this chapter we will use the local dependence structure described above in order to obtain a central limit theorem (CLT) for the norma1 lized sums ZN = N − 2 SN when the sequence ξ1 , ξ2 , ξ3 , ... satisfies some mixing and moment conditions and when F satisfies some regularity properties. Under rather general assumptions we will obtain almost optimal 1 convergence rate of order N − 2 ln2 (N + 1). These results improve the rates obtained in [26] and [30]. Our proofs require adaptation of the arguments ˜ n defined above are not indepenof [15] to the situation when Ξn and Ξ dent but only weakly dependent and the following section is devoted to formulation of such results. We will also obtain a functional CLT for the 1 random functions ZN (t) = N − 2 S[N t] , and the necessary background for Stein’s method in the situation of random functions is presented in Section 1.5. We will use two main probability metrics. Let X and Y be two random variables. The Wasserstein distance between their laws L(X) and L(Y ) is defined by dW (L(X), L(Y )) = sup{|Eh(X) − Eh(Y )| : h ∈ Lip1 } (1.1.1) where Lip1 is the class of Lipschitz functions with constant 1. The Kolmogorov (uniform) distance between L(X) and L(Y ) is defined by dK (L(X), L(Y )) = sup |P (X ≤ x) − P (Y ≤ x)|. (1.1.2) x∈R

Then by Theorem 3.1 in [3] (see also [24]) pwe have dK (L(X), L(Y )) ≤ (1 + sup |G0 |) dW (L(X), L(Y )) if Y has a distribution function G with density G0 . We will also use the following.

(1.1.3)

Lemma 1.1.1. Let X and Y be two random variables defined on the same probability space. Let Z be a random variable with density ρ bounded from above by some constant c > 0. Then, dK (L(Y ), L(Z)) ≤ 3dK (L(X), L(Z)) + 4ckX − Y kL∞ and for any b ≥ 1, 1−

1

dK (L(Y ), L(Z)) ≤ 3dK (L(X), L(Z)) + (1 + 4c)kX − Y kLb b+1 . The second inequality is proved in Lemma 3.3 in [30], while the proof of the first inequality goes in the same way as the proof of that Lemma 3.3, taking in (3.2) from there δ = kX − Y kL∞ .

page 4

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

1.2

LNcov

5

Stein’s method for normal aproximation

In the past 50 years Stein’s method has become one of the main methods for approximating a distribution on the real line by the standard normal distribution and in particular for proving central limit theorems with some convergence rate. For readers’ convenience, we begin this section with a short introduction to the method. 1.2.1

A short introduction

Consider the operator A which acts on differentiable functions f : R → R by the formula Af (w) = f 0 (w) − wf (w). Then, (see [3] and [62]) a random variable W has the standard normal distribution if and only if EAf (W ) = 0 for any bounded function f with a bounded derivative. The operator A is often referred to as the Stein operator associated with the standard normal distribution. The idea behind Stein’s method is to quantify this characterization of the normal distribution. Let h be a Liphshitz function with constant 1, and consider the ordinary differential equation Af (w) = h(w) − Eh(Z)

(1.2.1)

where Z is a standard normal random variable. Then (see [3] and [62]) there exists a twice differentiable function fh which solves (1.2.1) and (in particular) satisfies
max sup |fh |, sup |fh0 |, sup |fh00 | ≤ 4 where as usual sup |g| stands for the supremum of the absolute value of a real valued function g. Let W be a random variable whose first absolute moment E|W | is finite. Plugging in w = W in (1.2.1) we arrive at Eh(W ) − Eh(Z) = EAfh (W ) and therefore the following inequality holds true: dW (L(W ), N ) (1.2.2)  0 0 00 ≤ 4 sup |E[f (W ) − W f (W )]| : max(sup |f |, sup |f |, sup |f |) ≤ 1 where N stands for the standard normal law. We see then that estimating the difference of expectations of Lipschitz functions requires estimation of expectations of the form |EAf | for some class of differentiable functions. Many situations require estimation of the Kolmogorov distance between L(W ) and N . Such estimates, of course, follow from (1.1.3) and (1.2.2),

page 5

March 13, 2018 15:46

6

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

but this may cause a loss of accuracy. For instance (see [3]) when applied to sums of n i.i.d. random variable with finite third absolute moments, Stein’s 1 method yields optimal upper bound of order n− 2 in the Wasserstein metric which does not lead to an optimal upper bound in the Kolmogorov metric. The idea behind obtaining more accurate estimates of the Kolmogorov distance is to approximate the indicator function I(w ≤ x) of the ray (∞, x] by a piecewise linear function h and then approximating |EAfh | (see (1.4.17) and (1.4.19)). Such approximations, often referred to as “smoothing inequalities”, are used in the proof of (1.1.3), but the proof in [3] does not take into account the specific form of the function fh . This function satisfies some fine regularity properties (see (1.4.20) and (1.4.21)), and exploiting these properties requires some concentration inequality (see Proposition 1.4.2) which means controlling probabilities of the form P (a ≤ W ≤ b). The proof of Theorem 1.2.2 begins with a general description of this concentration inequality approach, so the readers who are interested to see some of the technical details are referred to the beginning of this proof. We also refer the readers to a more detailed discussion about this concentration inequality approach in [3]. 1.2.2

Normal approximation for graphical indexation

This section is devoted to formulation of abstract normal-approximation theorems in the situation of graphical indexation, which will be effective in the situation of local (strong) dependence. There are many ways to quantify the amount of dependence, and in Section 1.2.4 we will see that our results can be reformulated in terms of familiar mixing (weak dependence) coefficients. For the sake of readability, the results are not formulated under the most general (moment) assumptions, and the readers who are interested to see more general versions of these results are referred to (1.4.19) and (1.4.38) in Section 1.4. Let G = (E, V ) be a finite graph (where E is the set of edges and V is the set of vertices) and let dG (·, ·) be its associated shortest path distance (when v and u are not connected by a path we set dG (v, u) = ∞). Denote by B(v, r) a ball of radius r ≥ 0 around v ∈ V . For any v ∈ V set Nv = B(1, v) = {v} ∪ {u ∈ V : (u, v) ∈ E}.

(1.2.3)

Let {Xv , v ∈ V } be a collection of centered random variables defined on P some probability space (Ω, F, P ) and set W = v∈V Xv . For any A ⊂ V set XA = {Xi : i ∈ A} and Ac = V \ A. We denote by N the standard normal law on R and by |Γ| the cardinality of a set Γ.

page 6

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

7

Theorem 1.2.1. Let γ4 > 0, c0 , D ≥ 1 and 1 ≤ ρ ≤ 3 be such that E|Xv |4 ≤ γ44 , |Nv | ≤ D and |B(v, 3)| ≤ c0 Dρ for any v ∈ V . Then 3 X 1 ρ+1 1 δi + 4|EW 2 − 1| dW (L(W ), N ) ≤ 12c02 Dmax(2, 2 ) (γ43 |V | + γ42 |V | 2 ) + 4 i=1

where X X X  X EXv Xu |, Xu ) : sup |g| ≤ 1 , δ2 = | EXv g( δ1 = sup v∈V

v∈V u∈Nvc

u∈Nvc

(δ3 )2 = |

X

Cov(Xv1 Xu1 , Xv2 Xu2 )|

(v1 ,u1 ,v2 ,u2 )∈Γ

and Γ = {(v1 , u1 , v2 , u2 ) : d(v1 , v2 ) > 3 and ui ∈ Nvi , i = 1, 2} ⊂ V 4 = V × V × V × V which satisfies |Γ| ≤ |V |2 D2 . Note that we can always take c0 = 1 and ρ = 3, but for certain graphs the cardinality of balls B(v, 3) of radius 3 is of order D, and in this case we can choose ρ = 1. Relying on (1.1.3), Theorem 1.2.1 yields estimates of the Kolmogorov distance between L(W ) and N , as well, but (as mentioned earlier) this cannot yield close to optimal upper bounds. For instance, when W is a sum of i.i.d. random variables and EW 2 = 1 then considering a graph G with no edges we have Nv = {v}, and in this situation all δi ’s vanish and we get 1 an upper bound of order |V |− 2 , which only implies that dK (L(W ), N ) ≤ 1 C|V |− 4 for an appropriate constant C. Theorem 1.2.2. Let γ4 > 0, c0 , D ≥ 1 and 1 ≤ ρ ≤ 3 be as in Theorem 1.2.1. Then 5 X 1 1 dK (L(W ), N ) ≤ 16c0 D1+ρ (|V |γ43 + |V | 2 γ42 ) + 4|EW 2 − 1| + 8 δi + 3δ62 i=1

where δ1 , δ2 and X δ3 are defined in Theorem 1.2.1,
δ4 = Cov |W − Zv |, |Xv min(Yv2 , 1)| and for i = 5, 6, (1.2.4) v∈V

X δi =

X


EXv Xu gi (Yv , Yu ) − EXv Xu∗ gi (Yv , Yu∗ ) (1.2.5)

v∈V u:d(v,u)>3


2 where g5 (a, b) = I{(a,b):ab≥0} min(|a|, |b|, 1), g6 (a, b) = g5 (a, b) , IA stands P P for the indicator function of a set A, Yv = u∈Nv Xu , Zv = k:d(v,k)≤2 Xk and (Xu∗ , Yu∗ ) is a copy of (Xu , Yu ) which is independent of (Xv , Yv ). Both theorems follow from the arguments in the proof of Theorem 2.1 in [15] and their proofs are postponed until Section 1.4.

page 7

March 13, 2018 15:46

ws-book9x6

8

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

1.2.3

Weak local dependence coefficients

The terms δi , i = 1, ..., 6 in Theorem 1.2.2 vanish when XA and XB are independent for some specific pairs of sets A and B which are not connected by a single edge. In general, these terms are small when XA and XB are “weakly dependent” for the appropriate A’s and B’s. In the following we will estimate δi ’s in terms of various dependence coefficients. First, for any A, B ⊂ V and 1 ≤ p, q ≤ ∞ we will measure the dependence between XA and XB via the quantities εp,q (A, B) 
= sup |Cov g(XA ), h(XB ) | : max(kg(XA )kLp , kh(XB )kLq ) ≤ 1 ,  ∗ ∗ E1 (A, B) = sup |EG(XA , XB ) − EG(XA , XB )| : E|G(XA , XB )| ≤ 1  ∗ and E∞ (A, B) = sup |EG(XA , XB ) − EG(XA , XB )| : kGk∞ ≤ 1 (1.2.6) ∗ is a copy of XB which is independent of XA , and kGk∞ = where XB kGk∞,A,B stands here for essential supremum of the absolute value of a real valued function G(·, ·) with respect to the sum of the laws of (XA , XB ) ∗ and (XA , XB ). We remark that εp,q is nonincreasing in both p and q and that E∞ ≤ E1 . The first approximation of the δi ’s is given in the following simple lemma.

Lemma 1.2.3. Suppose that maxv∈V kXv kLr ≤ γr for any 1 ≤ r ≤ ∞ with some 0 < γr ≤ ∞. Then X X X δ1 ≤ γp εp,∞ ({v}, Nvc ), δ2 ≤ γp γq εp,q ({u}, {v}), (1.2.7) v∈V u∈Nvc

v∈V

X

(δ3 )2 ≤

γp2 γq2 ε p2 , q2 ({v1 , u1 }, {v2 , u2 }), (1.2.8)

(v1 ,u1 ,v2 ,u2 )∈Γ

and δ4 ≤

X

(kW kLp + D2 γp )γq εp,q (V \ B(v, 2), Nv ) (1.2.9)

v∈V

for any p, q ≥ 2, and δ5 ≤

X

X

Dγ33 E1 (Nv , Nu ) and

(1.2.10)

D2 γ44 E1 (Nv , Nu ).

(1.2.11)

v∈V u:d(v,u)>3

δ6 ≤

X

X

v∈V u:d(v,u)>3

page 8

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

9

Proof. Let u, v ∈ V . The inequalities in (1.2.7) follow directly from our assumption about the γr ’s. By the H¨older inequality for any r ≥ 2, kXv Xu kL r2 ≤ kXv kLr kXu kLr ≤ γr2 . Using this estimate with r = p and r = q and the definition of ε p2 , q2 we obtain (1.2.8). Next, (1.2.9) follows from the estimates kXv min(Yv2 , 1)kLq ≤ kXv kLq and kW − Zv kLp ≤ kW kLp + D2 γp , our assumption about the γr ’s and the assumption that |Nv | ≤ D. Finally, by the definitions of g5 and g6 , the H¨older inequality and the inequality |Nv | ≤ D, kXv Xu∗ g5 (Yv , Yu∗ )kL1 ≤ kXv Xu∗ Yv kL1 ≤ kXv kL3 kXu kL3 kYv kL3 ≤ Dγ33 and kXv Xu∗ g6 (Yv , Yu∗ )kL1 ≤ kXv Xu∗ Yv2 kL1 ≤ kXv Xu kL2 kYv2 kL2 ≤ kXv kL4 kXu kL4 kYv k2L4 ≤ γ44 D2 and inequalities (1.2.10) and (1.2.11) follow. Observe that E∞ (A, B) ≤ E1 (A, B). In certain situations it will be easier to estimate E∞ (Nv , Nv ) when d(u, v) > 3, and in this case the following lemma will be useful. Lemma 1.2.4. Let p > 2 and γp > 0 be such that kXv kLp ≤ γp for any v ∈ V . Then, X
1− p2 . (1.2.12) max(δ5 , δ6 ) ≤ 5γp2 E∞ (Nv , Nu ) v,u∈V :d(v,u)>3

Proof. For any random variable X and R > 0 set X (R) = XI{|X|≤R} , where I{|X|≤R} is the random variable which equals 1 when |X| ≤ R and 0 otherwise. Then by the H¨older and Markov inequalities, for any two random variables X and X0 , (R)

(R)

(R)

E|XX0 − X (R) X0 | ≤ E|X(X0 − X0 )| + E|X0 (X − X (R) )| (R)

= E|XX0 I{|X0 |≤R} | + E|X0 XI{|X|≤R} | 2

2

≤ kXX0 kL p2 (P (|X0 | > R))1− p + kXX0 kL p2 (P (|X| > R))1− p
−(p−2) p−1 ≤ kXkLp kX0 kp−1 . Lp + kXkLp kX0 kLp R Therefore, taking into account that sup |g5 | ≤ 1, for any v, u ∈ V and

page 9

March 13, 2018 15:46

10

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

R > 0 we have E|Xv Xu g5 (Yv , Yu ) − Xv(R) Xu(R) g5 (Yv , Yu )| ≤ E|Xv Xu − Xv(R) Xu(R) | ≤ 2γpp R−(p−2) and
(R) E|Xv Xu∗ g5 (Yv , Yu∗ ) − Xv(R) Xu∗ g5 (Yv , Yu∗ )| ≤ 2γpp R−(p−2) . Next, by the definition of E∞ (Nv , Nu ),
EXv(R) Xu(R) g5 (Yv , Yu ) − EXv(R) Xu∗ (R) g5 (Yv , Yu∗ )) ≤ R2 E∞ (Nv , Nu ).
− 1 Taking R = Rv,u = γp E∞ (Nv , Nu ) p , we derive from the above estimates that EXv Xu g5 (Yv , Yu ) − EXv Xu∗ g5 (Yv , Yu∗ ) ≤ R2 E∞ (Nv , Nu ) (1.2.13)
1− p2 . +4γpp R−(p−2) = 5γp2 E∞ (Nv , Nu ) Note that such choice of R is possible only when E∞ (Nv , Nu ) > 0, but when E∞ (Nv , Nu ) = 0 then XNv and XNu are independent which implies that the left-hand side of (1.2.13) vanishes, and so (1.2.13) trivially holds true. We conclude from the above estimates and the definition of δ5 that X
1− p2 δ5 ≤ 5γp2 E∞ (Nv , Nu ) . v,u∈V :d(v,u)>3

Repeating the above arguments with the function g6 in place of g5 we obtain the same upper bound for δ6 , and the lemma follows. 1.2.4

Relations with more familiar mixing coefficients

For any A ⊂ V let GA = σ{XA } = σ{Xv : v ∈ A} be the σ-algebra generated by XA . We first recall that the α, φ and ψ mixing (dependence) coefficients associated with two sub-σ-algebras G and H of F are defined by the formulas  α(G, H) = sup |P (Γ ∩ ∆) − P (Γ)P (∆)| : Γ ∈ G, ∆ ∈ H , (1.2.14) n P (Γ ∩ ∆) o φ(G, H) = sup − P (∆) : Γ ∈ G, ∆ ∈ H, P (Γ) > 0 (1.2.15) P (Γ) and n P (Γ ∩ ∆) o ψ(G, H) = sup − 1 : Γ ∈ G, ∆ ∈ H, P (Γ)P (∆) > 0 . P (Γ)P (∆) (1.2.16)

page 10

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

11

By Theorem A.5, Corollary A.1 and Corollary A.2 in [27] for any A, B ⊂ V and p > 1,
1− p1 ε∞,∞ (A, B) ≤ 4α(GA , GB ), εp,∞ (A, B) ≤ 6 α(GA , GB ) (1.2.17)
1− p1 − q1 and εp,q (A, B) ≤ 8 α(GA , GB ) for any q > 1 such that p1 + 1q < 1. In fact, it is clear that ε∞,∞ (A, B) ≥ α(GA , GB ) which makes these coefficients equivalent. Moreover, by Theorem A.6 in [27],
1 εp,q (A, B) ≤ 2 φ(GA , GB ) p for any 1 < q, p ≤ ∞ such that p1 + 1q = 1. In our application to nonconventional sums we will bound directly E∞ (A, B) for appropriate A’s and B’s, but still it is important to estimate these weak dependence coefficients in terms of the more familiar ones. Indeed, taking expectations in Lemma 3.2 in [26] we obtain that E∞ (A, B) ≤ 2φ(GA , GB ). Similarly, taking expectations in Lemma 3.1 in [39] we deduce that E1 (A, B) ≤ 2ψ(GA , GB ). Remark 1.2.5. In [56] a general upper bound on dK (L(W ), N ) was obtained when the summands Xv ’s are bounded while the arguments in [61] lead to such estimates assuming existence of finite eighth moments where in both situations the Xv ’s were normalized in a special way. In the resulting upper bounds obtained in either [56] or [61] the terms δ5 and δ6 are replaced with expressions which can be written as sums of covariances of the form Cov(g(XA ), h(XB )) for appropriate A’s and B’s which are not connected by a single edge and functions g and h with some polynomial growth. Thus, in these circumstances it is possible to obtain similar estimates to the ones in Theorem 1.2.2 which only involve the coefficients εp,q (A, B). Using (1.2.17), the resulting upper bounds will involve only mixing coefficients of the form α(GA , GB ). 1.3 1.3.1

Nonconventional CLT with convergence rates Assumptions and main results

Our setup consists of a ℘-dimensional stochastic process {ξn , n ≥ 0} on a probability space (Ω, F, P ) and a family of sub-σ-algebras Fk,l , −∞ ≤ k ≤

page 11

March 13, 2018 15:46

ws-book9x6

12

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

l ≤ ∞ such that Fk,l ⊂ Fk0 ,l0 ⊂ F if k 0 ≤ k and l0 ≥ l. We will impose restrictions on the mixing coefficients φn = sup{φ(F−∞,k , Fk+n,∞ ) : k ∈ Z}

(1.3.1)

where φ(·, ·) was defined in (1.2.15). In order to ensure some applications, in particular, to dynamical systems we will not assume that ξn is measurable with respect to Fn,n but instead impose restrictions on the approximation rate βq,r = sup kξk − E[ξk |Fk−r,k+r ]kLq .

(1.3.2)

k≥0

We do not require stationarity of the process {ξn , n ≥ 0}, assuming only that the distribution of ξn does not depend on n and that the joint distribution of (ξn , ξm ) depends only on n − m which we write for further reference by
d d ξn ∼ µ and ξn , ξm ∼ µm−n (1.3.3) d

where Y ∼ µ means that Y has µ for its distribution. For each θ > 0, set Z γθθ = kξn kθLθ = |x|θ dµ.

(1.3.4)

Let F = F (x1 , ..., x` ), xj ∈ R℘ be a function on (R℘ )` such that for some K, ι > 0, κ ∈ (0, 1] and all xi , zi ∈ R℘ , i = 1, ..., `, we have |F (x) − F (z)| ≤ K[1 +

` ` X X (|xi |ι + |zi |ι )] |xj − zj |κ i=1

(1.3.5)

i=1

and |F (x)| ≤ K[1 +

` X

|xi |ι ]

(1.3.6)

i=1

where x = (x1 , ..., x` ) and z = (z1 , ..., z` ). In fact, if ξn is measurable with respect to Fn,n then our results will follow with any Borel function F satisfying (1.3.6) without imposing (1.3.5), since the latter is needed only for approximation of ξn by conditional expectations E[ξn |Fn−r,n+r ] using (1.3.2). To simplify formulas we assume the centering condition Z ¯ F := F (x1 , ..., x` )dµ(x1 ) . . . dµ(x` ) = 0 (1.3.7) which is not really a restriction since we can always replace F by F − F¯ .

page 12

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

13

For each N ∈ N set SN =

N X

1

F (ξn , ξ2n , ..., ξ`n ) and ZN = N − 2 SN .

n=1

The main goal in this section is to prove a central limit theorem (CLT) with close to optimal convergence rates for the normalized sums ZN via Stein’s method. Stein’s method yields a functional CLT for the random function ZN (·) defined by [N t]

ZN (t) = N

− 12

X

F (ξn , ξ2n , ..., ξ`n ),

n=1

as well. These type of results require introduction of appropriate notations, and are delayed until Section 1.6. Our results will rely on the following assumption. Assumption 1.3.1. There exist b ≥ 2, q ≥ 1 and m > 0 such that 1 ι κ > + , γm < ∞ and γιb < ∞. b m q

(1.3.8)

We will also need either Assumption 1.3.2. There exist d ≥ 1 and c ∈ (0, 1) such that for any n ≥ 0, κ φn + βq,n ≤ dcn

(1.3.9)

κ = (βq,n )κ , where βq,n

or the following weaker Assumption 1.3.3. There exist d ≥ 1 and θ > 2 such that for any n ∈ N, κ φn + βq,n ≤ dn−θ

(1.3.10)

κ where βq,n = (βq,n )κ .

1.3.2

Asymptotic variance

The following theorem follows from arguments in [30]. Theorem 1.3.4. Suppose that Assumption 1.3.1 holds true and that Θ(b, q, κ) :=

∞ X

1− 1b

(n + 1)φn

n=0

+

∞ X

κ (n + 1)βq,n < ∞.

n=0

(1.3.11)

page 13

March 13, 2018 15:46

14

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

2 Then the limit σ 2 = limN →∞ EZN exists and for some c` > 0 which depends only on `, 1

2 |EZN − σ 2 | ≤ c` C0 N − 2

(1.3.12)

ι 2 for any N ∈ N, where C0 = K 2 (1 + γm ) Θ(b, q, κ). Moreover, σ 2 > 0 if and only if there exists no stationary in the wide sense process {Vn : n ≥ 0} such that (n)

(`)

F (ξn(1) , ξ2n , ..., ξ`n ) = Vn+1 − Vn , P-a.s. for any n ∈ N, where ξ (i) , i = 1, ..., ` are independent copies of the process ξ = {ξn : n ≥ 0} and a.s. stands for almost surely. Note that (1.3.11) holds true under Assumption 1.3.2. In fact, it holds true 2b , for instance, when when Assumption 1.3.3 is satisfied with some θ > b−1 5 θ > 2. (1)

(n)

(`)

Remark 1.3.5. Set Un = F (ξn , ξ2n , ..., ξ`n ), n ∈ N. Then the process U = {Un : n ≥ 1} is stationary in the wide sense and under Assumptions 1.3.1 and 1.3.3 the limit s2 = lim N −1 Var N →∞

N X

Un




n=1

exists and the above characterization of positivity of σ 2 is equivalent to the statement that s2 > 0 (see [11]). We refer the readers to [30] for the exact details. When, in addition, ξ is stationary then U is stationary and so by Theorem 18.2.1 in [33] (applied with the process U ) this characterization can be replaced with the condition that there exists no square integrable function g such that F = g ◦ T` − g, µ` − a.s. where T is the measure preserving map generating ξ, T` = T × T 2 × · · · × T ` and µ` = µ × µ × · · · × µ. This condition generalizes the usual coboundary condition in the “conventional” case (` = 1) to nonconventional sums above. Remark 1.3.6. When ξ1 , ξ2 , ξ3 , ... forms a stationary and sufficiently fast mixing Markov chain then under additional general conditions the asymptotic variance σ 2 is positive, unless F vanishes µ` -almost surely. In particular, this characterization of positivity of σ 2 is valid when ξn ’s are independent. See Theorem 2.4 in [29].

page 14

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

1.3.3

LNcov

15

CLT with convergence rate

We denote by N the standard normal law on R. Our main result is the following Berry-Esseen type theorem for nonconventional sums. Theorem 1.3.7. (i) Suppose that Assumptions 1.3.1 and 1.3.2 hold true with b = 5 and that σ 2 > 0. Then there exists z1 > 0 which depends only on ` such that for any N ∈ N,
max dW (L(σ −1 ZN ), N ), dK (L(σ −1 ZN ), N ) 3

5

1

≤ z1 C12 (1 + R) max(R 6 , R2 )N − 2 ln2 (N + 1) ι where R = σ −1 K(1 + γm ) and 4

C1 = 1 + d2 + (− ln c)−2 + d(1 − c 5 )−2 . (ii) Suppose that Assumptions 1.3.1 and 1.3.3 hold true with some θ 2b . Further assume that σ 2 > 0. Then there and b ≥ 5 such that θ ≥ b−4 exists z2 > 0 which depends only on ` such that for any N ∈ N,
max dW (L(σ −1 ZN ), N ), dK (L(σ −1 ZN ), N ) 3

1

1

≤ z2 C22 (1 + R) max(R1− b+1 , R2 )N −( 2 −2ζ) where R is defined as above, ζ =

3b θ(b−4)+4b

and with θb =

θ(b−1) , b

κ C2 = 1 + d2 + βq,0 + d(θb − 2)−1 . 2b Remark that under the conditions of Theorem 1.3.7 (ii) we have θ > b−1 and so the conditions of Theorem 1.3.4 hold true. We also note that by the contraction of conditional expectations, βq,0 ≤ 2γq . Moreover, when βr0 = 0 for some r0 , then Theorem 1.3.7 holds true when F is only a Borel function satisfying (1.3.6) and (1.3.7), namely, it is unnecessary to require any type of continuity of F . In particular, this takes place when our mixing assumptions hold true with the σ-algebras Fm,n = σ{ξm , ..., ξn } and r0 = 0 which yields the same convergence rates assuming only (1.3.6) and (1.3.7).

Remark 1.3.8. The purpose of Theorem 1.3.7 is to obtain close to optimal convergence rate in the CLT for the normalized sums ZN , under sufficient moment and mixing conditions. The assumption that b ≥ 5 in this theorem serves this cause, and the readers who are only interested to see under which moment conditions the CLT holds true are referred to Theorem 1.6 in which a functional CLT is established, in particular, when b = 2 + δ and θ > 4 + 8δ for some δ > 0.

page 15

March 13, 2018 15:46

16

1.3.4

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

The associated strong dependency graphs

Before proving Theorem 1.3.7 we will describe the graph associated with nonconventional sums discussed in Section 1.1. For any x, y ∈ R set d` (x, y) = min |ix − jy| = min |qi (x) − qj (y)| 1≤i,j≤`

1≤i,j≤`

(1.3.13)

where qi (x) = ix, i = 1, ..., `, and for any Γ, Λ ⊂ R set d` (Γ, Λ) = inf{d` (x, y) : x ∈ Γ, y ∈ Λ}.

(1.3.14)

Let A ⊂ R and set ΓA = {ja : a ∈ A, 1 ≤ j ≤ `} =

` [

qi (A).

(1.3.15)

i=1

Then by the definition of d` , for any A, B ⊂ R we have dist(ΓA , ΓB ) = d` (A, B)

(1.3.16)

where dist(E1 , E2 ) = inf{|e1 − e2 | : e1 ∈ E1 , e2 ∈ E2 } for any two sets E1 , E2 ⊂ R. Let N ∈ N, l ≥ 1 and set V = VN = {1, ..., N }. Consider the graph G = G` (N, l) given by G = (VN , EN,l ) where EN,l = {(n, m) ∈ VN ×VN : d` (n, m) < l}. (1.3.17) Namely, we will say that n and m are connected if d` (n, m) < l. Denote by dG the shortest path distance on the graph and by B(v, r) a ball of radius r around v ∈ V with respect to dG . Then the sets Nv = B(v, 1), v ∈ V satisfy [ Nv = {u ∈ V : d` (v, u) < l} = V ∩ Ii,j (v) (1.3.18) 1≤i,j≤`

where Ii,j (v) = ( ji v − jl , ji v + jl ), and therefore, |Nv | ≤ 3`2 l := D

(1.3.19)

since the length of each Ii,j (v) does not exceed 2l and l ≥ 1. The following lemma shows that the cardinality of balls of radius 3 is of order D. Lemma 1.3.9. There exists c0 > 1 which depends only on ` such that for any v ∈ V , |B(v, 3)| = |{u ∈ V : dG (v, u) ≤ 3}| ≤ c0 D.

page 16

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

Proof. Let v = v0 ∈ V . Then [ B(v0 , 3) =

Nv2 =

[

[

LNcov

17

Nv2

(1.3.20)

v1 ∈Nv0 v2 ∈Nv1

v2 ∈B(v0 ,2)

(1)

(2)

since Nu = B(u, 1) for any u ∈ V . Consider the linear maps Li,j and Li,j , 1 ≤ i, j ≤ `, given by (1)

Li,j (x) =

ix + l ix − l (2) and Li,j (x) = . j j

For each 1 ≤ i0 , j0 ≤ `, the interval Ii0 ,j0 (v0 ) in (1.3.18) can be written in the form (1)

(2)

Ii0 ,j0 (v0 ) = (Li0 ,j0 (v0 ), Li0 ,j0 (v0 )) := (vi0 ,j0 ,1 , vi0 ,j0 ,2 ) and hence the set of all neighbors of members of Ii0 ,j0 (v0 ) is contained in the union [
(2) (1) Li1 ,j1 (vi0 ,j0 ,1 ), Li1 ,j1 (vi0 ,j0 ,2 ) . 1≤i1 ,j1 ≤`

Applying the same reasoning to the latter intervals, we deduce from (1.3.18) and (1.3.20) that the set B(v0 , 3) is contained in the union [ Ii0 ,j0 ,i1 ,j1 ,i2 ,j2 (v0 ) 1≤i0 ,j0 ,i1 ,j1 ,i2 ,j2 ≤`

of the intervals given by
(2) (1) (2) (2) (1) (1) Ii0 ,j0 ,i1 ,j1 ,i2 ,j2 (v0 ) = Li2 ,j2 ◦ Li1 ,j1 ◦ Li0 ,j0 (v0 ), Li2 ,j2 ◦ Li1 ,j1 ◦ Li0 ,j0 (v0 ) (1.3.21) where ◦ stands for composition of maps. The length of each of these `6 intervals does not exceed 6`2 l, since each of them has the form (av0 − b, av0 + b) for some 0 ≤ b ≤ l(1 + ` + `2 ). Since l ≥ 1, we conclude that |B(v0 , 3)| ≤ `6 (6`2 l + 1) ≤ 7`8 l and the lemma follows. We conclude that the conditions in Theorems 1.2.1 and 1.2.2 about the cardinality of balls of radius 1 and 3 are satisfied with D = 2`2 l, ρ = 1 and the above c0 .

page 17

March 13, 2018 15:46

18

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

1.3.5

Expectation estimates

We first recall that (see [11], Ch. 4) for any two sub-σ-algebras G, H ⊂ F, 2φ(G, H) = sup{kE[g|G] − EgkL∞ : g ∈ L∞ (Ω, H, P ), kgkL∞ ≤ 1} (1.3.22) where φ(G, H) is defined by (1.2.15). The following result does not seem to be new but for readers’ convenience and completeness we will sketch its proof here. Lemma 1.3.10. Let G1 , G2 ⊂ F be two sub-σ-algebras of F and for i = 1, 2 let Vi be a Rdi -valued random Gi -measurable vector with distribution µi . Set d = d1 + d2 , µ = µ1 × µ2 , denote by κ the distribution of the random vector (V1 , V2 ) and consider the measure ν = 12 (κ + µ). Let B be the Borel σalgebra on Rd and H ∈ L∞ (Rd , B, ν). Then E[H(V1 , V2 )|G1 ] and EH(v, V2 ) exist for µ1 -almost any v ∈ Rd1 and |E[H(V1 , V2 )|G1 ] − h(V1 )| ≤ 2kHkL∞ (Rd ,B,ν) φ(G1 , G2 ), P − a.s. (1.3.23) where h(v) = EH(v, V2 ) and a.s. stands for almost surely. Proof. Clearly H is bounded µ and κ almost surely. Thus, E[H(V1 , V2 )|G1 ] exists and existence of EH(v, V2 ) (µ1 -a.s.) follows from the Fubini theorem. Relying on (1.3.22), inequality (1.3.23) follows easily for functions of the P form G(v1 , v2 ) = i I{v1 ∈Ai } gi (v2 ), where {Ai } is a measurable partition of the support of µ1 and I{v1 ∈Ai } = 1 when v1 ∈ Ai and equals 0 otherwise. Any uniformly continuous function H is a uniform limit of functions of the above form, which implies that (1.3.23) holds true for uniformly continuous functions. Finally, by Lusin’s theorem (see [57]), any function H ∈ L∞ (Rd , B, ν) is an L1 (and a.s.) limit of a sequence {Hn } of continuous functions with compact support satisfying kHn kL∞ (Rd ,B,ν) ≤ kHkL∞ (Rd ,B,ν) and (1.3.23) follows for any H ∈ L∞ (Rd , B, ν). Next, let Ui , i = 1, 2, ..., k be di -dimensional random vectors defined on the probability space (Ω, F, P ) from Section 1.3.1, and {Cj : 1 ≤ j ≤ s} be a partition of {1, 2, ..., k}. Consider the random vectors U (Cj ) = {Ui : i ∈ Cj }, j = 1, ..., s, and let (j)

U (j) (Ci ) = {Ui

: i ∈ Cj }, j = 1, ..., s

be independent copies of the U (Cj )’s. For each 1 ≤ i ≤ k let ai ∈ {1, ..., s} be the unique index such that i ∈ Cai , and for any bounded Borel function H : Rd1 +d2 +...+dk → R set (a ) (a ) (a ) D(H) = EH(U1 , U2 , ..., Uk ) − EH(U 1 , U 2 , ..., U k ) . (1.3.24) 1

2

k

page 18

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

19

The following result is a consequence of Lemma 1.3.10. Corollary 1.3.11. Suppose that each Ui is Fmi ,ni -measurable, where ni−1 < mi ≤ ni < mi+1 , i = 1, ..., k, n0 = −∞ and mk+1 = ∞. Then, for any bounded Borel function H : Rd1 +d2 +...+dk → R, D(H) ≤ 4 sup |H|

k X

φmi −ni−1

i=2

where sup |H| is the supremum of |H|. In particular, when s = 2 then k X
α σ{U (C1 )}, σ{U (C2 )} ≤ 4 φmi −ni−1 i=2

where σ{X} stands for the σ-algebra generated by a random variable X and α(·, ·) is given by (1.2.14). Proof. Denote by µi the distribution of the random vector U (Ci ), i = 1, ..., s. Then, we have to show that for any Borel function H : Rd1 +d2 +...+dk → R, Z EH(U1 , U2 , ..., Uk ) − H(u1 , u2 , ..., uk )dµ1 (u(C1 ) )dµ2 (u(C2 ) )...dµs (u(Cs ) ) ≤ 4 sup |H|

k X

φmi −ni−1

(1.3.25)

i=2

where u(Ci ) = {uj : j ∈ Ci }. In order to prove (1.3.25), denote by νi the distribution of Ui , i = 1, 2, ..., k. We first prove by induction on k that for any choice of H and Ui ’s with the required properties, Z |EH(U1 , U2 , ..., Uk ) − H(u1 , u2 , ..., uk )dν1 (u1 )dν2 (u2 )...dνk (uk )| ≤ 2 sup |H|

v X

φmi −ni−1 .

(1.3.26)

i=2

Indeed, suppose that k = 2 and set V1 = U1 , V2 = U2 , h(u1 ) = EH(u1 , U2 ), G1 = F−∞,n1 and G2 = Fm2 ,∞ . Taking expectation in (1.3.23) yields |EH(U1 , U2 ) − Eh(U1 )| ≤ 2 sup |H|φm2 −n1 which means that (1.3.26) holds true when k = 2. Now, suppose that (1.3.26) holds true for any k ≤ j − 1, U1 , ..., Uk with the required properties and any bounded Borel function H : Re1 +...+ek−1 → R, where e1 , ..., ek−1 ∈

page 19

March 13, 2018 15:46

20

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

N. In order to deduce (1.3.26) for k = j, set V1 = (U1 , ..., Uj−1 ), V2 = Uj , h(v1 ) = EH(v1 , Uj ), v1 = (u1 , ..., uj−1 ), G1 = F−∞,nj−1 and G2 = Fmj ,∞ . Taking expectation in (1.3.23) yields |EH(U1 , U2 , ..., Uj ) − Eh(U1 , U2 , ..., Uj−1 )| ≤ 2 sup |H|φmj −nj−1 . Applying the induction hypothesis with the function h completes the proof of (1.3.26), since sup |h| ≤ sup |H|. Next, we prove by induction on s that for any choice of k, H, Ui ’s with the required properties and C1 , ..., Cs , Z H(u1 , u2 , ..., uk )dµ1 (u(C1 ) )dµ2 (u(C2 ) )...dµs (u(Cs ) ) (1.3.27) Z −

k X φmi −ni−1 . H(u1 , u2 , ..., uk )dν1 (u1 )dν2 (u2 )...dνk (uk ) ≤ 2 sup |H| i=2

For s = 1 this is just (1.3.26). Now suppose that (1.3.27) holds true for any s ≤ j − 1, and any real valued bounded Borel function H defined on Rd1 +...+dk , where k and d1 , ..., dk are some natural numbers. In order to prove (1.3.27) for s = j, set u(I) = (u(C1 ) , u(C2 ) , ..., u(Cs−1 ) ) and let the function I be defined by Z Y (I) I(u ) = H(u1 , u2 , ...., uk ) dνj (uj ). (1.3.28) j∈Cs

Then Z

Z H(u1 , u2 , ..., uk )dν1 (u1 )dν2 (u2 )...dνk (uk ) =

I(u(I) )

Y

dνj (uj ).

j6∈Cs

(1.3.29) Let the function J be defined by Z (I) J(u ) = H(u1 , u2 , ...., uk )dµs (u(Cs ) ).

(1.3.30)

Then by (1.3.26), for any u(C1 ) , ..., u(Cs−1 ) , |I(u(I) ) − J(u(I) )| ≤ 2 sup |H|

X

φmi −ni−1 .

(1.3.31)

i∈Cs

It is clear that sup |J| ≤ sup |H|. Applying the induction hypothesis with the function J (considered as a function of the variable u) and taking into account (1.3.29) and (1.3.31) we obtain (1.3.27) with s = j and we complete the induction. Inequality (1.3.25) follows now by (1.3.26) and (1.3.27), and the proof of Corollary 1.3.11 is complete.

page 20

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

21

Remark 1.3.12. In the notations of Corollary 1.3.11, let Zi , i = 1, ..., s be a bounded σ{U (C
i )}-measurable random variable. Then each Zi has the form Zi = Hi U (Ci ) for some function Hi which satisfies sup |Hi | ≤ kZi kL∞ . Qs Considering the function H(u) = i=1 Hi (u(Ci ) ), we obtain from (1.3.25) that, s s s k Y Y Y
X E Zi − EZi ≤ 4 kZi kL∞ φmj −nj−1 . i=1

i=1

i=1

(1.3.32)

j=2

In general we can replace sup |H| in the right-hand side of (1.3.25) by some essential supremum of |H| with respect to an appropriate measure which has a similar but more complicated form as κ in Lemma 1.3.10. We also obtain the following result. Corollary 1.3.13. Suppose that each Ui is Fmi ,ni -measurable. Let H : Rd1 +d2 +...+dk → R satisfies |H(u)| ≤ K(1 +

k X

|ui |ι )

(1.3.33)

i=1

for some ι > 0 and all u = (u1 , u2 , ..., uk ) ∈ Rd1 +...+dk . Suppose that kUiι kLb ≤ Γb for some b ≥ 1, Γb ∈ (0, ∞) and all i = 1, 2, ..., k. Then 1− 1b

D(H) ≤ 6Tb ϕk where ϕk =

Pk

i=2

1− 1b

φmi −ni−1 , ϕk

1− 1b

= (ϕk )

(1.3.34) and Tb = K(1 + kΓb ).

Note that even though H is not bounded, the growth condition (1.3.33) and the moment assumptions in Corollary 1.3.13 guarantee that D(H) given by (1.3.24) is well defined, namely both expectations exist. The proof of Corollary 1.3.13 is a standard application of the H¨older and Markov inequalities and it goes as follows. Proof. First, when ϕk = 0 then U1 , ..., Uk are independent and the lefthand side of (1.3.34) vanishes, and so we assume without loss of generality that ϕk > 0. For any R > 0 let the function HR be defined by HR (u) = H(u)I(|H(u)| ≤ R). Then by (1.3.25), D(HR ) ≤ 4R

k X

φmi −ni−1 = 4Rϕk .

i=2

By the H¨ older and Markov inequalities, for any random variable X we have
1− 1b E|XI{|X|>R} | ≤ kXkLb P (|X| > R) ≤ kXkbLb R−(b−1)

page 21

March 13, 2018 15:46

ws-book9x6

22

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where I{|X|>R} is the random variable which equals 1 when |X| > R and equals 0 otherwise. By (1.3.33) and since kUiι kLb ≤ Γb , i = 1, ..., k, we have kH(U1 , U2 , ..., Uk )kLb ≤ K(1 + kΓb ) = Tb . Applying the previous inequality with X = H(U1 , U2 , ..., Uk ) we deduce that kH(U1 , U2 , ..., Uk ) − HR (U1 , U2 , ..., Uk )kL1 ≤ Tbb R−(b−1) . (a1 )

Using the same arguments with H(U1 H(U1 , U2 , ..., Uk ) we obtain (a1 )

kH(U1

(a2 )

, U2

(ak )

, ..., Uk

(a1 )

) − HR (U1

(a2 )

, U2

(a2 )

, U2

(ak )

, ..., Uk (ak )

, ..., Uk

) in place of

)kL1 ≤ Tbb R−(b−1) .

−1

Taking R = ϕk b Tb , we conclude that 1− 1b

D(H) ≤ D(HR ) + 2Tbb R−(b−1) ≤ 4Rϕk + 2Tbb R−(b−1) = 6Tb ϕk and the corollary follows.

In order to obtain an explicit constant C0 which satisfies (1.3.12), we will need the following consequence of Corollary 1.3.13. Corollary 1.3.14. Let H : Rd1 +d2 +...+dk → R satisfies (1.3.33) and (1.3.5) with H in place of F and with ui ’s in place of xi ’s. Let q, b > 1 and m > 0 be such that ι κ 1 > + b m q and set ri = [ 13 (mi − ni−1 )], i = 2, ..., k, r1 = r2 and rk+1 = rk . Suppose that kUi kLb < ∞ for any 1 ≤ i ≤ k. Then D(H) ≤ 6R0 Λb,κ

(1.3.35)

where Λb,κ =

k X i=2

φr i


1− 1b

+

k X




Ui − E[Ui |Fmi −ri ,ni +ri+1 ] q κ L i=1

ι R0 = K(1 + kγm ) and γm = max{kUi kLm : 1 ≤ i ≤ k}.

Proof. First, for any i = 1, ..., k set Ui,r = E[Ui |Fmi −ri ,ni +ri+1 ] and for any s > 0 set ∆i,r,s = kUi − Ui,r kLs and γs = max{kUi kLs : 1 ≤ i ≤ k}. Then, by the H¨ older inequality, for any i = 1, ..., k we have

|Ui − Ui,r |κ b ≤ ∆κi,r,κb ≤ ∆κi,r,q L

(1.3.36)

page 22

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

23

where in the second inequality we used that q > κb. Let 1 ≤ i, z ≤ k. Then ι + κq , by Lemma 3.1 (i) in [45] and our assumption that 1b > m

ι

|Uz |ι |Ui − Ui,r |κ b ≤ γm ∆κi,r,q (1.3.37) L and observe that the above inequality holds true with Uz,r in place of Uz , as well, since conditional expectation contracts Lp norms. We conclude from (1.3.5) and the above estimates that kH(U1 , U2 , ..., Uk ) − H(U1,r , U2,r , ..., Uk,r )kLb ι ≤ K(1 + 2kγm )

k X

(1.3.38)

∆κi,r,q .

i=1 (j)

(j)

Next, let Ur (Cj ) = {Ui,r : i ∈ Cj }, j = 1, ..., s be independent copies of Ur (Cj ) := {Ui,r : i ∈ Cj }, j = 1, ..., s. Applying Corollary 1.3.13 we deduce that (a )

(a )

(a )

|EH(U1,r , U2,r , ..., Uk,r ) − EH(U1,r1 , U2,r2 , ..., Uk,rk )| ≤ 6Tb

k X

φri

(1.3.39)


1− 1b

i=2

where we recall that ai is the unique index such that i ∈ Cai , and we have used again the contraction of conditional expectations which imply that ι kUi,r kLb ≤ kUiι kLb ≤ Γb for each i. Considering the product of the laws of the random vectors Vj = {U (Cj ), Ur (Cj )} = {Ui , Ui,r : i ∈ Cj }, j = 1, ..., s we can always assume that there exist (on a larger probability space) inde(j) (j) (j) pendent copies Vj = {Ui , Ui,r : i ∈ Cj }, j = 1, ..., s of the Vj ’s such that (a )

(a )

(a )

(a )

(1.3.36) holds true for any 1 ≤ i, z ≤ k with Ui i , Uz z , Ui,ri and Uz,rz in place of Ui , Uz , Ui,r and Uz,r , respectively. The appropriate version of (a ) (a ) (1.3.37) holds true with Uz,rz in place of Uz z , as well. Thus, similarly to (1.3.38) we obtain from (1.3.5) and (1.3.33) that

EH(U (a1 ) , U (a2 ) , ..., U (ak ) ) − EH(U (a1 ) , U (a2 ) , ..., U (ak ) ) b (1.3.40) 2,r 1 2 1,r k,r k L ι ≤ K(1 + 2kγm )

k X

∆κi,r,q .

i=1

The corollary follows now from (1.3.38), (1.3.39) and (1.3.40).

page 23

March 13, 2018 15:46

24

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Proof of Theorem 1.3.4. As claimed before its formulation, Theorem 1.3.4 follows from the arguments in [45], [44] and [30]. Indeed, the proof in [45] that σ 2 exists relies only on Lemma 4.3 from there which provides approximations of the left-hand side of (1.3.35), and so the proof that σ 2 exists in our circumstances proceeds in the same way as in [45], relying on Corollary 1.3.14 instead. Similarly, the proofs of the characterization of positivity of σ 2 and inequality (1.3.12) go exactly as in [30] and [44], respectively, relying on Corollary 1.3.14 instead of Lemma 3.4 in [30] and Proposition 3.2 (ii) in [44]. The appearance of an explicit constant C0 satisfying (1.3.12) is guaranteed from (1.3.35), since, contrary to the above Proposition 3.2 (ii), the right-hand side of (1.3.35) includes only explicit constants. The exact form of C0 given in Theorem 1.3.4 is easily recovered by keeping track of constants which come from the applications of Corollary 1.3.14. 1.3.6

Proof of Theorem 1.3.7

Set ι R = Rσ,K,m,ι = σ −1 K(1 + γm ).

Let n, N ∈ N, l ≥ 1 and set r = [ 3l ],

Ξn = ξn , ξ2n , ..., ξ`n and Ξn,r = ξn,r , ξ2n,r , ..., ξ`n,r where for any m ∈ N and r ≥ 0, ξn,r = E[ξm |Fm−r,m+r ]. 1 PN Then with these notations we have ZN = N − 2 n=1 F (Ξn ). Set 1

ZN,r = N − 2

N X

F (Ξn,r ) and

(1.3.41)

n=1 1 Z¯N,r = N − 2

N X


F (Ξn,r ) − EF (Ξn,r ) .

n=1

The first part of the proof is to approximate ZN by these normalized sums. Let 1 ≤ i, j ≤ `. Then by (1.3.38) applied with k = `, H = F and the random vectors Ui = ξin , i = 1, ..., `, ι κ κ kF (Ξn ) − F (Ξn,r )kLb ≤ K`(1 + `γm )βq,r ≤ σ`2 Rβq,r .

(1.3.42)

κ where βq,r = (βq,r )κ , and therefore, 1

κ . kZN − ZN,r kLb ≤ σ`2 RN 2 βq,r

(1.3.43)

page 24

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

25

Next, we have 1 kZN,r − Z¯N,r kL∞ = N − 2 |

N X

EF (Ξn,r )|.

(1.3.44)

n=1

In order to estimate the right-hand side in (1.3.44), let n > l, consider the random vectors Ui = ξin,r and set mi = in − r and ni = in + r, i = 1, ..., `. Then each Ui is Fmi ,ni measurable and l ≥ r. 3 Using the triangle inequality and the estimate (1.3.42) and then applying Corollary 1.3.14 with k = ` and the sets Ci = {in}, i = 1, ..., ` we deduce that EF (Ξn,r ) − EF (ξn(1) , ξ (2) , ..., ξ (`) ) 2n `n (2) (`) ≤ EF (Ξn,r ) − EF (Ξn )| + EF (Ξn ) − EF (ξn(1) , ξ2n , ..., ξ`n ) mi − ni−1 = n − 2r ≥ l − 2r ≥

1− 1b

κ ≤ σ`2 Rβq,r + 6σ`2 R(φr 1− 1b

where φr

κ + βq,r ) ≤ 7`2 σRcr,l

1

= (φr )1− b and 1− 1b

cr,l := φr

κ + βq,r .

(`) (1) (2) Notice that EF (ξn , ξ2n , ..., ξ`n ) = F¯ = 0 and therefore

|EF (Ξn,r )| ≤ 7σ`2 Rcr,l

(1.3.45)

when n > l. When n ≤ l we deduce from (1.3.6) and the contraction of conditional expectations that ι ) ≤ `σR kF (Ξn,r )kLb ≤ K(1 + `γιb

(1.3.46)

where we used that m > ιb, which implies that γιb ≤ γm . Since |EF (Ξn,r )| ≤ kF (Ξn,r )kLb , it follows from (1.3.44) and the above estimates that 1 1 kZN,r − Z¯N,r kL∞ ≤ `σRN − 2 l + 7σ`2 RN 2 cr,l ≤ 7σ`2 Rsr,l

(1.3.47)

where 1

1

1

1− 1b

κ + N 2 φr sr,l := N − 2 l + N 2 βq,r

.

(1.3.48)

Applying Lemma 1.1.1 we obtain from (1.3.43) and (1.3.47) that dK (L(σ −1 ZN ), N ) ≤ 3dK (L(σ −1 ZN,r ), N ) 1
1 κ 1− b+1 ≤ 9dK (L(σ −1 Z¯N,r ), N ) +4 `2 RN 2 βq,r 1

+4`2 R1− b+1 ar,b + 12 · 7`2 Rsr,l

(1.3.49)

page 25

March 13, 2018 15:46

26

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where 1

κ ar,b := N 2 βq,r

1
1− b+1

.

(1.3.50)

Moreover, for any two random variables X and Y and a Lipschitz function h with the constant 1 we have |Eh(X) − Eh(Y )| ≤ kX − Y kL1 . Thus by (1.3.43) and (1.3.47), dW (L(σ −1 ZN ), N ) ≤ σ −1 kZN − Z¯N,r kL1 (1.3.51)
1 κ +dW (L(σ −1 Z¯N,r ), N ) ≤ `2 R N 2 βq,r + 7sr,l + dW (L(σ −1 Z¯N,r ), N ). 1.3.7

Back to graphical indexation

Consider the graph G(N, l) defined in Section 1.3.4. Set
1 Xv = Xv,r,N = N − 2 σ −1 F (Ξv,r ) − EF (Ξv,r ) , v ∈ V = {1, 2, ..., N } (1.3.52) and consider their sum X W = Xv v∈V

which satisfies σW = Z¯N,r . In what follows we will estimate the remaining terms in Theorems 1.2.1 and 1.2.2. First, by (1.3.46) for any v we have 1

1

kXv kLb ≤ 2N − 2 σ −1 kF (Ξv,r )kLb ≤ 2N − 2 `R.

(1.3.53)

Next, we will estimate |EW 2 − 1|. First, for any two random variables X and Y defined on the same probability space, |EX 2 − EY 2 | ≤ kX − Y kL2 kX + Y kL2 ≤ kX − Y kL2 (kX − Y kL2 + 2kY kL2 ). Therefore, by (1.3.12), σ 2 |EW 2 − 1| = |E Z¯N,r


2

1

− σ 2 | ≤ c` C0 N − 2 + dr (dr + 2kZN kL2 ) (1.3.54)

where dr = kZ¯N,r − ZN kL2 . The right-hand side of (1.3.43) does not exceed 7σ`2 Rsr,l and so it follows from (1.3.43) and (1.3.47) that dr ≤ 14R`2 σsr,l . In order to estimate kZN kL2 we first use (1.3.12) in order to deduce that 1

1

kZN kL2 ≤ c`2 C02 + σ. Notice that, in fact, C0 = R2 σ 2 Θ, where Θ = 1

1

Θ(b, q, κ) is defined in (1.3.11), and it follows that kZN kL2 ≤ σ(c`2 Θ 2 R+1). Finally, by the definition of sr,l and the inequality (a+b+c)2 ≤ 3(a2 +b2 +c2 ) we obtain that 2− 2b

s2r,l ≤ 3(N −1 l2 + N φr

2κ + N βq,r ).

page 26

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

27

We conclude from (1.3.54) and the above estimates that there exists a constant a` > 0 which depends only on ` such that
2 |EW 2 − 1| = σ −2 |E Z¯N,r − σ 2 | (1.3.55) 1 1 1− 1b
κ 2 − 12 + A2 N 2 βq,r + A3 N 2 φr ≤ a` max(R, R ) A1 N where 1

1

A1 = Θ + l + l2 N − 2 + Θ 2 , 1

1

1− 1b

κ A2 = N 2 βq,r + 1 and A3 = N 2 φr

+ 1.

Next, the following result is the first step towards estimating the δi ’s appearing in Theorems 1.2.1 and 1.2.2. Lemma 1.3.15. There exists a constant c > 0 which depends only on ` such that 


(1.3.56) max α G{v} , G{u} , α G{v} , GNvc , α G{v1 ,u1 } , G{v2 ,u2 } ,


 α G{v1 ,u1 } , GNvc1 ∩Nuc1 , E∞ Nv1 , Nv2 , α V \ B(v, 2), Nv ≤ cφr for any v, u, v1 , u1 , v2 , u2 ∈ V such that d(v, u) > 1, d(v1 , v2 ) > 3 and d(v1 , u1 ) = d(v2 , u2 ) = 1, where d = dG is the shortest path distance associated with the graph G = G` (N, l) defined by (1.3.17). Proof. In the course of the proof we will use the following notations and abbreviations. For any B1 , ..., BL ⊂ R we will write B1 < B2 < ... < BL when b1 < b2 < ... < bL for any bi ∈ Bi , i = 1, ..., L. Next, for any v, u ∈ V we abbreviate G{v} = Gv , Γ{v} = Γv , G{v,u} = Gv,u and Γ{v,u} = Γv,u where for any A ⊂ V the set ΓA is defined by (1.3.15) and the σ-algebra GA is defined in the beginning of Section 1.2.4. In order to estimate the first two expressions on the left-hand side in (1.3.56), let v ∈ V and u ∈ Nvc = V \ Nv . Since Gu ⊂ GNvc we have α(Gv , Gu ) ≤ α(Gv , GNvc ). In order to estimate the above right-hand side, consider the set ΓNvc . Then by (1.3.16) and the definition of the graph G, dist(Γv , ΓNvc ) = d` ({v}, Nvc ) ≥ l.

page 27

March 13, 2018 15:46

28

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Considering the points in ΓNvc which lie in the intervals (iv, (i + 1)v), i = 0, ..., ` − 1 and (`v, ∞), we can write Γv ∪ ΓNvc =

L [

Bi , L ≤ 2` + 1

i=1

where each Bi is a subset of either Γv or ΓNvc , B1 < B2 < ... < BL and dist(Bi+1 , Bi ) ≥ l, i = 1, 2, ..., L − 1. (1.3.57) Consider the random vectors Ui = {ξb,r : b ∈ Bi }, i = 1, ..., L.

(1.3.58)

Since l ≥ 3r, there exist numbers mi ≤ ni such that Ui is Fmi ,ni -measurable and l ni ≤ mi+1 − (l − 2r) ≤ mi+1 − ≤ mi+1 − r, i = 1, ..., L − 1. (1.3.59) 3 Since Xv is a function of {Ui : Bi ⊂ Γv } and XNvc is a function of {Ui : Bi ⊂ ΓNvc } we obtain from Corollary 1.3.11 that
α Gv , GNvc ≤ 4(2` + 1)φr . Next, in order to estimate the third and fourth expressions on the left hand side in (1.3.56), let v1 , u1 ∈ V be such that d(v1 , u1 ) = 1. First notice that α(Gv1 ,u1 , Gv2 ,u2 ) ≤ α(Gv1 ,u1 , GNvc1 ∩Nuc1 ) for any v2 , u2 ∈ V such that d(v1 , v2 ) > 3 and d(v2 , u2 ) = 1, and therefore it is sufficient to estimate the fourth expression. Set Av1 ,u1 = Nvc1 ∩ Nuc1 . There exists no single edge connecting {v1 , u1 } and Av1 ,u1 and therefore d` ({v1 , u1 }, Av1 ,u1 ) ≥ l. Hence by (1.3.16), dist(Γv1 ,u1 , ΓAv

1 ,u1

) = d` ({v1 , u1 }, Av1 ,u1 ) ≥ l.

Thus, by considering the points in Av1 ,u1 which lie either between two consecutive points of Γv1 ,u1 or in one of the intervals (0, min(v1 , u1 )) and (` max(v1 , u1 ), ∞), we can write Γv1 ,u1 ∪ ΓAv

1 ,u1

=

L [

Bi , L ≤ 4` + 1

i=1

where each Bi is a subset of either Γv1 ,u1 or ΓAv1 ,u1 and Bi ’s satisfy (1.3.57). Consider the random vectors Ui , i = 1, ..., L given by (1.3.58) with the above Bi ’s. Since l ≥ 3r, there exist numbers ni ≤ mi , i = 1, ..., L satisfying (1.3.59) so that each Ui is Fmi ,ni -measurable. Since (Xv1 , Xu1 ) is a function

page 28

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

29

of {Ui : Bi ⊂ Γv1 ,u1 } and XAv1 ,u1 is a function of {Ui : Bi ⊂ ΓAv1 ,u1 }, we deduce from Corollary 1.3.11 that
α Gv1 ,u1 , GNvc1 ∩Nuc1 ≤ 4(4` + 1)φr . Next, let v1 and v2 be such that d(v1 , v2 ) > 3 and consider the sets Nv1 and Nv2 . In order to estimate E∞ (Nv1 , Nv2 ), let G = G(z1 , z2 ) be a bounded Borel function, where zi = {xs : s ∈ Nvi }, i = 1, 2. Consider the sets Mi = ΓNvi , i = 1, 2. Since Nv1 and Nv2 are not connected by a single edge we have d` (Nv1 , Nv2 ) ≥ l. Thus, by (1.3.16), dist(M1 , M2 ) = d` (Nv1 , Nv2 ) ≥ l. By (1.3.18) and the definition of the sets ΓA ’s, [ Ms = {1, 2, ..., N `} ∩ Ii,j,k (vs ), s = 1, 2 1≤i,j,k≤`

where k ki k ki v − l, v − l). j j j j As a consequence, since dist(M1 , M2 ) ≥ l we can write L [ M1 ∪ M2 = Bi , L ≤ 2`3 + 1 Ii,j,k (v) = (

i=1

where each Bi is contained in either M1 or M2 , and Bi ’s satisfy (1.3.57). Consider the random vectors Ui , i = 1, ..., L given by (1.3.58) with the above Bi ’s. Since l ≥ 3r, there exist numbers ni ≤ mi , i = 1, ..., L satisfying (1.3.59) so that each Ui is Fmi ,ni -measurable. Since XNvi is a function of {Ui : Bi ⊂ Mi }, i = 1, 2 it follows from Corollary 1.3.11 that ∗ |EG(XNv1 , XNv2 ) − EG(XNv1 , XN )| ≤ 4(2`3 + 1) sup |G|φr v2 ∗ where XN is a copy of XNv2 which is independent of XNv1 . Since the v2 above estimate holds true
for an arbitrary bounded Borel function G, we deduce that E∞ Nv1 , Nv2 ≤ 4(2`3 + 1)φr . Finally, let v ∈ V and set Bv = V \ B(v, 2). Then d` (Bv , Nv ) ≥ l and hence dist(ΓBv , ΓNv ) = d` (Bv , Nv ) ≥ l. Considering the points in ΓBv which lie either between two (non overlapping) intervals of the form Ii,j,k (v) or lie to the right (left) from the maximal (minimal) point in ΓNv , we see that the union ΓBv ∪ ΓNv can be written as a union of at most 2`3 + 1 sets B1 , ..., BL satisfying (1.3.57) such that each Bi is either a subset of ΓBv or a subset of ΓNv . Similarly to the above, we deduce that α Bv , Nv ) ≤ 4(2`3 + 1)φr and the proof of the lemma is complete.

page 29

March 13, 2018 15:46

30

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

The following corollary follows from Lemmas 1.2.3, 1.2.4, 1.3.15, (1.3.19) and (1.3.53). Corollary 1.3.16. With R = only on ` such that

ι K(1+γm ) , σ

1− 2b

δ2 , δ5 , δ6 ≤ c1 R2 N φr δ1 ≤ c1 RN

1 2

there exists c1 ≥ 1 which depends 1

, δ4 ≤ c1 Λ max(R, R2 )φr2

1− 1 φr b

and (δ3 )2 ≤

− 1b

,

1− 4 c1 l2 R4 φr b

1

where φsr = (φr )s for any s > 0 and Λ = kW kL2 N 2 + D2 . Proof of Theorem 1.3.7 (i). Suppose that Assumptions 1.3.1 and 1.3.2 1 4 hold true with b = 5 and set c5 = c1− 5 = c 5 . Then Θ(5, q, κ) ≤

2d (1 − c5 )2

(1.3.60)


0 P∞ x for any x ∈ (0, 1), where Θ(5, q, κ) is defined since n=0 (n+1)xn = 1−x in (1.3.11). Next, applying Corollary 1.3.16 with b = 5 and then using (1.3.9) yields 1

r

δ1 , δ2 , δ3 , δ4 , δ5 , δ62 ≤ c1 (1 + kW kL2 )d max(R, R2 ) max(l2 , N )c 10 . Taking l = A ln(N + 1) + 3 where A = we have, r

c 10 ≤ c

l−3 30

45 ln(c−1 )

+ 1 and recalling that r = [ 3l ] 3

≤ N−2

(1.3.61)

and since ln(x + 1) ≤ x for any x ≥ 1 we deduce that 1

1

δ1 , ..., δ5 , δ62 ≤ 4c1 d(1 + kW kL2 ) max(R, R2 )A2 N − 2 .

(1.3.62)

Next, using (1.3.55) with b = 5 and then (1.3.9) we derive that there exists a constant g` which depends only on ` such that 1

|EW 2 − 1| ≤ g` max(R, R2 )CN − 2 ln(N + 1)

(1.3.63)

where with Θ = Θ(5, q, κ), 1

C = 1 + Θ + Θ 2 + A2 + d2 and we used that derive that

1 2

1

≤ ln(N + 1) ≤ N 2 for any N ≥ 1. In particular, we kW kL2 ≤ 1 + g` max(R, R2 )C


12

.

(1.3.64)

Applying Theorems 1.2.1 and 1.2.2 with the Xv ’s and with ρ = 1, taking into account that |V | = N and the definition (1.3.19) of D, and then using

page 30

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

31

Lemma 1.3.9, (1.3.53), (1.3.62), (1.3.63) and (1.3.64) we deduce that there exists a constant z > 0 which depends only on ` such that dW (L(σ −1 Z¯N,r ), N ), dK (L(σ −1 Z¯N,r ), N ) (1.3.65) 1

3

≤ z(1 + R) max(R, R2 )C 2 N − 2 ln2 (N + 1) := D 1

where we used that 1 + max(R 2 , R) ≤ 2(1 + R) and max(R2 , R3 ) ≤ (1 + R) max(R, R2 ) for any R ≥ 0 and that ln(N + 1) ≥ 12 for any N ≥ 1, which implies that l ≤ 4A ln(N + 1). Finally, consider the terms sr,l and ar,b in (1.3.48) and (1.3.50). Then by (1.3.9), taking into account that l = A ln(N + 1) and b = 5 we obtain 1

1

1

sr,l ≤ 7AN − 2 ln(N + 1) + dN − 2 and ar,5 ≤ dN − 2

(1.3.66)

where in the first inequality we used that l ≤ 4A ln(N + 1). We conclude from (1.3.65), (1.3.51), (1.3.49) and (1.3.66) that there exists a constant z 0 > 0 which depends only on ` such that dW (L(σ −1 ZN ), N ), dK (L(σ −1 ZN ), N )
5 1 1 ≤ D + max(R 6 , R)z 0 dN − 2 + AN − 2 ln(N + 1) which completes the proof of Theorem 1.3.7, taking into account (1.3.60) 2 and that C ≤ 2 + 2Θ + A2 + d2 and A2 ≤ 2 + ln2·45 2 (c−1 ) , which follows from the inequalities x + x2 ≤ 1 + 2x2 and (x + y)2 ≤ 2(x2 + y 2 ), x, y ≥ 0. Proof of Theorem 1.3.7 (ii). Suppose that Assumptions 1.3.1 and 1.3.3 2b hold true, with θ and b such that θ > b−4 . Consider l’s of the form l = ζ 3N + 1, where ζ is yet to be determined. For such l’s we have κ φr , βq,r ≤ dr−θ ≤ dN −ζθ .

(1.3.67)

1 2,

which guarantees that

We assume now that ζθ ≥ 2

1 2(1− 1b )

and ζ ≤

A2 , A3 ≤ d + 1 and l ≤ N , with A2 and A3 defined right after (1.3.55). From this together with (1.3.67), (1.3.48), (1.3.55) and Corollary 1.3.16 we obtain the following estimates: 1

1

2

δ1 , ..., δ5 , δ62 ≤ (1 + kW kL2 ) max(R, R2 )N 1−( 2 − b )ζθ , sl,r ≤ 4N 2

2

|EW − 1| ≤ 4 max(R, R ) dN 2

l N

− 12

≤ 3N

2ζ− 12

− 21 +ζ

+ 2dN

1 1 2 −ζθ(1− b )

(1.3.68)

1 1 2 −ζθ(1− b )

+ QN

− 12 +ζ

1

,
,

1

and ar,b ≤ N ( 2 −ζθ)(1− b+1 )

where Q = 4 + Θ(b,
q, κ) + d. The powers of N appearing in (1.3.68) do not exceed 1 − 12 − 2b ζθ and therefore the sum of the expressions in (1.3.68) 1 2 1 does not exceed K0 N 1−( 2 − b )ζθ + N 2ζ− 2 ), where K0 = 4(1 + kW kL2 + d + Q) max(R, R2 ) + 10d.

page 31

March 13, 2018 15:46

32

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Let ζ = 3 θ(1 − 4b ) + 4


−1

be the solution of the equation

2ζ − This ζ indeed satisfies ζθ ≥

1 1 2
=1− − ζθ. 2 2 b

1 2(1− 1b )

1 2

and ζ ≤

since we have assumed that

2b b−4 , and 2ζ− 21

so the sum of the expressions in (1.3.68) does not exceed θ ≥ 2K0 N . Observe next that κ Θ(b, q, κ) ≤ 1 + βq,0 + 4d

∞ X

κ n−θb +1 ≤ 1βq,0 + 4d + 4d(θb − 2)−1 (1.3.69)

n=1

R∞ P∞ 1 where θb = θ(1 − 1b ) and we used that n=1 n−p ≤ 1 + 1 x−p dx = 1 + p−1 for any p > 1. Applying Theorems 1.2.1 and 1.2.2 we deduce from (1.3.51), (1.3.49), (1.3.53) and (1.3.68) that dW (L(σ −1 ZN ), N ), dK (L(σ −1 ZN )), N ) 1

1

≤ c(1 + kW k2 ) max(R1− b+1 , R2 )(Q + d2 )N −( 2 −2ζ) where c depends only on `. Now Theorem 1.3.7 (ii) follows, taking into account (1.3.69) and the inequality kW k22 ≤ 1 + d + Q which follows from (1.3.68). 1.4 1.4.1

General Stein’s estimates: proofs Proof of Theorem 1.2.1

In view of (1.2.2), in order to obtain an upper bound of dW (L(W ), N ), it is sufficient to estimate expressions of the form |E[f 0 (W ) − W f (W )]| for twice differentiable functions f : R → R which satisfy
max sup |f |, sup |f 0 |, sup |f 00 | ≤ 1. (1.4.1) We first show that for any such f , |E[f 0 (W ) − W f (W )]| ≤ δ1 + δ2 + r1 + 2r2 + r3 + |EW 2 − 1| P where with Yv = u∈Nv Xu , v ∈ V , X X r1 = E| (Xv Yv − EXv Yv )|, r2 = E|Xv Yv |I{|Yv |>1} v∈V

v∈V

and r3 =

X v∈V

E|Xv | min(Yv2 , 1)

(1.4.2)

(1.4.3)

page 32

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

33

and for any B ∈ F, IB denotes the random variable which equals 1 on B and 0 outside B. Before proving (1.4.2) we assume that it holds true and complete the proof of Theorem 1.2.1. First, by the definition of r3 and then by the Cauchy-Schwarz inequality, X X r3 ≤ E|Xv |Yv2 ≤ kXv kL2 kYv k2L4 ≤ |V |D2 γ43 (1.4.4) v∈V

v∈V

and by the Cauchy-Schwarz and Markov inequalities, X r2 ≤ kXv kL2 kYv I{|Yv |>1} kL2

(1.4.5)

v∈V

≤

X

kXv kL4 kYv kL4 kI{|Yv |>1} kL4 ≤

X

kXv kL4 kYv k2L4 ≤ |V |γ43 D2

v∈v

v∈V

γ44

where we used that kXv kL4 ≤ and |Nv | ≤ D, for any v ∈ V . In order to bound r1 , set ζv,u = Xv Xu − EXv Xu , v, u ∈ V . Then by the definition of Yv ’s, X X r1 = E| ζv,u |. v∈V u∈Nv

By the Cauchy-Schwarz inequality, X X X r12 ≤ E| ζv,u |2 =

X

(Av1 ,u1 + Bv1 ,u1 )

v1 ∈V u1 ∈Nv1

v∈V u∈Nv

where X

Au1 ,v1 =

X

Cov(ζv1 ,u1 , ζv2 ,u2 )

v2 ∈V :d(v1 ,v2 )>3 u2 ∈Nv2

and X

Bv1 ,u1 =

X

Cov(ζv1 ,u1 , ζv2 ,u2 ).

v2 ∈V :d(v1 ,v2 )≤3 u2 ∈Nv2

Notice that |

X

X

Au1 ,v1 | = δ32

v1 ∈V u1 ∈Nv1

where δ3 is defined in the statement of Theorem 1.2.1. On the other hand, let v1 , u1 , v2 , u2 ∈ V . By the H¨older inequality we have |Cov(ζv1 ,u1 , ζv2 ,u2 )| ≤ 2γ44 and therefore Bv1 ,u1 ≤ 2c0 Dρ+1 γ44 , where we used that |B(v1 , 3)| ≤ c0 Dρ . Hence, r12 ≤ δ32 + 2c0 |V |Dρ+2 γ44

(1.4.6)

page 33

March 13, 2018 15:46

ws-book9x6

34

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

and Theorem 1.2.1 follows now from (1.2.2), (1.4.2), (1.4.5), (1.4.4) and (1.4.6). For the purpose of proving (1.4.2) we will need the following notations from [15] which will be used in the proof of Theorem 1.2.2, as well. For any v ∈ V and t ∈ R set ˆ v (t) = Xv I{−Y ≤t1

Z



ˆ f (W ) − f 0 (W + t) K(t) − K(t) dt and Z
R4 = E f 0 (W ) − f 0 (W + t) K(t)dt. 0

R3 = E |t|≤1

|t|≤1

Proof. First we write EW f (W ) =

X


EXv f (W ) − f (W − Yv ) + δ1 (f ).

v∈V

The first expression in the above right-hand side can be written as Z 0 X
X EXv f (W ) − f (W − Yv ) = EXv f 0 (W + t)dt v∈V

=

−Yv

v∈V

X v∈V

Z

∞

E −∞

ˆ v (t)dt = E f 0 (W + t)K

Z

∞

−∞

ˆ f 0 (W + t)K(t)dt.

page 34

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

35

Next, by (1.4.8), Z ∞
Ef 0 (W ) = Ef 0 (W ) ∆2 + r + K(t)dt −∞ Z ∞
f 0 (W )K(t)dt + ∆2 + r Ef 0 (W ) =E −∞

and therefore E[f 0 (W ) − W f (W )] = E

Z

∞

f 0 (W )K(t)dt

−∞ ∞

Z

ˆ f 0 (W + t)K(t)dt + (∆2 + r)Ef 0 (W ) − δ1 (f )

−E −∞

and (1.4.9) follows by the definition of the Ri ’s. We will also use the following relations: Z ∞
ˆ r1 = E| K(t) − K(t) dt|, −∞ X Z X ˆ v (t)|dt = E |K E|Xv |(|Yv | − 1)I{|Yv |>1} ≤ r2 v∈V

|t|>1

(1.4.10) (1.4.11)

v∈V

and r3 = 2

X

Z E

v∈V

ˆ v (t)|dt |tK

(1.4.12)

|t|≤1

where r1 , r2 and r3 are defined in (1.4.3). In order to prove (1.4.2), let f : R → R be a twice differentiable function satisfying (1.4.1). We will estimate now the terms on the right-hand side of (1.4.9). First, by the definitions of δ1 , δ2 and r and since both sup |f | and sup |f 0 | do not exceed 1, |δ1 (f )| ≤ sup |f |δ1 ≤ δ1 and 0

0

2

(1.4.13)

2

|(∆2 + r)Ef (W )| ≤ sup |f |(δ2 + |EW − 1|) ≤ δ2 + |EW − 1|. Next, since sup |f 0 | ≤ 1 we deduce from (1.4.10) that |R1 | ≤ sup |f 0 |r1 ≤ r1 .

(1.4.14)

By the mean value theorem together with the inequality sup |f 0 | ≤ 1 we have |f 0 (W ) − f 0 (W + t)| ≤ 2 sup |f 0 | ≤ 2 for any t ∈ R. Therefore, Z X Z ˆ |R2 | ≤ 2E |K(t)|dt ≤2 E |t|>1

v∈V

|t|>1

ˆ v (t)|dt ≤ 2r2 |K

(1.4.15)

page 35

March 13, 2018 15:46

ws-book9x6

36

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where the last inequality follows from (1.4.11). Finally, set Z
ˆ S3 = R3 + R4 = E f 0 (W ) − f 0 (W + t) K(t)dt. |t|≤1

Since sup |f 00 | ≤ 1 by the mean value theorem we have |f 0 (W ) − f 0 (W + t)| ≤ |t| and therefore Z

ˆ |f 0 (W ) − f 0 (W + t)||K(t)|dt Z X Z ˆ ˆ v (t)|dt = 1 r3 ≤E |t||K(t)|dt ≤ E |t||K 2 |t|≤1 |t|≤1 |S3 | ≤ E

(1.4.16)

|t|≤1

v∈V

where the last equality follows from (1.4.12). Inequality (1.4.2) follows now from (1.4.9) and (1.4.13)-(1.4.16), and the proof of Theorem 1.2.1 is complete. 1.4.2

Proof of Theorem 1.2.2

The following proof is based on the proof of Theorem 2.1 in [15]. Let z ∈ R and α > 0. Consider the function hz,α : R → R given by the formula   if w ≤ z  1 (1.4.17) hz,α (w) = 1 + z−w if z < w ≤ z + α α   0 if w > z + α 1 R∞ 1 2 and set Nhz,α = (2π)− 2 −∞ hz,α (x)e− 2 x dx. Let f = fz,α be the unique bounded solution of the ordinary differential equation

Af (w) = f 0 (w) − wf (w) = hz,α (w) − Nhz,α which is given by the formula Z 1 2 w 2 f (w) = e

∞

(1.4.18)


1 2 e− 2 x h(x) − Nhz,α dx.

w

Since Ehz−α,α (W ) ≤ P (W ≤ z) ≤ Ehz,α (W ) and |Φ(z) − Nhz,α |, |Φ(z + α) − Nhz,α | ≤ Φ(z + α) − Φ(z) ≤ (2π)−1/2 α for any z ∈ R, where Φ is the standard normal distribution function, we deduce that sup |P (W ≤ z) − Φ(z)| ≤ sup |Ehz,α (W ) − Nhz,α | + 0.5α (1.4.19) z

z

= sup |EAfz,α (W )| + 0.5α. z

page 36

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

37

As in (4.5)-(4.8) in [15] the function f = fz,α satisfies the following properties (in fact, these estimates come from [62]): 0 ≤ f (w) ≤ 1, |f 0 (w)| ≤ 1, |f 0 (w) − f 0 (v)| ≤ 1 and 0

(1.4.20)

0

|f (w + s) − f (w + t)| (1.4.21) Z t ≤ (|w| + 1) min(|s| + |t|, 1) + α−1 | I{z≤w+u≤z+α} du| s

≤ (|w| + 1) min(|s| + |t|, 1) + I{z−max(s,t)≤w≤z−min(s,t)+α} for any w, v, s, t ∈ R, where I{a≤b≤c} = 1 when a ≤ b ≤ c and otherwise = 0, for any a, b, c ∈ R. In view of (1.4.19), our goal now is to bound |Ef 0 (W ) − EW f (W )| from above for functions f satisfying (1.4.20) and (1.4.21) where z ∈ R and α > 0 are fixed. First, by Lemma 1.4.1, |Ef 0 (W ) − EW f (W )| ≤

4 X

|Ri | + |δ1 (f )| + |Ef 0 (W )|(δ2 + |EW 2 − 1|).

i=1

(1.4.22) Since max(sup |f |, sup |f 0 |) ≤ 1, |δ1 (f )| ≤ δ1 and |Ef 0 (W )|(δ2 + |EW 2 − 1|) ≤ δ2 + |EW 2 − 1|

(1.4.23)

and since sup |f 0 | ≤ 1 and |f 0 (w) − f 0 (v)| ≤ 1 for any w, v ∈ R, similarly to (1.4.14) and (1.4.15) we obtain that |R1 | ≤ r1 and |R2 | ≤ r2 .

(1.4.24)

Estimating R3 and R4 is a more complicated task and it requires the following concentration inequality. Set X r4 = E|W Xv min(Yv2 , 1)|, (1.4.25) v∈V

r5 = |

X

EXv Xu g5 (Yv , Yu ) − EXv Xu∗ g5 (Yv , Yu∗ )| and (1.4.26)

v,u∈V

r6 = |

1 1 X EXv Xu g6 (Yv , Yu ) − EXv Xu∗ g6 (Yv , Yu∗ )| 2 2

(1.4.27)

v,u∈V

where the functions g5 and g6 are defined in Theorem 1.2.2, and recall that (Xu∗ , Yu∗ ) is a copy of (Xu , Yu ) which is independent of (Xv , Yv ). Proposition 1.4.2. For any a < b we have P (a ≤ W ≤ b) ≤ (0.125 + 0.5kW kL2 + 0.5δ1 )(b − a) + 2δ2 +(0.625 + 1.5kW kL2 + 1.5δ1 )r3 + 4r2 + 4r5 + 2|EW 2 − 1|.

page 37

March 13, 2018 15:46

ws-book9x6

38

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Before proving Proposition 1.4.2 we assume that it holds true and complete the proof of Theorem 1.2.2. Together with (1.4.8), (1.4.10), (1.4.11) and (1.4.12) we will use the following identities: Z X ˆ v (t)|dt (1.4.28) r4 = 2 E|W | |tK |t|≤1

v∈V

Z


ˆ Var K(t) dt and r62 =

r5 = |t|≤1

Z


ˆ |t|Var K(t) dt

(1.4.29)

|t|≤1

(see the proof of (1.4.29) after the statement of Theorem 2.1 in [15]). We estimate next R3 as in (4.11) in [15]. It follows from the definition of R3 and from (1.4.21), (1.4.12) and (1.4.28) that |R3 | ≤ r3 + r4 + R3,1 + R3,2

(1.4.30)

where Z R3,1 = E

1

ˆ I{z−t≤W ≤z+α} |K(t) − K(t)|dt and

0

Z

0

ˆ I{z≤W ≤z−t+α} |K(t) − K(t)|dt.

R3,2 = E −1

Set 3
3 δ 0 = D1 α + 2δ2 + 0.625 + kW kL2 + δ1 r3 + 4r2 + 4r5 + 2|EW 2 − 1| 2 2 where 1 1 D1 = 0.125 + kW kL2 + δ1 . 2 2 Then by Proposition 1.4.2 for any t ≥ 0, P (z − t ≤ W ≤ z + α) ≤ δ 0 + D1 t.

(1.4.31)

If either δ1 ≥ 1, δ2 ≥ 1 or kW kL2 ≥ 2 then the inequality in the statement of Theorem 1.2.2 clearly holds true, and so we can assume without loss of generality that max(δ1 , δ2 , 21 kW kL2 ) ≤ 1 and in this case δ 0 ≤ 2α + 2δ2 + 6r3 + 4r2 + 4r5 + 2|EW 2 − 1| := δ 00 .

(1.4.32)

In order to estimate R3,1 , we first notice that for any a, b ∈ R and c > 0, 2ab ≤ ca2 + c−1 b2 .

page 38

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

39

Fix some t ∈ [0, 1]. Similarly to (4.12) in [15], applying that inequality with ˆ a = I{z−t≤W ≤z+α} , b = |K(t) − K(t)| and c = c(t) = α(δ 00 + D1 t)−1 we deduce from (1.4.31) and (1.4.32) that Z 1 1 c(t)I{z−t≤W ≤z+α} dt R3,1 ≤ E 2 0 Z 1
−1 1 ˆ + E c(t) |K(t) − K(t)|2 dt 2 0 Z Z 1

1 1 1 −1 00 1 ˆ ˆ Var K(t) dt + α−1 D1 ≤ α+ α δ |t|Var K(t) dt 2 2 2 0 0 and a similar inequality holds true with R3,2 . Relying on (1.4.29), we deduce that
1 R3,1 + R3,2 ≤ α + α−1 δ 00 r5 + D1 r62 2
1 ≤ α + α−1 δ 00 r5 + 2r62 2 where in the last inequality we used our assumption that 1 max(δ1 , δ2 , kW kL2 ) ≤ 1. 2 Then by (1.4.30),
1 (1.4.33) |R3 | ≤ α + α−1 δ 00 r5 + r62 + r3 + r4 . 2 Next, similarly to (4.13) in [15] by (1.4.21) and Proposition 1.4.2 we obtain Z 1 |R4 | ≤ E (|W | + 1)|tK(t)|dt (1.4.34) −1

+α−1

Z

1

−1



Z

t

P (z ≤ W + u ≤ z + α)du |K(t)|dt

0

≤ (1 + kW kL2 )r3 + α−1 δ 00 r3 ≤ 3r3 + α−1 δ 00 r3 ˆ v (t)| ≤ E|K ˆ v (t)| and where in the first inequality we used that |Kv (t)| = |EK the identity (1.4.11), while in the second inequality we used our assumption that kW kL2 ≤ 2. It follows from (1.4.9), (1.4.23), (1.4.24), (1.4.33) and (1.4.34) that for any real z, |Ehz,α (W ) − Nhz,α | = |EAfz,α (W )| ≤ A + α + α−1 B where A = δ1 + δ2 + |EW 2 − 1| + r1 + r2 + 6r3 + r4 + 2r5 and B = (r5 + r3 )(δ2 + |EW 2 − 1| + 6r3 + 4r2 + 4r5 ) + 2r62 .

(1.4.35)

page 39

March 13, 2018 15:46

ws-book9x6

40

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Therefore, by (1.4.19) we have 3 dK (L(W ), N ) ≤ A + α + α−1 B. 2

(1.4.36)

p 2/3B 1/2 we conclude that p √ 3 dK (L(W ), N ) ≤ α + A + 3/2B 1/2 = A + 6B 1/2 . (1.4.37) 2 Note that when B vanishes then (1.4.37) directly by letting α → √ √ we obtain √ 0. Relying on the inequalities a + b ≤ a + b and 2ab ≤ a2 + b2 , a, b ≥ 0 and the definitions of A and B it follows that Taking α =

dK (L(W ), N ) ≤ r1 + 3r2 + 13r3 + r4 + 6r5 + 4r6 + δ1 + 4δ2 + 4|EW 2 − 1|. (1.4.38) We will prove now Theorem 1.2.2 relying on (1.4.38). Assume without loss of generality that |EW 2 −1| ≤ 1 since otherwise Theorem 1.2.2 trivially holds true. In order to estimate r4 we first have, X X r4 ≤ E|W − Zv ||Xv min(Yv2 , 1)| + E|Zv Xv min(Yv2 , 1)| := I1 + I2 . v∈V

v∈V

≤ kXv kL4 kZv kL2 kYv k2L4 for any v ∈ By the H¨ older inequality, V and therefore by the definitions of Yv and Zv , E|Xv Zv Yv2 |

I2 ≤ D4 |V |γ44 . Next, notice that I1 = δ 4 +

X

E|W − Zv |E|Xv min(Yv2 , 1)|

v∈V

where δ4 is defined in Theorem 1.2.2. Let v ∈ V . Then by the CauchySchwarz inequality and the definition of Zv , E|W − Zv | ≤ kW − Zv kL2 ≤ kW kL2 + kZv kL2 ≤ 1 + EW 2 + D2 γ4 ≤ 2 + |EW 2 − 1| + D2 γ4 and by the H¨ older inequality, E|Xv min(Yv2 , 1)| ≤ D2 γ43 . We conclude that r4 ≤ D4 |V |γ44 + δ4 + D2 |V |γ43 (2 + |EW 2 − 1| + D2 γ4 ) 4

≤ D |V

|γ44

2

+ δ4 + D |V

|γ43 (3

2

4

+ D γ4 ) = δ4 + 2D |V

|γ44

(1.4.39) 2

+ 3D |V |γ43

where in the last inequality our assumption that |EW 2 − 1| ≤ 1 is used. Next, we will estimate r5 . Let v, u ∈ V . Then by the H¨older inequality and since g5 (Yv , Yu ) ≤ |Yv |, E|Xv Xu g5 (Yv , Yu )| ≤ kXv kL3 kXu kL3 kYv kL3 ≤ Dγ43

page 40

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

41

and similar inequality holds true when Xu and Yu are replaced with Xu∗ and Yu∗ , respectively. Thus, by the definition of δ5 and our assumption that |B(v, 3)| ≤ c0 Dρ , X X r5 ≤ δ5 + 2Dγ43 ≤ δ5 + 2c0 D1+ρ |V |γ43 . (1.4.40) v∈V u:∈B(v,3)

Since g6 (Yv , Yu ) ≤ Yv2 and kXv Xu Yv2 kL1 ≤ kXv kL4 |kXu kL4 kYv k2L4 ≤ D2 γ44 it follows in a similar way that r62 ≤ δ6 + D4 |V |γ44 .

(1.4.41)

Theorem 1.2.2 is deduced now from (1.4.38) and the estimates (1.4.6), (1.4.5), (1.4.4), (1.4.39), (1.4.40) and (1.4.41) where we have taken into 1 1 account that D4 |V |γ44 ≤ D2 |V | 2 γ42 unless D2 |V | 2 γ42 > 1, while in the latter case the inequality in the statement of Theorem 1.2.2 trivially holds true. Proof of Proposition 1.4.2. First notice that EW 2 = 0 when r3 = 0 and in this situation Proposition 1.4.2 trivially holds true, and so we assume without loss of generality that r3 > 0. Set α = r3 and let f : R → R be the function defined by  b−a+α  if x ≤ a − α  − 2   b−a+α 1 2  (x − a + α) − if a − α < x ≤ a  2  2α a+b f (x) = x − 2 if a < x ≤ b   1 b−a+α  2  − 2α (x − b − α) + 2 if b < x ≤ b + α     b−a+α if x > b + α. 2

Then b−a+α 2 0 and f is a continuous function given by   if a ≤ x ≤ b  1 0 f (x) = 0 if x ≤ a − α or x ≥ b + α   linear otherwise. sup |f | =

(1.4.42)

(1.4.43)

By (1.4.42) and the Cauchy-Schwarz inequality, kW kL2 (b − a + α)/2 = kW kL2 sup |f | ≥ EW f (W ) X
= EXv f (W ) − f (W − Yv ) + δ1 (f ) v∈V

(1.4.44)

page 41

March 13, 2018 15:46

ws-book9x6

42

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where δ1 (f ) =

P

v∈V

EXv f (W − Yv ). By the definition of δ1 we have

b−a+α δ1 . (1.4.45) 2 ˆ v (t), K(t) and K(t) ˆ Next, with Kv (t), K defined in (1.4.7), Z 0 X X
EXv f (W ) − f (W − Yv ) = EXv f 0 (W + t)dt |δ1 (f )| ≤ sup |f |δ1 =

v∈V

−Yv

v∈V

=

X

Z E

ˆ v (t)dt f 0 (W + t)K

−∞

v∈V

Z

∞

∞

ˆ f 0 (W + t)K(t)dt := H1 + H2 + H3 + H4

=E −∞

where H1 = Ef 0 (W )

Z

Z


f 0 (W + t) − f 0 (W ) K(t)dt, |t|≤1 Z ˆ H3 = E f 0 (W + t)K(t)dt and |t|>1 Z
ˆ − K(t) dt. H4 = E f 0 (W + t) K(t)

K(t)dt, H2 = E |t|≤1

|t|≤1

The inequalities |H3 | ≤ r2 , |H4 | ≤

b − a + 2α + 2r5 and |H2 | ≤ 0.5L(α) 8

(1.4.46)

where L(α) = sup P (x ≤ W ≤ x + α) x∈R

are established exactly as in the proof of Proposition 3.1 in [15], relying only on (1.4.8), (1.4.10)-(1.4.12), (1.4.29) and the properties of f . To estimate H1 , set r = 1 − EW 2 . Then by (1.4.8) and (1.4.11), Z H1 = Ef 0 (W ) K(t)dt (1.4.47) |t|≤1

= Ef 0 (W )

Z

∞

Z K(t)dt −

−∞


K(t)dt

|t|>1

≥ Ef 0 (W )(1 − ∆2 − r − r2 ) ≥ P (a ≤ W ≤ b) − |∆2 | − r2 − |r| where the second inequality holds true since I{a≤x≤b} ≤ f 0 (x) ≤ 1 for any real x. We conclude from (1.4.44)-(1.4.47) that P (a ≤ W ≤ b) ≤ (0.125 + 0.5kW kL2 + 0.5δ1 )(b − a)

(1.4.48)

+(0.25 + 0.5kW kL2 + 0.5δ1 )α + 2r2 + 2r5 + |∆2 | + |r| + 0.5L(α).

page 42

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

43

Substituting a = x and b = x + α in (1.4.48), we obtain L(α) ≤ (0.375 + kW kL2 + δ1 )α + 2r2 + 2r5 + |∆2 | + |r| + 0.5L(α) and hence L(α) ≤ (0.75 + 2kW kL2 + 2δ1 )α + 4r2 + 4r5 + 2|∆2 | + 2|r|.

(1.4.49)

Finally, combining (1.4.48) and (1.4.49) we obtain (1.4.28), recalling that α = r3 . 1.5

Stein’s method for diffusion approximations

In the next section we will prove a functional central limit theorem for random functions of the form [N t] 1

ZN (t) = N − 2

X

F (ξn , ξ2n , ..., ξ`n )

n=1

and for similar ones. Stein’s method in the situation of a functional CLT requires some notations which will be introduced in the following section. 1.5.1

A functional CLT via Stein’s method

Let (Ω, F, P ) be a probability space and let D = D[0, 1] be the Skorokhod space of all functions w : [0, 1] → R which are continuous from the right and have left limits. Each element w of D is a bounded function, and we denote by sup |w| the supremum of |w|. Then the space D endowed with the supremum norm is a Banach space. Recall that a function f : D → R is twice Fr´echet differentiable if for any w ∈ D there exist a continuous linear functional Df (w)[·] : D → R and a continuous bilinear form D2 f (w)[·, ·] : D × D → R such that |f (w + h) − f (w) − Df (w)[h] − 12 D2 f (w)[h, h]| = 0. h→0 (sup |h|)2 lim

For any linear functional γ[·] : D → R and a bilinear form Γ[·, ·] : D×D → R let kγk =

|γ[h]|

sup h∈D: sup |h|=1

and kΓk =

sup

|Γ[h, h]|

h∈D: sup |h|=1

be their (operator) norms with respect to the supremum norm.

page 43

March 13, 2018 15:46

ws-book9x6

44

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

As in [1], let M be the space of all twice Fr´echet differentiable functions f : D → R such that kf kM := sup (1 + sup |w|3 )−1 |f (w)| + sup (1 + sup |w|2 )−1 kDf (w)k w∈D

w∈D

+ sup (1 + sup |w|)−1 kD2 f (w)k w∈D

+

sup

(sup |h|)−1 kD2 f (w + h) − D2 f (w)k < ∞.

w,h∈D,h6=0

Then k · kM is a norm on M . Let M0 be the space of all functions f ∈ M such that kf kM0 := kf kM + sup |f (w)| + sup kDf (w)k + sup kD2 f (w)k < ∞. w∈D

w∈D

w∈D

(1.5.1) Following [1], for any two probability measures F and G on D set Z Z f dF − f dG . dM0 (F, G) = sup f ∈M0 :kf kM0 ≤1

Then as in [1], dM0 (·, ·) is a probability metric on the set of probability measures on D. Next, let N ∈ N and let G = (V, E) be a graph on V = {1, ..., N } (where E is the set of edges and V is the set of vertices). Denote by dG the shortest path distance on the graph and for any n ∈ V set Nn = {m ∈ V : dG (n, m) ≤ 1}. Let {X1,N , X2,N , ..., XN,N }, N ∈ N be a triangular array of centered random variables defined on the probability space (Ω, F, P ) and consider the random function WN : [0, 1] → R given by [N t]

WN (t) =

X

Xn,N

n=1

which is a random element of D. We denote by cN (·, ·) the covariance function of WN (·), namely, the function given by the formula cN (t, s) = Cov(WN (t), WN (s)). Theorem 1.5.1. (i) Let GN (·) be a Gaussian process which has the covariance function cN (·, ·) and let p, q ≥ 1. Then there exists an absolute constant C such that dM0 (L(WN ), L(GN )) ≤ C(d1 + d2 + d3 + d4 ) := CτN

(1.5.2)

page 44

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

45

where with Xn = Xn,N and σn,m = EXn Xm , d1 =

N X

kXn kLp εp,∞ ({n}, Nnc ),

n=1

d2 =

N X X

c kXn Xm − σn,m kLq εq,∞ ({n, m}, Nnc ∩ Nm ),

n=1 m∈Nn

d3 =

N X

X


E|Xn Xm Xk | + E|Xn Xm |E|Xk | ,

n=1 m,k∈Nn

d4 =

N X X

kXn kLp kXm kLp εp,p ({n}, {m})

c n=1 m∈Nn

and the (mixing) coefficients εa,b (·, ·) are defined in (1.2.6). (ii) Suppose that there exists Γ > 0 such that [N s] − [N t] (1.5.3) kWN (s) − WN (t)kL2 ≤ Γ N for any N ∈ N and 0 ≤ t ≤ s ≤ 1. Further assume that the limits limN →∞ cN (t, s) = c(t, s), s, t ∈ [0, 1] exist and that lim τN ln2 N = 0.

N →∞

Then WN and GN weakly converge in the Skorokhod space D to a continuous centered Gaussian process G with covariance function c(·, ·). Theorem 1.5.1 (i) follows from the arguments in the proof of Lemma 3.1 in [1] (see the remark proceeding it). Not all the details are given in [1], and in the proof below we will provide the missing details and refer to [1] when the corresponding arguments go exactly as in [1]. The proof of Theorem 1.5.1 (ii) is a consequence of Proposition 3.1 in [2] and a standard tightness criterion from [6], and it is given here for readers’ convenience, as well. Proof of Theorem 1.5.1. Let B = B(·) be a standard Brownian motion on [0, ∞) and let Z be a standard normal random variable which is independent of B. Let Y be the Ornstein-Uhlenbeck process with standard normal equilibrium distribution. Namely, Y is the unique strong solution of the stochastic differential equation √ dY (t) = −Y (t)dt + 2dB(t), Y (0) = Z which is given by √ Z t −(t−s) Y (t) = e Z + 2 e dB(s). −t

0

(1.5.4)

page 45

March 13, 2018 15:46

46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

By (1.5.4) and the Itˆ o isometry, for any u, v ≥ 0 we have d

Y (u + v) − e−v Y (u) = σ(v)Z

(1.5.5)

√ d where σ(v) = 1 − e−2v and X = Y means that two random variables X and Y have the same distribution. Let Y1 , ..., YN be N independent copies of Y and consider the random vector Y¯ = Y¯N = {Y1 , ..., YN }. Let Σ = ΣN be the covariance matrix of the random vector {Xn,N : 1 ≤ n ≤ N }. Then we can write Σ = AA∗ for some matrix A = AN , where A∗ stands for the transpose of A. Let Y˜ = Y˜N be the random vector given by Y˜ = AY¯ := {Y˜n : 1 ≤ n ≤ N }. Then by (1.5.5), for any u, v ≥ 0, d Y˜ (u + v) − e−v Y˜ (u) = σ(v)ZΣ

(1.5.6)

where ZΣ = {Z1 , ..., ZN } is a centered Gaussian vector whose covariance matrix is Σ. Consider the D-valued Markov process {W (·, u) : u ≥ 0} given by W (t, u) =

N X

[N t]

Y˜n (u)J Nn (t) =

X

Y˜n (u), t ∈ [0, 1]

n=1

n=1

where Ja (t) = 1 if a ≤ t and equals 0 otherwise. Then by (1.5.6), for any u, v ≥ 0, d

W (·, u + v) − e−v W (·, u) = σ(v)GN (·).

(1.5.7)

Here GN (·) is a centered Gaussian process with the covariance function cN (·, ·) which can be represented in the form d

GN (t) =

N X n=1

[N t]

Zn J (t) = n N

X

Zn .

(1.5.8)

n=1

Consider the semigroup {Tu : u ≥ 0} associated with {W (·, u) : u ≥ 0} which acts on functions f : D → R and is given by Tu f (w) = E[f (W (·, u))|W (·, 0) = w]. Since Y (0) and Y (u) − e−u Y (0) are independent, it follows from (1.5.7) that Tu f (w) = Ef (we−u + σ(u)GN ).

(1.5.9)

Relying on (1.5.9), the proof of (2.9) in [1] is carried out here in the same way with the Gaussian processes GN (·) in place of the Brownian motion

page 46

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

47

B(·) (denoted there by Z), and it follows that the infinitesimal generator A of {W (·, u) : u ≥ 0} is given for any f ∈ M by Tu f (w) − f (w) = −Df (w)[w] (1.5.10) u N X +ED2 f (w)[GN , GN ] = Df (w)[w] + ] EXn,N Xm,N D2 f (w)[J Nn , J m N Af (w) := lim

u→0

n,m=1

where in the second equality we used (1.5.8) and the definition of Σ. Let g ∈ M0 satisfy Eg(GN ) = 0 and consider the (differential) equation Af = g.

(1.5.11)

Then the function f = fg : D → R given by Z ∞ fg (w) = − Tu g(w)du 0

is well defined, in the domain of A, solves (1.5.11) and satisfies kfg kM0 ≤ C0 kgkM0 for some absolute constant C0 . Indeed, the first three properties follow relying on (1.5.9) in a similar way to Section 4 of [17]. We remark that these properties were originally proved in [1], but there was a mistake in the proof, and this issue was sorted out in [17]. The upper bound kfg kM0 ≤ C0 kgkM0 is obtained similarly to [1]. Note that in [1] a similar upper bound was obtained only for the norm k·kM relying on the dominated convergence theorem, but the same arguments yield the upper bound for the norm k·kM0 . Finally, similarly to Lemma 3.1 in [1] and the approximations preceding Theorem 3 from there, for any function f ∈ M0 we have X EDf (WN )[WN ] − ] (1.5.12) EXn,N Xm,N D2 f (WN )[J Nn , J m N 1≤n,m≤N

≤ 16(d1 + d2 + d3 + d4 )kf kM0 where the definition of the mixing coefficients εs,t is used to bound from above the expressions from (3.4) in [1]. Taking f = fg , for which EAfg (WN ) = Eg(WN ) = Eg(WN ) − Eg(GN ) and then using (1.5.10) and (1.5.12) we complete the proof of Theorem 1.5.1 (i). For the proof of Theorem 1.5.1 (ii), let G = G(t) be a centered Gaussian process on [0, 1] with a covariance function c(·, ·). We will prove next that GN converges in distribution to G and that G has a continuous modification. This is indeed sufficient in order to derive Theorem 1.5.1 (ii) by virtue of Proposition 3.1 in [2], Theorem 1.5.1 (i) and our assumption

page 47

March 13, 2018 15:46

48

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

that limN →∞ τN ln2 N = 0. Since the covariance function of GN converges to the covariance function of G, any finite dimensional distribution of GN weakly converges to the corresponding finite dimensional distribution of G. We show now that {GN : N ≥ 1} forms a tight sequence of D-valued random variables as N → ∞. Indeed, by (1.5.3), E|GN (s) − GN (t)|2 = cN (s, s) − 2cN (t, s) + cN (t, t) (1.5.13) [N s] − [N t] = E|WN (s) − WN (t)|2 ≤ Γ N for any N ∈ N and 0 ≤ t ≤ s ≤ 1. Since GN is Gaussian we have E|GN (s) − GN (t)|4 (1.5.14)  [N s] − [N t] 2
2 = 3 E|GN (s) − GN (t)|2 ≤ 3Γ2 . N We conclude from the Cauchy-Schwarz inequality that for any 0 ≤ t ≤ s ≤ u ≤ 1, E|GN (s) − GN (t)|2 |GN (u) − GN (s)|2 (1.5.15)  [N s] − [N t] 2 p ≤ E|GN (s) − GN (t)|4 E|GN (u) − GN (s)|4 ≤ 3Γ2 N and therefore by Ch. 15 of [6], the family of distributions {L(GN ) : N ≥ 1} is tight and it follows that GN converges in distribution to G. In order to show that G has a continuous modification, taking into account that G is Gaussian, for any 0 ≤ t, s ≤ 1 we have
2 E|G(s) − G(t)|4 = 3 E|G(s) − G(t)|2 . Since c(t, s) = limN →∞ cN (t, s), the above right-hand side is the limit as
2 N → ∞ of 3 E|GN (s) − GN (t)|2 and therefore by (1.5.14), E|G(s) − G(t)|4 ≤ 3Γ2 |s − t|2 and G indeed has a continuous modification by the Kolmogorov continuity theorem, which completes the proof of Theorem 1.5.1 (ii). 1.5.2

Finite dimensional convergence rate

In some situations we can consider (versions of) G and GN on one probability space and obtain that E sup0≤t≤1 |GN (t) − G(t)| converges to 0 when N → ∞ with some convergence rate. This type of estimates together with Theorem 1.5.1 yield explicit estimates of dM0 (WN , G). One simple example is the case when Xn,N ’s are uncorrelated with variance 1. In this situation

page 48

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

we have Af (w) = −Df (w)[w] +

PN

n=1

LNcov

49

D2 f (w)[J Nn , J Nn ] which is the infi-

] nitesimal generator of the random function BN given by BN (t) = B( [tN N ). Since (see Theorem 1 in [1]), 1

E sup |B(t) − BN (t)| ≤ CN − 2 ln N 0≤t≤1

for some absolute constant C, we obtain from Theorem 1.5.1 (i) an explicit convergence rate. The situation when Xn,N ’s are “asymptotically correlated” as N → ∞ can be considered, as well. See Section 4 in [2] for another example in which similar realization exists. In general, it is unclear whether two Gaussian processes whose covariance functions are “close” admit such a realization, and in our application to nonconventional sums the weak limit (in general) will not have independent increments. Still, the following estimates hold true. Lemma 1.5.2. Let d, N ∈ N and let kt : Rd → R, t = (t1 , ..., td ) ∈ [0, 1]d be a family of functions. Let µ be a probability measure on [0, 1]d by g(w) = Rand consider the function g = gk,µ,d : D → R given d kt (w(t1 ), ..., w(td ))dµ(t). Suppose that for any t ∈ R the function kt and its Fourier transform are absolutely integrable. Then Z |Eg(GN ) − Eg(G)| ≤ sup CN,t,d |kˆt (x)|kxk21 dx t∈supp(µ)

Pd

where kxk1 = i=1 |xi |, CN,t,d = sup1≤i,j≤d |cN (ti , tj )−c(ti , tj )| and c(t, s) are the limits defined in Theorem 1.5.1 (ii). Proof. By the Fubini theorem, it suffices to prove the lemma in the case when µ assigns unit mass to a single point t = (t1 , ..., td ) ∈ Rd . Let h : Rd → R be an absolutely integrable function with absolutely integrable Fourier transform. Let ϕt,d and
ϕt,d,N be the characteristic
functions of the random vectors G(t1 ), ..., G(td ) and GN (t1 ), ..., GN (td ) , respectively, which are given by
1 X c(ti , tj )xi xj and ϕt,d (x) = exp − 2 1≤i,j≤d

ϕt,d,N (x) = exp −

1 2

X


cN (ti , tj )xi xj .

1≤i,j≤d

By the mean value theorem, |ϕt,d,N (x) − ϕt,d (x)| ≤ CN,t,d kxk21

(1.5.16)

page 49

March 13, 2018 15:46

ws-book9x6

50

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

and we conclude from the Fourier inversion formula that

|Eh GN (t1 ), ..., GN (td ) − Eh G(t1 ), ..., G(td ) | Z Z
2 ˆ ˆ = h(x) ϕt,d,N (x) − ϕt,d (x) dx ≤ CN,t,d |h(x)|kxk 1 dx Rd

Rd

and the lemma follows. Note that d may depend on N and so, when sup0≤t,s≤1 |cN (t, s) − c(t, s)| converges to 0 as N → ∞ then Lemma 1.5.2 provides estimates on the difference |Eg(GN ) − Eg(G)| for functions g which can be approximated sufficiently well by functions of the form gκ,µ,d , d ∈ N such that the integrals R |kˆt (x)|kxk21 dx have some regularity as a function of t and d. Next let h : Rd → R be a twice differentiable function and let v ∈ Rd . Denote by Hh (v)[·, ·] the Hessian form of h at v which is given by X ∂ 2 h(v) . Hh (v)[u, p] = ui pj ∂vi ∂vj 1≤i,j≤d

2

For any u ∈ R set kuk∞ = max{|ui | : 1 ≤ i ≤ d} and let kHv k∞ = sup{|Hv [u, u]| : kuk∞ = 1} and L∞ (Hh ) =

kHh (v + u) − Hh (v)k∞ kuk∞ v,u∈Rd ,u6=0 sup

be the (operator) norm and Lipschitz constant of Hv with respect to the supremum norm, respectively. We consider the following two norms of h: khkM,d = ρ∞,3 (h) + ρ∞,2 (k∇h(·)k1 ) + ρ∞,1 (kHh (·)k∞ ) + L∞ (Hh ) and khkM0 ,d = khkM,d + 2ρ∞,0 (h) + ρ∞,0 (k∇h(·)k1 ) + ρ∞,0 (kHh (·)k∞ ) where for any k ≥ 0 and f : Rd → R, ρ∞,k (f ) = sup v∈Rd

|f (v)| . 1 + (kvk∞ )k

The following result follows. Corollary 1.5.3. Let g = gk,µ,d : D → R be as in Lemma 1.5.2. Suppose, in addition, that the functions kt (·), t ∈ supp(µ) are twice differentiable. Then there exists an absolute constant c0 such that

≤ c0 τN

sup t∈supp(µ)

kkt (·)kM0 ,d +

sup t∈supp(µ)

|Eg(WN ) − Eg(G)| Z
CN,t,d |kˆt (x)|kxk21 dx .

page 50

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

51

Proof. By Theorem 1.5.1 (i) and Lemma 1.5.2 it is sufficient to show that kgk,µ,d kM0 ≤ c0

sup

kkt (·)kM0 ,d .

(1.5.17)

t∈supp(µ)

By the dominated convergence theorem it is sufficient to consider the case when µ assigns unit mass to some point t ∈ [0, 1]d . Let t ∈ [0, 1]d and consider the map πt : D → Rd given by πt (w) = (w(t1 ), ..., w(td )). Let h : Rd → R be a twice differentiable function such that khkM0 ,d < ∞. In order to prove (1.5.17), it is sufficient to show that the lifted function ht given by ht = h ◦ πt satisfies kht kM ≤ khkM,d and kht kM0 ≤ khkM0 ,d

(1.5.18)

and this is a consequence of the equalities Dht (w)[w1 ] = ∇h(πt (w)) · πt (w1 ) and D2 ht (w)[w1 , w2 ] = Hh (πt (w))[πt (w1 ), πt (w2 )] where u · v stands for the standard inner product of two vectors u, v ∈ Rd . Remark 1.5.4. Let d ∈ N and t = (t1 , ..., td ) ∈ [0, 1]d . Then Corollary 1.5.3 provides, in particular, rates in the multidimensional CLT for the
random vectors WN (t¯) := WN (t1 ), ..., WN (td ) , namely, upper bounds of expressions of the form Eh(WN (t¯)) − Eh(G(t1 ), G(t2 ), ..., G(td )) for some family of twice differentiable functions h. It is customary to have explicit dependence on d in such rates, under some restrictions on the growth rates of some norms of the function h and its derivatives with respect to the Euclidean norm k · k2 . Our √ estimates, of course, can be reformulated in this way since kvk∞ ≤ kvk2 ≤ dkvk∞ for any v ∈ Rd . 1.6

A nonconventional functional CLT

Let (Ω, F, P ), Fn,m , {ξn : n ≥ 0} and F be as described in Section 1.3.1. Consider the random function ZN given by [N t]

ZN (t) = N

− 21

X

F (ξn , ξ2n , ..., ξ`n ).

n=1

The main goal of this section is to prove a functional CLT for these random functions. We need first the following lemma. Lemma 1.6.1. Suppose that the assumptions of Theorem 1.3.4 hold true. Let t, s > 0 and N ∈ N and set bN (t, s) = EZN (t)ZN (s). Then the limit limN →∞ bN (t, s) = b(t, s) exists and for any N ∈ N, 1

|bN (t, u) − b(t, u)| ≤ c` C0 N − 2

(1.6.1)

page 51

March 13, 2018 15:46

52

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where C0 is defined in Theorem 1.3.4 and c` is a constant which depends only on `. The proof of this lemma follows from the arguments in [44] similarly to the proof of (1.3.12) in Theorem 1.3.4 (see [45] for an explicit formula for the limits b(t, s) and for other properties of these limits see [31]). Now we can state our nonconventional functional CLT. Theorem 1.6.2. (i) Suppose that Assumptions 1.3.1 and 1.3.3 hold true 4b . Then the random function ZN (·) conwith θ and b > 0 such that θ > b−2 verges in distribution to a centered Gaussian process η(·) whose covariance function is b(·, ·). (ii) Moreover, let k, µ and d be as in Lemma 1.5.2 and consider the R function2 g = gk,µ,d . Further assume that kκt (·)kM0 ≤ 1 and |kˆt (x)|kxk1 dx ≤ 1 for any t ∈ supp(µ). Then 1

|Eg(ZN ) − Eg(η)| ≤ a` C2 max(B, B 3 )N −( 2 −2θ1 )

(1.6.2)

ι ), C2 is defined in Theorem 1.3.7, where B = K(1 + γm

θ1 =

3b 1 < 4b + 2θ(b − 2) 4

and a` is a constant which depends only on `. When Assumption 1.3.2 holds true then the above right-hand side can be replaced with 1 a` C1 max(B, B 3 )N − 2 ln2 (N + 1), where C1 is defined in Theorem 1.3.7. As in Theorem 1.3.7, when βr0 = 0 for some r0 then all the above results hold true when F is only a Borel function which satisfies (1.3.6). In particular, when our mixing assumptions hold true with the σ-algebras Fm,n = σ{ξm , ..., ξn } then we can take r0 = 0 and we obtain a functional nonconventional CLT when F only satisfies (1.3.6). Proof of Theorem 1.6.2. Let l ≥ 1, set r = [ 3l ] and consider the random functions ZN,r (·) and Z¯N,r (·) given by [N t]

ZN,r (t) = N

− 12

X

F (ξn,r , ξ2n,r , ..., ξ`n,r ) and

n=1

Z¯N,r (t) = ZN,r (t) − EZN,r (t). Let g ∈ M0 be such that kgkM0 ≤ 1. We first estimate the difference |Eg(ZN (·)) − Eg(Z¯N,r (·))|.

page 52

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

53

By the definition of M0 , for any w, h ∈ D we have |g(w + h) − g(w)| ≤ sup |h| and therefore |Eg(ZN (·)) − Eg(Z¯N,r (·))| ≤ E sup |ZN (·) − Z¯N,r (·)|.

(1.6.3)

In order to estimate the right-hand side of (1.6.3) we apply first (1.3.45) and (1.3.46) to obtain sup |Z¯N,r (·) − ZN,r (·)| 1

≤ N−2

N X

|EF (Ξn,r )| ≤ 7σ`2 Rsr,l := R1 , P -a.s.

n=1

where sr,l is defined in (1.3.48). Secondly, we apply (1.3.42) and since m > ιb we have

sup |ZN (·) − ZN,r (·)|

Lb

1

≤ N−2

N X

kF (Ξn ) − F (Ξn,r )kLb

n=1 1

ι κ ≤ N 2 `2 K(1 + γm )βq,r := R2

concluding from (1.6.3) that |Eg(ZN (·)) − Eg(Z¯N,r (·))|

≤ sup |ZN (·) − Z¯N,r (·)| Lb ≤ R1 + R2 := R(N, r).

(1.6.4)

Next, consider l’s of the form l = 3N ζ1 + 3, where ζ1 is a parameter yet to be chosen. Consider the triangular array {X1,N , X2,N , ..., XN,N } where the Xn,N ’s are given by (1.3.52). The purpose of the following arguments is to show that Theorem 1.5.1 can be applied with the random functions WN = WN (·) = Z¯N,r (·). We begin with showing that condition (1.5.3) holds true when ζ1 is chosen appropriately. We first claim that there exists C > 0 such that Var(ZN (s) − ZN (t)) ≤ C 2

[N s] − [N t] N

(1.6.5)

for any N ∈ N and 0 ≤ t ≤ s ≤ 1. Indeed, by Corollary 1.3.14 there exists a constant c0 > 0 such that for any n, m ∈ N, 1 |Cov(F (Ξn ), F (Ξm ))| ≤ c0 τ ([ d` (n, m)]) 3

page 53

March 13, 2018 15:46

ws-book9x6

54

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics 1− 1

1

κ ≤ dk −(1− b )θ , k ≥ 0 and d` (n, m) is given by where τ (k) = φk b + βq,k (1.3.13). For any k ≥ 0 and n ∈ N there exist at most 2`2 natural m’s such that d` (n, m) = k. Therefore since θ(1 − 1b ) > 1, ∞ X

|Cov(F (Ξn ), F (Ξm ))| ≤ 6`2 c0

m=1

∞ X

τ (s) := A < ∞

s=0

for any natural n, and so Var(ZN (s) − ZN (t)) ≤ N −1

[N s]

[N s]

X

X

|Cov(F (Ξn ), F (Ξm ))|

n=[N t] m=[N t]

≤A

[N s] − [N t] N

implying (1.6.5). Next, by (1.3.42),

1

≤ N−2


kZN,r (s) − ZN,r (t) − ZN (s) − ZN (t) kL2 r [N s] X [N s] − [N t] [N s] − [N t] κ c2 βq,r ≤ c2 c3 ≤ c2 c3 N N

n=[N t]

1

κ ι ≤ c2 N − 2 for some c2 > 0. ), assuming that βq,r where c3 = K`(1 + `γm l −θ κ κ ≤ Finally, we recall that βq,r ≤ dr and r = [ 3 ] ≥ N ζ1 . Therefore βq,r 1 1 −2 dN when ζ1 ≥ 2θ and in this case we can take c2 = d. It follows that (1.6.5) holds true with ZN,r in place of ZN , possibly with a different constant C 0 in place of C. Since
E|Z¯N,r (s) − Z¯N,r (t)|2 = Var ZN,r (s) − ZN,r (t)

this completes the proof that condition (1.5.3) holds true for the random functions WN = Z¯N,r , N ≥ 1. Next, let ¯bN (·, ·) be the covariance function of WN . We claim that, under additional restrictions on ζ1 , the limiting covariances of ¯bN (·, ·) are b(·, ·) appearing in Lemma 1.6.1. Indeed, for any 0 ≤ t, s ≤ 1 we have |¯bN (t, s) − bN (t, s)| = |EZ¯N,r (t)Z¯N,r (s) − EZN (t)ZN (s)| ≤ kZN (t)kL2 kZN (s) − Z¯N,r (s)kL2 + kZ¯N,r (s)kL2 kZN (t) − Z¯N,r (t)kL2 1 1
≤ R(N, r) 2c`2 C02 + σ + R(N, r) := B(N, r). In the first inequality we used the Cauchy-Schwarz inequality while in the second inequality we used (1.6.4) and (1.3.12) with [N t] and [N s] in place 1 of N . Let ζ1 ≥ 2θ be such that τN ln2 N and B(N, r) converge to 0 as N → ∞, where τN is defined in Theorem 1.5.1 (for some choice of p and q).

page 54

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

55

Then by Theorem 1.5.1 the random function WN converges in distribution on the Skorokhod space D to a centered continuous Gaussian process η with the covariance function b(·, ·). Finally, by Proposition 3.1 in [2] we deduce that ZN (·) weakly converges to η, assuming that R(N, r) ln2 N converges to 0 as N → ∞. By (1.2.17), Lemma 1.3.15 and similarly to (1.3.53) the di ’s defined in Theorem 1.5.1 (i) satisfy 1

1− 1b

d1 ≤ cN 2 Bφr

1− 2b

, d2 ≤ cB 2 Dφr

1

, d3 ≤ cD2 N − 2 B 3 1− 2b

and d4 ≤ cN B 2 φr

ι where c depends only on `, B = K(1 + γm ), D is given by (1.3.19) and we l have chosen here p = 2q = b. Since r = [ 3 ] we have

φr ≤ dr−θ ≤ dN −θζ1

(1.6.6)

and, taking into account (1.3.19), we deduce that there exists a constant c4 which depends only on ` such that τN ≤ c4 d max(B, B 3 )N −a1

(1.6.7)

where
1 2 a1 = min (1 − )ζ1 θ − 1, − 2ζ1 . b 2
−1 Taking θ(1 − 2b ) < ζ1 < 14 we have a1 > 0 and τN ln2 N converges to 0 as N → ∞. Such choice of ζ1 is possible by our assumption that 1 4b and note that such ζ1 indeed satisfies ζ1 ≥ 2θ . Finally, by (1.6.6) θ > b−2 the definition of sr,l in (1.3.48), (1.3.19) and similarly to (1.3.68) we have 1

k R(N, r) = R1 + R2 ≤ c2 B(sr,l + N 2 βq,r ) ≤ c5 dBN −a2

where c5 is a constant which depends only on ` and 1 1 1 a2 = min − ζ1 , (1 − )θζ1 − ) ≥ a1 . 2 b 2 Therefore, for the above choice of ζ1 the sequences R(N, r) ln2 N and B(N, r) converge to 0 as N → ∞, as well, and we have completed the proof of the first statement of Theorem 1.6. 3b In order to prove the second statement, we take ζ1 = 4b+2θ(b−2) which solves the equation 2 1 ζ1 θ(1 − ) − 1 = − 2ζ1 . b 2 This choice of ζ1 together with Corollary 1.5.3 and the previous estimates yield (1.6.2), taking into account (1.3.69). We note that a1 > 0 for this 4b ζ1 in view of our assumption that θ > b−2 . When Assumption 1.3.2 holds true then as in the proof of Theorem 1.3.7 (i) we take l = A ln(N + 1) + 3, 45 where A = − ln(c) and obtain the desired upper bound.

page 55

March 13, 2018 15:46

56

1.7 1.7.1

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Extensions to nonlinear indexes Preliminaries

Let q1 , ..., q` be strictly increasing functions which map N to N and are ordered so that q1 (n) < q2 (n) < ... < q` (n) for any sufficiently large n. For any N ∈ N consider the random function [N t] 1

ZN (t) = N − 2

X

F (ξq1 (n) , ξq2 (n) , ..., ξq` (n) ) − F¯




(1.7.1)

n=1

where F¯ is given by (1.3.7). We further assume that the difference qi (n) − qi−1 (n) tends to ∞ as n → ∞ for any i = 1, 2, ..., `, where q0 ≡ 0, though the situation when some of these differences are nonnegative constants can be considered, as well (see Section 3 in [31]). By disregarding some of the first summands in (1.7.1), we can also consider the case when qi ’s are strictly increasing on some ray [Q, ∞), and then we can consider polynomial qi ’s with positive leading coefficients which map N to itself. Next, for any n, m ∈ N set d˜` (n, m) = min |qi (n) − qj (m)| 1≤i,j≤`

˜ ˜N,l ) on VN = {1, 2, ..., N } where and consider the graph G(N, l) = (VN , E the set of edges is given by ˜N,l = {(n, m) ∈ VN × VN : d˜` (n, m) < l}. E We begin with imposing restrictions on qi ’s which guarantee that the cardinalities of the sets Nn = {1 ≤ m ≤ N : d˜` (n, m) < l} are of order l. First notice that [ Nn = VN ∩ Ji,j (n) 1≤i,j≤`


where Ji,j (n) = [qj−1 (qi (n) − l)], [qj−1 (qi (n) + l)] . Assumption 1.7.1. There exists Q ≥ 1 such that for any a, b ≥ 1, |qj−1 (a) − qj−1 (b)| ≤ Q(1 + |a − b|) for any 1 ≤ j ≤ `.

(1.7.2)

page 56

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

57

When (1.7.2) holds true then |Ji,j (n)| ≤ Q(1 + 2l) and we deduce that |Nn | ≤ 2`2 Q(1 + l). From the above inequality we derive that the order of the cardinality of ˜ balls of radius 3 in the graph G(N, l) is at most l3 . Condition (1.7.2) holds true, for instance, when all qj ’s have the form qj (x) = [pj (x)] where each pj is a strictly increasing function whose inverse p−1 has bounded derivative j on some ray [K, ∞). For example we can take pj ’s to be a polynomials with positive leading coefficient, exponential functions etc. The following theorem is proved similarly to Theorem 1.3.7 (i). Theorem 1.7.2. Suppose that Assumptions 1.3.1 and 1.3.2 are satisfied with b = 5. Furthermore, assume that (1.7.2) holds true and that the limit 2 σ 2 = limN →∞ EZN , ZN = ZN (1) exists and is positive. Then ZN converges in distribution to a centered normal random variable with variance σ 2 and there exists a constant c which depends only on ` so that for any N ≥ 1,
max dW (L(σ −1 ZN ), N ), dK (L(σ −1 ZN ), N ) 3

5

1

2 − σ2 | ≤ cQ4 C12 (1 + R) max(R 6 , R2 )N − 2 ln4 (N + 1) + 4σ −2 |EZN

where C1 and R are defined in Theorem 1.3.7 (i). Furthermore, if the limits b(t, s) appearing in Lemma 1.6.1 exist then the random function ZN (·) weakly converges in the Skorokhod space D toward a centered continuous Gaussian process η with covariance function b(·, ·). A corresponding version of Theorem 1.3.7 (ii) follows, as well, but the purpose of this section is only to demonstrate that Stein’s method in the nonconventional setup is not restricted to linear indexes. Some convergence rate in the functional CLT can be given, as well. In [45] existence of the limits b(t, s) was proved when for some 1 ≤ k ≤ ` we have qi (n) = in for any i ≤ k and n ∈ N, while for any k < i ≤ ` and ε > 0, lim (qi (n + 1) − qi (n)) = ∞ and

n→∞

lim inf (qi (εn) − qi−1 (n)) > 0 n→∞

which imply that lim (qi (εn) − qi−1 (n)) = ∞.

n→∞

page 57

March 13, 2018 15:46

58

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

In [31] the authors showed that the limits b(t, s) exist when all qj ’s are polynomials. In the situation of [45], convergence rate can be obtained assuming that there exist constants c, α, γ1 > 0 and 0 < γ2 < 1 such that for any i > k and n ∈ N, (1.7.3) qi (n + 1) − qi (n) ≥ cnα and qi (nγ2 ) − qi−1 (n) ≥ cnγ1 while in the setup of [31] convergence rates toward these limits can be easily recovered from the arguments in the proof of their existence. 1.7.2

Another example with nonlinear indexes

For general indexes q1 (n), q2 (n), ..., q` (n) existence of the limits b(t, s) (and thus the CLT) depends on properties of solutions of equations of the form qi (n) = qj (m) + u, where u ∈ N and the pair (n, m) ∈ N × N satisfy some restriction of the form m = cn + k for some c and k. When all qi ’s are polynomials then the related number theory problem is solved in Section 4 of [31]. Instead of providing technical conditions for the existence of these limits, we will provide a demonstrative example. Recall that two positive numbers a a, b are multiplicatively independent if ln ln b 6∈ Q. Let λ1 , λ2 > 1 and consider the functions q1 (x) = λx1 and q2 (x) = λx1 λx2 . In order to avoid complications, we will not replace qi ’s with their integer part and instead we consider a family of random variables {ξu : u ≥ 0} on the probability space (Ω, F, P ) which satisfies the conditions from Section 1.3.1. Namely, we assume that the distribution of ξu does not depend on u, that distribution of (ξu , ξv ) depends only on u − v and that there exists a nested family Fu,v , 0 ≤ u ≤ v of sub-σ-algebras of F such that (1.3.10) holds true for any ρ > 0 in place of n where φρ and βq,ρ are defined similarly to φn and βq,n but with nonnegative ρ’s in place of natural n’s. R Let µ be the distribution of ξu and for any a ≥ 1 set γaa = kξu kaLa = |x|a dµ. Suppose that F¯ in (1.3.7) vanishes, and consider the normalized random function [N t] X − 21 F (ξλn1 , ξλn1 λn2 ). ZN (t) = N n=1

Theorem 1.7.3. In addition to the latter assumptions, suppose that Assumption 1.3.1 holds true with b = 5. (i) The random function ZN (·) converges in the Skorokhod space D to a centered Gaussian process η(·). When λ1 and λ2 are multiplicatively independent then the covariances of η are given by Z Cov(η(t), η(s)) = lim EZN (t)ZN (s) = min(t, s) N →∞

F 2 (x, y)dµ(x)dµ(y)

page 58

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Stein’s method in the nonconventional setup

59

and in particular η has stationary and independent increments. When λ1 and λ2 are not multiplicatively independent then the covariances of η are given by Z Cov(η(t), η(s)) = lim EZN (t)ZN (s) = min(t, s) F 2 (x, y)dµ(x)dµ(y) N →∞

+aq1 ,p1 (t, s) + aq1 ,p1 (s, t) where

ln λ1 ln λ2

=

p1 q1

for some coprime integers q1 and p1 and aq1 ,p1 (t, s)

1 t s = min( , , ) q1 p1 p1 + q1

Z F (x1 , x2 )F (x2 , x3 )dµ(x1 )dµ(x2 )dµ(x3 ).

(ii) Moreover, for any 0 ≤ t, s ≤ 1 and N ≥ 1, 1

1

|EZN (t)ZN (s) − b(t, s)| ≤ CN − 2 ln 2 (N + 1)

(1.7.4)

and when σ 2 = Var(η(1)) > 0 then with ZN = ZN (1),
max dW (L(σ −1 ZN ), N ), dK (L(σ −1 ZN ), N ) 5

1

≤ C(1 + σ −1 ) max(σ − 6 , σ −2 )N − 2 ln4 (N + 1) where N is the standard normal law and C is a constant which depends only ι on K, θ, c, d, `, γm (and this dependence can be recovered from the proof ). Proof. Similarly to the situation of Theorem 1.7.2, in order to prove Theorem 1.7.3 we only have to show that the limits b(t, s) exist and obtain convergence rates towards them. We first claim that there exists a constant C > 0 such that 2 EZN ≤C

(1.7.5)

for any N ∈ N. Indeed, for any natural n and m set m ). En,m = EF (ξλn1 , ξλn1 λn2 )F (ξλm , ξλm 1 1 λ2

When m = n + k for some k > 0 then the minimal difference between the m m numbers λn1 , λn1 λn2 , λm 1 and λ1 λ2 equals
λn1 min λn2 − 1, |λk1 − λn2 | := ρ(n, k). n n When ρ(n, k) vanishes then λm 1 = λ1 λ2 and otherwise these numbers are distinct. It follows from Corollary 1.3.14 that
2 |En,m | = |En,m − F¯ | ≤ C1 τρ(n,k) (1.7.6)

page 59

March 13, 2018 15:46

60

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics 1− 1b

for some constant C1 > 0, where τρ = φρ our version of Assumption 1.3.2 we have

κ + βq,ρ and we recall that under

τρ ≤ 2dρ−θb , ∀ρ > 0

(1.7.7)

where θb = θ(1 − 1b ). Next, write 2 EZN = N −1

N X

X

En,n + 2N −1

n=1

En,m := I1 + I2 .

1≤n

|J1 | ≤ 2cN −1

N X

M1 ≤ 2cM1 .

n=1

On the other hand, by (1.7.6) and (1.7.7) we have |J2 | ≤ 2

∞ X

b τc0 λn1 < 4dc−θ 0

n=1

∞ X

b λ−nθ < R1 1

n=1

for some constants R1 > 0 and c0 > 0, and the proof of (1.7.5) is complete. Next, let N ∈ N, t, s ∈ [0, 1] and let M be of the form M = A ln(N + 1) where A > 0 is a constant. For the proof of Theorem 1.7.3 (i) we can take any constant A > 0 but in order to prove Theorem 1.7.3 (ii) we will consider A as a parameter. Set X 1 Z˜N (u) = N − 2 F (ξλn1 , ξλn1 λn2 ), u ∈ [0, 1]. M 0, and therefore R1,N (t, s) converges to min(t, s) F 2 (x1 , x2 ) as N → ∞. Next, for any ε ∈ (0, 1) we have R2,N (t, s) = N −1

N X

X

En,m

k=1 (n,m)∈JN,k,t,s,ε

+N

−1

N X

X

En,m := Q1,N (ε, t, s) + Q2,N (ε, t, s)

k=1 (n,m)∈RN,k,t,s,ε

where JN,k,t,s,ε = IN,k,t,s ∩ {|λk1 − λn2 | ≥ ε}

page 61

March 13, 2018 15:46

62

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

and RN,k,t,s,ε = IN,k,t,s ∩ {|λk1 − λn2 | < ε}. Now, by (1.7.6) there exist constants C3 , C2 , c2 > 0 such that |Q1,N (ε, t, s)| ≤ N −1

N X

X

|En,m | ≤ C2 N −1

N X

τc2 ελn1 ≤ 2dC2 (c2 ε)−θb

n=M

τc2 ελn1

k=1 n=M

k=1 (n,m)∈JN,k,t,s,ε

= C2

N X N X

(1.7.12)

∞ X

θb b λ−nθ = C3 (c2 ε)−θb λ−M . 1 1

n=M

Thus, when ε is fixed then Q1,N (ε, t, s) converges to 0 as N → ∞. Next, let (n, m) ∈ RN,k,t,s,ε , k ∈ N, and assume that 1 1 ε < min(1, λ1 , λ1 ln λ2 ). 2 2

(1.7.13)

The inequality |λk1 − λn2 | < ε together with the mean value theorem imply that ε ε k ln λ1 − k < n ln λ2 < k ln λ1 + k λ1 − ε λ1 and so with Λ =

ln λ1 ln λ2 ,

|kΛ − n| ≤

(λk1

ε 2ε := ε0 < 1. ≤ λ1 ln λ2 − ε) ln λ2

(1.7.14)

Suppose next that Λ 6∈ Q. Then the sequence {kΛ : k ∈ N} is equidistributed modulo 1, and therefore by (1.7.8) and (1.7.14) the upper limit as N → ∞ of N

−1

N X

X

|En,m |

(1.7.15)

k=1 (n,m)∈RN,k,t,s,ε

does not exceed M1 ε0 and hence the upper limit of R2,N (t, s) as N → ∞ does not exceed M1 ε0 . Letting ε → 0 we deduce that R2,N (t, s) converges to 0 as N → ∞. Exchanging (n, t) and (m, s) and repeating the above arguments we deduce similarly that R3,N (t, s) converges to 0 as N → ∞. We conclude that the limit b(t, s) exists and that Z b(t, s) = min(t, s) F 2 (x1 , x2 )dµ(x1 )dµ(x2 ).

page 62

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Stein’s method in the nonconventional setup

LNcov

63

Next, suppose that Λ ∈ Q and write Λ = pq11 for some coprime positive 1 integers p1 and q1 . Since rp q1 , r = 1, 2, ..., q1 − 1 is not an integer, there exists a constant c1 > 0 such that either k divides q1 or |kΛ − n| > c1 for any n ∈ N. When k = jq1 for some natural j, then |kΛ − n| < ε0 only if n = p1 j and in this case n = kΛ and λn+k = λn1 λn2 . We conclude that when 1 0 ε < c1 and ε satisfies (1.7.13), X Nt Ns
Q2,N (ε, t, s) = N −1 EΛk,Λk+k I k ≤ min( , ) (1.7.16) Λ Λ+1 1≤k≤N :q1 |k

which converges as N → ∞ to aq1 ,p1 (t, s) defined in the statement of the theorem. Exchanging n and m and repeating the above arguments, it follows that R3,N (t, s) converges to aq1 ,p1 (s, t) as N → ∞ and Theorem 1.7.3 (i) follows. Next we explain how to obtain convergence rates in Theorem 1.7.3 (ii). First, by (1.7.10), 1 1 1 |bN (t, s) − ˜bN (t, s)| ≤ A 2 CN − 2 ln 2 (N + 1). Suppose that Λ ∈ Q and let ε > 0 be sufficiently small but fixed. Then we obtain from Corollary 1.3.14 and (1.7.7) that the convergence rates in all the limits computed earlier in this proof are of order λ1−c0 M = (N + 1)−Ac0 ln λ1 for an appropriate c0 and when taking a sufficiently large A we obtain (1.7.4). When Λ 6∈ Q, we first recall that the inequalities |λk1 − λn2 | < ε and (1.7.13) imply that 2ε |n − Λk| < = ε0 . λ1 ln λ2 By the Erd˝ os-Tur´ an inequality (see Theorem 2.5 in [52]) for any ε0 > 0 and N ∈ N, 1 ln N |B(Λ, ε0 ) ∩ [1, N ]| ≤ 2ε0 + C4 N N where B(Λ, ε0 ) = {k ∈ N : Λk mod 1 ∈ [0, ε0 ) ∪ [1 − ε0 , 1)} 1

and C4 > 0 is an absolute constant. Taking ε = εN = aN − 2 for some sufficiently small a > 0, the rate of convergence of |Q2,N (εN )| to 0 in 1 θb (1.7.12) is of order N 2 θb λ−M which for a sufficiently large A is at most 1 1 of order N − 2 . We conclude that (1.7.4) holds true when Λ is irrational, as well. The convergence rates in the CLT stated in Theorem 1.7.3 follow now using the arguments in Section 1.3.6.

page 63

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

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Chapter 2

Nonconventional local central limit theorem

2.1

Introduction

The classical De Moivre-Laplace theorem states that if X1 , X2 , X3 , ... are independent identically distributed (i.i.d.) 0 − 1 Bernoulli random variables PN taking on 1 with probability p and SN = n=1 Xn , then the probability P {SN = k} is equivalent as N → ∞ to 1

(2πN pq)− 2 exp(−(k − N p)2 /2N pq), q = 1 − p uniformly in k such that |k − N p| = o(N pq)2/3 . The latter expression is the density of a normal distribution with mean N p = ESN and variance N pq = VarSN evaluated at the point k, and so the De Moivre-Laplace theorem can be viewed as a local (central) limit theorem (LLT) for the sums SN . Modern versions of the local limit theorem include the situation when the summands Xn ’s are not lattice valued, where in this situation the asymptotics of expectations of the form Eg(SN − u) is determined for continuous functions g with compact support. Among the main situations when an LTT holds true is the case when PN SN = n=1 g(ζn ) where ζ1 , ζ2 , ζ3 , ... is a stationary Markov chain whose transition operator has a spectral gap when acting on a Banach space B which contains the function g and satisfies certain conditions (see [28]) such as the Lasota-Yorke inequality. The situation when the transition probability of ζ1 , ζ2 , ζ3 , ... satisfies certain regularity condition (a version of the Doeblin condition) can be considered, as well (see [50] and [51]). As an application of the first situation we can consider sums of the form PN −1 SN = n=0 g(T n Z0 ), where T ranges over a special class of dynamical systems (e.g. one sided topologically mixing subshift of finite type), Z0 is distributed according to a special T -invariant Gibbs measure µ and g is a H¨ older continuous function. For instance, when T is a (locally) distance ex65

page 65

March 13, 2018 15:46

66

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

PN −1 panding map on some compact space then the sums SN = n=0 g(T n Z0 ) PN and S˜N = n=1 g(ζn ) are identically distributed, where ζ1 , ζ2 , ζ3 , ... is a stationary Markov chain with initial distribution µ whose transition operator is the dual of the Koopman operator f → f ◦T with respect to the space L1 (µ) (such operators are often referred to as “transfer operators”). Relying on the Ruelle-Perron-Frobenius theorem for such operators (see [10] and [53]), these transfer operators indeed have a spectral gap when acting on spaces of (locally) H¨ older continuous functions. Extensions to other (invertible) dynamical system such as two sided topologically mixing subshifts of finite type and C 2 Axiom A diffeomorphisms in a neighborhood of an attractor follow, as well (see Lemma 1.6 and Sections 3-4 in [10]). In [29] extensions of the LLT to nonconventional sums of the form N X SN = F (ζn , ζ2n , ..., ζ`n ) n=1

were obtained in the case of a Markov chain whose transition probability satisfies a version of the Doeblin condition, where F is a square integrable function. This condition guarantees that the Markov chain under consideration is sufficiently well ψ-mixing, and, in fact, under certain conditions the proof from [29] can be modified to the case of a general ψ-mixing Markov chain (see Remark 2.4.2). The main result of this chapter is an extension of the above nonconventional LLT to the case when ζ = {ζn : n ≥ 0} is a stationary Markov chain generated by a (locally) distance expanding map T on a compact space X considered with a special T -invariant initial (Gibbs) distribution µ and F is a (locally) H¨older continuous function which satisfies some additional regularity conditions related to a prescribed periodic point of T . Namely, we will prove an LLT for these (nonconventional) sums when ζ is a stationary Markov chain whose transition probabilities are given by X P (ζ1 ∈ Γ|ζ0 = x) = ef (y) y∈Γ:T y=x

where f : X → R is an appropriately chosen function, assuming that the process {T n Z0 : n ≥ 0} satisfies certain mixing and approximation conditions when Z0 is distributed according to µ. Extensions to dynamical system will be discussed, as well. More precisely, let T be either a two sided subshift of finite type or a C 2 Axiom A diffeomorphism (in particular, Anosov) in a neighborhood of an attractor. Then we derive an LLT for sums of the form N X SN (x) = F (T n x, T 2n x, ..., T `n x) (2.1.1) n=1

page 66

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

67

when x is drawn at random with respect to an appropriate Gibbs measure. The structure of this chapter is as follows. In the next section we will describe the general Fourier analysis approach for proving an LLT, and in Section 2.3 we will show that, in general, a nonconventional LLT for Markov chains follows from appropriate estimates of norms of products of certain random Fourier operators together with some quantitative mixing properties. For readers’ convenience, in Section 2.4 we will repeat the main arguments of the proof from [29] and then in Section 2.5 we will describe the setup of locally distance expanding maps, present the corresponding thermodynamic formalism theory and additional mixing assumptions and state and prove our main results. The proof relies on some (soft) version of the random complex Ruelle-Perron-Frobenius theorem for iterates of random transfer operators, which will follow from the more general results of Part 2. The proof will involve also a “periodic point approach” to random dynamics which will be discussed in Section 2.10 and will be elaborated in Chapter 7. In the last section of this chapter we will explain how to modify the proof from Section 2.5 in order to prove the nonconventional LLT in the dynamical systems case (i.e. for the sums (2.1.1)). 2.2

Local central limit theorem via Fourier analysis

Let S1 , S2 , S3 , .. be a sequence of random variables. We will say that the sequence satisfies the local (central) limit theorem (LTT) with respect to a measure ν on R, a centralizing constant m ∈ R and a normalizing constant σ > 0 if for any real continuous function g on R with compact support, Z √ (u−mN )2 gdν = 0. (2.2.1) lim sup σ 2πN Eg(SN − u) − e− 2N σ2 N →∞ u∈supp ν

Let ϕN : R → C be the characteristic function of SN given by ϕN (t) = EeitSN . As in many expositions of the LLT, we will distinguish between lattice and non-lattice cases. We call the case a lattice one when there exists h > 0 such that with probability one SN , N ∈ N take values on the lattice hZ := {hk : k ∈ Z}, or equivalently when ϕN (t) = ϕN (t + 2π h ) for any t ∈ R and N ∈ N. When there exists no h with this property then we call the case non-lattice one. The Fourier-analysis proof of the LLT is based on the following growth properties of the characteristic functions ϕN . Assumption 2.2.1. For any δ > 0, 1

lim N 2 sup |ϕN (t)| = 0

N →∞

t∈Jδ

page 67

March 13, 2018 15:46

68

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where in the lattice case Jδ = [− πh , πh ] \ (−δ, δ) while in the non-lattice case Jδ = [−δ −1 , −δ] ∪ [δ, δ −1 ]. Assumption 2.2.2. There exist δ0 ∈ (0, 1), positive constants c0 and d0 1 2 and a sequence (bn )∞ n=1 of real numbers such that limN →∞ n bn = 0 and 2

|ϕN (t)| ≤ c0 e−d0 N t + bN for any N ∈ N and t ∈ [−δ0 , δ0 ]. 1

Theorem 2.2.3. Suppose that N − 2 (SN − mN ) converges in distribution as N → ∞ to a centered normal random variable with variance σ 2 > 0 and that Assumptions 2.2.1 and 2.2.2 hold true. Then in the lattice case the LLT holds true with the measure νh which assigns mass h to each point of the lattice hZ = {hk : k ∈ Z}, while in the non-lattice the LLT holds with the Lebesgue measure ν0 . Proof. First, by Theorem 10.7 in [12] (see also Section 10.4 there and Lemma IV.5 together with arguments of Section VI.4 in [28]) it suffices to prove (2.2.1) for all continuous complex-valued functions g on R such that Z ∞ |g(x)|dx < ∞ (2.2.2) −∞

and having Fourier transform Z gˆ(λ) =

∞

e−iλx g(x)dx, λ ∈ R

(2.2.3)

−∞

vanishing outside a finite interval [−L, L]. Then, in particular the inversion formula Z ∞ 1 eiλx gˆ(λ)dλ (2.2.4) g(x) = 2π −∞ holds true. Let g be a function with the above properties and let u ∈ R. Then by (2.2.4) we have Z ∞ 1 Eg(SN − u) = ϕN (λ)e−iλu gˆ(λ)dλ (2.2.5) 2π −∞ where ϕN (λ) = √EeiλSN is the characteristic function of SN . Changing variables s = λσ N we obtain Z ∞ √ 1 s
− isu s
√ σ 2πN Eg(SN − u) = √ ϕN √ e σ N gˆ √ ds. (2.2.6) 2π −∞ σ N σ N

page 68

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

69

On the other hand, from the formula for the characteristic function of the Gaussian distribution and the Fourier inversion formula it follows that R Z Z (u−mN )2 gdνh ∞ iλ(u − mN ) λ2
− 2N σ2 √ gdνh = √ exp − dλ (2.2.7) e − 2 2π −∞ σ N where in the lattice case νh is the measure which assign mass h to each point of the lattice hZ = {hk : k ∈ Z}, while in the non-lattice case we set h = 0 and denote the Lebesgue measure by ν0 . Now, in the non-lattice case by (2.2.6) and (2.2.7) for any δ, T > 0 we can write Z √ (u−mN )2 σ 2πN Eg(SN − u) − e− 2N σ2 gdν0 (2.2.8) ≤ I1 (N, T ) + I2 (T ) + I3 (N, δ) + I4 (N, δ, T ), where Z

T

Z 2 λ
λ
− λ2 −e gdν0 |dλ, |e ϕN √ gˆ √ σ N σ N −T R Z | gdν0 | λ2 I2 (T ) = √ e− 2 dλ, 2π |λ|>T Z kˆ g k∞ λ
√ |dλ and I3 (N, δ) = √ √ √ |ϕN 2π δσ N ≤|λ|≤Lσ N σ N Z kˆ g k∞ λ
√ |dλ I4 (N, δ, T ) = √ √ |ϕN 2π T T

(2.2.20) assuming that δ < min( πh , δ0 ), where g k∞ < ∞. R = sup |r(t)| ≤ (2 + hL)kˆ |t|≤ π h

The proof of the theorem is complete in both cases by first taking a sufficiently small δ, then letting N → ∞ and then T → ∞. 2.3

Nonconventional LLT for Markov chains by reduction to random dynamics

Our setup consists of a probability space (Ω, F, P ) together with a stationary Markov chain ζ = {ζn : n ≥ 0} evolving on a compact space X equipped with the Borel σ-algebra BX . Let P (x, Γ) = P (ζ1 ∈ Γ|ζ0 = x), x ∈ X , Γ ∈ BX be the transition probabilities of the R Markov chain ζ and let µ be its stationary distribution which satisfies P (x, Γ)dµ(x) = µ(Γ) for any Γ ∈ BX . Let F = F (x1 , ..., x` ), ` ≥ 1 be a Borel function on X ` = X × · · · × X such that Z 2 b = F 2 (x1 , ..., x` )dµ(x1 )...dµ(x` ) < ∞. (2.3.1) Consider the sums SN =

N X

F (ζn , ζ2n , ..., ζ`n ), N ∈ N

n=1

and set F¯ :=

Z F (x1 , x2 , ..., x` )dµ(x1 )dµ(x2 )...dµ(x` ).

(2.3.2)

page 71

March 13, 2018 15:46

ws-book9x6

72

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

In what follows the centralizing constant m will always be m = F¯ . As we have seen, the LLT follows from the CLT together with some control on the decay rates of ϕN as N → ∞. In this section we will reduce the problem of approximation of the characteristic functions ϕN of SN to approximation of norms of certain products of random Fourier operators. Let P (`, ·, ·) be the ` step transition probability which is given by P (`, x, Γ) = P (ζ` ∈ Γ|ζ0 = x). It will be more convenient to write P (`, ·, ·) = P` (·, ·). The first step of the reduction goes as follows. Let 1 ≤ M ≤ N and t ∈ R. Then |ϕN (t)| = |EeitSN | = |EE[eitSN |ζ1 , ζ2 , ..., ζ`M ]| it(SN −SM )

≤ E|E[e Set M = M` (N ) = N −

` 2[ N −N ], 2

(2.3.3)

|ζ1 , ζ2 , ..., ζ`M ]|.

where N` = [N (1 −

`M − (` − 1)N ≥

1 2` )]

+ 1. Then

N 2

and therefore by the Markov property, it(SN −SM )

E[e

N Y

|ζ1 , ζ2 , ..., ζ`M ] =

Qt,ζk ,ζ2k ,...,ζ(`−1)k 1(ζ`M )

(2.3.4)

k=M +1

where 1 is the function which takes the constant value 1 and for any x ¯= (x(1) , ..., x(`−1) ) ∈ X `−1 = X × · · · × X and t ∈ R the Fourier operator Qt,¯x is given by Z Qt,¯x g(y) = eitF (¯x,z) g(z)P` (y, dz) = E[eitF (¯x,ζ` ) g(ζ` )|ζ0 = y] for any bounded Borel measurable function g : X → C. The right-hand side of (2.3.4) consists of a product of random Fourier operators, and Assumptions 2.2.1 and 2.2.2 will follow from appropriate estimates of the norms of these random products. Still, the process {(ζn , ζ2n , ..., ζ(`−1)n ) : n ≥ 1} is not stationary (unless ` = 2) and thus, in contrary to the classical situation of products of random operators, these random products are not taken along paths of a measure preserving system, namely they do not have the M +1 M +2 N ω1 ω1 form Aϑ ◦ Aϑ ◦ · · · ◦ Aϑ ω1 for some family {Aω1 } of random operators and a measure preserving system (Ω1 , F1 , P1 , ϑ). (i) In order to overcome this difficulty, let ζ (i) = {ζn : n ≥ 0}, i = 1, 2, ..., ` − 1 be ` − 1 independent copies of the process ζ = {ζn : n ≥ 0} and set (2)

(`−1)

Θn = (ζn(1) , ζ2n , ..., ζ(`−1)n ), n ≥ 0.

(2.3.5)

page 72

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

73

Then the process Θ = {Θn : n ≥ 0} is stationary and, in fact, forms a Markov chain. Consider the sets of indexes Mi = Mi,N = {in : M < n ≤ N }, i = 1, 2, ..., ` − 1. Then for each i, the random variables {ζm : m ∈ Mi } are the only ones appearing in (2.3.4) in the i-th coordinate inside F . For the sake of convenience set M` = M`,N = {`M }. Observe that N jM − (j − 1)N ≥ , j = 1, 2, ..., ` (2.3.6) 2 which means that the sets M1 , ..., M`−1 are ordered so that mi + N2 ≤ mi+1 for any i = 1, 2, ..., ` − 1 and mi ∈ Mi , i = 1, ..., ` − 1. Under appropriate mixing conditions and regularity assumptions on F (which will be introduced later depending on the case) this large distance between the Mi ’s implies that the random variables ζ(Mi ) = {ζm : m ∈ Mi }, i = 1, 2, ..., ` are weakly dependent and as a consequence we will derive that N N Y Y (`) E| Qt,ζk ,ζ2k ,...,ζ(`−1)k 1(ζ`M )| − E| Qt,Θk 1(ζ`M )| ≤ w(t)cN k=M +1

k=M +1

(2.3.7) where ζ (`) is another copy of ζ which is independent of the rest of the copies, w : R → R is a continuous function and (cN )∞ N =1 is a sequence satisfying 1 limN →∞ N 2 cN = 0. Assuming the validity of (2.3.7) and relying on the previous estimates, the goal now is to estimate N Y (`) E| Qt,Θn 1(ζ`M )|. k=M +1

It will be convenient in Section 2.8 to represent this expression in the following way. Let (ΩΘ , FΘ , PΘ , ϑ) be an invertible measure preserving system corresponding to the process Θ (where ϑ is the path shift) and let p0 : ΩΘ → X `−1 be a measurable function so that the processes Θ and {p0 ◦ ϑn : n ≥ 0} have the same distribution (see Section 2.7.1). Then Z Z N Y M +1 (`) E| Qt,Θn 1(ζ`M )| = |Qϑit ω,N −M 1(x)|dµ(x)dPΘ (ω) (2.3.8) k=M +1

Z Z =

−M |Qω,N 1(x)|dµ(x)dPΘ (ω) it

where Qω it = Qit,p0 (ω) and for any n ∈ N, n−1

ω ϑω ϑ Qω,n it = Qit ◦ Qit ◦ · · · ◦ Qit

ω

and we arrive at the classical situation of products of random operators, which in our situation are evaluated at the function 1.

page 73

March 13, 2018 15:46

74

2.4

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Markov chains with densities

In this section we will describe the proof of the LLT from [29] in the case when the `-step transition probabilities P` (·, ·) = P (`, ·, ·) have densities with respect to some probability measure, which are bounded and bounded away from 0. 2.4.1

Basic assumptions and CLT

We assume here that there exists a Borel measure η on X and a constant γ ∈ (0, 1) such that for any Borel measurable set Γ ⊂ X and x ∈ X , γη(Γ) ≤ P (`, x, Γ) ≤ γ −1 η(Γ).

(2.4.1)

This assumption is a specific form of the two sided Doeblin condition. The right-hand side already implies that there exists a stationary distribution µ and that the geometric ergodicity condition kP (n, x, ·) − µk ≤ β −1 e−βn , β > 0 holds true, where k · k is the total variation norm. Inequality (2.4.1) implies that the measures P (`, x, ·) are absolutely continuous with respect to η, and that the corresponding densities dP (`, x, ·) (2.4.2) p(`) (x, ·) := dη satisfy γ ≤ p(`) (x, y) ≤ γ −1

(2.4.3)

for any x and η-a.a. y, where a.a. stands for almost all. Next, in [29] the case when F is a function of the first `−1 variables µ` = µ × · · · × µ-almost surely is excluded according to the following reasoning. Consider the function F` : X ` → R given by Z F` (x) = F (x) − F (x1 , ..., x`−1 , y)dµ(y), x = (x1 , ..., x` ). (2.4.4) Then F (x) depends only on x1 , ..., x`−1 for µ` -a.a. x if and only if F` vanishes µ` -a.s., where a.s. stands for almost surely. Since µ is the stationary measure of P (·, ·) the distribution µn,` of (ζn , ζ2n , ..., ζ`n ) is absolutely continuous with respect to µ` , for any n ∈ N. We refer the readers to the end of the proof of Theorem 2.4 in [29] for the details. As a consequence, when F` vanishes µ` -a.s. then with G` = F − F` for any N ∈ N we have SN =

N X n=1

G` (ζn , ζ2n , ..., ζ(`−1)n ), P -a.s.

page 74

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

page 75

75

This leads to the reduction from ` to ` − 1 and we can assume from the beginning that ` = j is the maximal number of variables for which F is a function of j variables, µ` -a.s. Consider the family Fm,n , m ≤ n of σ-algebras given by Fm,n = σ{ζm , ..., ζn } and set Fm,∞ = σ{ζs : s ≥ m}. The ψ-mixing coefficients associated with the family Fm,n is given by ψn

(2.4.5)  P (A ∩ B) − 1| : A ∈ F0,k , B ∈ Fk+n,∞ , P (A)P (B) > 0, k ≥ 0 . = sup | P (A)P (B)

By [4] inequality (2.4.1) implies the exponentially fast mixing rate ψn ≤ abn , a > 0, b ∈ (0, 1).

(2.4.6)

In fact (see [4]) this mixing condition implies that (2.4.1) holds true with some n0 in place of ` and for µ-a.a. x. By Theorems 2.2 and 2.3 in [29] (which rely on (2.4.6)) the limit σ 2 = lim

N →∞


2 1 E SN − F¯ N

1

exists and N − 2 (SN −N F¯ ) converges in distribution to the centered normal distribution with variance σ 2 . For a different proof of the CLT, see Chapter 1. Henceforth, we assume that σ 2 > 0, and we refer the readers to Theorems 2.3 and 2.4 in [29] for equivalent conditions for positivity of σ 2 . Next, we will consider here the following particular lattice and nonlattice cases. For any v = (v1 , ..., v`−1 ) ∈ X `−1 consider the set Bv = {h ≥ 0 : F (v, x) − F (v, y) ∈ hZ for µ2 -a.a. (x, y) ∈ X 2 }

(2.4.7)

where hZ = {hk : k ∈ Z}. We call the case a lattice one if there exists h > 0 such that F (x) ∈ hZ for µ` -a.a. x, and h = sup{u ≥ 0 : u ∈ Bv } for µ`−1 -a.a. v. Since the distribution of (ζn , ζ2n , ..., ζ`n ) is absolutely continuous with respect to µ` for any n ∈ N, it follows in this lattice case that SN takes values on hZ for any N . We call the case a non-lattice case if µ`−1 {v : Bv = ∅} > 0. Note that there are other cases beyond what we designated as a lattice and a non-lattice case.

March 13, 2018 15:46

ws-book9x6

76

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

2.4.2

Characteristic functions estimates

Relying on (2.4.6), it is shown in [29] that (2.3.7) holds true with w ≡ 1 and cN = (` − 1)ψ[ N ]−2 . In fact, this approximation follows also from Corollary 2 1.3.11, taking into account (2.3.6) and relying on the relation φn ≤ ψn , where φn is defined by (1.3.1). Consider the Fourier operators Qx¯,t from (2.3.5) where x ¯ = (x1 , ..., x`−1 ). For each x ¯ ∈ X `−1 and t ∈ R consider the function ρ : X `−1 → [0, 1] given by Z Z
x, z) dη(z) ρt (¯ x) = sup dη(v) p(`) (y, z)p(`) (z, v) exp itF (¯ y∈X

where p(`) (·, ·) is the `-th step transition density defined by (2.4.2). Then for any x ¯1 , x ¯2 ∈ X `−1 and t ∈ R, Z
kQx¯1 ,t Qx¯2 ,t k∞ = sup sup dη(v) exp itF (¯ x2 , v) f (v) (2.4.8) f :kf k∞ =1 y∈X

Z


x1 ) p(`) (y, z)p(`) (z, v) exp itF (¯ x1 , z) dη(z) ≤ ρt (¯

×

and therefore by the submultiplicativity of norms of operators, E|

N Y

(`) Qt,Θn 1(ζ`M )|

k=M +1 N −N` 2

Y ]

≤E [

j=1

N Y

≤ Ek

Qt,Θn k∞ (2.4.9)

k=M +1 N −N`

MY +2j

[ 2 ] Y

Qt,Θn ∞ ≤ E ρt (ΘM +2j−1 ).

k=M +2(j−1)+1

j=1

Next, we check that Assumptions 2.2.1 and 2.2.2 hold true we will use the following. Let κ be a probability measure on some measure space and let f be a measurable function. Then by (5.2) from [29], Z Z Z 1 if (x) 1−| e dκ(x)| ≥ |eif (x) − eif (y) |2 dκ(x)dκ(y). (2.4.10) 4 In order to show that Assumption 2.2.1 holds true, in the above notations, let Γ1 and Γ2 be two subsets of the unit circle such that inf

γi ∈Γi ,i=1,2

|γ1 − γ2 | = δ > 0 and min κ{x : eif (x) ∈ Γi } = ε > 0. (2.4.11) j=1,2

Then with Gj = {x : e

Z

if (x)

Z

≥ G1

G2

∈ Γj }, j = 1, 2, Z Z |eif (x) − eif (y) |2 dκ(x)dκ(y)

|eif (x) − eif (y) |2 dκ(x)dκ(y) ≥ δ 2 ε2 .

(2.4.12)

page 76

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

77

Next, fix x ¯ ∈ X `−1 and for any t ∈ R consider the function gx¯,t given by gx¯,t (y) = exp(itF (¯ x, y)). Then either gx¯,t (y) does not depend on y η-a.s. or there exist Borel subsets Γ1 = Γ1,t,¯x and Γ2 = Γ2,t,¯x of the unit circle such that (2.4.11) holds true with κ = η and some δ = δx¯,t and ε = εx¯,t . In the latter case we consider the Borel probability measures κy,v , y, v ∈ X on X given by Z 1 κy,v (G) = (2`) p(`) (y, z)p(`) (z, v)dη(z) p (y, v) G where p(2`) (·, ·) are the densities of transition probabilities of P (2`, ·, ·) with respect to the measure η. Then by (2.4.1) and the definition of κy,v for j = 1, 2 we have κy,v {z : gx¯,t (z) ∈ Γj } ≥

γ2 p(2`) (y, v)

η{z : gx¯,t (z) ∈ Γj } ≥

γ 2 εx¯,t . (2`) p (y, v)

(2.4.13) This together with (2.4.10)-(2.4.12) applied now with the measure κy,v yields Z ct (¯ x, y, v) := p(2`) (y, v) − p(`) (y, z)p(`) (z, v)gx¯,t (z)dη(z) (2.4.14) Z
γ 4 δx2¯,t ε2x¯,t = p(2`) (y, v) 1 − gx¯,t (z)dκy,v (z) ≥ (2`) . 4p (y, v) R Set Uy = {v : p(2`) (y, v) > 2}. Then η(Uy ) ≤ 12 Uy p(2`) (y, v)dη(v) ≤ 12 , and so Z Z dη(v) dη(v) 1 (2.4.15) ≥ ≥ . (2`) (2`) 4 (y, v) (y, v) X p X \Uy p Combining this with (2.4.14) we deduce that Z 1 ρt (¯ x) ≤ 1 − inf ct (¯ x, y, v)dη(v) ≤ 1 − γ 4 δx2¯,t ε2x¯,t . y 16

(2.4.16)

In the non-lattice case gx¯,t (y) cannot be η-a.s. constant in y if t 6= 0 and Bx¯ = ∅, where Bx¯ is defined in (2.4.7), while in the lattice case it cannot be η-a.s. constant in y if 0 < |t| < 2π h . Let δ > 0. In the non-lattice case set −1 −1 Jδ = [−δ , −δ] ∪ [δ, δ ] while in the lattice case set Jδ = [− πh , πh ] \ (−δ, δ). Since ρt (¯ x) is continuous in t, it follows from (2.4.16) that there exists cδ (¯ x) > 0 such that sup ρt (¯ x) ≤ 1 − cδ (¯ x) t∈Jδ

(2.4.17)

page 77

March 13, 2018 15:46

78

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where in the non-lattice case we assume that Bx¯ = ∅. We conclude that for any δ > 0 there exists cδ > 0 and a Borel set G ⊂ X `−1 such that for any t ∈ Jδ , ρt (¯ x) ≤ 1 − cδ for all x ¯ ∈ G and µ`−1 (G) ≥ ε > 0.

(2.4.18)

We return now to (2.4.9). The transition probabilities PΘ (·, ·) of the stationary Markov chain Θ = {Θn : n ≥ 0} on X `−1 are determined by the formula ¯ = PΘ (¯ x, Γ)

`−1 Y

P (j, x(j) , Γj )

j=1

¯ = Γ1 × · · · × Γ`−1 is a Borel set of where x ¯ = (x(1) , ..., x(`−1) ) ∈ X `−1 and Γ `−1 `−1 X . The measure µ = µ × · · · × µ is the stationary distribution of Θ and its `-th step transition probabilities PΘ (`, x ¯, ·) have transition densities (`) `−1 pΘ (·) with respect to η satisfying (`)

x, ·) ≤ γ −(`−1) γ `−1 ≤ pΘ (¯ where γ is the same as in (2.4.1). Next, observe that by (2.4.18), P {Θn ∈ G} = P {Θ1 ∈ G} = µ`−1 (G) ≥ ε

(2.4.19)

for any n ∈ N. Consider the counting function [ 21 (N −N` )]

V (N ) =

X

IG (Θ2n−1 )

n=1

and the events Γ(N ) = {V (N )
0 independent of N . We refer the readers to the paragraph preceding (5.18) from [29] for the exact details. We conclude by (2.4.18) that [

N −N` 2

Y j=1

] Nε

ρt (ΘM +2j−1 ) ≤ IΓ(N ) + (1 − cδ ) 9`2

page 78

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

79

which together with (2.4.9), (2.4.20) and the estimates from Section 2.3 complete the proof that Assumption 2.2.1 holds true. We show now that Assumption 2.2.2 holds true, as well. Similarly to the above arguments, we start with the estimate (2.4.8) and the first inequality from (2.4.16), but now employing the Taylor reminder formula we can represent ct (¯ x, y, v) from there for |t| small enough in the following way (cf. Ch. 8 in [12]), Z 1 2 ct (¯ x, y, v) = t p(`) (y, z)p(`) (z, v)D(¯ x, y, z, v)dη(z) + t2 ϕˆx¯,y,v (t) 2 X (2.4.21) where for some constant C > 0, ϕˆx¯,y,v (t) ≤ C ϕˆx¯ (t) → 0 as t → 0 and Z
2 1 p(`) (y, z)p(`) (z, v)F (¯ x, z)dη(z) . D(¯ x, y, z, v) = F (¯ x, z) − (2`) p (y, v) Now, either F (¯ x, z) does not depend on z η-a.s., i.e. F` (¯ x) = 0 η-a.s. which is excluded, or there exist Borel subsets U1 = U1,¯x and U2 = U2,¯x of the real line R such that inf

z∈U1 ,w∈U2

|z − w| = δx¯ > 0 and min η(Gj (¯ x)) = εx¯ j=1,2

where Gj (¯ x) = {z : F (¯ x, z) ∈ Uj }, j = 1, 2. In this case Z p(`) (y, z)p(`) (z, v)D(¯ x, y, z, v)dη(z) X Z ≥ γ2 D(¯ x, y, z, v)dη(z) X Z
2 ≥ γ 2 inf F (¯ x, z) − c dη(z) c

≥ γ 2 εx¯ inf

inf

c a∈U1 ,b∈U2

(2.4.22)

G1 (¯ x)∪G2 (¯ x)


1 (a − c)2 + (b − c)2 = γ 2 εx¯ δx2¯ . 2

Now by (2.4.16), (2.4.21) and (2.4.22) for |t| small enough, ρt (¯ x) ≤ 1 −

t2 2 γ εx¯ δx2¯ + Cϕx¯ (t). 4

Observe that by (2.3.1), Z ˆ D(¯ x) := sup D(¯ x, y, z, v)dη(z)dη(v) y Z ≤ 2(1 + γ −3 ) F 2 (¯ x, z)dη(z) < ∞, µ`−1 -a.s.

(2.4.23)

page 79

March 13, 2018 15:46

80

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Hence ˆ x) ≤ L} = 1. lim µ`−1 {¯ x : D(¯

L→∞

This together with (2.4.23) yield that there exist c > 0 and a Borel set G ⊂ X `−1 such that µ`−1 (G) = ε > 0 and for all |t| small enough, 2

ρt (¯ x) ≤ e−ct whenever x ¯ ∈ G.

(2.4.24)

Considering, again, the counting function [ 21 (N −N` )]

W (N ) =

X

IG (Θ2n−1 )

n=1

and the events Γ(N ) = {W (N ) < ε(

N − 2)}. 8`2

we complete the proof that Assumption 2.2.2 holds true relying again on the large deviation results from [8] together with [20]. Remark 2.4.1. In the proofs that Assumptions 2.2.1 and 2.2.2 hold true we could have avoided using large deviation estimates. Indeed, inequalities of the form VarV (N ) ≤ CN and VarW (N ) ≤ CN together with the Markov inequality are sufficient for obtaining estimates of the form P (V (N )) ≤ C1 N −1 and P (W (N )) ≤ C2 N −1 which is sufficient for Assumptions 2.2.1 and 2.2.2 to hold. Such upper bounds of the variances of V (N ) and W (N ) follow from the mixing condition (2.4.6), see Remark 5.2 from [29]. Remark 2.4.2. The situation that (2.4.1) holds true with some n0 in place of ` can also be considered. In this case (2.4.1) will hold true with `n0 , as well, and imposing appropriate restrictions on functions of the form Pn0 xj , yj ) our proof proceeds similarly. In fact, the assumption that j=1 F (¯ (2.4.1) holds true for any x can be relaxed to µ-a.a. x, and so our proof can be modified to the general situation of a ψ-mixing stationary Markov chain, taking into account Theorem 5 in [4].

page 80

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

2.5 2.5.1

LNcov

81

Markov chains related to dynamical systems Locally distance expanding maps and transfer operators

Let (X , ρ) be a compact metric space normalized in size so that diamρ (X ) ≤ 1. We consider X with its Borel σ-algebra BX and denote by B(x, r) an open ball of radius r > 0 around x ∈ X . Let T : X → X be a continuous and surjective map. We further assume that there exist ξ, η > 0 and γ > 1 such that the following conditions hold true. Assumption 2.5.1 (Uniform Openness). For any z ∈ X , B(T z, ξ) ⊂ T B(z, η). Assumption 2.5.2 (Topological exactness). There exists a constant nξ ∈ N such that T nξ B(z, ξ) = X for any z ∈ X . Assumption 2.5.3 (Locally distance expanding). For any z1 , z2 ∈ X, ρ(T z1 , T z2 ) ≥ γρ(z1 , z2 ), if ρ(z1 , z2 ) < η. We remark that the locally distance expanding condition implies that T |B(z,η) is injective for any z ∈ X . Together with the compactness of the space X this yields that deg T := sup |T −1 {x}| < ∞ x∈X −1

where |T {x}| is the number of preimages of a point x ∈ X . Given f : X → C, we can define the transfer operator Lf acting on functions g : X → C by the formula X Lf g(x) = ef (y) g(y). (2.5.1) y∈T −1 {x}

When f takes real values and Lf 1 = 1 then Lf is a Markov operator which defines transition probabilities via the formula X Pf (x, Γ) = Lf IΓ (x) = ef (y) , x ∈ X , Γ ∈ BX . y∈Γ:T y=x

In what follows we will prove a nonconventional LLT for Markov chains generated by transition probabilities of the above form together with a natural stationary measure µ = µ(f ) (see Section 2.5.3).

page 81

March 13, 2018 15:46

82

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Next, let 0 < α ≤ 1. For any g : X → C set vα,ξ (g) = inf{R : |g(x) − g(x0 )| ≤ Rρα (x, x0 ) if ρ(x, x0 ) < ξ} and kgkα,ξ = kgk∞ + vα,ξ (g) where kgk∞ stands for the supremum of the absolute value of a function g : X → C. Denote by Hα,ξ = Hα,ξ (X , ρ) the space of all functions g : X → C such that kgkα,ξ < ∞. In what follows we will always assume that the function f in the definition of the transfer operator is a member of Hα,ξ . 2.5.2

Inverse branches, the pairing property and periodic points

The following two properties of the iterates T n , n ≥ 1 of the map T are discussed in [53] in the more general setup of iterates of random expanding maps, and are given here for readers’ convenience. First, the locally distance expanding property guarantees that the map T is injective when it is restricted to open balls with radius η. This together with the uniform openness property implies that for any y ∈ X there exists a unique continuous inverse branch Ty−1 : B(T y, ξ) → B(y, η) of T sending T (y) to y. By the locally distance expanding property we have ρ(Ty−1 z1 , Ty−1 z2 ) ≤ γ −1 ρ(z1 , z2 ) for any z1 , z2 ∈ B(T y, ξ)

(2.5.2)

and thus, in fact, Ty−1 B(T y, ξ) ⊂ B(y, min(γ −1 ξ, η)) ⊂ B(y, ξ). As a consequence, for any n ∈ N the map n −n Ty−n := Ty−1 ◦ · · · ◦ TT−1 ξ) n−1 y : B(T y, ξ) → Bω (y, γ

is well defined and is a continuous inverse branch of T n sending T n y to y such that T n ◦ Ty−n = Id B(T n y,ξ) , Ty−n (T n z) = z and ρ(Ty−n z1 , Ty−n z2 ) ≤ γ −n ρ(z1 , z2 ) for any z1 , z2 ∈ B(T n y, ξ).

(2.5.3)

page 82

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

83

Secondly, let x, x0 ∈ X be such that ρ(x, x0 ) < ξ and let n ∈ N. For any y ∈ T −n {x} the point y 0 = Ty−n x0 satisfies T n y 0 = x0 and so by (2.5.3), ρ(y, y 0 ) = ρ(Ty−n x, Ty−n x0 ) ≤ γ −n ρ(x, x0 ). 0 Since Ty−n 0 x = y, exchanging the roles x and x we can write

T −n {x0 } = {y 0 = Ty−n x0 : y ∈ T −n {x}}. Henceforth we will refer the property described above as the paring property. Next, the following lemma describes the structure of periodic points of the map T . Lemma 2.5.4. Let z0 ∈ X and 0 < r ≤ ξ. Let kr be the smallest nonnegative integer k such that γ −k ξ ≤ r. Then for any M ≥ nξ + kr there exists x0 ∈ X such that rγ . T M x0 = x0 and ρ(z0 , x0 ) ≤ γ−1 In particular periodic points are dense in X . Proof. Let z0 ∈ X . We first consider the case when r = ξ. Let M ≥ nξ . Then by the topological exactness property, T M B(z0 , ξ) = X and we have used that T is surjective, as well. Therefore, there exists z1 ∈ B(z0 , ξ) such that T M z1 = z0 . Set z2 = Tz−M z1 , which is well defined 1 since z1 ∈ B(z0 , ξ) = B(T M z1 , ξ). Then by (2.5.3), ρ(z2 , z1 ) ≤ γ −M ρ(z1 , z0 ) < γ −M ξ < ξ z0 . Thus, we can define recursively a sequence since z1 = Tz−M T M z1 = Tz−M 1 1 ∞ {zk }k=0 such that ρ(zk , zk−1 ) < ξ and zk+1 = Tz−M zk , k ∈ N. Then k T M zk+1 = zk and we claim that {zk }∞ k=0 is a Cauchy sequence. Indeed, for any k ≥ 1 we have ρ(zk+1 , zk ) = ρ(Tz−M zk , Tz−M T M zk ) k k =

ρ(Tz−M zk , Tz−M zk−1 ) k k

≤γ

−M

(2.5.4)

ξ

where in the last inequality we used (2.5.3) and that ρ(zk , zk−1 ) < ξ. This completes the proof that {zk }∞ k=0 is a Cauchy sequence, since γ > 1. We conclude that limk→∞ zk = x0 exists, and since T M zk+1 = zk , it follows by the continuity of T M that T M x0 = x0 . The inequality ρ(z0 , x0 ) ≤

γξ γ−1

page 83

March 13, 2018 15:46

84

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

is a consequence of (2.5.4). Next, let r < ξ and set k = kr . The above proof will proceed similarly with r in place of ξ if we show that T nξ +kr B(z0 , r) = X

(2.5.5)

for any z0 ∈ X . Indeed, since Tz−k (B(T k z0 , ξ)) ⊂ B(z0 , γ −k ξ) ⊂ B(z0 , ξ) 0 we derive that B(T k z0 , ξ) ⊂ T k B(z0 , γ −k ξ) = T k B(z0 , γ −kr ξ) and (2.5.5) follows by substituting both sides in T nξ and using the topological exactness assumption and that r ≥ γ −kr ξ. Example 2.5.5. Let (X , T ) be a one sided topologically mixing subshift of finite type (see [10] and Section 2.11). Namely, let A = (Ai,j ) be a matrix of size n × n with 0 − 1 entries such that AM has only positive entries for some M ≥ 1. Consider the space N X = XA = {x = (xi )∞ i=1 ∈ A : Axi ,xi+1 = 1 ∀i ∈ N}, A = {1, 2, ..., n} (2.5.6) and let T : X → X be the one sided shift operator. The distance between two distinct points x, y ∈ X is given by

ρ(x, y) = 2− min{i∈N: xi 6=yi } . Let x ∈ X . For any K ≥ M and x1 ∈ A we have AK x1 ,x1 > 0 and so there exist x2 , ..., xK ∈ A such that Axi ,xi+1 = 1 for any i = 1, 2, ..., K, where xK+1 = x1 . Then the periodic point x ¯ ∈ X given by x ¯ = (x1 , x2 , ..., xK , x1 , x2 , ...., xK , ....) satisfies T K x ¯=x ¯ and ρ(x, x ¯) ≤ 2−K . 2.5.3

Thermodynamic formalism constructions and the associated Markov chains

We will state here the main results from [53] in the case of deterministic distance expanding maps. Note that, in fact, those results are a particular case of the results from Part 2. Let f ∈ Hα,ξ be a real valued function and consider the transfer operator Lf given by (2.5.1). Since f is bounded and deg T < ∞ the operator Lf acts continuously on the space B(X ) of all bounded Borel measurable functions on X . As a consequence, the dual operator L∗f which acts on the space of measures on X by the formula L∗f κ = κ ◦ Lf

page 84

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

85

is well defined. We begin with the following Ruelle-Perron-Frobenius type theorem (see, for instance, [53]). Theorem 2.5.6. There exists a unique triplet (λ, h, ν) consisting of a positive number λ = λ(f ) , a strictly positive function h = h(f ) ∈ Hα,ξ and a probability measure ν = ν (f ) on X such that Lf h = λh, L∗f ν = λν and ν(h) = 1. An exponential convergence result is established, as well (see [53]). Theorem 2.5.7. There exist constants A > 0 and c ∈ (0, 1) such that for any g ∈ Hα,ξ and n ∈ N,

−n n


λ Lf (g) − ν(g)hk∞ ≤ A ν|g| + vα,ξ (g) cn . (2.5.7) We remark that a stronger version of Theorem 2.5.7 follows from Theorem 4.2.2 in Chapter 4 of Part 2. Next, consider the (Gibbs) measure µ = µ(f ) = hν. Then µ is T invariant since for any bounded Borel function g : X → R, µ(g ◦ T ) = ν(h · (g ◦ T )) = λ−1 L∗f ν(h · (g ◦ T )) = λ−1 ν(Lf (h · (g ◦ T ))) = λ−1 ν(gLf (h)) = ν(hg) = µ(g) where we used that Lf (g1 · (g2 ◦ T )) = g2 · Lf g1 for any g1 , g2 : X → C. The following decay of correlations and Gibbs property are established in [53], as well. Proposition 2.5.8. There exist constants A > 0 and c ∈ (0, 1) such that for any g1 ∈ Hα,ξ , g2 ∈ L1 (X , µ) and n ∈ N, |µ(g1 · (g2 ◦ T n )) − µ(g1 )µ(g2 )| ≤ Akg1 kα,ξ µ(|g2 |)cn . As a consequence, for any g1 , g2 ∈ L1 (X , µ), lim µ(g1 · (g2 ◦ T n )) = µ(g1 )µ(g2 )

n→∞

and therefore the measure preserving system (X , BX , µ, T ) is mixing. Proposition 2.5.9. There exist constants C1 , C2 > 0 such that for any y ∈ X and n ∈ N,
µ Ty−n B(T n y, ξ) ≤ C2 . (2.5.8) C1 ≤ λn eSn f (y)

page 85

March 13, 2018 15:46

ws-book9x6

86

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

The following important property of ν and µ follows from here. Let y ∈ X and n ∈ N. Since Ty−n B(T n y, ξ) ⊂ B(y, γ −n ξ) we deduce that for any 0 < r < ξ, µ(B(y, r)) ≥ C1 e−kr kf k∞ λkr where kr is the smallest nonnegative integer k satisfying r ≥ ξγ −k , and in particular µ and ν assign positive mass to open sets. Next, given a real valued f ∈ Hα,ξ we consider the function f˜ = f + ln h − ln h ◦ T − ln λ (2.5.9) where h = h(f ) and λ = λ(f ) . Then ˜

˜

˜

˜

Lf˜1 = 1, λ(f ) = 1, h(f ) ≡ 1 and ν (f ) = µ(f ) = µ(f ) . Henceforth, we will always replace f by f˜, namely, it will be assumed that Lf 1 = 1, λ(f ) = 1 and h(f ) ≡ 1. In these circumstances the transfer operator Lf defines transition probabilities via the formula X Pf (x, Γ) = Lf IΓ (x) = ef (y) , x ∈ X , Γ ∈ BX . (2.5.10) y∈Γ:T (y)=x

Let {ζn : n ≥ 0} be the Markov chain with initial distribution µ = µ(f ) and transition probabilities Pf (·, ·) and let ` be a positive integer. In this section we will prove a nonconventional local (central) limit theorem for sums of the form N X SN = F (ζn , ζ2n , ..., ζ`n ) n=1 `

where F : X → R is a function with some regularity conditions yet to be specified. 2.5.4

Relations between dynamical systems and Markov chains

In the “conventional” case ` = 1 the topic of limit theorems for sums of the Pn−1 form k=0 G ◦ T k taken with the measure µ = µ(f ) is well studied (see, for instance [23] and [28]). One of the main strategies for proving such limit theorems is as follows. Let Z0 be a random element of X which is drawn according to µ, and consider the stationary X -valued process Z = {Zk , k ≥ 0}, where Zk = T k Z0 . Then by Lemma XI.3 in [28] for any n ∈ N, d

(ζn−1 , ζn−2 , ..., ζ0 ) = (Z0 , Z1 , ..., Zn−1 )

(2.5.11)

page 86

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

87

d

where X = Y means that X and Y have the same distribution. As a consequence, n−1 X

d

G(Zk ) =

k=0

n X

G(ζk )

(2.5.12)

k=1

and therefore any distributional limit theorem for the sums on the lefthand side will follow from the corresponding limit theorem for the sums on the right-hand side. Now the general theory of limit theorems for quasicompact Markov operators (see Chap. XII in [28]) can be applied, taking into account Theorems 2.5.6 and 2.5.7. Consider now the nonconventional situation when ` > 1. First, it follows from (2.5.11) that SN =

N X

d

F (ζn , ζ2n , ..., ζ`n ) =

N X

F (Z`N −n , Z`N −2n , ..., Z`N −n` ) (2.5.13)

n=1

n=1

and so an LLT for the sums from the above right-hand side follows from an LLT for the sums on the left-hand side. In fact, since Z is stationary there exists a two sided stationary process Z = {Zn : n ∈ Z} on some probability space (Ω, F, P ) and a measurable function π : Ω → X such that d

{π ◦ Zn : n ≥ 0} = Z.

(2.5.14)

Therefore, for any N ∈ N, SN =

N X

d

F (ζn , ζ2n , ..., ζ`n ) =

n=1

N X

F ◦ π(Z−n , Z−2n ..., Z−n` )

(2.5.15)

n=1

and again an LLT for the sums from the above right-hand side follows from the corresponding LLT for SN . Still, in Section 2.11 we will extend this type of nonconventional LLT to sums of the form N X

F (T n x, T 2n x, ..., T `n x)

n=1

when T is either a topologically mixing two sided subshift of finite type or an Axiom A diffeomorphism (in a neighborhood of an attractor) and x is drawn according to a Gibbs measure corresponding to some H¨older continuous (potential) function.

page 87

March 13, 2018 15:46

88

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

2.5.5

Mixing and approximation assumptions

Let Fm,n , −∞ ≤ m ≤ n ≤ ∞ be a family of sub-σ-algebras of BX such that Fk,l ⊂ Fk0 ,l0 if k 0 ≤ k and l0 ≥ l. Consider the X -valued stationary process {Zk : k ≥ 0}, where Z0 is distributed according to µ and Zk = T k Z0 . For any n, r ≥ 0 let Zn,r be a Fn−r,n+r -measurable random element of X . Let d ∈ N and consider the metric on X d = X × X × · · · × X given by ρd,∞ (x, y) = max ρ(xi , yi ) 1≤i≤d

for any x = (x1 , ..., xd ) and y = (y1 , ..., yd ) in X d . For any H : X d → C set vα,ξ (H) := inf{R ≥ 0 : |H(x) − H(y)| ≤ R(ρd,∞ (x, y))α for all x, y with ρd,∞ (x, y) < ξ} and let Hd = Hα,ξ (X d , ρd,∞ ) be the space of all functions H : X d → C so that vα,ξ (H) < ∞. Note that H is continuous when vα,ξ (H) < ∞ and therefore k · kα,ξ = k · k∞ + vα,ξ (·) defines a norm on Hd . For each q ≥ 1 consider the approximation coefficients βq,α,ξ (r), r ≥ 0 given by βq,α,ξ (r) (2.5.16)  kH(Z1 , Z2 , ..., Zd ) − H(Z1,r , Z2,r , ..., Zd,r )kLq : H ∈ Hd , d ∈ N . = sup vα,ξ (H) In most of this section we will assume that q = ∞ and this case
α β∞,α,ξ (r) ≤ β∞ (r) if β∞,α,ξ (r) < ξ, where β∞ (r) = sup kρ(Zn , Zn,r )kL∞ .

(2.5.17)

n≥0

Our results will rely on the following Assumption 2.5.10. ∞ X

n(φn + β∞,α,ξ (n)) < ∞

n=1

where φn was defined in (1.3.1). Assumption 2.5.10 holds true in the situation when T is a one sided topologically mixing subshift of finite type, as well as in the case when T is an expanding C 2 endomorphism of a Riemannian manifold. In the first case Fm,n is the σ-algebra generated by cylinder sets with fixed coordinates.

page 88

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

89

Similarly, in the second case Fm,n is generated by partitions of the form Tn −j (M) where M is a Markov partition with a sufficiently small j=m T diameter. We remark that the results stated in this section hold true under some conditions involving the Lq -approximation coefficients βq,α,ξ for some 1 ≤ q < ∞ rather than β∞,α,ξ (see Remark 2.8.7 for the precise details). Next, let F : X ` → R satisfy that |F (x) − F (y)| ≤ CF ρ`,∞ (x, y))α

(2.5.18)

for some constant CF > 0 and all x = (x1 , ..., x` ) and y = (y1 , ..., y` ) in X ` so that ρ`,∞ (x, y) < ξ. For any N ∈ N consider the sum SN =

N X

F (ζn , ζ2n , ..., ζ`n ).

n=1

Our main result is a local limit theorem for these sums (and we refer the readers to Section 2.11 for extensions to certain dynamical systems). 2.5.6 (i)

Asymptotic variance and the CLT (i)

Let ζ = {ζn : n ≥ 0}, i = 1, ..., ` be independent copies of ζ = {ζn : n ≥ 0} and set UN =

N X

(2)

(`)

F (ζn(1) , ζ2n , ..., ζ`n ).

n=1

Theorem 2.5.11. Suppose that Assumption 2.5.10 holds true. Then the limits
2
2 1 1 E SN − N F¯ and s2 = lim E UN − N F¯ σ 2 = lim N →∞ N N →∞ N ¯ exist, where F is defined in (2.3.2). Moreover, σ 2 > 0 if and only if s2 > 0 and the latter two are positive if and only if there exists no g ∈ L2 (X ` , µ` ) such that F ◦ (T × T 2 × · · · × T ` ) = F¯ + g ◦ (T × T 2 × · · · × T ` ) − g, µ` -a.s. (2.5.19) 1 where a.s. stands for almost surely. Furthermore, N − 2 (SN − N F¯ ) converges in distribution towards a centered normal random variable with the variance σ 2 .

Proof. Existence of σ 2 is proved similarly to [45] relying on (2.5.11) and on Corollary 1.3.14 instead of Lemma 4.3 from there. The characterization

page 89

March 13, 2018 15:46

90

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

of positivity of σ 2 follows from the arguments in the proof of Theorem 2.3 in [30], relying on (2.5.11) and on Corollary 1.3.14 instead of Lemma 3.4 from there. The proof of the CLT proceeds similarly to Chapter 1 first by using (2.5.11) and then using Assumption 2.5.10 in order to obtain an appropriate version of Corollary 1.3.16. We remark that when (2.5.19) holds true then we can replace g there by a member of Hα,ξ (and in this case the equality will hold true for any x ∈ X ` ). Indeed, this is a consequence of the arguments in the proof of Lemma III.1 in [23] and the appropriate versions of Theorems 2.5.6 and 2.5.7 for the map T × T 2 × · · · × T ` . Next, consider the function F` : X ` → R given by Z F` (x1 , ..., x` ) = F (x1 , ..., x` ) − F (x1 , ..., x`−1 , y)dµ(y). (2.5.20) Then F` satisfies (2.5.18) with some constant CF` and Z F` (y1 , ..., y`−1 , x)dµ(x) = 0 ∀y1 , ..., y`−1 ∈ X and in particular F¯` = 0 where F¯` is defined similarly to F¯ in (2.3.2) but with F` in place of F . Theorem 2.5.11 holds true also with F` in place of F and in particular the limit N X
2 1 F` (ζn , ζ2n , ..., ζ`n ) E N →∞ N n=1

σ`2 = lim

(2.5.21)

exists. If g satisfies (2.5.19) then the function g` which is defined similarly to F` but with g in place of F satisfies (2.5.19) with F` in place of F . Therefore, σ 2 > 0 when σ`2 > 0. Henceforth, we will assume that σ`2 > 0 which is equivalent to (2.5.19) not being satisfied with any g ∈ L2 (X ` , µ` ) and F replaced with F` . 2.6

Statement of the local limit theorem

Let x0 ∈ X be a periodic point of T and let m0 ∈ N be so that T m0 x0 = x0 . Consider the point x ¯0 = (x0 , ..., x0 ) ∈ X `−1 and the points Tˆk x ¯0 = (T k x0 , T 2k x0 , ..., T k(`−1) x0 ) ∈ X `−1 , 0 ≤ k < m0 . For any u ∈ X `−1 let the function Fu : X → R be given by Fu (x) = F (u, x). Our results will rely on the following regularity assumption on Fu around the above points.

page 90

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

91

Assumption 2.6.1. The function u → Fu is continuous at the points u = Tˆk x ¯0 , k = 0,
1, ..., m0 − 1 when considered as a function from X `−1 to α,ξ H (X ), k · kα,ξ . Since F is a H¨ older continuous function this assumption is equivalent to the assumption that the function u → vα,ξ (Fu − Fv ) is continuous at these points. This continuity assumption holds true when the following assumption is satisfied. Assumption 2.6.2. There exist constants b, KF > 0 such that with any v = Tˆk x ¯0 , k = 0, 1, ..., m0 − 1 we have



Fu (r) − Fv (r) − Fu (r0 ) − Fv (r0 ) ≤ KF ρ`−1,∞ (u, v) b ρ(r, r0 ) α (2.6.1) 0 `−1 0 for any r, r ∈ X and u ∈ X such that ρ(r, r ) < ξ and ρ`−1,∞ (u, v) < ξ. Next, consider the map T` = Tˆ × T ` = T × T 2 × · · · × T ` and let the function Fx0 ,m0 : X → R be given by Fx0 ,m0 (x) =

m 0 −1 X

F (T k x0 , T 2k x0 , ..., T (`−1)k x0 , T `k x)

(2.6.2)

k=0

=

m 0 −1 X

F ◦ T`k (¯ x0 , x).

k=0

We call the case a non-lattice one if for any t ∈ R \ {0} there exist no nonzero g ∈ Hα,ξ and λ ∈ C, |λ| = 1 such that eitFx0 ,m0 g = λg ◦ T `m0 , µ-a.s.

(2.6.3)

where we recall that µ = µ(f ) . The non-lattice condition means that the function Fx0 ,m0 is non-arithmetic (or aperiodic) with respect to the map τ0 = T m0 ` in the classical sense of [23] and [28], which according to Lemma XII.7 in [28] means that the spectral radius of the corresponding Fourier operator defined in (2.9.5) is strictly less than 1 for any real t 6= 0. Theorem 2.6.3. Suppose that Assumption 2.5.10 holds true, and let F be a non-arithmetic function satisfying (2.5.18) and Assumption 2.6.1. Assume in addition that σ`2 > 0. Then for any continuous function g : R → R with a compact support, Z √ ¯ )2 (u−N F g(x)dx = 0. lim sup σ 2πN Eg(SN − u) − e− 2N σ2 N →∞ u∈R

page 91

March 13, 2018 15:46

92

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Next, we call the case a lattice one if the function F takes values on some lattice hZ = {hk : k ∈ Z}, h > 0 and the following assumption holds true. Assumption 2.6.4. The function Fx0 ,m0 cannot be written in the form Fx0 ,m0 = a + β − β ◦ T `m0 + h0 k, µ-a.s.

(2.6.4)

for some h0 > h, a ∈ R, β : X → R such that eiβ ∈ Hα,ξ and an integer valued function k : X → Z. 2π Assumption 2.6.4 is equivalent to the statement that for any t ∈ (− 2π h , j )\ {0} there exists no g ∈ Hα,ξ \ {0} and a ∈ R such that

geitFx0 ,m0 = eia g ◦ T `m0 , µ-a.s.

(2.6.5)

Indeed, when such g and a exist then by ergodicity of (X , BX , µ, T `m0 ) the function |g| is µ-a.s. constant, and dividing by |g| we arrive at (2.6.4) with h0 = 2π |t| > h, while the other direction is trivial. Theorem 2.6.5. Suppose that Assumption 2.5.10 holds true, and let F be a lattice valued function as described above so that (2.5.18) and Assumption 2.6.1 hold true. Assume in addition that σ`2 > 0. Then for any continuous function g : R → R with a compact support, √ X ¯ )2 (u−N F lim sup σ 2πN Eg(SN − u) − he− 2N σ2 g(hk) = 0. N →∞ u∈hZ

hk∈hZ

Remark that, in general, the natural choice of m0 is m0 = nξ , where nξ is specified in Assumption 2.5.2. Still, in some situations there exist periodic points with smaller order. For instance, when (X , T ) is a (topologically mixing) one sided subshift of finite type and Ai,i = 1 for some i then T ¯i = ¯i, where ¯i = (iii...) is the word which consists only of the letter i, and the above conditions concern the function F (¯i, ¯i, ..., ¯i, ·) and the `-th step shift T ` . In the following sections we will show that Assumptions 2.2.1 and 2.2.2 hold true in both lattice and non-lattice cases, which in view of Theorems 2.2.3 and 2.5.11 will imply that the statements of both Theorems 2.6.3 and 2.6.5 hold true. 2.7

The associated random transfer operators

In this section we will make the first step towards the proof that Assumption 2.2.2 holds true.

page 92

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

2.7.1

LNcov

93

Random complex RPF theorem

Let F : X ` → R be a function satisfying (2.5.18). In this section we provide precise details of the construction of the random Fourier operators Qω it defined after (2.3.8) which in our circumstances are random transfer (i) operators denoted by Lω it . Let ζ , i = 1, ..., ` − 1 be independent copies of ζ. Consider the process Θ = {Θn : n ≥ 0} given by (2)

(`−1)

Θn = (ζn(1) , ζ2n , ..., ζ(`−1)n ). Then Θ is the stationary Markov chain with initial distribution µ`−1 = µ×· · ·×µ and transition operator Lˆfˆ which is the transfer operator defined by the map Tˆ = T × T 2 · · · × T `−1 and the function fˆ(x1 , ..., x`−1 ) = P`−1 j=1 f (xj ). Let M (Θ) = (ΩΘ , BΘ , PΘ , ϑ) be the natural invertible measure preserving system (MPS) associated with the process Θ. This means that ΩΘ = (X `−1 )Z ϑ the left shift map and the processes Θ and {p0 ◦ ϑn , n ≥ 0} have the same distribution, where p0 is the 0-th coordinate projection given by p0 (s) = s0 , s = {sn : n ∈ Z} ∈ (X `−1 )Z . Note that by Theorem 2.5.8 the MPS (X , BX , µ, T ) is mixing. Therefore, the MPS’s (X `−1 , BX `−1 , µ`−1 , Tˆ) and M (Θ) are mixing, as well. Consider now the random function Fω (·) on X given by Fω (·) = F (p0 (ω), ·), ω ∈ ΩΘ .

(2.7.1)

ω The operators Qω it = Lit introduced below (2.3.8) are the random transfer ω operators Lz defined by the map T ` and the (potential) function S` f +zFω which act on functions g : X → C by the formula X eS` f (y)+zFω (y) g(y). Lω z g(x) = y∈T −` {x}

The following theorem plays a crucial role in Section 2.8 and it concerns only the above random transfer operators where additional assumptions needed for more general results of Chapters 4 and 5 are automatically satisfied (see Section 5.4). Theorem 2.7.1. (i) There exists a constant r > 0 such that for PΘ almost every ω ∈ ΩΘ and any z ∈ C with |z| < r there exists a triplet λω (z), hω (z) and νω (z) consisting of a nonzero complex number λω (z), a

page 93

March 13, 2018 15:46

94

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

complex function hω (z) ∈ Hα,ξ and a complex continuous linear functional νω (z) : Hα,ξ → C such that
∗ ω Lω = λω (z)νϑω (z) and (2.7.2) z hϑω (z) = λω (z)hω (z), Lz νω (z) νω (z)hω (z) = νω (z)1 = 1. When z = t ∈ R then λω (t) > 0, the function hω (t) is strictly positive, νω (t) is a probability measure and the equality νω (t) Lω t g) = λω (t)νϑω (t)(g) holds true for any bounded Borel function g : X → C. (ii) Moreover, the above triplet is measurable. Namely, for each z ∈ B(0, r) = {α ∈ C : |α| < r} the maps ω → λω (z), ω → hω (z) and ω → νω (z) are measurable. (iii) Furthermore, this triplet is (strongly) analytic and uniformly bounded. Namely, the maps
∗ λω (·) : B(0, r) → C, hω (·) : B(0, r) → Hα,ξ and νω (·) : B(0, r) → Hα,ξ are analytic, where (Hα,ξ )∗ is the dual space of Hα,ξ , and for any k ≥ 0 there exists Ck > 0 such that PΘ -a.s. for any z ∈ B(0, r),
(k) (k) max |λ(k) ω (z)|, khω (z)kα,ξ , kνω (z)kα,ξ ≤ Ck where g (k) stands for the k-th derivative of a function on the complex plane which takes values in some Banach space. When z = 0 then Lω = L`f and therefore by the uniqueness part of Theorem 2.5.6 P -a.s., hω (0) ≡ 1, λω (0) = 1 and νω (0) = µω (0) = µ(f ) = µ (2.7.3) (f ) where we recall our assumption that Lf 1 = 1 which implies that λ = 1, h(f ) ≡ 1 and ν (f ) = µ(f ) . The following proposition is established in Part 2 in a more general situation, as well. Proposition 2.7.2. There exist constants r, A > 0 and c ∈ (0, 1) such that PΘ -a.s. for any n ∈ N, g ∈ Hα,ξ and z ∈ C with |z| < r,

Lω,n


z g − νϑn ω (z)g hω (z) ≤ Acn α,ξ λω,n (z) where n−1 Y ω ϑω ϑn−1 ω Lω,n = L ◦ L ◦ · · · ◦ L and λ (z) = λϑk ω (z). ω,n z z z z k=0

In Part 2 we will obtain several additional results concerning the random transfer operators Lω z such as exponential decay of correlations and results concerning Lyapunov exponents.

page 94

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

2.7.2

LNcov

95

Distortion properties

We will need the following. Lemma 2.7.3. Let {gω }ω∈ΩΘ be a family of functions from Hα,ξ . Let ω ∈ Ω, n ∈ N and consider the function n−1 X ω Sn,` g= gϑk ω ◦ T k` k=0

on X . Let x, x0 ∈ X be such that ρ(x, x0 ) < ξ. Then for any y ∈ T −n` {x}, n−1 X ω ω |Sn,` g(y) − Sn,` g(y 0 )| ≤ ρα (x, x0 ) vα,ξ (gϑj ω )γ −α`(n−j) (2.7.4) j=0 0

Ty−n` {x0 }

Ty−n`

where y = and is the inverse branch of T n` around T n` y = x introduced in Section 2.5.2. Proof. Let x, x0 ∈ X be such that ρ(x, x0 ) < ξ and let y and y 0 be as in the statement of the lemma. Observe that for any j < n, −`(n−j)

T `j Ty−`n x = TT `j y Since y =

Ty−`n x,

−`(n−j) 0

x and T `j Ty−`n x0 = TT `j y

x.

it follows that

ω ω |Sn,` g(y) − Sn,` g(y 0 )| ≤

n−1 X

|gϑj ω (T `j y) − gϑj ω (T `j y 0 )|

j=0

=

n−1 X

−`(n−j)

|gϑj ω TT `j y


−`(n−j) 0
x − gϑj ω TT `j y x |

j=0

≤

n−1 X

−`(n−j)

vα,ξ (gϑj ω )ρα TT `j y

−`(n−j) 0


x, TT `j y

x

j=0

and (2.7.4) follows from (2.5.3). The following random Lasota-Yorke type inequality follows. Lemma 2.7.4. Suppose that Lf 1 = 1. Then there exist constants B ≥ 1 and c ∈ (0, 1) such that for any ω ∈ ΩΘ , n ∈ N, g ∈ Hα,ξ and z ∈ C,
n| 0 is a constant which depends only on `. We conclude that for any t ∈ [−t0 , t0 ] and N ≥ J0 := max(4, (4c0 R)(c1 a0 )−1 ), Z Z 2 −M |Lω,N 1(x)|dµ(x)dPΘ (ω) ≤ A1 e−da0 t N + PΘ (ΓcN −M,c1 ). (2.8.15) it Next, we claim that there exists c1 > 0 such that √ lim nPΘ (Γcn,c1 ) = 0. n→∞

(2.8.16)

Before proving this claim we complete the proof that Assumption 2.2.2 holds true relying on (2.8.16). Indeed, by (2.7.10) and (2.8.15) for any t ∈ [−t0 , t0 ] and N ≥ J0 , Z Z √ −M |ϕN (t)| ≤ N cN + |Lω,N 1(x)|dµ(x)dPΘ (ω) it √ √ 2 ≤ N cN + A1 e−da0 t N + N PΘ (ΓcN −M,c1 )

page 104

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

105

where (cn )∞ is the sequence on the right-hand side of (2.7.8), which san=1 √ tisfies limn→∞ ncn = 0. Since N − M ≥ a0 N for any N > 3, the third term in the above right-hand side converges to 0 as N → ∞ and the proof that Assumption 2.2.2 holds true is complete. In order to prove (2.8.16), let k ∈ N and consider the random variable Vˆk given by k−1 X
ω Vˆk (ω) = Varµ (Sk,` F) = Varµ Fϑk ω ◦ T j` . j=0 ω Then Vˆk (ω) = µ(Sk,` F` )2 where the random function F` is given by F`,ω (·) = F` (p0 (ω), ·). For any n ≥ 1, ε0 > 0 and c > 0 set n 1X I ˆ jk < ε0 } Γn,ε0 ,c,k = {ω ∈ ΩΘ : n j=1 {Vk (ϑ ω)≥ck}

and for the sake of convenience we set Γ0,ε0 ,c,k = ΩΘ . Equality (2.8.16) will follow from the following two claims. Claim 2.8.4. For any c2 > 0 and ε0 > 0 there exists k0 such that for any natural n and k > k0 , ΩΘ \ Γ[ nk ],ε0 ,c2 ,k ⊂ Γn,c1 where c1 =

(2.8.17)

1 4 ε0 c2 .

Claim 2.8.5. There exist c2 , ε0 > 0 such that for any sufficiently large k, √ lim nPΘ (Γn,ε0 ,c2 ,k ) = 0. (2.8.18) n→∞

Indeed, we obtain (2.8.16) by applying Claim 2.8.4 with sufficiently large k and using (2.8.18) with [ nk ] in place of n on the left-hand side. Proof of Claim 2.8.4. First, by the definitions of F and Vˆk , m X Vˆm (ω) = Var F (p0 (ϑj ω), ζ`j )

(2.8.19)

j=1

for any m ≥ 1. Next, the process {Zm : m ≥ 0} (defined in Section 2.5.4) is stationary since T preserves µ, and for any k, s ≥ 0 the pairs (ζs , ζs+k ) and (Zk , Z0 ) have the same distribution (see (2.5.11)). Thus, for any j, m ≥ 0 and ω ∈ ΩΘ ,
Cov F (p0 (ϑj ω), ζ`j ), F (p0 (ϑj+m ω), ζ`(j+m) )
= Cov F (p0 (ϑj ω), Zm` ), F (p0 (ϑj+m ω), Z0 )
= µ F (p0 (ϑj ω), T m` (·))F (p0 (ϑj+m ω), ·)

−µ F (p0 (ϑj ω), ·) · µ F (p0 (ϑj+m ω), ·) .

page 105

March 13, 2018 15:46

ws-book9x6

106

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

For any ω ∈ ΩΘ we have kF (p0 (ω), ·)kα,ξ ≤ kF kα,ξ , and therefore by Proposition 2.5.8 there exist constants C > 0 and a ∈ (0, 1) such that for any j, m ≥ 0 and ω ∈ ΩΘ , Cov(F (p0 (ϑj ω), ζ`j ), F (p0 (ϑj+m ω), ζ`(j+m) ) ≤ Ca`m . (2.8.20) Next, let ε0 , c2 > 0 and ω ∈ ΩΘ . In order to prove (2.8.17) let n, k ∈ N be such that n ≥ k > 1 and set rk X Br = F (p0 (ϑj ω)), ζ`j ) j=(r−1)k+1

r = 1, ..., [ nk ] and B[ nk ]+1 =

n X

n

j

F (p(ϑ ω), ζ`j ) −

[k] X

Br .

r=1

j=1

We assert that there exists a constant A1 > 0 which does not depend on n, k and ω such that for any r = 1, ..., [ nk ] + 1, X Cov(Br , Bs ) ≤ A1 . (2.8.21) s>r

Indeed, set h(m) = Ca`m , m ∈ N where C and a come from (2.8.20). Let s > r. Then by (2.8.20), k X
|Cov(Br , Bs )| ≤ h (s − 1)k + j1 − (r − 1)k − j2 j1 ,j2 =1

=

k−1 X

X
k−1
(k − m)h (s − r)k + m + (k − m)h (s − r)k − m

m=0

m=1

=

k X

X
k−1
uh (s − r)k + k − u + uh (s − r)k + u − k

u=1

u=1

≤2

k X


uh (s − r − 1)k + u

u=1

where in the last inequality we used that h is non-increasing. We conclude that [n k k] X X X
Cov(Br , uh (j − 1)k + u Bs ) ≤ 2 s>r

≤2

∞ X k X j=1 u=1

j=1 u=1

∞ X

(j − 1)k + u h (j − 1)k + u = 2 mh(m) := A1 < ∞ m=1

page 106

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

107

and (2.8.21) follows. P[n/k]+1 ω Finally, since Sn,` F = r=1 Br we deduce from (2.8.21) that n

Vˆn (ω) =

ω Varµ (Sn,` F)

≥

[k] X


n Vˆk (ϑjk ω) − 2A1 [ ] + 1 . k j=1

By the definition of the set Γ[n/k],ε0 ,c2 ,k , if ω 6∈ Γ[n/k],ε0 ,c2 ,k then the first sum in the above right-hand side is not less than c2 εk[n/k], and hence ω Varµ (Sn,` F) ≥ c2 ε0 k[n/k] − 2A1 ([n/k] + 1)

≥ c2 ε0 k[n/k] − 4A1 [n/k] = [n/k](c2 ε0 k − 4A1 ) where we used that [n/k] ≥ 1. If in addition k > k0 := n 1 ω Varµ (Sn,` F) ≥ k[ ] · c2 ε0 . k 2 Claim 2.8.4 follows now with c1 = 41 c2 ε0 since 2[ nk ] ≥ 1.

8A1 c2 ε0

n k

then

for any n ≥ k ≥

In order to prove Claim 2.8.5 we will need the following general result. Lemma 2.8.6. Let Vi = (Ai , Bi ) : (X , BX ) → Xi × Yi , i = 1, 2, ..., m be m pairs of measurable functions taking values in measurable spaces Xi and Yi , respectively, so that each Ai is F−∞,ai -measurable and each Bi is (i) (i) (i) Fai +bi ,∞ -measurable for some ai ∈ Z and bi ∈ N. Let Vi = (Ai , Bi ), i = 1, 2, ..., m be independent copies of the Vi ’s with respect to µ. Then Qm Qm for any measurable functions G1 : j=1 Xi → [−1, 1] and G2 : j=1 Yi → [−1, 1], m X
(1) (1) (m) α(bi ) |Covµ G1 (A1 , ..., A(m) ), G (B , ..., B ) | ≤ 4 2 m m 1

(2.8.22)

i=1

where Covµ stands for the covariance with respect to µ and α(n) is defined similarly to φn (see (1.3.1)) but with the mixing coefficient α defined in (1.2.14) in place of φ. Proof. The lemma goes by induction on m. First, the case when m = 1 follows from Theorem A.5 in [27]. Next, suppose that (2.8.22) holds true for m = j, all measurable spaces Xi , Yi , i = 1, 2, ..., j, all measurable functions G1 and G2 as in the statement of the lemma and all couples of measurable functions Vi = (Ai , Bi ) : (X , BX ) → Xi × Yi which satisfy the conditions in the statement of the lemma with some ai ∈ Z, bi ∈ N, i = 1, 2, ..., j. Let Xi , Yi , i = 1, 2, ..., j + 1 be measurable spaces and for each i let Ui =

page 107

March 13, 2018 15:46

ws-book9x6

108

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

(Ai , Bi ) : (X , BX ) → Xi × Yi be a measurable function so that each Ai is F−∞,αi -measurable and each Bi is Fαi +βi ,∞ -measurable for some αi ∈ Z Qj+1 Qj+1 and βi ∈ N. Let H1 : i=1 Xi → [−1, 1] and H2 : i=1 Yi → [−1, 1] (i) (i) (i) be measurable functions and let Ui = (Ai , Bi ), i = 1, 2, ..., j + 1 be independent copies of the Ui ’s with respect to µ. Then by the Fubini theorem we can write (1)

(j+1)

(j)

(1)

(1)

(j+1)

Eµ H1 (A1 , ..., Aj+1 )H2 (B1 , ..., Bj+1 ) Z =

(1)

(j)

Eµ H1 (A1 , ..., Aj , x)H2 (B1 , ..., Bj , y)dm(x, y)

where m is the distribution of Uj+1 . Applying the induction hypothesis we deduce that j X
Covµ H1 (A(1) , ..., A(j) , x), H2 (B (1) , ..., B (j) , y) ≤ 4 α(βi ) 1 1 j j i=1

for any x and y. On the other hand, by Theorem A.5 in [27] we have Z (1) (j) (1) (j) Eµ H1 (A1 , ..., Aj , x)Eµ H2 (B1 , ..., Bj , y)dm(x, y) Z (1) (j) (1) (j) − Eµ H1 (A1 , ..., Aj , x)Eµ H2 (B1 , ..., Bj , y)dk(x, y) ≤ 4α(βj+1 ) where k is the product of the distributions of Aj+1 and Bj+1 . The second integral in the above left-hand side is just the product of the expectations (1) (j+1) (1) (j+1) of H1 (A1 , ..., Aj+1 ) and H2 (B1 , ..., Bj+1 ) with respect to µ, and the proof of the induction step is complete. Proof of Claim 2.8.5. First, by the definitions of Vˆk (·) and the process {Θn : n ≥ 0} and since PΘ is ϑ-invariant, EPΘ Vˆk ◦ ϑjk = EPΘ Vˆk = E

k X

(1)

(2)

(`)

F` (ζi , ζ2i , ..., ζ`i )


2

i=1 (i) {ζn

for any j, k ∈ N, where : n ≥ 0}, i = 1, 2, ..., ` are independent copies of the Markov chain {ζn : n ≥ 0}. Moreover, it follows from (2.8.20) that there exists a constant C1 > 0 such that Vˆk ≤ C1 k, PΘ -a.s. and therefore EPΘ Vˆk2 ≤ C12 k 2 . Applying Theorem 2.5.11 with the function F` we obtain that k 1 X (1) (2) (`)
2 lim E F` (ζi , ζ2i , ..., ζ`i ) > 0 k→∞ k i=1

(2.8.23)

page 108

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

109

since we assumed that σ`2 > 0. It follows that there exists a constant b > 0 such that EPΘ Vˆk ≥ bk

(2.8.24)

for any sufficiently large k. Before we proceed with the proof we need the following, so-called, Paley-Zygmund inequality. Let X be a nonnegative random variable and assume that A21 ≤ (EX)2 ≤ EX 2 ≤ A22 for some A1 , A2 > 0. Then for any ρ ∈ (0, 1), 2 P (X ≥ ρA1 ) ≥ A21 A−2 2 (1 − ρ) .

(2.8.25)

Indeed, by the Cauchy-Schwarz inequality, A1 ≤ EX = EXI{X≥ρA1 } + EXI{X 0. PΘ (Vˆk ≥ bk) ≥ ε1 := 2 4C12

(2.8.26)

Next, for any n ≥ 0 set
ˆ n = (Θn , Θn+1 , ..., Θn+k−1 , Zn = (Zn(1) , Z (2) , ..., Z (`−1) ) Θ 2n (`−1)n (i)

and Zˆn = (Zn+k−1 , ..., Zn ), where {Zm : m ≥ 0}, i = 1, 2, ..., ` − 1 are independent copies of the process {Zm : m ≥ 0}. We note that the process {Zn : n ≥ 0} is the stationary process generated by the map Tˆ := T × T 2 × · · · × T `−1 and the Tˆ-invariant measure µ`−1 . We assume that our probability space is large enough so that both the process {Zn , n ≥ 0} and other processes below can be constructed on it and we denote everywhere by
k P the corresponding probability on it. Consider the function Vk : X `−1 → R given by Vk (y) = Var

k X


(`−1) F (ys(1) , ..., ys(`−1) , ζs` ) ,

s=1

y=

(2) (`−1) {(ys(1) , y2s , ..., y(`−1)s )

: 1 ≤ s ≤ k} ∈ X `−1


k

.

We observe that for any n ∈ N the random vectors {Vˆk ◦ ϑjk }nj=0 , ˆ jk )}n and {Vk (Zˆ(n−j)k )}n have the same distribution. Indeed, {Vk (Θ j=0 j=0

page 109

March 13, 2018 15:46

ws-book9x6

110

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

the first two are identically distributed since (ΩΘ , BΘ , PΘ , ϑ) is the invertible MPS corresponding to the processes {Θn : n ≥ 0}, while the second and third random vectors are identically distributed in view of (2.5.11). Therefore, for any c2 , ε0 > 0, PΘ (Γn,ε0 ,c2 ,k ) = P

n 1 X I{Vk (Zˆ(n−j)k )≥c2 k} < ε0 . n j=1

(2.8.27)

Furthermore, by (2.8.26) and since PΘ is ϑ-invariant, for any 0 ≤ j ≤ n,   1 1 (2.8.28) P Vk (Zˆ(n−j)k ) ≥ bk = PΘ Vˆk ◦ ϑjk ≥ bk 2 2  1 = PΘ Vˆk ≥ bk ≥ ε1 . 2 1 1 Next, fix some c2 < 3 b and ε0 < 4 ε1 , where b > 0 and ε1 appear in (2.8.26). By considering the product space of the distributions of the processes {(Zin , Zin,s ) : n, s ≥ 0}, i = 1, 2, ..., ` − 1 we can always assume (i) that there exist independent copies {Zin,s : n, s ≥ 0} of {Zin,s : n, s ≥ 0}, i = 1, 2, ..., ` − 1 so that (i)

(i)

kρ(Zin , Zin,s )kL∞ = kρ(Zin , Zin,s )kL∞ for any n, s ≥ 0 and i = 1, 2, ..., ` − 1. Notice that
Vk ∈ Hα,ξ X , ρ∞,(`−1)k
since F ∈ Hα,ξ X `−1 , ρ∞,`−1 . Therefore, there exists a constant c0 = c0 (k), which depends only on k and CF in (2.5.18), such that for any s ∈ N, n ∈ N and 0 ≤ j ≤ n, kVk (Zˆ(n−j)k ) − Vk (Zˆ(n−j)k,s )kL∞ ≤ c0 β∞,α,ξ (s)

(2.8.29)

(i) where Zˆm,s , m ∈ N is defined similarly to Zˆm but with the Zj,s ’s in place (i)

of the Zj ’s. Fix some s ∈ N such that c0 β∞,α,ξ (s) < 14 c2 k and set Ws,j,n = I{Vk (Zˆ(n−j)k,s )> 5 c2 k} , 0 ≤ j ≤ n. 4

Then P

n n 1 X 1 X I{Vk (Zˆ(n−j)k )≥c2 k} < ε0 ≤ P Ws,j,n < ε0 . n j=1 n j=1

(2.8.30)

In order to estimate the above right-hand side we will estimate first the Pn expectation and variance of the sums j=1 Ws,j,n . It follows from Lemma 2.8.6 that
Cov(Ws,j1 ,n , Ws,j2 ,n ) ≤ 4(` − 1)α (j2 − j1 − 1)k − 2s + 1

page 110

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

111

for any 1 ≤ j1 , j2 ≤ n such that (j2 − j1 − 1)k > 2s − 1. Since α(m) is decreasing in m, we infer that n X X Var( Cov(Ws,j1 ,n , Ws,j2 ,n ) (2.8.31) Ws,j,n ) ≤ j=1

X

+

1≤j1 ≤j2 ≤n:j2 −j1 ≤2s

4(` − 1)α(j2 − j1 − 2s) ≤ 2ns + 4(` − 1)n

1≤j1 ≤j2 ≤n:j2 −j1 >2s

∞ X

α(j)

j=1

and observe that the last sum converges by Assumption 2.5.10 since α(n) ≤ φn for any n ≥ 0. Next, by (2.8.29), taking into account that c0 β∞,α,ξ (s) < b 1 4 c2 k and c2 < 3 , for any 0 ≤ j ≤ n we have  5 EWs,j,n = P Vk (Zˆ(n−j)k,s ) > c2 k (2.8.32) 4  1 ≥ P Vk (Zˆ(n−j)k ) ≥ bk ≥ ε1 2 where in the last inequality (2.8.28) is used. We conclude from (2.8.31) and (2.8.32) and the Markov inequality that there exists a constant C2 which is independent of k such that for any n ∈ N, n n n X 1 1 X 1 X Ws,j,n − E Ws,j,n ≥ ε0 Ws,j,n < ε0 ≤ P P n j=1 n j=1 4 j=1 P∞ 2ns + 4(` − 1)n j=1 α(j) C2 ≤ 16 · ≤ 2 2 ε0 n n 1 where in the first inequality we used that ε0 < 4 ε1 and Claim 2.8.5 follows, taking into account (2.8.27). 2.8.5

Remarks

We conclude this section with several remarks whose purpose is to discuss weaker assumptions under which the above proof of Claim 2.8.5 proceeds similarly, and the possibility of proving this claim without using Assumption 2.5.10, relying on some large deviation argument instead (see Remark 2.8.10). Note that Assumption 2.5.10 is still needed in order to obtain (2.7.8), which is an estimate which requires stronger mixing conditions than just the decay of correlations obtained in Proposition 2.5.8 which was used in the proof of Claim 2.8.4. Remark 2.8.7. The proof of Claim 2.8.5 proceeds similarly under some restrictions on the Lq -approximation rate βq (s) := sup kρ(Zn , Zn,s )kLq n≥0

page 111

March 13, 2018 15:46

ws-book9x6

112

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

for some q ≥ 1. For instance, in place of (2.8.30) we have P

n n 1 X 1 X I{Vk (Zˆ(n−j)k )≥c2 k} < ε0 ≤ P Ws,j,n < ε0 n j=1 n j=1

+

n X  1 P |Vk (Zˆ(n−j)k ) − Vk (Zˆ(n−j)k,s )| ≥ c2 k 4

k=1

and a similar inequality holds true in place of (2.8.32). Using the Markov inequality in order to estimate the probabilities appearing in the sum in the above right-hand side, the proof of Claim 2.8.5 will proceed similarly if we assume that there exists a sequence of natural numbers (sn )∞ n=1 so that 1 3 n 2 βqq (sn ) and n− 2 sn converge to 0 as n → ∞. Moreover, an inequality similar to (2.7.8) will hold true under similar conditions, as well and therefore the nonconventional local limit theorem itself holds true under conditions concerning the rate of decay of Lq -approximation and mixing coefficients. Remark 2.8.8. Another way to deduce (2.8.28) goes as follows. By Theoω rem 2.3 in [38] the limit σ 2 = limn→∞ n1 Varµ (Sn,` F) exists PΘ -a.s., it does 2 not depend on ω and σ > 0 if and only if F does not admit a (random) coboundary representation. In our situation this coboundary representation is equivalent to F` having an appropriate coboundary representation, which is excluded. Thus, for any sufficiently large n we have  n 1 ω PΘ ω : Varµ (Sn,` F) > σ 2 > . 2 2 Remark 2.8.9. The proof that Assumption 2.2.2 holds true proceeds similarly without approximating the Zi ’s by the Zi,s ’s, assuming some rate of decay of correlation for bounded observables. For instance, an inequality of the form |Covµ`−1 (g, h ◦ Tˆk )| ≤ ckgk∞ khk∞ k −2−δ for some c, δ > 0, all k ≥ 1 and bounded Borel functions g and h is sufficient, where we recall that Tˆ = T × T 2 × · · · × T `−1 . Nevertheless, this condition implies that the process {Tˆm : m ≥ 0} taken with the initial distribution µ`−1 is sufficiently well α-mixing, which, in some sense, is stronger than our assumptions and it does not hold in important examples such as one sided topologically mixing subshifts of finite type and expanding C 2 endomorphisms of a Riemmanian manifold.

page 112

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

113

Remark 2.8.10. A different strategy to prove (2.8.18) is to rely on some large deviation results instead of mixing and approximation conditions. In many dynamical systems such results are available, while the mixing and approximation conditions usually hold true when Markov partitions exist. Still, there are several difficulties in applying such large deviation results, as described below. Consider the functions V˜k : X `−1 → R, k ∈ N given by V˜k (¯ x) = Var

k X

k X

2 F (Tˆk−j x ¯, ζ`j ) = E F` (Tˆk−j x ¯, ζ`j ) .

j=1

j=1

Then each V˜k is a (locally) H¨older continuous and by (2.5.11) when x ¯ is `−1 drawn according to µ then the stationary processes kn ˜ ˆ {Vk ◦ T : n ≥ 0} and {Vˆk ◦ ϑkn : n ≥ 0} have the same distribution. Relying on the exponential decay of correlations of the system (X , BX , µ, T ) obtained in Theorem 2.5.7 we derive that the limits n X 1 V˜k ◦ Tˆkj , k ∈ N σk2 = lim Varµ`−1 n→∞ n j=1 exist and that σk2 = 0 if and only if the function V˜k admits an appropriate coboundary representations (see [33]). When σk2 > 0 then we can apply Theorem E (i) in [28] in order to estimate the µ`−1 -probabilities of sets of the form Z X n n o n X
jk ˜ ˆ V˜k (Tˆjk x ¯) dµ`−1 (¯ x) ≥ nε V k (T x ¯) − Γn = x ¯: j=1

j=1

which is sufficient for showing that Assumption 2.2.2 holds true. The disadvantage here is that in order to use such estimates we have to assume that σk2 > 0 for some unknown sufficiently large k which is an assumption that is hard to verify. 2.9

Decay of characteristic functions for large t’s

In this section we will show that Assumption 2.2.1 holds true in the circumstances of Theorem 2.6.3 in the non-lattice case and Theorem 2.6.5 in the lattice case. We will use directly the stationary process Θ = {Θn : n ≥ 0} and not its associated MPS M (Θ) = (ΩΘ , BΘ , PΘ , ϑ) and note that we will not use here Assumption 2.5.10 but only the decay of correlations established in Proposition 2.5.8.

page 113

March 13, 2018 15:46

114

2.9.1

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Basic estimates and strategy of the proof


n For any n ∈ N, t ∈ R and v¯ = (v (0) , v (1) , ..., v (n−1) ) ∈ X `−1 consider the ¯,n transfer operator Lvit which acts on function g on X by the formula X Pn−1 (k) k` v ¯,n eSn` f (y)+it k=0 F (v ,T y) g(y) (2.9.1) Lit g(x) = y∈T −n` {x}

and notice that (n−1)

¯,n Lvit = Lvit

,1

(n−2)

◦ Lvit

,1

(0)

◦ · · · ◦ Lvit

,1

.

(2.9.2)

Next, by (2.7.9) and the definition of the MPS M (Θ), for any t ∈ R and N ≥ 1, (ΘM +1 ,...,ΘN ),N −M

|ϕN (t)| = |EeitSN | ≤ cN + EkLit

1k∞ (2.9.3) √ for some sequence which satisfies limn→∞ ncn = 0. Thus, in order to show that Assumption 2.2.1 holds true it is sufficient to estimate the expectations (cn )∞ n=1

(ΘM +1 ,...,ΘN ),N −M

EkLit

1k∞

appropriately. Consider again the map Tˆ = T × T 2 × · · · × T `−1 : X `−1 → X `−1 . Then by stationarity of Θ and (2.5.11), (ΘM +1 ,...,ΘN ),N −M

EkLit

Z =

(Θ1 ,...,ΘN −M ),N −M

1k∞ = EkLit

1k∞

(2.9.4)

v ¯ˆ (¯ x),N −M kLitT ,N −M −1 1k∞ dµ`−1 (¯ x)

where for any x ¯ = (x1 , .., x`−1 ) ∈ X `−1 and n ≥ 0, v¯ ˆ (¯ x) = (Tˆn x ¯, Tˆn−1 x ¯, ..., x ¯) T ,n

is the inversion of the truncated orbit O0,n (¯ x) = {Tˆk x ¯ : 0 ≤ k ≤ n} of x ¯. Let x0 ∈ X be a periodic point of T , i.e. T m0 x0 = x0 for some m0 ∈ N, and set x ¯0 = (x0 , x0 , ..., x0 ) ∈ X `−1 which satisfies Tˆm0 x ¯0 = x ¯0 . Next, for each t ∈ R, consider the transfer operator Rit generated by the map τ0 = T `m0 and the potential S`m0 f + itFx0 ,m0 , which acts on functions g : X → C by the formula 0 Rit g(x) = L`m (eitFx0 ,m0 g)(x) f

=

X y∈τ0−1 {x}

eS`m0 f (y)+it

Pm0 −1 k=0

F (Tˆ k x ¯0 ,T `k y)

g(y)

(2.9.5)

page 114

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

115

Pm−1 where Sm f = k=0 f ◦ T m for any m ∈ N. Set Y = X `−1 , let n0 ∈ N and consider the periodic element ux¯0 ,n0 of (Y m0 )n0 given by x0 , Tˆx ¯0 , ..., Tˆm0 −1 x ux¯0 ,n0 = {¯ ¯0 , x ¯0 , Tˆx ¯0 , ..., Tˆm0 −1 x ¯0 , ...}

(2.9.6)

which consists of n0 consecutive copies of O0,m0 −1 (¯ x0 ). Note that, in fact, ux¯0 ,n0 = O0,n0 m0 −1 (¯ x0 ) and observe that x ¯0 ,n0

n0 = Rit ◦ Rit ◦ · · · ◦ Rit = Luit Rit

,n0 m0

.

(2.9.7)

The idea behind the proof is to show that for any N ∈ N, on a set whose µ`−1 probability is sufficiently close to 1, the sequence v¯Tˆ,N −M −1 (¯ x) contains proportional to N number of Y-valued subsequences of length n0 m0 which are close to ux¯0 ,n0 (see Corollary 2.9.3 for the precise formuvˆ

lation). Relying on Section 2.9.4, this will insure that LitT ,N −M −1 can be represented in the form vˆ

LitT ,N −M −1

(¯ x),N −M

(¯ x),N −M

= A1 ◦ B1 ◦ A2 ◦ B2 ◦ · · · ◦ Bk ◦ Ak+1

with operators Ai and Bi depending on t, k ≥ c1 (N − M ) for some c1 > 0, kAi k ≤ D(1 + |t|) for some constant D and all i’s and each one of the Bi ’s n0 is sufficiently close to Rit in the norm k · kα,ξ (see (2.9.27) for the precise formulation). This representation reduces the problem of approximating the norm of the transfer operator inside the integral on the right-hand side n0 of (2.9.4) to approximation of norms of the powers Rit of a single transfer operator which is a well studied problem (see Section 2.9.5). Since we can take x’s from a set whose probability is sufficiently close to 1 this yields (see Section 2.9.6) appropriate upper bounds on the integral in (2.9.4). 2.9.2

Probabilities of large number of visits to open sets

Let (Yi , di ), i = 1, ..., k be metric spaces equipped with the Borel σ-algebras BYi . For each i let Si : Yi → Yi be a Borel measurable map and νi be a Si -invariant Borel probability measure on Yi which assigns positive probability to open sets. Suppose that each measure preserving system (Yi , BYi , Si , νi ), i = 1, 2, ..., k satisfies ∞ X m=0

|Covνi (gi , gi ◦ Sim )| < ∞

(2.9.8)

page 115

March 13, 2018 15:46

ws-book9x6

116

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

for any bounded Lipschitz function gi : Yi → R. Consider the product measure preserving system (Y, BY , ν, S) where Y = Y1 × Y2 × · · · × Yk , ν = ν1 × ν2 × · · · × νk and S = S1 × S2 × · · · × Sk . Let B ⊂ Y be an open set and let n ∈ N. Consider the function Nn,B : Y → R which counts the number of visits to B from time 0 to time n − 1 given by Nn,B (y) =

n−1 X

IB (S m y)

m=0

where IB is the indicator function of the set B. For any c > 0 set Bn,c (B) = {y ∈ Y : Nn,B (y) ≥ cn}. Lemma 2.9.1. There exist constants b > 0 and c ∈ (0, 1), which may depend on B, such that for any n ∈ N, 1 − ν(Bn,c (B)) ≤

b . n

(2.9.9)

Proof. The set B is open and thus there exist open balls Bi ⊂ Yi , i = 1, ..., k such that B1 × B2 × · · · × Bk ⊂ B. Write Bi = B(yi , si ) for some yi ∈ Yi and si > 0 and set ri = 21 si . For each i, let gi : Yi → [0, 1] be a bounded Lipschitz function such that IB(yi ,si ) ≥ gi ≥ IB(yi ,ri ) .

(2.9.10)

For instance, the function gi given by gi (x) = max 0, 1 − max(0,

di (x, yi ) − ri
) ri

satisfies the above conditions and has Lipschitz constant ri−1 . Consider the function g : Y → [0, 1] given by g(y) =

k Y

gi (yi ), y = (y1 , ..., yk )

i=1

and set Gn = n ∈ N,

Pn−1

m=0

g ◦ S m , n ∈ N. Then by (2.9.10), for any y ∈ Y and Nn,B (y) ≥ Gn (y)

and therefore for any c > 0, 1 − ν(Bn,c (B)) ≤ 1 − ν(Bn,c (g))

(2.9.11)

page 116

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

117

where Bn,c (g) = {y ∈ Y : Gn (y) ≥ cn}. By (2.9.10), for each i = 1, ..., k we have Z Eνi gi = gi (yi )dνi (yi ) ≥ νi (B(yi , ri )) := ai > 0 where ai is positive since we have assumed that each νi assigns positive mass to open sets. Since Si preserves νi , i = 1, ..., k we conclude that k

k

Y Y 1 Eν Gn = Eν g = Eνi gi ≥ ai := a > 0. n i=1 i=1

(2.9.12)

Next, for any m ≥ 0 we have Covν (g, g ◦ S m ) =

k Z Y

gi · (gi ◦ Sim )dνi −

i=1

Z k Y


2 gi dνi .

i=1

Using the inequality k k k Y X Y αi − βi ≤ |αi − βi | i=1

i=1

i=1

which holds true for any αi , βi , i = 1, ..., k such that |αi |, |βi | ≤ 1, we deduce that k X |Covνi (gi , gi ◦ Sim )| |Covν (g, g ◦ S m )| ≤ i=1

where we have used that each gi takes values on the interval [0, 1]. We conclude from this and (2.9.8) that there exists M > 0 which depends only on g such that for any j ≥ 0, X X |Covν (g ◦ S j , g ◦ S m )| = |Covν (g, g ◦ S m )| m≥j

m≥0

≤

k X X

|Covν (gi , gi ◦ Sim )| ≤ M < ∞

i=1 m≥0

and therefore Varν (Gn ) ≤ 2nM. a 2,

(2.9.13)

where a comes from (2.9.12). Then by (2.9.12), (2.9.13) and the Set c = Chebyshev inequality, 1 8M n 8M a−2 ν(Gn < nc) ≤ ν(|Gn − Eν Gn | > an) ≤ 2 2 = . (2.9.14) 2 a n n The lemma follows with the above c and with b = 8M a−2 , taking into account (2.9.11).

page 117

March 13, 2018 15:46

ws-book9x6

118

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Remark 2.9.2. In many situations large deviation type results exist for the map S and, in fact, the arguments in the proof above show that we could Pn−1 have used some large deviation results for sums Gn = j=0 g◦S j generated by a Lipschitz continuous function g and the map S. Still, as in Remark 2.8.10, it is unclear whether the corresponding large deviation upper bound is strictly negative. For instance, in the situation of nonconventional sums consider the map S = Tˆ = T × T 2 × · · · × T `−1 . Applying Theorem E (i) in [28] with the function g we will get exponentially fast converges to 0 of probabilities of sets of the form {Gn ≥ ε0 n}, ε0 > 0 assuming that limn→∞ n−1 VarGn > 0, or, equivalently, that g does not admit a coboundary representation with respect to S, which in general is hard to verify. 2.9.3

Segments of periodic orbits

Let (Yi , Si , νi ), i = 1, ..., k be as in Section 2.9.2. It will be convenient to have the following notation of concatenation. Let m1 , ..., ml ∈ N and (i) w(i) = {wj : 1 ≤ j ≤ mi } ∈ Y mi , i = 1, ..., l, where we recall that Y = Pk Y1 × Y2 × · · · × Yk . Set Mk = i=1 mi , M0 = 0 and let the concatenation of the w(i) ’s w = w(1) w(2) ...w(l) ∈ Y Ml be given by (j)

wi = wi−Mj−1 if Mj−1 + 1 ≤ i ≤ Mj . For notational convenience, we will allow concatenation with the empty symbol, that is when w0 = ∅ we write w0 w = ww0 = w. Next, suppose that each Si is continuous. Consider the metric d = d∞ on Y given by d(y, z) = max di (yi , zi ), y = (y1 , ..., yk ), z = (z1 , ..., zk ) 1≤i≤k

and for any m ≥ 1 consider the metric dm,∞ on Y m = Y × Y × · · · × Y given by dm,∞ (¯ y , z¯) = max d(¯ yi , z¯i ) 1≤i≤m

where y¯ = (¯ y1 , ..., y¯m ) and z¯ = (¯ z1 , ..., z¯m ). Fix some y = (y1 , ..., yk ) ∈ Y, m ≥ 1 and δ > 0 and consider the Bowen ball Bm (y, δ) = {z ∈ Y :

max

0≤i≤m−1

d(S i z, S i y) < δ}.

page 118

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

119

Since each Si is continuous the set Bm (y, δ) is open. For any 0 ≤ a ≤ b ≤ ∞ and z ∈ Y consider the orbit segment Oa,b (z) = {S i z : a ≤ i ≤ b} ∈ Y b−a+1 of z from time a to time b. Lemma 2.9.1 provides estimate on the νprobability of the set of all z’s such that the truncated orbit O0,n−1 (z) contains at least nc orbit segments of the form Oa,a+m−1 (z), a ≤ n − m which satisfy dm,∞ (Oa,a+m−1 (z), O0,m−1 (y)) < δ. If z satisfies this condition then there exist numbers {ai : 1 ≤ i ≤ L + 1}, L ≥ [ nc m ] so that for any 1 ≤ i ≤ L, ai + m ≤ ai+1 ≤ n and d(S ai +j z, S j y) < δ for any 0 ≤ j ≤ m − 1. (2.9.15) Therefore on a set whose probability is not less than 1 − nb we can write nc O0,n−1 (z) = w(1) v (1) w(2) v (2) ...v (L) w(L+1) , L ≥ [ ] m (i)

(i)

where each v (i) = (v0 , ..., vm−1 ) ∈ Y m satisfies dm,∞ (v (i) , O0,m−1 (y)) =

max

0≤j≤m−1

(i)

d(vj , S j y) < δ

and some of the w(i) ’s are allowed to be empty. Suppose now that y is a periodic point and let my ∈ N be so that S my y = y. For any n0 ∈ N consider the orbit segment uy,n0 = O0,n0 my −1 (y). Then uy,n0 = ww...w, w = O0,my −1 (y) namely it is periodic and consists of n0 consecutive copies of O0,my −1 (y). Considering m’s of the form m = n0 my , k ∈ N it follows that up to a set whose ν-probability does not exceed nb we can write nc O0,n−1 (z) = w(1) v (1) w(2) v (2) w(3) ...v (L) w(L+1) , L ≥ [ ] m (i)

(i)

where each v (i) = (v0 , ..., vn0 my −1 ) ∈ Y n0 my satisfies dn0 my ,∞ (v (i) , uy,n0 ) < δ and some of the w(i) ’s are allowed to be empty. We return now to our situation of nonconventional sums. In this case we take Si = T i and νi = µ for any 1 ≤ i ≤ k = ` − 1. Then (2.9.8) holds true by Proposition 2.5.8, taking into account that the norms k · kα,ξ and

page 119

March 13, 2018 15:46

120

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

k · k1,ξ are equivalent since the space X is compact. For any s ∈ N and u ¯, v¯ ∈ (X `−1 )s set ρs,∞ (¯ u, v¯) = max ρ(ui,j , vi,j ) 1≤i≤s, 1≤j≤`−1

where u ¯ = (¯ u1 , ..., u ¯s ), v¯ = (¯ v1 , ..., v¯s ), u ¯i = {ui,j : 1 ≤ j ≤ ` − 1} ∈ X `−1 `−1 and v¯i = {vi,j : 1 ≤ j ≤ ` − 1} ∈ X for each i. Let x0 be a periodic point of T and let m0 be so that T m0 x0 = x0 . Consider the point x ¯0 = (x0 , ..., x0 ) ∈ X `−1 , which satisfies Tˆm0 x ¯0 = x ¯0 , and for any natural n0 the periodic sequence ux¯0 ,n0 = ww...w, w = O0,m0 −1 (¯ x0 ) which consists of n0 consecutive copies of O0,m −1 (¯ x0 ) = {¯ x0 , Tˆx ˆ0 , ..., Tˆm0 −1 x ¯0 }. 0

We summarize the above discussion in the following corollary. Corollary 2.9.3. For any n0 ∈ N and δ0 > 0 there exist constants b1 = b1 (n0 , δ0 ) > 0 and c1 = c1 (n0 , δ0 ) > 0 and measurable sets Tn = Tn,n0 ,δ0 ⊂ X `−1 , n ∈ N such that for any n ∈ N, b1 (2.9.16) µ`−1 (Tn ) ≤ n and for any y¯ = (y1 , ..., y`−1 ) ∈ X `−1 \ Tn we can write {¯ y , Tˆy¯, ..., Tˆn−1 y¯} = w(1) v (1) w(2) v (2) w(3) ...v (L) w(L+1) , L ≥ c1 n (2.9.17) where some of the w(i) ’s can be empty and each v (i) satisfies ρn0 m0 ,∞ (v (i) , ux¯0 ,n0 ) < δ0 . In other words, there exist indexes ai , i = 1, ..., L + 1, L ≥ c1 n such that for any 1 ≤ i ≤ L, ai + km0 ≤ ai+1 ≤ n and ρ(T s(ai +jm0 +l) ys , T sl x0 ) < δ0 (2.9.18) for any 0 ≤ j ≤ n0 − 1, 0 ≤ l ≤ m0 − 1 and 1 ≤ s ≤ ` − 1. Remark 2.9.4. Consider the situation when (X , T ) is a one sided topologically mixing subshift of finite type (see [10] and Section 2.11). A Bowen ball B is just a cylinder set and its indicator function IB is Lipshitz continuous, and it is easy to check directly that the corresponding asymptotic variance is positive. A power and a product of one sided (topologically mixing) subshifts of finite type is a one sided (topologically mixing) subshift of finite type, and in this case as discussed in the end of Remark 2.9.2 we can find sets Γn with the properties specified in the above corollary so that µ`−1 (Γn ) ≤ b1 e−c1 n for some b1 , c1 > 0.

page 120

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

121

Example 2.9.5. Consider the above situation of a one sided topologically mixing subshift of finite type. Let a be a periodic point with period K ≥ 1 and write a = ααα... where α is a word of length K. Then the above corollary shows that for any k there exist c1 , b1 > 0 such that up to a probability of bn1 the truncated word x0 x1 , ..., xn−1 of an element x = x0 x1 x2 ... of X contains at least c1 n disjoint blocks of the form αα...α of length K. In fact, according to Remark 2.9.4 this happens on a set whose probability is not less than 1 − b1 e−nc1 . 2.9.4

Parametric continuity of transfer operators

¯,n Consider the transfer operators Luit defined in (2.9.1) and recall (2.9.2). Then the arguments in the proof of Lemma 2.7.4
show that there exists a n constant D > 0 so that for any n ∈ N, u ¯ ∈ X `−1 and t ∈ R, ¯,n kLuit kα,ξ ≤ D(1 + |t|) := Dt . (2.9.19)
`−1 n (0) (n−1) Next, for any v¯, u ¯ ∈ X write v¯ = (v , ..., v ), u ¯ = (u(0) , ..., u(n−1) ) and set

ρn,∞ (¯ v, u ¯) =

max ρ`−1,∞ (v (i) , u(i) ).

0≤i≤n−1


k0 Lemma 2.9.6. Let k0 ∈ N and u ¯ ∈ X `−1 . Suppose that the function u → Fu = F (u, ·) is continuous at the points u = u(0) , u(1)
, ..., u(k0 −1) when considered as a function from X `−1 to Hα,ξ (X ), k · kα,ξ . Then for any ε0 > 0 there exists δ0 = δ0 (k0 , ε0 ) > 0 which may depend only on ε0 and k0 (and the function F ) such that ¯,k0 ¯,k0 kLvit − Luit kα,ξ ≤ B 2 k0 (|t| + t2 )(1 + |t|)2 ε0
k0 for any t ∈ R and v¯ ∈ X `−1 so that ρk0 ,∞ (¯ v, u ¯) < δ0 , where B > 0 is a constant which does not depend on k0 u ¯, v¯, ε0 and t.

Proof. Let k0 and u ¯ be as in the statement of the lemma and let ε0 > 0 ¯,k0 ¯,k0 and t ∈ R. We recall first the decompositions (2.9.2) of Lvit and Luit . w,0 Then, using the convention that L is the identity map, we can write ¯,k0 ¯,k0 Lvit − Luit

=

kX 0 −1

(v (k0 −1) ,...,v (j+1) ),k0 −j−1

Lit

(j)

Lvit

,1

(j)

− Luit

,1
(u(j−1) ,...,u(0) ),j Lit

j=0

and therefore by submultiplicativity of the operator norm and (2.9.19), ¯,k0 ¯,k0 kLvit − Luit kα,ξ ≤ D2 (1 + |t|)2

kX 0 −1 j=0

(j)

kLvit

,1

(j)

− Luit

,1

kα,ξ .

page 121

March 13, 2018 15:46

ws-book9x6

122

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Now, let 0 ≤ j < k0 . Then (j)

Lvit

,1

(j)

− Luit

,1

= L`f (eitFv(j) − eitFu(j) )

and therefore (j)

v kLit

,1

(j)

− Luit

,1

kα,ξ ≤ kL`f kα,ξ keitFv(j) − eitFu(j) kα,ξ .

In order to estimate the above right-hand side, we apply first the mean value theorem and the H¨ older continuity of F (see (2.5.18)) to obtain
α keitFv(j) − eitFu(j) k∞ ≤ 2|t|kFv(j) − Fu(j) k∞ ≤ 2CF |t| ρ`−1,∞ (v (j) , u(j) ) assuming that ρ`−1,∞ (v (j) , u(j) ) < ξ. Next, we will estimate
vα,ξ eitFv(j) − eitFu(j) . Let x, x0 ∈ X be such that ρ(x, x0 ) < ξ, and consider the functions U and V given by V (x) = Vj (x) = Fv(j) (x) and U (x) = Uj (x) = Fu(j) (x). Then we can write
0
0 eitFv(j) (x) − eitFu(j) (x) − eitFv(j) (x ) − eitFu(j) (x ) 0 0
= eitU (x) eit(V (x)−U (x)) − eit(V (x )−U (x ))
0 0 0
+ eit(U (x)−U (x )) − 1 eitV (x ) − eitU (x ) := I1 + I2 . By the mean value theorem we have

|I1 | ≤ 2|t| V (x) − U (x) − V (x0 ) − U (x0 ) and |I2 | ≤ 4|t|2 |U (x) − U (x0 )| · kV − U k∞ . Using now (2.5.18) we derive that |U (x) − U (x0 )| ≤ CF ρα (x, x0 ) and that
α kV − U k∞ ≤ CF ρ`−1,∞ (v (j) , u(j) ) assuming that ρ`−1,∞ (v (j) , u(j) ) < ξ. Next, it is clear from the definition of V and U that

V (x) − U (x) − V (x0 ) − U (x0 ) ≤ kFv(j) − Fu(j) kα,ξ ρα (x, x0 ). By the continuity assumption in the statement of the lemma, there exists δ1 = δ1 (k0 , ε0 ) > 0 so that for any 0 ≤ j < k0 , kFv(j) − Fu(j) kα,ξ ≤ ε0 (j)

`−1

for any v ∈ X such that ρ`−1,∞ (v (j) , u(j) ) < δ1 . Let 0 < δ0 < min(δ1 , ξ) be so that (δ0 )α < ε0 . The lemma follows now (with such δ0 ) from the above estimates.

page 122

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

2.9.5

123

Quasi compactness of Rit , lattice and non-lattice cases

We first claim that for any t ∈ N the transfer operator Rit satisfies the Lasota-Yorke type inequality with respect to the norm k · kα,ξ and the (semi) norm k · k∞ . More precisely, we assert that there exist constants c ∈ (0, 1) and A > 0 such that
n kRit gkα,ξ ≤ A cn kgkα,ξ + (1 + |t|)kgk∞ (2.9.20) for any n ∈ N, t ∈ R and g ∈ Hα,ξ = Hα,ξ (X ). Indeed, (2.9.20) follows by arguments similar to the ones in the proof of (2.7.5) in Lemma 2.7.4 since in that proof we have only used the pairing property described in Section 2.5.2, the H¨ older continuity of F and some elementary inequalities. Observe now that by (2.9.20) the spectral radius r(it) of Rit with respect to the norm k · kα,ξ does not exceed 1. We conclude from Theorem II.5 in [28] (applied with the (semi) norm k|·|k = k·k∞ ) that Rit is quasi-compact with respect to the norm k · kα,ξ when r(it) = 1. Note that conditions K1 and K2 in Chapter XI of [28] hold true with the map τ0 = T `m0 , the transfer 0 operator Q = R0 = L`m and the space B = (Hα,ξ , k · kα,ξ ). We conclude f that all the results from Chapter XI in [28] (cited below) hold true for the family of transfer operators Rit , t ∈ R. Next, in the non-lattice case let J ⊂ R be a compact set not containing 0. We claim that there exist constants A > 0 and c ∈ (0, 1), which may depend on J, such that for any n ∈ N and t ∈ J, n kRit kα,ξ ≤ Acn . (2.9.21) Indeed, since Fm0 ,x0 is non-arithmetic, Proposition XI.7 in [28] implies that r(it) < 1 for any t ∈ R \ {0}, and (2.9.21) follows from Corollary III.13 in [28]. 2π Now, in the lattice case, let J ⊂ (− 2π h , h ) be a compact set not containing the origin. As in the non-lattice case, we claim that there exist constants A > 0 and c ∈ (0, 1), which may depend on J, such that for any n ∈ N and t ∈ J, n kRit kα,ξ ≤ Acn . (2.9.22) Indeed, by Proposition XI.7 in [28] for any t 6= 0 the spectral radius r(it) equals 1 if and only if there exist g ∈ Hα,ξ \ {0} and α ∈ R such that geitFx0 ,m0 = eiα g ◦ T `m0 , µ-a.s. where we recall that Rit is indeed quasi-compact when r(it) = 1. Since h is maximal (see Assumption 2.6.4) the spectral radius r(it) of Rit is strictly 2π less than 1 when t ∈ (− 2π h , h )\{0}, and (2.9.22) also follows from Corollary III.13 in [28].

page 123

March 13, 2018 15:46

124

2.9.6

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Norms estimates

We complete here the proof that Assumption 2.2.1 holds true in both nonlattice and lattice cases. Recall first that by Assumption 2.6.1 the function u → Fu is continuous at the points u = Tˆk x ¯0 , 0 ≤ k < m0 . Set Y = X `−1 and let n0 ∈ N and ε0 > 0. Consider the periodic element ux¯0 ,n0 of (Y m0 )n0 given by (2.9.6). Then the function u → Fu is continuous at each of the Y-valued components of ux¯0 ,n0 . Consider the number δ0 = δ0 (n0 m0 , ε0 ) > 0 appearing in Lemma 2.9.6, and let v¯ ∈ (Y m0 )n0 be so that ρn0 m0 ,∞ (¯ v , ux¯0 ,n0 ) < δ0 .

(2.9.23)

Using (2.9.7) and then applying Lemma 2.9.6, we deduce that ¯,n0 m0 n0 kLvit − Rit kα,ξ (2.9.24) ¯0 ,n0 ux ,n0 m0

¯,n0 m0 = kLvit − Lit

kα,ξ ≤ Bn0 m0 (|t| + t2 )(1 + |t|)2 ε0

for any t ∈ R. Next, let δ ∈ (0, 1). In the non-lattice case set J = Jδ = [−δ −1 , −δ] ∪ [δ, δ −1 ] while in the lattice case we set π π J = Jδ = [− , ] \ (−δ, δ). h h By (2.9.21) in the non-lattice case and by (2.9.22) in the lattice case there exist constants A > 0 and c ∈ (0, 1), which may depend on δ, such that for any n ∈ N and t ∈ J = Jδ , n kRit kα,ξ ≤ Acn .

(2.9.25)

Set Dδ = max{Dt : t ∈ Jδ } where Dt is defined in (2.9.19). Fix some 1 n0 ∈ N so large such that Acn0 < 4D and then fix some ε0 > 0 so small δ such that 1 for any t ∈ J. Bn0 m0 (|t| + t2 )(1 + |t|)2 ε0 < 4Dδ Then by (2.9.7), (2.9.24) and (2.9.25), ¯,n0 m0 kLvit kα,ξ ≤

1 2Dδ

for any v¯ satisfying (2.9.23) with these fixed n0 and δ0 = δ0 (n0 m0 , ε0 ). Consider now the sets Tn = Tn,n0 ,δ0 , n ∈ N and the numbers b1 = b1 (n0 , δ0 ) > 0

page 124

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Local limit theorem

125

and c1 = c1 (n0 , δ0 ) > 0 from Corollary 2.9.3. Then by (2.9.4), for any N ∈ N and t ∈ R, (ΘM +1 ,...,ΘN ),N −M

EkLit

≤ µ`−1 (TN −M ) +

Z

vˆ

1k∞ = Eµ`−1 kLitT ,N −M −1 vˆ

ITNc −M (¯ x)kLitT ,N −M −1

(·),N −M

(¯ x),N −M

1k∞ (2.9.26)

kα,ξ dµ`−1 (¯ x)

where Tnc = X \ Tn for any n ∈ N. By Corollary 2.9.3, on the set Tnc we can write vTˆ,N −M −1 (¯ x) = w(1) v(1) w(2) v(2) w(3) ...v(L) w(L+1) , L ≥ c1 (N − M ) where each v(i) ∈ (Y m0 )n0 satisfies ρm0 n0 ,∞ (v(i) , ux¯0 ,n0 ) < δ0 while some of the w(i) ’s can be empty. We remark that Corollary 2.9.3 guarantees that we can decompose the truncated orbit O0,N −M −1 (¯ x) of x ¯ in such a way, but since ux¯0 ,n0 is periodic it is invariant under inversion and so the reverse truncated orbit vTˆ,N −M −1 (¯ x) inherits this decomposition from the truncated orbit. We conclude that for any x ¯ ∈ Tn and t ∈ Jδ , the vTˆ ,N −M (¯ x),N −M −1 random product Lit can be represented in the form vˆ

LitT ,N −M −1

(¯ x),N −M

= A1 ◦ B1 ◦ A2 ◦ B2 ◦ · · · ◦ Bk ◦ Ak+1

(2.9.27)

where k ≥ c1 (N − M ), the norms kAj kα,ξ , 1 ≤ j ≤ k + 1 are bounded from above by Dδ and the norms kBj kα,ξ , 1 ≤ j ≤ k are bounded from 1 . There exists a constant a0 which depends only on ` such above by 2D δ that N − M ≥ a0 N for any N > 3. We conclude that here exists a constant d > 0 such that vˆ

kLitT ,N −M −1

(¯ x),N −M

kα,ξ ≤ 2−N d

for any N > 3 and x ¯ ∈ TNc −M , which together with (2.9.3), (2.9.26) and (2.9.16) completes the proof that Assumption 2.2.1 holds true. Remark 2.9.7. At first glance it seems that the arguments in the proof that Assumption 2.2.1 holds true can be modified in order to show that Assumption 2.2.2 holds. Indeed, for any c > 0 the set {V˜k > ck} is open and thus contains an open ball, where V˜k is defined in Remark 2.8.10. The problem here is that in general we have no control on the radius of these balls and so their µ`−1 -measures cannot be controlled uniformly over k. As a consequence, it is not possible to choose ε0 independently of k as we did in the proof of Claim 2.8.5.

page 125

March 13, 2018 15:46

126

2.10

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

The periodic point approach to quenched and annealed dynamics

The idea behind the proof (in Section 2.9) that Assumption 2.2.1 holds true is not restricted to random transformations arising in the nonconventional setup. In this section we present a scheme to prove a fiberwise (and annealed) local limit theorems using the ideas of Section 2.9, and we refer the readers to Chapter 7 for more details. Let (Ω, B, P, τ ) be an ergodic measure preserving system (not necessarily invertible) on a topological space Ω where B is the Borel σ-algebra. Let Bω , ω ∈ Ω be a family of complex Banach spaces. Assumption 2.10.1. The probability measure P assigns positive measure to open sets, τ is continuous and there exist ω0 ∈ Ω and m0 ∈ N so that τ m0 ω0 = ω0 . Moreover, the map ω → Bω is locally constant around the points τ i ω0 , 0 ≤ i < m0 . Let I ⊂ R be an interval around 0 (possibly I = R) and {Aω it : ω ∈ Ω}, t ∈ I : B → B . Consider the random be families of continuous linear maps Aω ω θω t n products Aω,n : B → B given by ω θ ω it n−1

τ Aω,n it := Ait

ω

◦ · · · ◦ Aτitω ◦ Aω it , t ∈ I, n ∈ N.

Assumption 2.10.2. For each compact set J ⊂ I the family of maps ω → Aω it , where t ranges over J, is equicontinuous at the points ω = i τ ω0 , 0 ≤ i < m0 with respect to the operator norm and there exists a constant B = B(J) ≥ 1, which may depend on J, such that P -a.s., kAω,n it k ≤ B

(2.10.1)

for any n ∈ N and t ∈ J. Note that the equicontinuity requirement in Assumption 2.10.2 makes sense when Assumption 2.10.1 is satisfied. Consider the deterministic family of 0 ,m0 operators Ait , t ∈ I given by Ait = Aω : Bω0 → Bω0 , and note that it 0 ,sm0 Asit = Ait ◦ Ait ◦ · · · ◦ Ait = Aω it

for any s ∈ N and t ∈ I. Assumption 2.10.3. For any compact J ⊂ I which does not contain the origin there exist constants c = c(J) > 0 and b = b(J) ∈ (0, 1) such that kAsit k ≤ cbs for any s ∈ N and t ∈ J.

page 126

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

127

This assumption holds true when the map t → Ait is continuous and the spectral radius of all Ait ’s is strictly less than 1 except for the spectral radius of A0 . Lemma 2.10.4. Under Assumptions 2.10.1, 2.10.2 and 2.10.3, for each compact subset J of I which does not contain the origin there exist a constant d = dJ > 0 and a random variable rω such that P -a.s., −nd kAω,n it k ≤ rω 2

for any n ∈ N and t ∈ J. Proof. Fix some compact subset J of I which does not contain the origin. 1 where c = For P -a.a. ω, let n0 be sufficiently large so that cbn0 < 4B c(J), b = b(J) and B = B(J). Relying on Assumptions 2.10.1 and 2.10.2, there exists an open set Un0 = Un0 (J) ⊂ Ω containing ω0 such that for any ω ∈ Un0 we have Bτ i ω = Bτ i ω0 , 0 ≤ i < m0 and 1 ω0 ,n0 m0 0 m0 0 m0 kAω,n − Anit0 k = kAω,n − Ait k< (2.10.2) it it 4B for any t ∈ J. Set p0 = P (Un0 ) > 0. Applying the mean ergodic theorem with the map τ and the indicator function of Un0 we see that P -a.s. ∞ there exists an infinite strictly increasing sequence (nk )∞ k=1 = (nk (ω))k=1 of natural numbers such that for any m ≥ 1, τ m ω ∈ Un0 if and only if m ∈ {nk : k ≥ 1} and lim

k→∞

For each k ∈ N set mk = nkm0 n0

1 nk = > 0. k p0 and

kn = kn (ω) = max{k ∈ N : mk ≤ n}. Then p0 kn = , P -a.s.. n m 0 n0 Finally, P -a.s. for each n and t ∈ J we can write

(2.10.3)

lim

n→∞

m1

τ Aω,n it = B1 ◦ Ait

ω,n0 m0

m2

◦ B2 ◦ Aτit

ω,n0 m0

mk n

◦ · · · ◦ Bl ◦ Aτit

ω,n0 m0

◦ Bkn +1

where kBi k ≤ B = B(J) for each i. Since τ mi ω ∈ Un0 for each i, we obtain from (2.10.2) that 1 1 1 τ mi ω,n0 m0 ≤ bcn0 + ≤ kAit k ≤ kAnit0 k + 4B 4B 2B for each i, and the lemma follows, taking into account (2.10.3).

page 127

March 13, 2018 15:46

128

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Note that when, in addition, kAω t k ≤ 1 for any t ∈ I, then bounds of the form 2

kAst k ≤ ce−dt

for some d > 0 and t’s sufficiently close to 0 will yield in a similar way estimates of the form 2

−d1 t kAω,n , d1 > 0 t k ≤ c(ω)e

while in general bounding the random norms kAω,n t k for small t’s requires a different approach such as the random complex RPF theorem in Part 2, and we refer again to Chapter 7 for details. When, in addition to the above assumptions, the measure preserving system (Ω, B, P, τ ) satisfies some decay of correlations then using similar arguments to the ones from Sections 2.8 and 2.9 we obtain appropriate R estimates on the norms of iterates of the operators Ait = Aω dP (ω) which it will yield a local central limit theorem for a Markov chain whose transition operator is A0 . 2.11

Extensions to dynamical systems

In this section we will explain how to prove a local limit theorem for sums of the form SN (x) = SN F (x) =

N X

F (T x, T 2 x, ..., T ` x)

(2.11.1)

n=1

in the case when T is either a topologically mixing two (one) sided subshift of finite type or a C 2 Axiom A diffeomorphism (in particular, Anosov) in a neighborhood of an attractor (see [10]), x is distributed according to a Gibbs measure and F is a H¨older continuous function with additional regularity around the elements of the orbit of one periodic point, where we recall that periodic points are dense in both cases. 2.11.1

Subshifts of finite type

Let A be a 0 − 1 matrix of size n × n and set A = {1, 2, ..., n}. We assume that there exists M ∈ N such that AM has only positive entries. Set ΣA = {(xi )i∈Z : Axi ,xi+1 = 1, ∀i ∈ Z} ⊂ AZ , N Σ+ and A = {(xi )i≥0 : Axi ,xi+1 = 1, ∀i ≥ 0} ⊂ A N Σ− A = {(xi )i≤0 : Axi ,xi−1 = 1, ∀i ≥ 0} ⊂ A .

page 128

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

129

− Z N Then ΣA , Σ+ A and ΣA are compact subsets of the product spaces A , A N and A , respectively. We equip ΣA with the metric

ρ(x, y) = 2−n(x,y) where n(x, y) = min{n ≥ 0 : xn 6= yn or x−n 6= y−n } where we use the conventions min ∅ = ∞ and 2−∞ = 0. Similarly, we equip − Σ+ A and ΣA with the metrics ρ+ and ρ− which are defined similarly but with n+ (x, y) = min{n ≥ 0 : xn 6= yn } and n− (x, y) = min{n ≥ 0 : x−n 6= y−n }. in place of n(x, y). Then the corresponding Borel σ-algebras B, B+ and B− are generated by appropriate cylinder sets. Next, consider the metric ρ`,∞ on (ΣA )` given by ρ`,∞ (¯ x, y¯) = max ρ(x(i) , y (i) ) 1≤i≤`

where x = (x(1) , ..., x(`) ) and y = (y (1) , ..., y (`) ). Let (ΣA , T ) be the corresponding two sided togologically mixing subshift of finite type, where T : ΣA → ΣA is the left shift given by (T x)k = xk+1 , ∀ k ∈ Z. − Denote by T+ : → the one sided left shift and by T− : Σ− A → ΣA the one sided right shift. We equip ΣA with a T -invariant Gibbs measure µ = µ(f ) (see [10]) associated with some H¨older continuous (potential) function f . Recall that µ is constructed first as a T+ -invariant measure T+ on Σ+ older continuous function f + : Σ+ A associated with some H¨ A → R (which, when lifted to ΣA , differs from f by a coboundary term) and then extended to ΣA . Using the reflection map R given by (xi )i≥0 → (xi )i≤0 , the measure µ+ defines an appropriate T− -invariant measure µ− on Σ− A which is the Gibbs measure corresponding to the function f − = f + ◦ R−1 . Note that the maps T+ and T− are locally distance expanding in the sense of Section 2.5.1 with the metrics ρ+ and ρ− , respectively, while the invertible map T is not locally distance expanding.

Σ+ A

2.11.2

Σ+ A

The strategy of the proof

The proof of the LLT in the conventional situation (` = 1) when F depends only on the coordinates with nonnegative indexes is a direct consequence of the relation (2.5.12), where the Markov chain {ζn : n ≥ 0} from there

page 129

March 13, 2018 15:46

ws-book9x6

130

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

is the one defined in Section 2.5 using the map T+ and the function f + in place of T and f . When F depends also on other coordinates then (see Lemma 1.6 from [10]) it differs by a coboundary term from a function G+ which depends only on coordinates with nonegative indexes. This makes it possible to use the above Markov chain and obtain the desired LLT, taking into account that the measurable map p+ : ΣA → Σ+ A given by p+ (x) = (xi )i≥0 is a factor map between the measure preserving systems (ΣA , B, µ, T ) and (Σ+ A , B+ , µ+ , T+ ), namely p+ is bijective, p+ ◦T = T+ ◦p+ −1 and µ ◦ p+ = µ+ . In the nonconventional situation (` > 1) the distribution of SN F (x) d

when x ∼ µ is no longer invariant under the reflection map (T x, T 2 x, ..., T N ` x) → (T N ` x, T N `−1 x, ..., T x) and instead of (2.5.12) the relations (2.5.14) and (2.5.15) from the end of Section 2.5.4 hold true. In view of (2.5.15), in order to obtain the LLT when F depends only on the coordinates with nonpositive indexes we should consider the process Z generated by the map T −1 and the Markov chain ζ − = {ζn− : n ≥ 0} defined in Section 2.5 using the map T− and the function f − . Indeed, the measurable map p− : ΣA → Σ− A given by p(x) = (xi )i≤0 , where x = (xi )i∈Z is a factor map between the measure preserving systems (ΣA , B, µ, T −1 ) and (Σ− A , B− , µ− , T− ), and thus we derive from (2.5.15) that for any N ∈ N, N X n=1

d

F (T n x, T 2n x, ..., T `n x) =

N X

− − F − (ζn− , ζ2n , ..., ζ`n )

n=1

when F depends only on the coordinates with nonpositive indexes, where d the function F − is defined by the relation F = F − ◦ p− and x ∼ µ. When F depends also on other coordinates then an appropriate version of the above Lemma 1.6 holds true (see Lemma 2.11.2), and so the desired LLT will essentially follow from the arguments in Sections 2.8 and 2.9 applied with the Markov chain ζ − . 2.11.3

Local limit theorem: one sided case

Consider the variable x ¯ = (x(1) , ..., x(`) ) ∈ (ΣA )` . Let F : (ΣA )` → R be a function which depends only on the coordinates with nonnegative indexes (i) (i.e. on xj , i = 1, 2, ..., ` − 1, j ≥ 0). Let {ζn+ : n ≥ 0} be the associated Markov chain defined in Section 2.5 using the map T+ and the function f + .

page 130

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

131

Then similarly to (2.5.13) (or (2.5.15)) and Section 2.11.2, the sums RN (x) =

N X

F (T −n x, T −2n x, ..., T −`n x)

n=1 d

PN + + when x ∼ µ have the same distribution as the sums n=1 F (ζn+ , ζ2n , ..., ζ`n ) from Section 2.5, and therefore appropriate LLT’s for the sums RN follow directly from Theorems 2.6.3 and 2.6.5. Replacing T+ with T− and Σ+ A with Σ− A we obtain LLT’s for the sums QN (x) =

N X

F (T n x, T 2n x, ..., T `n x)

n=1

when F depends only on the coordinates with nonpositive indexes. Consequently, using the symbolic representation via Markov partitions (see Sections 3 and 4 from [10]), we derive appropriate LLT’s for sums of the form ˜ N (x) = R

N X

F (g −n x, g −2n x, ..., g −`n x)

n=1

and ˜ N (x) = Q

N X

F (g n x, g 2n x, ..., g `n x)

n=1 2

when g is a C Axiom A diffeomorphism (in particular, Anosov) in a neighborhood of an attractor, x is distributed according to an appropriate Gibbs measure and in the first case F is constant on the atoms of the partition T∞ −j (M) while in the second case it is constant on the atoms of the j=0 g T∞ partition j=0 g j (M), where M is a Markov partition with a sufficiently small diameter (see [10]). 2.11.4

Two sided case

The above LLT’s can be extended to the case when F depends only on the coordinates in places from −s to ∞ (or from −∞ to s) for some fixed s > 0 in the subshift case, and when F is constant on atoms of partitions T∞ T∞ −j (M) (or j=s g j (M)) in the Axiom A case. Extensions of these j=−s g LLT’s to the case when F is allowed to depend on all the coordinates is done similarly to the conventional case (` = 1), and for readers’ convenience in this section we provide the main details. We first consider the situation of a

page 131

March 13, 2018 15:46

132

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

two sided topologically mixing subshift of finite type and the corresponding extensions in the Axiom A maps case will follow. Before discussing an LLT, note first that all the results from [45] and [30] hold true for the sequence of random variables {Xn : n ≥ 0} where Xn = T n X0 and X0 is distributed according to µ. Namely, Theorem 2.5.11 holds true for the sums SN F and the characterization (2.5.19) for positivity of the asymptotic variance holds true. Replacing F with F` we deduce, in particular, that the limit 1 Eµ SN F` )2 σ`2 = lim N →∞ N exists and that it vanishes if and only if (2.5.19) holds. Therefore, the limit σ 2 = limN →∞ N −1 VarSN F is positive when σ`2 > 0. We distinguish here between lattice and non-lattice cases in the same way as in Section 2.5 but with the function F˜x0 ,m0 given by m0 X ˜ Fx0 ,m0 (x) = F (Tˆk x ¯0 , T k` x) (2.11.2) k=1

where Tˆ = T × T × · · · × T , x ¯0 = (x0 , ..., x0 ) ∈ Σ`−1 and x0 ∈ ΣA is a A T -periodic point of period m0 . Now we can state the main result of this section. 2

`−1

Theorem 2.11.1. Suppose that F is a H¨ older continuous function which satisfies (2.6.1) at any point v. Further assume that σ`2 > 0. Then in the non-lattice case the sequence {SN : N ≥ 1} satisfies the LLT with the centralizing constant m = F¯ , the normalizing constant σ and the Lebesgue measure while in the lattice case it satisfies the LLT with m, σ and the measure assigning mass h to each point of the lattice hZ. 2.11.5

Reduction to one sided case

As in the conventional case (` = 1), the following version of Lemma 1.6 from [10] is the key to deduce limit theorems for two sided subshifts of finite type from the corresponding limit theorems for the one sided subshifts. Lemma 2.11.2. Let F : (ΣA )` → R be a H¨ older continuous function. Then there exist H¨ older continuous functions G+ , H + : (ΣA )` → R such that G+ depends only on coordinates with nonnegative indexes and F = G+ − H + + H + ◦ T¯ (2.11.3) where T¯ = T × T 2 × · · · × T ` . Moreover, the function G+ satisfies (2.6.1) at any point v when F does. The same thing holds true with a function G−

page 132

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

133

which depends only on coordinates with nonpositive indexes, a function H − and with T¯−1 = T −1 × T −2 × · · · × T −` in place of T¯. The proof of the first part of the lemma goes essentially as the proof of Lemma 1.6 from [10] with the map T¯ in place of T , while the proof of the second part (about G+ satisfying (2.6.1)) is a direct consequence of the arguments from there. Next, consider the function G− from Lemma 2.11.2, namely the function satisfying (2.11.3) with T¯−1 in place of T¯ which depends only on the coordinates with nonpositive indexes. Consider the random variables SN G− (defined on ΣA ) given by −

SN G (x) =

N X

G− (T n x, T 2n x, ..., T `n x), N ∈ N

n=1 ` where x is drawn according to µ. Let F − : (Σ− A ) → R be the unique − − function satisfying G = F ◦ p− and consider the Markov chain {ζn− : n ≥ 0} generated as in Section 2.5 by the map T− and the H¨older continuous (potential) function f − : Σ− A → R (corresponding to the measure µ− ). Define the function G− similarly to (2.5.20) but with G− in place of F and ` − the function F` similarly to (2.5.20) but with F − in place of F and µ− in place of µ. Set N X
1 − − 2 Eµ− F`− (ζn− , ζ2n , ..., ζ`n ) N →∞ N n=1

σ`2 (F`− ) := lim

where we recall that this limit exists by Theorem 2.5.11. In order to make the reduction to the one sided case we need first the following. Lemma 2.11.3. The inequality σ`2 > 0 implies that σ`2 (F`− ) > 0. Proof. Suppose that σ`2 > 0. Then F` does not admit a coboundary representation of the form (2.5.19). Since G− and F differ by the coboundary term H − − H − ◦ T¯−1 the functions G− ` and F` differ by a coboundary term, as well and so the function G− also cannot admit a coboundary re` presentation of this form. Using the map p− defined in Section 2.11.2, we conclude that the function F`− cannot admit a coboundary representation of the form (2.5.19) with T− , and therefore Theorem 2.5.11 implies that σ`2 (F`− ) > 0. We also need the following.

page 133

March 13, 2018 15:46

ws-book9x6

134

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Lemma 2.11.4. Suppose that the non-lattice (lattice) condition holds true with some T -periodic point x0 ∈ ΣA . Then the function F − satisfies the non-lattice (lattice) condition from Section 2.5 with the map T− and the (T− -periodic) point p− (x0 ). Proof. Let x0 be a periodic point of period m0 as in the statement of the lemma. First, since G− and F differ only by the coboundary term H − − H − ◦ T¯−1 then G− satisfies the non-lattice (lattice) condition with the point x0 , as well. Secondly, any solution of (2.6.3) (or (2.6.5)) with the function Fp−− (x0 ),m0 and the map T− extends (after pulling it back with the factor map p− and applying the map T −m0 ` ) to a solution of ˜− (2.6.3) (or (2.6.5)) with the function G x0 ,m0 defined similarly to (2.11.2) − with G in place of F , where we also used that T preserves µ and that (since T− is ergodic) any solution g of the equation (2.6.3) (or (2.6.5)) must have a constant nonzero absolute value, implying that 1/g is H¨older continuous. We show now that Assumptions 2.2.1 and 2.2.2 hold true in both lattice and non-lattice cases considered above. First observe that N X F (T n x, T 2n x, ..., T `n x) = SN G(x) SN (x) := SN F (x) = n=1

+H(x, ..., x) − H ◦ T¯N (x, ..., x), where we abbreviate G = G− and H = H − . For any r ≥ 0, let Hr be a function which depends only on the coordinates with indexes from −∞ to r such that kHr kκ,1 ≤ kHkκ,1 and kH − Hr k∞ ≤ c2−κr where κ is the H¨ older exponent of H and c > 0 is a constant which depends only on H. For instance, we can consider the function Z
−1 Hr (¯ x) = µ` (Er (¯ x)) H(¯ y )dµ` (¯ y) Er (¯ x)

where with x ¯ = (x Er (¯ x) =

` \

(1)

, ..., x (j)

{¯ y : yi

(`)

) and y¯ = (y

(j)

= xi

(1)

, ..., y (`) ),

for all |i| ≤ r} = {¯ y : ρ`,∞ (¯ x, y¯) ≤ 2−r }

j=1

which, in fact, depends only on the coordinates with indexes from −r to r. It follows that
SN (x) − SN G(x) − Hr ◦ T¯−1 (x, x, ..., x) − Hr ◦ T¯N −1 (x, ..., x) ≤ 2c2−κr (2.11.4)

page 134

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Local limit theorem

LNcov

135

for any x ¯. Set ur = Hr ◦ (T −r × · · · × T −r ). Then ur depends only on the coordinates in places from −∞ to 0 and for any x ¯, SN (x) − SN G(x) − ur (T r x, T r x, ..., T r x) (2.11.5)
N +r 2N +r `N +r −κr −ur (T x, T x, ..., T x) ≤ 2c2 . Since µ is T −1 invariant we can replace x with T −`N −r x without changing the distributions. Recall that ζ − is the Markov chain whose transition probabilities are defined by the transfer operator generated by the map T− , and the H¨ older continuous function f − : Σ− A → R. This together with (2.5.11) and the discussion in Section 2.11.2 yields that SN G(x) + ur (T r x, T r x, ..., T r x) −ur (T N +r x, T 2N +r x, ..., T `N +r x) d

=

N X

F − (T−N `+r−n x− , T−N `+r−2n x− , ..., T−N `+r−n` x− )

n=1

+ur ◦ d

=

N X

p− (T−N ` x− , ..., T−N ` x− )

(`−1)N −

− ur ◦ p− (T−

(`−2)N −

x , T−

x , ..., x− )

− − − − F − (ζn− , ζ2n , ..., ζ`n ) + ur ◦ p− (ζr− , ..., ζr− ) − ur ◦ p− (ζN +r , ...ζ`N +r )

n=1 d

d

when x ∼ µ and x− ∼ µ− . Next, let r < N4 . Repeating the arguments from Section 2.5 we obtain the upper bound  PN − − − − (2.11.6) |EeitSN | ≤ E E eit k=M +1 F (ζk ,ζ2k ,...,ζ`k )  − − − − − . ×ur ◦ p− (ζ ,ζ , ..., ζ )|ζ , ..., ζ N +r

2N +r

`N +r

1

`M

Now, using the Markov property and the identity − − − − − − E[h1 (ζk− , ..., ζk+l )h2 (ζk+l+s )|ζk−1 ] = E[h1 (ζk− , ..., ζk+l )Lsf − h2 (ζk+l )|ζk−1 ]

which holds true for any k ∈ Z, l, s > 0 and bounded measurable functions h1 and h2 , we derive that the expression inside the absolute value from the right-hand side of (2.11.6) equals N  Y

L`f − +itF − (ζ − ,ζ − ,...,ζ − k

k=M +1

2k

(`−1)k

 ,·)

vr,N

where the random function vr,N is given by − − itur ◦p− (ζN +r ,...,ζ(`−1)N +r ,·)

vr,N (x) = L`r f−e

(x).

page 135

March 13, 2018 15:46

136

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Next, the map T −r can, at most, expand distance by a factor of 2r . Therefore, ur ◦ p− is H¨ older continuous with exponent κ and a constant vκ,1 (ur ) which does not exceed 2κr vκ,1 (Hr ). Since Σ− A is compact the norms k · kκ,ξ and k·kκ,1 are equivalent and since kLsf − kκ,ξ is bounded in s then kLsf − kκ,1 is bounded in s, as√well. Let r = rN = C ln N for an appropriate constant C, so that 2−κrN N converges to 0 as N → ∞. Then 2rN grows polynomially fast in N , and so the norms kurN kα,1 and the random norms kvrN ,N (·)kκ,1 grow polynomially fast in N , as well. Moreover, for this choice of rN the estimate (2.11.5), which comes from replacing H with HrN , does not affect the validity of Assumptions 2.2.1 and 2.2.2. Furthermore, when iturN (¯ y ,·) x is fixed then the function y¯ → L`r (x) is H¨older continuous f−e with exponent κ and a constant which does not exceed vκ,1 (urN ), where we used that Lf − 1 = 1. Relying on this together with the above estimates of kvrN ,N (·)kκ,1 , the proof of a corresponding version of (2.7.8) proceeds similarly since in the situation of a topologically mixing subshift of finite type the approximation and mixing coefficients defined in Section 2.5.5 decay exponentially fast to 0 (see [10]). Using the above estimates, similar arguments to the ones in Sections 2.8 and 2.9, but with the random function + F+ ω = F (p0 (ω), ·) in place of F, show that Assumptions 2.2.1 and 2.2.2 hold true in our situation, and a nonconventional LLT theorem for the sums SN = SN F follows. Finally, employing symbolic representations via Markov partitions (see Sections 3 and 4 from [10]) we derive this limit theorem for the sums PN n 2n x, ..., T `n x) in the Axiom A case. n=1 F (T x, T

page 136

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Chapter 3

Limit theorems for nonconventional arrays

We extend the nonconventional law of large numbers, the central limit theorem and Poisson limit theorems from [42], [45]
and [43] to sums of PN the form n=1 F ξp1 n+q1 N , ξp2 n+q2 N , . . . , ξp` n+q` N where {ξn , n ≥ 0} is a sufficiently fast mixing stochastic process with some moment conditions and stationarity properties while F is a continuous function with polynomial growth and certain regularity properties. We discuss also the crucial question on positivity of the variance in this central limit theorem. 3.1

Introduction

In this chapter we study limit theorems for nonconventional sums of the form X SN = F (ξq1 (n,N ) , ..., ξq` (n,N ) ) (3.1.1) N ≥n≥1

where ξm , m = 0, 1, ... is a sequence of random variables with sufficiently weak dependence, qi (n, N ), i = 1, ..., ` are functions taking on integer values on integers and F is a Borel measurable function which depending on conditions imposed on ξm ’s will also be assumed to be sufficiently regular, for instance, H¨ older continuous. Since the summands in (3.1.1) depend on the number of summands N it is natural according to the probabilistic terminology to call such expressions by the name (triangular) arrays. To avoid extensive technicalities we will restrict ourselves to the case of linear functions qi (n, N ) = pi n + qi N, i = 1, ..., ` which includes, in particular, the symmetric case ` = 2k, qi (n, N ) = (k − i + 1)(N − n), i = 1, ..., k and qi (n, N ) = in, i = k + 1, ..., 2k. Thus, we will be dealing here with expressions of the form X SN = F (ξp1 n+q1 N , ξp2 n+q2 N , ..., ξp` n+q` N ) (3.1.2) N ≥n≥1

137

page 137

March 13, 2018 15:46

138

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where pi , qi , i = 1, ..., ` are integers. In Section 3.2 we will show that under appropriate weak dependence conditions on the sequence ξm , m = 0, 1, ... the strong law of large numbers for normalized sums (3.1.2) holds true, i.e. with probability one Z 1 SN = F (x1 , x2 , ..., x` )dµ(x1 )dµ(x2 ) · · · dµ(x` ) (3.1.3) lim N →∞ N where µ is the distribution of ξ(m) which is supposed to be the same for all m. Unlike in the nonconventional sums case considered in [42] and [45], where qj = 0 for all j, in the nonconventional arrays (3.1.2) some pi n + qi N and pj n + qj N , i 6= j can be close to each other (and even coincide) for large n and N , and so the corresponding entries ξpi n+qi N and ξpj n+qj N can be strongly dependent which complicates the use of the weak dependence condition imposed on the sequence ξm , m ≥ 0. When the stochastic sequence ξm , m = 0, 1, 2, ... is generated by a dynamical system, i.e. ξm = T m g, for some measure preserving transformation T and a function g, then for F (x1 , ..., x` ) = f1 (x1 ) · · · f` (x` ) the L2 convergence in (3.1.3) can be proved under weak mixing condition on T (see [48]) but the almost sure convergence obtained here is new for such limits. In Section 3.3 we will study the central limit theorem type results for expressions of the form (3.1.2). It is clear that if N −1/2 (SN − ESN ) converges in distribution as N → ∞ to the normal distribution then the normalized variance N −1 VarSN must converge as N → ∞. It turns out that the latter convergence cannot be ensured for general coefficients pi , qi , i = 1, ..., `, as above, but we will prove this here under certain assumption (see Assumption 3.3.1) which holds true, for instance, in the symmetric case X SN = F (ξk(N −n) , ξ(k−1)(N −n) , ..., ξN −n , ξn , ξ2n , ..., ξkn ). (3.1.4) N ≥n≥1

In fact, the martingale approximation technique will work here for any coefficients pi , qi assuming appropriate weak dependence in the stochastic sequence ξm , m = 0, 1, ..., and so the central limit theorem can be proved here similarly to [45] whenever convergence of variances holds true. We will give also conditions which ensure positivity of the limiting variance since without knowing this central limit theorem is not very meaningful. The conditions imposed in Sections 3.2 and 3.3 are valid for sufficiently fast mixing stochastic processes, in particular, Markov chains satisfying some form of the Doeblin condition while on the dynamical systems side our results can be applied to topologically mixing subshifts of finite type taken with a Gibbs measure constructed by a H¨older continuous function,

page 138

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional arrays

139

and so we can deal also with expanding transformations and Axiom A diffeomorphisms. In Section 3.4 we will study Poisson type limit theorems for nonconventional arrays. The classical Poisson limit theorem can be expressed in the following way. Let {ξn , n ≥ 0} be independent identically distributed (i.i.d.) random variables having a distribution µ and ΓN , N ≥ 1 be a sequence of measurable sets of real numbers such that limN →∞ N µ(ΓN ) = λ > 0. Then PN the sum SN = n=1 IΓN (ξn ), where IΓ is the indicator of a set Γ, converges in distribution to a Poisson random variable with the parameter λ. An extension of such type of limit theorems to the nonconventional setup was obtained in [43] and [47]. We will extend here some of the results from [43] and [47] to the arrays case. Namely, we will consider nonconventional arrays of the form N Y ` X SN = IΓN (ξpi n+qi N ) (3.1.5) n=1 j=1

where ξm , m = 0, 1, ... is a sequence of random variables, ΓN is a sequence of shrinking with N sets and IΓ is the indicator of a set Γ. We will show that if ξm , m = 0, 1, ... is a ψ-mixing stationary sequence then under certain conditions on pi ’s the sums SN weakly converge to a Poisson random variable with a parameter λ provided lim N (µ(ΓN ))` = λ.

N →∞

We will also obtain more involved results which concern arrivals to shrinking cylinder sets by subshifts. Namely, we will obtain Poisson type limit theorems for expressions of the form SNm =

Nm Y ` X

IBm ◦ T pi n+qi N

(3.1.6)

n=1 j=1

where Bm is a cylinder set of length m built by a fixed nonperiodic sequence from a space of sequences Ω, T is the left shift, P is a T -invariant ψ-mixing measure and the sequence {Nm : m ≥ 1} satisfies lim Nm (P (Bm ))` = λ.

m→∞

3.2 3.2.1

(3.1.7)

Strong law of large numbers for nonconventional arrays Setup and the main result

Our setup consists of a ℘-dimensional stochastic process {ξn , n = 0, 1, ...} on a probability space (Ω, F, P ) and of a family of σ-algebras

page 139

March 13, 2018 15:46

ws-book9x6

140

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Fkl ⊂ F, −∞ ≤ k ≤ l ≤ ∞ such that Fkl ⊂ Fk0 l0 if k 0 ≤ k and l0 ≥ l. It is often convenient to measure the dependence between two sub-σ-algebras G, H ⊂ F via the quantities   $q,p (G, H) = sup{kE g|G − E[g]kp : g is H-measurable and kgkq ≤ 1}, (3.2.1) where the supremum is taken over real functions and k · kr , r ∈ [1, ∞) is the Lr (Ω, F, P )-norm. Then more familiar α, ρ, φ and ψ-mixing (dependence) coefficients can be expressed via the formulas (see [11], Ch. 4), α(G, H) = 14 $∞,1 (G, H), ρ(G, H) = $2,2 (G, H) φ(G, H) = 12 $∞,∞ (G, H) and ψ(G, H) = $1,∞ (G, H). We set also $q,p (n) = sup $q,p (F−∞,k , Fk+n,∞ )

(3.2.2)

k≥0

and accordingly 1 1 $∞,1 (n), ρ(n) = $2,2 (n), φ(n) = $∞,∞ (n), ψ(n) = $1,∞ (n). 4 2 Our setup also includes conditions on the approximation rate α(n) =

β(p, r) = sup sup kξk − E[ξk |Fk−r0 ,k+r0 ]kp .

(3.2.3)

r 0 ≥r k≥0

If p = 2 the exterior supremum here is not needed but for general p it is taken to obtain β(p, r) nonincreasing in r. In what follows we can always extend the definitions of Fkl given only for k, l ≥ 0 to negative k by defining Fkl = F0l for k < 0 and l ≥ 0. Furthermore, we do not require stationarity of the process {ξn , n ≥ 0} assuming only that the distribution of ξn does not depend on n and the joint distribution of {ξn , ξn0 } depends only on n − n0 which we write for further references by d

d

ξn ∼ µ and (ξn , ξn0 ) ∼ µn−n0 for all n, n0

(3.2.4)

d

where Y ∼ µ means that Y has µ for its distribution. Next, let F = F (x1 , ..., x` ), xj ∈ R℘ be a function on R℘` such that for some ι, K > 0, κ ∈ (0, 1] and all xi , yi ∈ R℘ , i = 1, ..., `, we have |F (x1 , ..., x` ) − F (y1 , ..., y` )| ≤ K 1 +

` X j=1

|xj |ι +

` X j=1

|yj |ι

`
X

|xj − yj |κ

j=1

(3.2.5)

page 140

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Nonconventional arrays

LNcov

141

and |F (x1 , ..., x` )| ≤ K 1 +

` X


|xj |ι .

(3.2.6)

j=1

It will be clear from the proof that if ξn is Fn−k,n+k -measurable for a fixed k and all n then the regularity conditions (3.2.5) and (3.2.6) are not needed and it suffices to assume that F is a Borel measurable and bounded. To simplify formulas we assume a centering condition Z F¯ = F (x1 , ..., x` ) dµ(x1 ) · · · dµ(x` ) = 0 (3.2.7) which is not really a restriction since we can always replace F by F − F¯ . Our main result relies on Assumption 3.2.1. With d = (` − 1)℘ there exist ∞ > p, q ≥ 1 and δ, m > 0 with δ < κ − dp satisfying ∞ X


$q,p (n) + (β(q, r))δ < ∞,

(3.2.8)

n=0

γm + γ2q(ι+2) < ∞ and where γθθ = kξn kθθ = E|ξn |θ =

R

3 ι+2 δ 1 ≥ + + 4 p m q

(3.2.9)

|x|θ dµ.

Our strong law of large numbers for nonconventional arrays is represented by the following result. Theorem 3.2.2. Suppose that (3.2.7) and Assumption 3.2.1 hold true, the integers p1 , p2 , ..., p` in the sum SN defined by (3.1.2) are nonzero, distinct and pi n + qi N ≥ 0 for all i, n, N ≥ 1. Then with probability one, lim

N →∞

3.2.2

1 SN = 0. N

(3.2.10)

Auxiliary lemmas

We will rely on the following general result which appears in [45] as Corollary 3.6. Lemma 3.2.3. Let G and H1 ⊂ H2 be σ-subalgebras on a probability space (Ω, F, P ), X and Y be d-dimensional random vectors and fi = fi (x, ω), i = 1, 2 be collections of random variables that are continuously (or separable)

page 141

March 13, 2018 15:46

142

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

dependent on x ∈ Rd for almost all ω, measurable with respect to Hi , i = 1, 2, respectively, and satisfy kfi (x, ω) − fi (y, ω)kq ≤ C1 (1 + |x|ι + |y|ι )|x − y|κ (3.2.11) and kfi (x, ω)kq ≤ C2 (1 + |x|ι ). Set f˜i (x, ω) = E[fi (x, ·)|G](ω) and gi (x) = E[fi (x, ω)]. (i) Assume that q ≥ p, 1 ≥ κ > θ > dp and a1 ≥ p1 + ι+1 m . Then for i = 1, 2, d 1− d kf˜i (X(ω), ω) − gi (X)ka ≤ c $q,p (G, Hi )(C1 + C2 ) pθ C pθ (1 + kXkι+1 ), 2

m

(3.2.12) where c = c(ι, κ, θ, p, q, a, δ, d) > 0 depends only on the parameters in brackets. δ (ii) Next, assume that a1 ≥ p1 + ι+2 m + q . Then for i = 1, 2, δ kE[fi (X, ·)|G] − gi (X)ka ≤ R + 2c(C1 + C2 )(1 + 2kXkι+2 m )kX − E[X|G]kq (3.2.13) where R denotes the right-hand side of (3.2.12). (iii) Furthermore, let x = (v, z) and X = (V, Z), where V and Z are d1 and d − d1 -dimensional random vectors, respectively, and let fi (x, ω) = fi (v, z, ω) satisfy (3.2.11) in x = (v, z). Set g˜i (v) = Efi (v, Z(ω), ω). Then for i = 1, 2, 0 (3.2.14) kE[fi [V, Z, ·]|G] − g˜i (V )ka ≤ c (1 + kXkιm ) d1 d1
1− pθ × $q,p (G, Hi )(C1 + C2 ) pθ C2 + kV − E[V |G]kδq + kZ − E[Z|Hi ]kδq . (iv) Finally, for a, p, q, ι, m, δ satisfying conditions of (ii), kf˜1 (X(ω), ω) − f˜2 (Y (ω), ω) − g1 (X) + g2 (Y )ka (3.2.15)
ι+2 ι+2 δ ≤ c $q,p (G, H2 ) 1 + kXkm + kY km kX − Y kq where c = c(ι, κ, θ, p, q, a, δ, d) > 0 depends only on the parameters in brackets.

We will also need the following lemmas from Section 6 of the unpublished version of [45] (see arXiv:1012.2223v2). Lemma 3.2.4. Suppose that {an : n ≥ 1} is a sequence of nonnegative numbers such that for some integer l ≥ 1 and any integer n ≥ 1, n X 2l X 2l−r an+1 ≤ c C r aj 2l . j=1 r=2

Then an ≤ A nl with A = max{2l cl C 2l , C 2l , a1 }.

page 142

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional arrays

143

Proof. We derive the above inequality by induction. It is clearly valid for n = 1. Assume it is valid for j = 1, 2, . . . , n. Then Pn P2l 2l−r an+1 ≤ c j=1 r=2 C r (Aj l ) 2l l r−2 Pn 1 P2l l−1 ≤ c C 2 A1− l r=2 C r−2 A− 2l ≤ A0 (n+1) j=1 j l where 1

A0 = c C 2 A1− l

2l−2 X

r

C r A− 2l

r=0 0

≤ A. In particular, A = and we need to pick A so that Al 1 1 l l 2l 2l − 2l max{2 c C , C , a1 } will do because CA ≤ 1, 2 c C 2 A− l ≤ 1 and 1

c C 2 A1− l

2l−2 X

r

1

1

C r A− 2l ≤ c C 2 A1− l (2l − 1) ≤ c C 2 A1− l 2l ≤ l A.

r=0

Now let (Ω, F, P ) be a probability space with a filtration of σ-algebras Gj , j ≥ 1. Suppose that random variables Xj , j ≥ 1 are Gj measurable and for each 1 ≤ p < ∞ set X kE(Xj |Gi )kp ≥ sup kXj kp := γp . (3.2.16) Ap = sup i

j≥i

j

Lemma 3.2.5. Let the sequence {Xj : i ≥ j} of random variables be as Pn above and Sn = j=1 Xj . Then for each l ≥ 1 there is a constant cl depending only on l such that l ESn2l ≤ cl A2l 2l n .

Proof. Without loss of generality we assume that A2l < ∞ since otherwise 2l the above inequality is trivial. We begin by expanding Sj+1 = (Sj +Xj+1 )2l by the binomial theorem, 2l   X 2l 2l−r r 2l−1 2l 2l Sj+1 = Sj + 2lSj Xj+1 + Sj Xj+1 r r=2 and expressing Sj2l−1 =

j 2l−2 X X X 2l−1 2l−2−r (Si2l−1 − Si−1 )= Xi Sir Si−1 i=1

1≤i≤j

r=0

page 143

March 13, 2018 15:46

ws-book9x6

144

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where S0 = 0. This enables us to rewrite 2l Sj+1

=

Sj2l

+ 2l

X

Zi Xj+1 +

r=2

1≤i≤j

P2l−2

r 2l−2−r . r=0 Si Si−1

where Zi = Xi

2l ESn+1

=

EX12l

+ 2l

X 1≤i≤j≤n

X

= 2l

r

r Sj2l−r Xj+1

Then,

n X 2l   X 2l r E(Zi Xj+1 ) + E(Sj2l−r Xj+1 ) r j=1 r=2

E(Zi Wi ) +

1≤i≤n

2l   X 2l

n X 2l   X 2l r E(Sj2l−r Xj+1 ) r j=1 r=2

Pn

where Wi = j=i E(Xj+1 |Fi ). We note that kXi k2l ≤ γ2l ≤ A2l and 2l−2−r 2l−2 kWi k2l ≤ A2l . Taking into account that |Sir Si−1 | ≤ max(Si2l−2 , Si−1 ) together with the Cauchy–Schwarz inequality we obtain

E|Zi Wi | ≤ k

2l−2 X

2l−2−r Sir Si−1 k

l l−1

kXi k2l kWi k2l

r=0

≤ (2l − 1)A22l (ESi2l )

l−1 l

2l + (ESi−1 )

l−1 l


.

Next, for r ≥ 2, r |E(Sj2l−r Xj+1 )| ≤ kSj k2l−r kXj+1 kr2l ≤ Ar2l kSj k2l−r . 2l 2l

It follows that 2l E[Sn+1 ] ≤ (2l)!

2l n X X j=1

≤ cl

n X 2l X


Ar2l kSj k2l−r + A22l Sj k2l−2 + A22l kSj−1 k2l−2 2l 2l 2l

r=2

Ar2l kSj k2l−r 2l

j=1 r=2

where cl = 3(2l)!. The sequence an = E[Sn2l ] satisfies the condition of 2l Lemma 3.2.4 with c = cl , C = A2l and a1 ≤ γ2l and the result follows. 3.2.3

Ordering and decompositions

It will be important for our method to order the numbers pi n + qi N, i = 1, ..., ` but, in general, the same ordering is not possible for all n between 1 and N and we will have to consider several stretches of n’s with different ordering.

page 144

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Nonconventional arrays

LNcov

145

Without loss of generality we assume that p1 < p2 < ... < p` and since pi n + qi N ≥ 0 always then qi ≥ 0 for i = 1, ..., `. Let NN be the set of n ∈ {1, 2, ..., N } such that all pi n + qi N, i = 1, ..., ` are distinct. For each n ∈ NN we define distinct integers εi (n, N ), i = 1, 2, ..., ` such that pεi (n,N ) n + qεi (n,N ) N < pεi+1 (n,N ) n + qεi+1 (n,N ) N for all i = 1, 2, ..., ` − 1. (3.2.17) Let E` be the set of all permutations of (1, 2, ..., `). For each ε = (ε1 , ..., ε` ) ∈ E` set Nε,N = {n ∈ {1, 2, ..., N } : εj (n, N ) = εj for each j = 1, ..., `}. Some of the sets Nε,N can be empty and for each n ∈ Nε,N , pεi n + qεi N < pεi+1 n + qεi+1 N, i.e. n > (qεi − qεi+1 )(pεi+1 − pεi )−1 N if εi+1 > εi and n < (qεi − qεi+1 )(pεi+1 − pεi )−1 N if εi+1 < εi . Hence, Nε,N = {n : aε N < n < bε N } for some (not unique) aε ≥ 0 and bε ≤ 1, Nε,N are disjoint for different ε ∈ E` and, clearly, NN = ∪ε∈E` Nε,N . (3.2.18) There is always ε = (ε1 , ..., ε` ) with aε = 0 and then ε1 = min{i : qi = ˆN = {1, 2, ..., N } \ min1≤j≤` qj } and ε` = max{i : qi = max1≤j≤` qj }. Set N NN . Then ˆN ⊂ {(qi − qj )(pj − pi )−1 N : for some i, j = 1, ..., `, i 6= j}, N ˆN does not exceed `2 . and so the cardinality of N Lemma 3.2.6. If aε N < n < bε N then (pεi+1 − pεi )n + (qεi+1 − qεi )N ≥ min(n − aε N − 1, bε N − n − 1). (3.2.19) Proof. Let integers m1 , m2 ≥ 1 satisfy aε N < n − m1 and n + m2 < bε N . Then pεi (n − m1 ) + qεi N < pεi+1 (n − m1 ) + qεi+1 N and pεi (n + m2 ) + qεi N < pεi+1 (n + m2 ) + qεi+1 N. Hence, (pεi+1 − pεi )n + (qεi+1 − qεi )N > m1 (pεi+1 − pεi ) ≥ m1 if εi+1 > εi and if εi > εi+1 then (pεi+1 − pεi )n + (qεi+1 − qεi )N > m2 (pεi − pεi+1 ) ≥ m2 , and so the assertion follows.

page 145

March 13, 2018 15:46

ws-book9x6

146

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Next, for each ε = (ε1 , ..., ε` ) ∈ E` we define Z F`,ε (xε1 , ..., xε` ) = F (x1 , ..., x` ) − F (x1 , ..., x` )dµ(xε` ) and for all j = ` − 1, ` − 2, ..., 1, R Fj,ε (xε1 , ..., xεj ) = F (x1 , ..., x` )dµ(xε` )dµ(xε`−1 ) · · · dµ(xεj+1 ) R − F (x1 , ..., x` )dµ(xε` ) · · · dµ(xεj ). Observe that EFj,ε (xε1 , ..., xεj−1 , ξn ) = 0. For j = 1, ..., ` set X
Fj,ε ξpε1 n+qε1 N , ξpε2 n+qε2 N , ..., ξpεj n+qεj N . Sj,ε (N ) = n∈Nε,N

Then SN =

X

F (ξp1 n+q1 N , ..., ξp` n+q` N ) =

N ≥n≥1

` XX

Sj,ε (N ) + SˆN

(3.2.20)

ε∈E` j=1

where SˆN =

X

F (ξp1 n+q1 N , ..., ξp` n+q` N ).

ˆN n∈N

3.2.4

Proof of Theorem 3.2.2

For any k, n, m, N ∈ N, r ≥ 0 and 1 ≤ j ≤ ` set ξk,r = E[ξk |Fk−r,k+r ], ρεj (n, N ) = pεj n + qεj N , Yj,ε,ρεj (n,N ) = Fj,ε (ξρε1 (n,N ) , ..., ξρεj (n,N ) ) and Yj,ε,m = 0 if m 6= ρεj (n, N ), Fj,ε,n,r = Fj,ε,n,r (x1 , ..., xj−1 , ω) = E[Fj,ε (x1 , ..., xj−1 , ξn )|Fn−r,n+r ]
and Yj,ε,ρεj (n,N ),r = Fj,ε,ρεj (n,N ),r ξρε1 (n,N ),r , ..., ξρεj−1 (n,N ),r , ω . Recall that by the construction, EFj,ε,n,r (x1 , ..., xj−1 , ω) = 0. 0

(3.2.21)

Hence, for any r ≥ r and l + r ≤ ρεj−1 (n, N ) + r ≤ ρεj (n, N ) − r we obtain from Lemma 3.2.3(iv) with G = F−∞,ρεj−1 (n,N )−r , H1 = Fρεj (n,N )−r0 ,ρεj (n,N )+r0 , H1 = Fρεj (n,N )−r,ρεj (n,N )+r , f1 = Fj,ε,ρεj (n,N ),r0 and f2 = Fj,ε,ρεj (n,N ),r together with the contraction property of conditional expectations that   kE Yj,ε,ρεj (n,N ),r0 − Yj,ε,ρεj (n,N ),r F−∞,l+r k4 (3.2.22)   ≤ kE Yj,ε,ρεj (n,N ),r0 − Yj,ε,ρεj (n,N ),r F−∞,ρεj−1 (n,N )+r k4 ≤ C1 $q,p (ρεj (n, N ) − ρεj−1 (n, N ) − 2r)(β(q, r0 ))δ

page 146

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Nonconventional arrays

LNcov

147

where C1 > 0 depends on parameters in Assumption 3.2.1 but it is independent of n, N, r, r0 , ε and j. Similarly, if ρεj−1 (n, N )+r ≤ l+r ≤ ρεj (n, N )−r then Lemma 3.2.3(iv) yields   (3.2.23) kE Yj,ε,ρεj (n,N ),r0 − Yj,ε,ρεj (n,N ),r F−∞,l+r k4 ≤ C2 $q,p (ρεj (n, N ) − l − 2r)(β(q, r0 ))δ where, again, C2 > 0 does not depend on n, N, r, r0 , ε and j. Thus, if ρεj (n, N ) ≥ l+2r then one of estimates (3.2.22) or (3.2.23) is valid. Observe that there are at most 2r numbers n with l ≤ ρεj (n, N ) < l + 2r for which we estimate the left-hand sides in (3.2.22) and (3.2.23) just relying on the contraction property of conditional expectations bounding them by kYj,ε,ρεj (n,N ),r0 − Yj,ε,ρεj (n,N ),r k4 ≤ C3 (β(q, r0 ))δ

(3.2.24)

where we used Lemma 3.2.3(iv) with G = F (on our probability space (Ω, F, P )) and C3 > 0 does not depend on n, N, r, r0 , ε and j. Next, replacing r by 2r+1 and r0 by 2r we obtain from Assumption 3.2.1, Lemma 3.2.6 and (3.2.22)-(3.2.24) that  P (3.2.25) supl n∈Nε,N , ρε (n,N )≥l kE Yj,ε,ρεj (n,N ),2r j  −Yj,ε,ρεj (n,N ),2r+1 Fl+2r+1 k4 ≤ C4 2r (β(q, 2r ))δ where C4 > 0 does not depend on ε, j and r, N ≥ 1. Since 2r (β(q, 2r ))δ is bounded in view of Assumption 3.2.1 we can apply Lemma 3.2.5 to obtain that E(Sj,ε,2r (N ) − Sj,ε,2r+1 (N ))4 ≤ C5 N 2 24r (β(q, 2r ))4δ

(3.2.26)

where Sj,ε,m (N ) =

X

Yj,ε,ρεj (n,N ),m

n∈Nε,N

and C5 > 0 does not depend on ε, j and r, N ≥ 1. Observe that by similar estimates employing Lemma 3.2.3(i) in place of Lemma 3.2.3(iv) together with Lemma 3.2.5 we obtain that E(Sj,ε,1 (N ))4 ≤ C6 N 2

(3.2.27)

where C6 > 0 does not depend on ε, j and r, N ≥ 1. It follows from Assumption 3.2.1 that X 2r (β(q, 2r ))δ < ∞, r≥0

page 147

March 13, 2018 15:46

ws-book9x6

148

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

and so we obtain from (3.2.26) and (3.2.27) that X √ kSj,ε,2r (N ) − Sj,ε,2r+1 (N )k4 ≤ C7 N kSj,ε (N )k4 ≤ kSj,ε,1 (N )k + r≥0

(3.2.28) where C7 > 0 does not depend on ε, j and N ≥ 1. By Chebyshev’s inequality P{

1 kSj,ε (N )k44 ≤ C74 N −3/2 |Sj,ε (N )| ≥ N −1/8 } ≤ N N 7/2

(3.2.29)

which together with the Borel–Cantelli lemma yields that with probability one, lim

N →∞

1 Sj,ε (N ) = 0. N

(3.2.30)

Observe also that for any n ≤ N it follows from Assumption 3.2.1 and (3.2.26) that kFj,ε (ξpε1 n+qε1 N , ..., ξpεj n+qεj N ) k4
P` ≤ C8 1 + i=1 (E|ξpεi n+qεi N |4ι )1/4 < ∞

(3.2.31)

where C8 > 0 does not depend on ε, j and N ≥ 1. Since the sum SˆN in (3.2.20) contains no more than `2 terms we see that kSˆN k4 ≤ C9

(3.2.32)

for some C9 > 0 independent of N ≥ 1. Applying again the Chebyshev inequality and the Borel–Cantelli lemma we obtain that with probability one 1 ˆ SN = 0 N →∞ N lim

which together with (3.2.20) and (3.2.30) completes the proof of Theorem 3.2.2. Remark 3.2.7. It is possible to provide a proof of Theorem 3.2.2 relying on the strong law of large numbers for mixingale arrays from [34] verifying mixingale conditions by means of Lemma 3.2.3 (cf. [42]) though it is not clear whether this approach will yield Theorem 3.2.2 under substantially weaker conditions.

page 148

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Nonconventional arrays

3.3

LNcov

149

Central limit theorem

3.3.1

I.i.d. case

We start with considering sums SN in (3.1.2) with a sequence of independent identically distributed (i.i.d) random variables {ξn , n ≥ 0}. Clearly, if N −1/2 (SN − ESN ) converges in distribution to a normal random variable then n−1 VarSN must converge as N → ∞. It turns out that this is not the case, in general. Consider the sum SN =

N X

F (ξ2n+N , ξ2N −2n )

(3.3.1)

n=1

where ξn , n ≥ 0 are i.i.d. random variables each having a distribution µ and, say, F is a bounded measurable function on R2 . In notations of (3.1.2) this corresponds to ` = 2, p1 = −p2 = 2, q1 = 1 and q2 = 2. Observe also that in the notations of Section 3.2.3 we have here E2 consisting of two points ε(1) and ε(2) with Nε(1) ,N = {n : 0 < n < N/4} and Nε(2) ,N = {n : N/4 < n < N }. Assume as before that Z F¯ = F (x, y)dµ(x)dµ(y) = 0. (3.3.2) Then EF (ξ2n+N , ξ2N −2n ) = 0 provided n 6= N/4.

(3.3.3)

Set S˜N =

X

F (ξ2n+N , ξ2N −2n )

1≤n≤N, n6=N/4

then the limiting behavior as N → ∞ of N −1/2 SN and of N −1/2 S˜N , as well as of their variances N −1 VarSN and N −1 VarS˜N , is the same. Since ES˜N = 0 we have X 2 (3.3.4) VarS˜N = ES˜N = E F (ξ2n+N , ξ2N −2n ) 1≤n,m≤N ;n,m6=N/4


(1) (2) (3) (4) ×F (ξ2m+N , ξ2N −2m ) = UN + UN + UN + UN . Here (1)

UN :=

X 1≤n≤N,n6=N/4

EF 2 (ξ2n+N , ξ2N −2n ) = D(N )

Z

F 2 (x, y)dµ(x)dµ(y)

page 149

March 13, 2018 15:46

ws-book9x6

150

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where D(N ) = N − 1 if N is divisible by 4 and D(N ) = N if not. Next, X
(2) UN := E F (ξ2n+N , ξ2N −2n )F (ξ2m+N , ξ2N −2m ) (3.3.5) 1≤n,m 2m here for some εj ’s we deduce from (3.4.37) and Lemma 3.2.6 that r Y j=1

(Nm )

bij

=

r Y

(Nm )

Eηij

a ≤ (1 + ψ(m))r` (P (Bm ))r` .

(3.4.47)

j=1

Finally, we conclude from (3.4.4), (3.4.46) and (3.4.47) together with the counting argument that as m → ∞, P (N ) (3.4.48) b m (k,l) (i1 ,...,ir )∈∪1≤k 0 such that P -a.s.,
0 0 ω,j 0 Aω,j (4.1.6) z Cω ⊂ Cθ j ω and ∆Cθj ω Az Cω < d0 for any z ∈ U and j0 ≤ j ≤ 2j0 . Some of our results can be obtained only under the following assumption. Assumption 4.1.3. The complex cone Cω is P -a.s. reproducing of order k = kω ∈ N, namely there exists a random variable rω > 0 such that P -a.s. for any f ∈ Bω there exist f1 , ..., fk ∈ Cω0 so that f = f1 + f2 + ... + fk and kf1 k + kf2 k + ... + kfk k ≤ rω kf k.

(4.1.7)

Example 4.1.4. Let X be a topological space together with the Borel σalgebra B and a set E ⊂ Ω × X measurable with respect to the product σ-algebra F × B. Denote by Eω = {x ∈ X : (ω, x) ∈ E}, ω ∈ Ω the fibers of E and consider the situation when each Bω is a (Banach) space of functions on Eω . Let P be the σ-algebra induced on E from the product σ-algebra F × B. In this situation a family b = {bω : ω ∈ Ω} is naturally identifies with a function ¯b : E → C given by ¯b(ω, x) = bω (x) and we say that b is measurable if the function ¯b is P-measurable. Suppose now that the (evaluation) functionals given by x ˆ(bω ) = bω (x), x ∈ Eω are continuous and set [ Y= {ω} × Yω ⊂ B ∗ ω∈Ω

where Yω = {ˆ x : x ∈ Eω } for each ω. Identifying Yω with Eω we endow Y with the σ-algebra P. Under this identification the function ˆb defined in (4.1.2) coincides with the function ¯b on the set Y. We can extend P to B ∗ by adding all the subsets of B ∗ \ Y and their unions with elements of P, i.e. say that a set Γ ⊂ B ∗ is measurable if Γ ∩ Y ∈ P. Denote this extension by G. Then a family b is G-measurable in the sense described at the beginning of this section if and only if the function ¯b is P-measurable. 4.2

Main results

Our main result is the following Ruelle-Perron-Frobenius (RPF) theorem. Theorem 4.2.1. Suppose that Assumptions 4.1.1 and 4.1.2 hold true. Then P -a.s. for any z ∈ U there exists a unique triplet (λω (z), hω (z), νω (z))

page 175

March 13, 2018 15:46

176

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

consisting of a complex number λω (z) 6= 0 and random elements hω (z) ∈ Cω , νω (z) ∈ Cω∗ such that κω νω (z) = 1, ω ∗ Aω z hω (z) = λω (z)hω (z), (Az ) νθω (z) = λω (z)νω (z) and νω (z)hω (z) = 1 (4.2.1) where Cω and κω are specified in Assumption 4.1.1. Moreover, λω (z), hω (z) and νω (z) are analytic functions of z, the function ω → λω (z) is measurable and the families {hω (z) : ω ∈ Ω} and {νω (z) : ω ∈ Ω} are G-measurable.

The above uniqueness statement is restricted to elements of the cones Cω and Cω∗ . Nevertheless, when the cones Cω are reproducing and θ is ergodic then we prove that the unique triplet from Theorem 4.2.1 is the only triplet satisfying (4.2.1) such that κω νω (z) = 1 and νω hω (z) 6= 0. In particular, it is the unique triplet which satisfies these conditions under the restriction that νω (z) ∈ Cω∗ . Next, consider the random variables λω,n (z) =

n−1 Y

λθk ω (z)

k=0

where n ∈ N and z ∈ U . Let V be a neighborhood of 0 whose closure V¯ is contained in U . The second result we prove here is the following exponential convergence theorem. Theorem 4.2.2. Suppose that Assumptions 4.1.1, 4.1.2 and 4.1.3 hold true. (i) There exists a random variable k1 (ω) such that P -a.s. for any n ≥ k1 (ω), z ∈ V¯ , q ∈ Bθ−n ω and µ ∈ Bω∗ ,

θ−n ω,n

Az
n q

≤ Cθ−n ω Dω kqk · c j0

−n − ν (z)q h (z) (4.2.2) ω θ ω

λ −n (z) θ ω,n and

θ−n ω,n ∗

(Az
n ) µ j

λ −n (z) − µhω (z) νθ−n ω (z) ≤ Cθ−n ω Dω kµk · c 0 θ ω,n

(4.2.3)

where Cω = rω Mω kκω k−1 ,
Dω = c−3 dKω cω + Kω Mω klω k−1 kκω k−1 , c = tanh( 14 d0 ) and cω = supz∈V¯ |lω hω (z)| which is measurable in ω and P -a.s. finite. (ii) Let δ > 0 be so that B(0, 2δ) ⊂ V . Assume that the random variables |lω hω (0)|, Kω klω k−1 and Mω kκω k−1 are bounded. Then there exist

page 176

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

177

constants δ1 > 0 and Q > 0 which depend only on δ and the essential supremum of the latter random variables (and this dependence can be recovered explicitly from the proof ) such that (4.2.2) and (4.2.3) hold true P -a.s. with rθ−n ω Q in place of Cθ−n ω Dω for any n ≥ j0 and z ∈ B(0, δ1 ). In the course of the proof of Theorem 4.2.2 we prove additional types of exponential convergences, some of which without assuming that the cones Cω are reproducing while the others require ergodicity of θ. We refer the readers to Section 4.4. Note also that in certain situations |lω hω (0)| can be controlled by means of the random variables Kω , klω k etc. (see Chapters 5 and 6). 4.3

Block partitions and RPF triplets

The first step towards proving Theorems 4.2.1 and 4.2.2 is the following. 4.3.1

Reverse block partitions

For any ω ∈ Ω, z ∈ U and a natural n ≥ j0 consider the following “block partition”, Azθ

−n

ω,n

= Azθ

−j0

ω,j0

· Aθz

−2j0

ω,j0

◦ · · · ◦ Aθz

−(k−1)j0

ω,j0

◦ Azθ

−n

ω,j0 +r

(4.3.1)

where n = kj0 + r for some integers k ≥ 1 and 0 ≤ r < j0 . By Assumption 4.1.6, Aθz

−n

ω,j0 +r 0 Cθ−n ω

⊂ Cθ0 −(k−1)j0 ω and ∆Cθ−(k−1)j0 ω (Azθ

−n

ω,j0 +r 0 Cθ−n ω )

< d0 (4.3.2)

and for any 1 ≤ m ≤ k − 1, Azθ

−mj0

ω,j0 0 Cθ−mj0 ω

∆Cθ−(m−1)j0 ω (Aθz

⊂ Cθ0 −(m−1)j0 ω and

−mj0

ω,j0 0 Cθ−mj0 ω )

(4.3.3)

< d0 .

In particular, it follows that Aθz

−n

ω,n

Cθ−n ω ⊂ Cω .

(4.3.4)

Next, set c = tanh

1
d0 ∈ (0, 1). 4

We claim that P -a.s., δCω Aθz

−n

ω,n

fn , Aθz

−s

ω,s


n gs ≤ d0 c[ j0 ]−2

(4.3.5)

page 177

March 13, 2018 15:46

ws-book9x6

178

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

for any s ≥ n ≥ j0 , z ∈ U and sequences {fm : m ≥ 1} and {gm : m ≥ 1} with fm , gm ∈ Cθ0 −m ω for any m ≥ 1. Before proving (4.3.5) we need the following notations. For any finite sequence G1 , ..., Gm of linear operators we write m Y Gi = G1 ◦ G2 ◦ · · · ◦ Gm i=1

and for the sake of convenience set Y G = Identity, G∈∅

i.e. the empty product will always be equal to the identity map, no matter which spaces are under consideration. In order to prove (4.3.5), we first write n = kj0 + r and s = qj0 + p, for some 1 ≤ k ≤ q and 0 ≤ r, p < j0 . Then, Aθz

−n

ω,n

k−2 Y

=

Aθz

−mj0

ω,j0




◦ Aθz

−(k−1)j0

ω,j0

◦ B1 and

(4.3.6)

m=1

Aθz

−s

ω,s

k−2 Y

=

Aθz

−mj0

ω,j0




◦ Aθz

−(k−1)j0

ω,j0

◦ B2

m=1

where B1 = B1 (n, ω, z) = Aθz B2 = B2 (n, s, ω, z) =

q−1 Y

Aθz

−mj0

ω,j0

−n




ω,j0 +r

◦ Aθz

−s

and ω,j0 +r

.

m=k

Applying repeatedly Theorem A.2.3 (iii) and taking into account (4.3.2) and (4.3.3) we derive that δCω

 k−2 Y

Aθz

−mj0

ω,j0


h1 ,

k−2 Y

Aθz

−mj0

ω,j0


 h2 ≤ ck−2 δCθ−(k−2)j0 ω (h1 , h2 )

m=1

m=1

(4.3.7) for any h1 , h2 ∈ Cθ0 −(k−2)j0 ω . Let fn ∈ Cθ0 −n ω and gs ∈ Cθ0 −s ω . It follows from the inclusions in (4.3.2) and (4.3.3) that B1 fn , B2 gs ∈ Cθ0 −(k−1)j0 ω and that the functions h1 and h2 given by h1 = Azθ

−(k−1)j0

ω,j0

(B1 fn ) and h2 = Aθz

−(k−1)j0

ω,j0

(B2 gs )

are members of Cθ0 −(k−2)j0 ω . Applying (4.3.3) we derive that δCθ−(k−2)j0 ω (h1 , h2 ) < d0

page 178

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

179

and (4.3.5) follows from (4.3.6), (4.3.7) and the above estimates. Since lω ∈ Cω∗ , we conclude from (4.3.5), Theorem A.2.3 (ii) and (4.1.4) that n n −s −n 1 kA¯θz ω,s gs − A¯zθ ω,n fn k ≤ Kω d0 c[ j0 ]−2 ≤ R(ω)c j0 (4.3.8) 2 for any s ≥ n ≥ j0 , z ∈ U and sequences {fm : m ≥ 1} and {gm : m ≥ 1} with fm , gm ∈ Cθ0 −m ω for any m ≥ 1. Here R(ω) = 12 c−3 d0 Kω and −j A¯θz ω h =

Aθz

−j

ω,j

h

(4.3.9)

−j lω (Aθz ω,j h)

for any j ≥ j0 , h ∈ Cθ0 −j ω and z ∈ U . By considering the case when fm = gm for any m ∈ N, we deduce that the sequence {A¯θz

−n

ω

fn : n ∈ N} ⊂ Bω

is a Cauchy sequence, and therefore P -a.s. the limits ˆ ω (z) := lim A¯θ−n ω,n fn , z ∈ U h z n→∞

(4.3.10)

exist in the Banach space Bω . Considering now the situation when n = s ˆ ω (z), z ∈ U and then letting n → ∞ in (4.3.8), we deduce that the limits h do not depend on the choice of the sequence {fm : m ≥ 1}. Moreover, fixing n and letting s → ∞ it follows from (4.3.8) that for any choice of fn ∈ Cθ0 −n ω and n ≥ j0 , kA¯θz

−n

ω,n

n

ˆ ω (z)k ≤ R(ω)c[ j0 ] . fn − h

(4.3.11)

ˆ ω (z) ∈ C 0 = Furthermore, by (4.3.5) and Theorem A.2.3 (i) we have h ω ˆ Cω \ {0} and the limits hω (z) exist also in the projective metric δCω . Since lω is continuous, for any z ∈ U we have that
ˆ ω (z) = 1. lω h Therefore by (4.1.4) from Assumption 4.1.1 P -a.s. for any z ∈ U , 1 ˆ ω (z)k ≤ Kω ≤ kh klω k klω k

(4.3.12)

ˆ ω (z) ∈ Cω . using that h ˆ ω (z) is G-measurable and analytic in z. Indeed, Finally, we show that h by Assumption 4.1.1 there exists a G-measurable family b = {bω : ω ∈ Ω} so that bω ∈ Cω0 for each ω. Choosing fn = bθ−n ω the desired measurability ˆ ω (z) follows. Since the operators Aω are analytic in z the limit h ˆ ω (z) of h z is analytic in z ∈ U as it is a uniform limit of analytic functions.

page 179

March 13, 2018 15:46

ws-book9x6

180

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Remark 4.3.1. Applying the continuous operator Aω z to both sides of (4.3.10) we derive that the limit
−n lθω Aθz ω,n+1 fn ˆ λω (z) = lim
−n ω,n n→∞ l Aθ fn z ω exists and satisfies ˆ ˆ ˆ Aω z hω (z) = λω (z)hθω (z).

(4.3.13)

ˆ θω (z)) = 1, λ ˆ ω (z) can also be written as Since lθω (h
ˆ ω (z) = lθω Aω h ˆ λ z ω (z)

(4.3.14)

ˆ ω (z) does not depend on the which, in particular, implies that the limit λ ˆ ω (z) is both measurable in ω choice of the sequence {fk : k ≥ 1} and that λ ω ˆ ω (z), h ˆ ω (z)) is not and analytic in z since the operators Az are. The pair (λ the one from the resulting RPF triplet (λω (z), νω (z), νω (z)). Still, it will ˆ ω (z) and hω (z) are follow (after the construction of these triplets) that h linearly dependent, namely that hω (z) is (possibly) a different representaˆ ω (z), and so in the sense of tive from the projective class of the function h convergence in the projective metric δCω we have arrived to the right limit. ˆ ω (z) and λω (z) differ by a (multiplicative) coboundary term, As a result, λ namely P -a.s. for any z ∈ U we can write ˆ ω (z) = aω (z)(aθω (z))−1 λω (z) λ

(4.3.15)

ˆ ω (z)) which, in fact, satisfies h ˆ ω (z) = aω (z)hω (z). where aω (z) = νω (z)(h 4.3.2

Forward block partition and dual operators

Let n ≥ j0 and write n = kj0 + r for some natural k ≥ 1 and 0 ≤ r < j0 . For any z ∈ U consider the following “block partition”, Aω,n = Azθ z

(k−1)j0

ω,j0 +r

◦ Aθz

(k−2)j0

ω,j0

◦ · · · ◦ Aθz

2j0

j0

ω,j0

0 ◦ Aω,j . z (4.3.16)

ω,j0 0 Cθmj0 ω )

< d0 , (4.3.17)

ω,j0

◦ Aθz

By Assumption 4.1.6 for any 0 ≤ m ≤ k − 2, Azθ

mj0

ω,j0 0 Cθmj0 ω

Aθz

⊂ Cθ0 (m+1)j0 ω , ∆Cθ(m+1)j0 ω (Aθz (k−1)j0

ω,j0 +r 0 Cθ(k−1)j0 ω

∆Cθn ω (Aθz

(k−1)j0

mj0

⊂ Cθ0 n ω and

ω,j0 +r 0 Cθ(k−1)j0 ω )

(4.3.18)

< d0 .

In particular, it follows that 0 0 Aω,n z Cω ⊂ Cθ n ω , P -a.s.

(4.3.19)

page 180

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

181

for any z ∈ U and n ≥ j0 . Next, for
the sake of convenience, we denote sometimes the dual ope∗ also by Aω,n,∗ . Consider P -a.s. the dual block partition of rator Aω,n z z (4.3.16),
∗
∗
∗

j0 (k−2)j0 (k−1)j0 ω,j0 ∗ ω,j0 +r ∗ 0 Aω,n = Aω,j ◦ Azθ ω,j0 ◦ · · · ◦ Aθz ◦ Aθz z z (4.3.20) where n = kj0 + r, k ∈ N and 0 ≤ r < j0 . Now we begin with construction of νω (z) and λω (z). Note that this construction will not depend on the results from Section 4.3.1. First, by (A.2.6) from Section A.2.5, taking into account (4.3.17) and (4.3.18), it follows that P -a.s. for any z ∈ U and n ≥ j0 , Aθz Azθ

mj0

(k−1)j0

ω,j0 +r,∗ ∗ Cθ n ω

ω,j0 ,∗ ∗ Cθ(m+1)j0 ω

⊂ Cθ∗(k−1)j0 ω and

(4.3.21)

⊂ Cθ∗mj0 ω , m = 0, 1, 2, ...

and the corresponding δ-diameters of the appropriate images (with respect to the appropriate dual cones) do not exceed d0 . Note that in this application of (A.2.6) we have used that the cones Cω are proper and linearly convex. In particular, it follows that Aω,n,∗ Cθ∗n ω ⊂ Cω∗ , P -a.s. z

(4.3.22)

for any n ≥ j0 and z ∈ U . Next, let s ≥ n ≥ j0 , µ1 ∈ Cθ∗s ω and µ2 ∈ Cθ∗n ω for ω ∈ Ω satisfying (4.3.21). We claim that
n (4.3.23) δCω∗ Aω,s,∗ µ1 , Aω,n,∗ µ2 ≤ d0 c[ j0 ]−2 . z z Indeed, write n = kj0 +r and s = qj0 +p where 1 ≤ k ≤ q and 0 ≤ r, p < j0 . Consider the following block partition
∗
∗
∗
j0 (k−2)j0 ω,j0 ∗ 0 Aω,s = Aω,j ◦ Aθz ω,j0 ◦ · · · ◦ Aθz ◦ Bs z z where Bs = Bs (n, ω, z) = Aθz

(k−1)j0


ω,s−(k−1)j0 ∗

.

Using (4.3.21) with n and s we see that Bn µ1 , Bs µ2 ∈ Cθ∗(k−1)j0 ω for any z ∈ U . Therefore, by (4.3.21) and the diameter estimates following it,
(k−2)j0 (k−1)j0 ω,j0 ,∗ ω,j0 ,∗ δC ∗(k−2)j Azθ (Bs µ1 ), Aθz (Bn µ2 ) < d0 . θ

0ω

Similarly to (4.3.5), (4.3.23) follows by a repetitive use of Theorem A.2.3 (iii), taking into account (4.3.21) and the diameter estimates following it. Next, we derive from Theorem A.2.3 (ii) and (4.3.23) that n 1 (4.3.24) kA¯ω,s,∗ µ1 − A¯ω,n,∗ µ2 k ≤ Mω d0 c[ j0 ]−2 z z 2

page 181

March 13, 2018 15:46

182

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where Mω is specified in Assumption 4.1.1 and for any ν ∈ Cω∗ and m ≥ j0 , A¯ω,m,∗ ν= z

Aω,m,∗ ν z
κω Aω,m,∗ ν z

with κω coming from (4.1.5) in Assumption 4.1.1. Similarly to (4.3.10), we conclude that for any choice of {µn : n ≥ 1} with µn ∈ Cθ∗n ω the limits νω (z) := lim A¯ω,n,∗ µn , z ∈ U z n→∞

(4.3.25)

exist and are independent of the choice of {µn : n ≥ 1}. The operators Aω z are analytic and so νω (z) is a uniform limit of analytic functions which ∗ makes it analytic in z. Note that we have used here that (Aω z ) is analytic in z which holds true since the duality map A → A∗ is an isometry. In order to show that νω (z) is G-measurable, let ν = {νω : ω ∈ Ω} be a G-measurable family so that νω ∈ Cω∗ for any ω (for instance we can take νω = lω ). Taking µn = νθn ω , the required measurability follows by (4.3.25). Next, applying Theorem A.2.3 (i) and using that any dual cone is linearly convex (see Appendix A) it follows that νω (z) ∈ Cω∗ . Moreover, by (4.3.24) and (4.3.25), n 1 kA¯ω,n,∗ µn − νω (z)k≤ Mω d0 c[ j0 ]−2 z 2

for any n ≥ j0 . Furthermore, since νω (z) ∈ Cω∗ and κω deduce from (4.1.5) that 1 Mω ≤ kνω (z)k ≤ . kκω k kκω k

(4.3.26)
νω (z) = 1 we

(4.3.27)

Replacing ω with θω, making the choice µn = lθn (θω) and
∗ then plugging in both sides of (4.3.25) into the continuous operator Aω , we derive that z the limits
κω Aω,n+1,∗ lθn+1 ω z (4.3.28) λω (z) := lim
, z ∈U θω,n,∗ n→∞ κ lθn+1 ω θω Az P -a.s. exist and satisfy ∗ (Aω z ) νθω (z) = λω (z)νω (z).

Aω z

(4.3.29)

are analytic in z and therefore the map z → λω (z) is The operators analytic since it is a uniform limit of analytic functions. Since the operators Aω z are G-measurable the limit λω (z) is measurable in ω. Substituting both sides of (4.3.29) in κω we deduce that

∗ λω (z) = λω (z)κω νω (z) = κω (Aω (4.3.30) z ) νθω (z)

page 182

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

183

which, in particular, provides a different proof that λω (z) is analytic in z and measurable in ω. Next, applying (4.3.29) repeatedly with ω ¯ = θn−1 ω, ....θω, ω it follows that for any n ∈ N, Aω,n,∗ νθn ω (z) = λω,n (z)νω (z), z

(4.3.31)

where λω,n (z) :=

n−1 Y


λθi ω (z) = κω Aω,n,∗ νθn ω(z) . z

i=0

Cθ∗n ω

Since νθn ω (z) ∈ we deduce from (4.3.22) that Aω,n,∗ νθn ω (z) ∈ Cω∗ for z ∗ ∗ any n ≥ j0 . This together with the fact that κω ∈ (Cω ) implies that λω,n (z) 6= 0 for any sufficiently large n, and as a consequence P -a.s. we have λω (z) 6= 0 for any z ∈ U . 4.3.3

RPF triplets

For any z ∈ U and P -a.a. ω set hω (z) =

ˆ ω (z) h . ˆ ω (z)) νω (z) h

(4.3.32)

ˆ ω (z) ∈ C 0 and νω (z) ∈ C ∗ , and The denominator does not vanish since h ω ω so hω (z) is well defined,
νω (z) hω (z) = 1 and hω (z) ∈ Cω0 (4.3.33) where the inclusion follows since Cω is invariant under multiplication of nonzero complex numbers (i.e. it is a complex cone). Since νω (z) and ˆ ω (z) are G-measurable and analytic in z so is hω (z). We claim that the h
triplet λω (z), hω (z), νω (z) is the RPF triplet from Theorem 4.2.1. The missing ingredient is to show that P -a.s. we have Aω z hω (z) = λω (z)hθω (z) for any z ∈ U . This will follow from the following claim. Claim 4.3.2. For a.a. ω ∈ Ω, for each z ∈ U and any choice of {qn : n ≥ 1} with qn ∈ Cθ0 −n ω , −n

Aθz ω,n qn . n→∞ λθ −n ω,n (z)νθ −n ω (z)qn

hω (z) = lim

(4.3.34)

Before proving Claim 4.3.2 we assume it to hold true and complete the proof that
λω (z), hω (z), νω (z)

page 183

March 13, 2018 15:46

ws-book9x6

184

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

satisfy the conditions specified in Theorem 4.2.1. Indeed, let qn = hθ−n ω (z) and recall that νθ−n ω (z)hθ−n ω (z) = 1 and that hθ−n ω (z) ∈ Cθ0 −n ω (see (4.3.33)). Plugging in both sides of (4.3.34) into the continuous operator Aω z we deduce that Aθz

−(n+1)

θω,n+1

hθ−(n+1) θω (z) = λω (z)hθω (z) λθ−(n+1) θω,n (z) (4.3.35) where the second equality follows by applying (4.3.34) with θω in place of ω and with the sequence {˜ qn : n ≥ 1} given by q˜n = qn−1 = hθ−n θω (z) = hθ−(n−1) ω (z) in place of {qn : n ≥ 1} and using again (4.3.33). We remark that replacing ω with θω is indeed possible P -a.s. since θ is P -invariant. Aω z hω (z)

= λω (z) lim

n→∞

Proof of Claim 4.3.2. First, by (4.3.31) P -a.s., λθ−n ω,n (z)νθ−n ω (z)qn = νω (z)Aθ

−n

ω,n

qn

(4.3.36)

for any n ∈ N, qn ∈ Bθ−n ω and z ∈ U . Therefore, when n ≥ j0 and qn ∈ Cθ0 −n ω we can write −n −n −n A¯θz ω,n qn Aθz ω,n qn Aθz ω,n qn = = −n −n λθ−n ω,n (z)νθ−n ω (z)qn νω (z)Aθz ω,n qn νω (z)A¯θz ω,n qn −n where A¯θz ω,n is given by (4.3.9). By (4.3.10) the numerator converges to ˆ ω (z) while the denominator converges to νω (z)h ˆ ω (z) and therefore by the h definition (4.3.32) of hω (z), −n A¯θz ω,n qn = hω (z) n→∞ ν (z)A ¯θz−n ω,n qn ω

lim

which completes the proof of the claim. The proof of the statement about uniqueness of this triplet is postponed to Section 4.5 and meanwhile we will prove Theorem 4.2.2. 4.4

Exponential convergences

We will assume in this section that all the conditions of Theorem 4.2.2 hold true. Then λω (z), hω (z), νω (z) are both analytic in z and measurable in ω. We need first the following estimates. Let V be a neighborhood of 0 whose closure V¯ is contained in U .

page 184

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

4.4.1

LNcov

185

Taylor reminders and important bounds

For P -a.a. ω consider the analytic function αω (·) : U → C given by ˆ ω (z) αω (z) = νω (z)h

Then by (4.3.12) and

Kω Mω := vω . klω kkκω k

(4.4.2)

which can also be written as αω (z) = (4.3.27), |αω (z)| ≤

(4.4.1)

1 lω (hω (z)) .

¯ On the other hand, since this
function does not vanish on V it follows that ˆ βω := inf z∈V¯ |νω (z) hω (z) | > 0. Since the latter function is continuous then this infimum can be taken over a countable dense set which makes ˆ ω (z) are G-measurable. It follows from βω measurable since νω (z) and h (4.3.32) and (4.3.12) that Kω kκω k ≤ khω (z)k ≤ Kω Mω klω kβω

(4.4.3)

for any z ∈ V¯ . The disadvantage of bounds with βω is that the latter is not defined by means of the random variables Kω , klω k, Mω and kκω k appearing in our assumptions which makes it hard to estimate. In what follows we will bound khω (z)k from above around z0 = 0 by means of the latter random variables. Applying Lemma 2.8.2 with k = 0 and the analytic function αω (·) it follows that 4|z|vω δ ¯ 2δ) ⊂ for any z ∈ B(0, δ) = {ζ ∈ C : |ζ| < δ} where δ > 0 satisfies that B(0, V . Consider the set Vω = B(0, δω ) ⊂ C where
1 δω = δ min 1, |αω (0)|(vω )−1 . 8 Then for any z ∈ V¯ω , 1 3 |αω (0)| ≤ |αω (z)| ≤ |αω (0)| (4.4.4) 2 2 and therefore by (4.3.12), |αω (z) − αω (0)| ≤

2 2Kω ≤ khω (z)k ≤ := ζω . 3klω k|αω (0)| klω k|αω (0)|

(4.4.5)

In many situations (e.g. our applications in the next chapters) we will have a positive lower bound on |αω (0)|, which will make the above upper bound effective.

page 185

March 13, 2018 15:46

ws-book9x6

186

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

4.4.2

Exponential convergences (2)

For the sake of convenience set V (1) = V and V (2) = Vω = Vω . We first show that there exist random variables k1 (ω) and k2 (ω), such that P -a.s. for any n ≥ ki (ω), z ∈ V¯ (i) , i = 1, 2 and q ∈ Cθ−n ω , −n

n Aθz ω,n q

− hω (z) ≤ Rω(i) R(ω)c j0 (4.4.6) D(ω, q, z, n) := λθ−n ω,n (z)νθ−n ω (z)q where R(ω) is defined below (4.3.8), Rω(1) = 2(βω−1 + Wω ), Rω(2) = 4(|αω (0)|−1 + 2Wω ) and Wω = Kω Mω klω k−1 kκω k−1 . The random variable k2 (ω) will depend only on ζω , Mω kκω k−1 and R(ω), while the random variable k1 (ω) will depend also on βω . Before proving (4.4.6), we assume that it holds true and complete the proof of Theorem 4.2.2. Set Cω = rω Mω kκω k−1 . We claim that P -a.s. for any n ≥ ki (ω), z ∈ V¯ (i) , i = 1, 2 and q ∈ Bθ−n ω ,

θ−n ω,n

Az
n q (i) j

(4.4.7)

λ −n (z) − νθ−n ω (z)q hω (z) ≤ Cθ−n ω Rω R(ω)kqkc 0 . θ ω,n Indeed, in the circumstances of Theorem 4.2.2, the cones Cθ−n ω are reproducing of order k = kθ−n ω with a constant rθ−n ω . Therefore, for any q ∈ Bθ−n ω we can write q = q1 + ... + qk where each qi is a member of the cone Cθ−n ω and kq1 k + kq2 k + ... + kqk k ≤ rθ−n ω kqk. Inequality (4.4.7) follows by first applying (4.4.6) with the functions qi , i = 1, ..., k, then multiplying the resulting inequality by νθ−n ω (z)qi and then using the equality q = q1 + q2 + ... + qk and the inequality k X

|νθ−n ω (z)qi | ≤

i=1

k X

kνθ−n ω (z)kkqi k ≤ Mθ−n ω kκθ−n ω k−1 rθ−n ω kqk

i=1

where we use (4.3.27). In order to prove (4.4.6), first we apply (4.3.36) and obtain that for P -a.a. ω and any n ≥ j0 , z ∈ U and q ∈ Cθ−n ω , λθ−n ω,n (z)νθ−n ω (z)q = νω (z)Aθz

−n

ω,n

q.

Next, set bn (q, z) = bn (ω, q, z) =

lω (Aθz

−n

ω,n

q)

θ −n ω,n

νω (z)Az

q

=

1 . −n νω (z)A¯θz ω,n q

page 186

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

187

Then,

−n D(ω, q, z, n) = bn (q, z) · A¯θz ω,n q − hω (z) (4.4.8)



−n 1 ˆ ω (z) + bn (q, z) − ˆ ω (z) h ≤ bn (q, z) A¯θz ω,n q − h ˆ ω (z) νω (z)h where we use the definition of hω (z) in (4.3.32). By (4.3.11) and (4.3.27), ˆ ω (z) (bn (q, z))−1 − νω (z)h (4.4.9) −n θ ω,n ˆ ω (z) = νω (z)A¯z q − νω (z)h n

−n ˆ ω (z)k ≤ Mω kκω k−1 R(ω)c j0 ≤ kνω (z)kkA¯θz ω,n q − h

where R(ω) was defined below (4.3.8). Using (4.3.11) again, we deduce from (4.4.9) and the definition of αω (z) that for any z ∈ V¯ ,
n ˆ ω (z)k· |αω (z)|−1 D(ω, q, z, n) ≤ R(ω)c j0 D−1 1 + Mω kκω k−1 kh
n = R(ω)c j0 D−1 1 + Mω kκω k−1 khω (z)k assuming that n

D := |αω (z)| − Mω kκω k−1 R(ω)c j0 > 0. When z ∈ V¯ then by the definition of βω , n

D ≥ βω − Mω kκω k−1 R(ω)c j0 and when in addition z ∈ V¯ω then by (4.4.4), D≥

n 1 |αω (0)| − Mω kκω k−1 R(ω)c j0 . 2

Let k1 (ω) be the smallest natural number k1 such that k1

Mω kκω k−1 R(ω)c j0 ≤

1 βω 2

(4.4.10)

and k1 ≥ j0 . Similarly, let k2 (ω) be the smallest natural number k2 such that k2

Mω kκω k−1 R(ω)c j0 ≤

1 |αω (0)| 4

(4.4.11)

and k2 ≥ j0 . It is clear that k1 (ω) and k2 (ω) are measurable, and the proof of (4.4.6) is complete, taking into account (4.4.3) and (4.4.5).

page 187

March 13, 2018 15:46

188

4.4.3

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Additional types of exponential convergences

We begin with proving (4.2.3) in Theorem 4.2.2. First, for any ω ∈ Ω, µ ∈ Bω∗ , n ∈ N and q ∈ Bθ−n ω the duality relation µ(Aθz

−n

ω,n

q) = (Aθz

−n

ω,n,∗

µ)q

holds true. Applying µ to the expression inside the norm in (4.4.7) we deduce that P -a.s. for any n ≥ ki (ω) and z ∈ V¯ (i) , i = 1, 2,

θ−n ω,n,∗

(Az
n µ)q

≤ Cθ−n ω Rω(i) R(ω)kqkkµkc j0

−n − µh (z) ν (z)q ω θ ω

λ −n (z) θ ω,n (4.4.12) for any µ and q as above. We conclude by taking supremum over all q’s with kqk = 1 that

θ−n ω,n,∗

Az
n µ

≤ Cθ−n ω Rω(i) R(ω)kµkc j0 . (4.4.13)

−n − µh (z) ν (z) ω θ ω

λ −n (z) θ ω,n Suppose next that θ
is ergodic. Consider the random variable δω = 1, |αω (0)|(vω )−1 from the definition of the nieghborhood Vω , and recall that |αω (0)|−1 = |lω hω (0)|. Let a ∈ (0, 1) be sufficiently small so that pa = P (Ga ) > 0, where 1 8 δ min

Ga = {ω : a ≤ kκω k, klω k ≤ a−1 } ∩ {ω : |lω hω (0)|, Mω , Kω ≤ a−1 }. (4.4.14) Applying the mean ergodic theorem with the indicator function of the set G = Ga we see that P -a.s. there exists a strictly increasing sequence {li : 5 i ≥ 1}, li = li (ω) of natural numbers such that δθli ω ≥ a8δ for any i ≥ 1 and limn→∞ lnn = p1a . We observe that the random variable k2 (ω) is P -a.s. bounded from above by some constant k¯2 (which depends only on a) when ω ∈ Ga , since k2 (ω) is the minimal natural number k2 satisfying (4.4.11) and k2 ≥ j0 . Plugging in θn ω in place of ω in (4.4.7) and then considering n’s of the 5 form n = li it follows that P -a.s. for any z ∈ B(0, a8δ ) := U2 , g ∈ Bω and s ≥ k¯2 ,

Aω,ls g

ls

z

− hθls ω (z)νω (z)g ≤ 8C1 c−3 rω Mω kκω k−1 kgka−5 c j0 . (4.4.15)

λω,ls (z) When the random variables Kω , Mω are bounded from above, |αω (0)| is bounded away from 0 and kκω k and klω k are bounded and bounded away

page 188

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

LNcov

189

from 0 then pa = 1, assuming that a is sufficiently small. In this case li = i for any i and we obtain appropriate estimates on the differences

Aω,n g

z − hθn ω (z)νω (z)g .

λω,n (z) In fact, these estimates hold in this situation without assuming that θ is ergodic since in this case the random neighborhood Vω contains a deterministic neighborhood of 0. Corresponding results for the dual operators follow similarly, relying on (4.4.13). 4.5

Uniqueness of RPF triplets


˜ ω (z), h ˜ ω (z), ν˜ω (z) be a triplet consisting of a nonzero Let z ∈ U and let λ ˜ ω (z) an element h ˜ ω (z) ∈ Bω and a functional ν˜ω (z) ∈ B ∗ complex number λ ω satisfying P -a.s., ω ∗ ˜ ˜ ˜ ˜ ω (z)˜ Aω ˜θω (z) = λ νω (z) z hω (z) = λω (z)hω (z), (Az ) ν ˜ and ν˜ω (z)hω (z) = κω ν˜ω (z) = 1.

Iterating these equalities we obtain that for any natural n, ˜ ω,n (z)˜ Aω,n,∗ ν˜θn ω (z) = λ νω (z), P -a.s. z

(4.5.1)

˜ ˜ ˜ Aω,n z hω (z) = λω,n (z)hθ n ω (z), P -a.s.

(4.5.2)

and

where ˜ ω,n (z) = λ

n−1 Y

˜ ω (z). λ

k=0

We begin with proving the uniqueness statement from Theorem 4.2.1. Suppose that ˜ ω (z) ∈ C 0 and ν˜ω (z) ∈ C ∗ , P -a.s. h ω ω Substituting µn = ν˜θn ω (z) in (4.3.25) and using the equalities (4.5.1), A¯ω,n,∗ ν˜θn ω (z) = ν˜ω (z) and κω ν˜ω (z) = 1 it follows that ν˜ω (z) = νω (z), P -a.s.

(4.5.3)

Using the equality κω νω (z) = κω ν˜ω (z) = 1 we obtain that λω (z) = ∗ ω ∗ ˜ κω (Aω ˜θω (z) and therefore z ) νθω (z) and λω (z) = κω (Az ) ν ˜ ω (z), P -a.s. λω (z) = λ

(4.5.4)

page 189

March 13, 2018 15:46

ws-book9x6

190

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

˜ θ−n ω (z) in (4.3.34) and taking into account Finally, substituting qn = h (4.5.2), (4.5.3) and (4.5.4) we derive that ˜ ω (z) = hω (z), P -a.s. h

(4.5.5)

Next, we prove a stronger type of uniqueness under the assumption that the cones Cω are reproducing and that θ is ergodic. We claim that (4.5.3)(4.5.5) hold true assuming that ν˜ω (z)hω (z) 6= 0 and that k˜ νω (z)k ≤ Jω for some random variable Jω (e.g. when ν˜ω (z) ∈ Cω∗ ). Indeed, let L > 0 be such that P (TL ) > 0, where TL = {ω : rω , Mω kκω k−1 , kκω k, Jω ≤ L} and rω , Mω and κω were introduced in Assumptions 4.1.1 and 4.1.3. By ergodicity of θ−1 we have P

∞ [


θi TL = 1.

i=1

Let ιL (ω) be the first backward visiting time to the set TL , namely the smallest index i ∈ N such that θ−i ω ∈ TL . Consider the conditional probability measure PL on TL which is given by PL (Q) =

P (Q ∩ TL ) , Q∈F P (TL )

and the map ϑ = ϑL = θ−ιL . Then ϑ preserves PL and the measure preserving system (Ω, F, PL , ϑ) is ergodic (see Section 5 in Chapter 1 of [14]). Plugging in µ = ν˜ω (z) in (4.4.13) and then using (4.5.1) yields ˜ ϑn ω,n (z)

λ
ν˜ϑn ω (z) − ν˜ω (z)hω (z) νϑn ω (z) = 0, lim n→∞ λϑn ω,n (z)

PL -a.s.

(4.5.6)

Next, denote by ln,ω (z) the linear functional inside the norm in the above left hand-side. Since κϑn ω νϑn ω (z) = κϑn ω ν˜ϑn ω (z) = 1 we have κϑn ω ln,ω (z) =

˜ ϑn ω,n (z) λ − ν˜ω (z)hω (z) λϑn ω,n (z)

and, taking into account that limn→∞ kln,ω (z)k = 0 and kκϑn ω k ≤ L, it follows that |κϑn ω ln,ω (z)| ≤ Lkln,ω (z)k → 0 as n → ∞ and therefore ˜ ϑn ω,n (z) λ = ν˜ω (z)hω (z). n→∞ λϑn ω,n (z) lim

(4.5.7)

page 190

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

191

Since ν˜ω (z)hω (z) 6= 0, we conclude from (4.5.6) and (4.5.7) that lim k˜ νϑn ω (z) − νϑn ω (z)k = 0
where we also used that max k˜ νθn ω (z)k, kνθn ω (z)k ≤ L (recall (4.3.27)). Applying the mean ergodic theorem with the function g(ω) = k˜ νω (z) − νω (z)k and the ergodic map ϑ we conclude that n→∞

ν˜ω (z) = νω (z), PL -a.s. Letting L → ∞ it follows that ν˜ω (z) = νω (z), P -a.s. Similarly to the begin˜ ω (z) = λω (z), P -a.s. In order to show ning of this section, we deduce that λ ˜ that hω (z) = hω (z), P -a.s. we consider again the systems (Ω, F, PL , ϑL ), ˜ ω (z) = λω (z) and ν˜ω (z) = νω (z) in order to deduce use (4.4.7) and that λ ˜ ω (z) = hω (z), PL -a.s. We complete the proof by letting L → ∞. that h 4.6

The largest characteristic exponents

We recall that when θ is ergodic then by Kigman’s subadditive ergodic theorem, for any z ∈ C the so-called largest characteristic exponent of the cocycle {Aω z : ω ∈ Ω}, 1 ln kAω,n z k n exists P -a.s. and does not depend on ω. In this section we will show that there exists a ball B(0, δ0 ), δ0 > 0 around 0 such that with probability one all the limits Λ(z), z ∈ B(0, δ0 ) exist and we will describe the behavior of the function Λ(·). These results will be obtained under the following assumptions. Λ(z) = lim

n→∞

Assumption 4.6.1. The map θ is ergodic and the random variables kκω k, klω k, kκω k−1 , klω k−1 , Mω , Kω and |lω hω (0)| are bounded from above by some a−1 > 1, i.e. P (Ga ) = 1 where Ga is the set defined in (4.4.14). Assumption 4.6.2. There exists a constant C > 1 such that sup kAω z k ≤ C|λω (0)|, P -a.s.

z∈V¯

where V is the neighborhood of 0 specified before Theorem 4.2.2. We will show that (under reasonable conditions) these assumptions hold true in our application to random transfer operators in Chapter 5.

page 191

March 13, 2018 15:46

192

4.6.1

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Analyticity of the largest characteristic exponent around 0

Our main result here is the following. Theorem 4.6.3. Suppose that Assumptions 4.6.1 and 4.6.2 and the assumptions of Theorem 4.2.2 hold true. (i) There exists a constant δ0 > 0 such that P -a.s. for any z ∈ B(0, δ0 ) we have 21 ≤ ln |λω (z)/λω (0)| ≤ 32 and Z 1 lim ln kAω,k k/|λ (0)| = ln |λω (z)/λω (0)|dP (ω) := L0 (z). ω,k z k→∞ k As a consequence, 1 (4.6.1) lim ln |λω,k (0)| = Λ(0), P -a.s. k→∞ k and Λ(z) = Λ(0) + L0 (z)

(4.6.2)

where we use the convention −∞ + c = −∞ for any c < ∞. The function L0 (·) is harmonic function of the variable ( 0. When the while limn→∞ kAω,n z hω (z)k spaces Bω = B and the operators Aω z = Az are not random, the above decomposition of B means that Az has an isolated eigenvector hω (z) = h(z) with eigenvalue r(z) and that the rest of its spectrum is contained in the 1 closed disk around 0 with radius (1 − c j0 )r(z) (see [28]), namely Az has a spectral gap. In the random case we have decompositions of more than one operator and so the results obtained in this section can be viewed as a generalization of spectral gap to the case of decomposition of random operators.

Before starting to prove Theorem 4.6.3, we need the following. 4.6.2

The pressure function

Proposition 4.6.5. Suppose that the Assumptions of Theorem 4.2.1 hold ¯ ω (·) : true. Then, P -a.s. there exist s = sω > 0 and an analytic function Π B(0, s) → C satisfying ¯

eΠω (z) =

λω (z) ¯ , Πω (0) = 0 λω (0)

(4.6.6)

and ¯ ω (z)| ≤ ln 2 + π. |Π

(4.6.7)

Moreover, when also Assumptions 4.6.1 and 4.6.2 hold true then there exists a constant δ1 > 0 such that δ1 ≤ sω , P -a.s. Proof. For P -a.a. ω set ¯ ω (z) = λω (z) , z ∈ U. λ λω (0) Let V1 = B(0, 2ε) be an open ball around 0 ∈ C whose closure V¯1 is contained in U . The map z → kAω z k is continuous and therefore achieves a maximum on V¯1 , which we denote by S = S(ω). By (4.2.1), ∗ λω (z) = κω (Aω z ) νθω (z)

which together with (4.3.27) yields ∗ −1 |λω (z)| ≤ kκω k · k(Aω := ω z ) k · kνθω (z)k ≤ kκω k · S · Mθω · kκθω k

¯ ω (z)| ≤ ¯ω := and it follows that |λ k = 0 we deduce that

ω |λω (0)| .

¯ ω (z) − 1| ≤ |λ

Applying Lemma 2.8.2 with

4¯ ω |z| δ

page 193

March 13, 2018 15:46

194

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

for any z ∈ B(0, ε). Set s =

ε 8¯ ω .

Then for any z ∈ B(0, s),

1 ¯ ω (z)| ≤ 3 ≤ |λ (4.6.8) 2 2 and therefore we can define on B(0, s) a branch of the logarithm of the ¯ ω (·), i.e. an analytic function Π ¯ ω : B(0, s) → C satisfying (4.6.6) function λ and (4.6.7). Finally, suppose that Assumptions 4.6.1 and 4.6.2 hold true. Let a be as specified in Assumption 4.6.1 and C as specified in Assumption 4.6.2. 3 Then sω ≥ a8Cε , P -a.s. 4.6.3

Proof of Theorem 4.6.3

Suppose that all the conditions of Theorem 4.6.3 (i) hold true. Consider 5 the neighborhood U2 = B(0, a8δ ) := B(0, δ2 ) of 0 specified before (4.4.15). By taking a sufficiently small ε in the proof of Proposition 4.6.5 we can always assume that δ1 < δ2 . Next, by (4.6.8), 1 ¯ ω (z)| ≤ 3 ≤ |λ (4.6.9) 2 2 ¯ ω (z)| is integrable. Set δ0 = 1 δ1 . for any z ∈ B(0, δ1 ). In particular, ln |λ 2 We claim that P -a.s., k−1 1X ¯ ln |λθj ω (z)| = L0 (z) k→∞ k j=0

lim

(4.6.10)

for any z ∈ B(0, δ0 ). Indeed, applying the mean ergodic theorem with the random variables λω (z) we deduce that (4.6.10) holds true P -a.s. for z’s belonging to a countable dense subset of B(0, δ0 ). Applying Lemma 2.8.2 with k = 0 and then using (4.6.9) we infer that there exists a constant d > 0 such that P -a.s. for any z0 ∈ B(0, δ0 ) and h ∈ B(0, 21 δ0 ), λ ¯ ω (z0 + h) − 1 ≤ d|h|. ¯ λω (z0 ) We conclude that the function L0 (·) is continuous and that the family of functions k−1 1X ¯ ln |λθj ω (·)| : B(0, δ0 ) → C k j=0

is equicontinuous, where ω belongs to a full P -probability set and k ∈ N, and our claim follows.

page 194

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Random complex Ruelle-Perron-Frobenius theorem via cones contractions

195

Next, in the case when li = i, taking supremum in (4.4.15) over all g’s such that kgk = 1 and taking into account (4.3.27), the lower bound from (4.4.3) and the upper bound from (4.4.5), we deduce that for any sufficiently large k, C1 (ω)|λω,k (z)| ≤ kAω,k z k ≤ C2 (ω)|λω,k (z)|

(4.6.11)

for any z ∈ U2 , where C1 (ω), C2 (ω) > 0. Taking the logarithms of the expressions in (4.6.11) and observing that ¯ ω,k (z)| = ln |λ

k−1 X

¯ θj ω (z)| ln |λ

j=0

it follows that Z 1 ω,k −1 ¯ ω (z)|dP (ω0 ) = L0 (z), P -a.s. lim ln kAz k|λω,k (0)| = ln |λ 0 k→∞ k (4.6.12) for any z ∈ B(0, δ0 ), where we also used (4.6.10). It is clear now from (4.6.12) that equalities (4.6.1) and (4.6.2) hold true. In order to complete the proof of Theorem 4.6.3 (i), we show now that the function L0 (·) is harmonic. Indeed, since Z
¯ ω (z)dP (ω) Π L0 (z) = < R ¯ ω (z)dP (ω) is analytic. it is sufficient to show that the function z → Π ¯ Indeed, applying Lemma 2.8.2 with Q = Πω , U = B(0, 2δ0 ), k = 1, z0 = 0 ¯ ω (z)| ≤ ln 2 + π, and δ = δ0 and taking into account that supz∈B(0,2δ0 ) |Π ¯ we obtain uniform in ω estimates on Taylor reminders (around 0) of Πω which guarantees that L0 is analytic in B(0, δ0 ). In order to prove Theorem 4.6.3 (ii), suppose that ln |λω (0)| is integrable. Then by the mean ergodic theorem, k−1

1 1X ln |λω,k (0)| = lim ln |λθj ω (0)| k→∞ k k→∞ k j=0 Z = ln |λω (0)|dP (ω), P -a.s. lim

and we conclude from (4.6.12) that P -a.s., Z 1 ω,k lim ln kAz k = ln |λω (z)|dP (ω) := L(z) k→∞ k for any z ∈ B(0, δ0 ).

(4.6.13)

page 195

March 13, 2018 15:46

196

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Next, we prove the rest of Theorem 4.6.3 (ii). Let z ∈ B(0, δ0 ) and q ∈ Bω be such that |νω (z)(q)| > 0. Taking into account (4.3.27), the lower bound in (4.4.3) and the upper bound in (4.4.5), we deduce from (4.4.15) in the case when P (Ga ) = 1 that C1 (q, ω, z)|λω,k (z)| ≤ kAω,k z qk ≤ C2 (q, ω, z)|λω,k (z)|

(4.6.14)

for any sufficiently large k, where C1 (q, ω, z), C2 (q, ω, z) > 0. This together with (4.6.11) implies that ω,k ω,k D1 (q, ω, z)kAω,k z k ≤ kAz qk ≤ D2 (q, ω, z)kAz k

for any sufficiently large k, for some D1 (q, ω, z), D2 (q, ω, z) > 0, and (4.6.3) follows from (4.6.13). On the other hand, if νω (z)q = 0 then by (4.4.15), k

j0 kAω,k z q/λω,k (z)k ≤ C(ω)kqkc

for some C(ω) > 0. This together with the previous arguments shows that 1 −1 lim sup ln Aω,k z q/λω,k (0)k ≤ L0 (z) + j0 ln c < L0 (z) k k→∞ and since ln |λω (0)| is integrable we derive that lim sup n→∞

1 −1 ln kAω,n z qk ≤ L(z) + j0 ln c < L(z). n

page 196

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Chapter 5

Application to random locally distance expanding covering maps

5.1

Random locally expanding covering maps

In this chapter we apply the results from Chapter 4 in the setup which consists of a complete probability space (Ω, F, P ) together with an invertible P -preserving transformation θ : Ω → Ω, of a compact metric space (X , ρ) normalized in size so that diamX ≤ 1 together with the Borel σ-algebra B, and of a set E ⊂ Ω × X measurable with respect to the product σ-algebra F × B such that the fibers Eω = {x ∈ X : (ω, x) ∈ E}, ω ∈ Ω are compact. The latter yields (see [16] Chapter III) that the mapping ω → Eω is measurable with respect to the Borel σ-algebra induced by the Hausdorff topology on the space K(X ) of compact subspaces of X and the distance function d(x, Eω ) is measurable in ω for each x ∈ X . Furthermore, the projection map πΩ (ω, x) = ω is measurable and it maps any F × B-measurable set to a F-measurable set (see “measurable projection” Theorem III.23 in [16]). Let {Tω : Eω → Eθω , ω ∈ Ω} be a collection of continuous bijective maps between the metric spaces Eω and Eθω so that the map (ω, x) → Tω x is measurable with respect to the σ-algebra P which is the restriction of F × B on E. Consider the skew product transformation T : E → E given by T (ω, x) = (θω, Tω x).

(5.1.1)

For any ω ∈ Ω and n ∈ N consider the n-th step iterates Tωn given by Tωn = Tθn−1 ω ◦ · · · ◦ Tθω ◦ Tω : Eω → Eθn ω .

(5.1.2)

Our additional requirements concerning the family of maps {Tω : ω ∈ Ω} are collected in the following assumptions. 197

page 197

March 13, 2018 15:46

198

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Assumption 5.1.1 (Topological exactness). There exist a constant ξ > 0 and a random variable nω ∈ N such that P -a.s., Tωnω (Bω (x, ξ)) = Eθnω ω for any x ∈ Eω

(5.1.3)

where for any ω ∈ Ω, x ∈ Eω and r > 0, Bω (x, r) denotes a ball in Eω around x with radius r. Assumption 5.1.2 (The pairing property). There exist random variables γω > 1 and Dω ∈ N such that P -a.s. for any x, x0 ∈ Eθω with ρ(x, x0 ) < ξ we can write Tω−1 {x} = {y1 , ...., yk } and Tω−1 {x0 } = {y10 , ..., yk0 }

(5.1.4)

where ξ is specified in Assumption 5.1.1, k = kω,x = |Tω−1 {x}| ≤ Dω and ρ(yi , yi0 ) ≤ (γω )−1 ρ(x, x0 )

(5.1.5)

for any 1 ≤ i ≤ k. Next, for any n ∈ N set γω,n =

n−1 Y i=0

γθi ω and Dω,n =

n−1 Y

Dθi ω .

(5.1.6)

i=0

When the pairing property holds true then it follows by induction on n that for P -a.a. ω and for any x, x0 ∈ Eθn ω with ρ(x, x0 ) < ξ we can write (Tωn )−1 {x} = {y1 , ...., yk } and (Tωn )−1 {x0 } = {y10 , ..., yk0 }

(5.1.7)

where k = kω,x,n = |(Tωn )−1 {x}| ≤ Dω,n , |Γ| denotes the cardinality of a finite set Γ and
ρ Tωj yi , Tωj yi0 ≤ (γθj ω,n−j )−1 ρ(x, x0 )

(5.1.8)

for any 1 ≤ i ≤ k and 0 ≤ j < n. We will also use the following. By Lemma 4.11 in [53] (applied with r = ξ), there exists an integer valued random variable Lω ≥ 1 and Fmeasurable functions ω → xω,i ∈ X , i = 1, 2, 3, ... so that xω,i ∈ Eω for each i and Lω [ Bω (xω,k , ξ) = Eω , P -a.s. (5.1.9) k=1

page 198

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Application to random locally distance expanding covering maps

199

Next, let g : E → C be a measurable function. For any ω ∈ Ω consider the function gω : Eω → C given by gω (x) = g(ω, x). For any 0 < α ≤ 1 set vα,ξ (gω ) = inf{R : |gω (x) − gω (x0 )| ≤ Rρα (x, x0 ) if ρ(x, x0 ) < ξ} and kgω kα,ξ = kgω k∞ + vα,ξ (gω )
α where k · k∞ is the supremum norm and ρα (x, x0 ) = ρ(x, x0 ) . These norms are F-measurable as a consequence of the following result (cf. Lemma 4.1 in [53]) together with Theorem III.23 in [16]. Lemma 5.1.3. Let (Ω, F) be a measurable space, X be another space and Q be a σ-algebra on the product space Ω × X so that for any Γ ∈ Q, πΩ Γ = {ω ∈ Ω : (ω, x) ∈ Γ for some x ∈ X } ∈ F.

(5.1.10)

Then for any Q-measurable function g = gω (x) = g(ω, x) on Ω × X the supremum norm kgω k∞ = sup |g(ω, x)| x∈X

is F-measurable as a function of ω ∈ Ω. (n)

Proof. Since |g| is also Q-measurable, each set Γk = {(ω, x) : nk ≤ P∞ k (n) |g(ω, x)| < k+1 = , k=0 n IΓ(n) n } belongs to Q. Then the function g k where IΓ is the indicator of Γ, is Q-measurable. Consider the supremum (n) norms kgω k∞ = supx∈X g (n) (ω, x). Then {ω : kgω(n) k∞ ≤

k [ k (n) }= πΩ Γ k ∈ F n j=0

(n)

and so kgω k∞ is F-measurable. Since |g(ω, x)| − g (n) (ω, x) ≤ 1 sup n (ω,x)∈Ω×X we have also for ω ∈ Ω that kgω k∞ − kgω(n) k∞ ≤ 1 . n (n)

Hence kgω k∞ = limn→∞ kgω k∞ implying the assertion of the lemma. Next, consider the Banach spaces (Bω , k · k) = (Hωα,ξ , k · kα,ξ ) α,ξ of all functions h : Eω → C such that khkα,ξ < ∞ and denote by Hω,R the space of all real valued functions in Hωα,ξ . Let H = Hω ≥ 1 be a

page 199

March 13, 2018 15:46

200

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

R random variable so that ln Hω dP (ω) < ∞ and let Hα,ξ (H) be the set of all measurable functions g : E → C such that vα,ξ (gω ) ≤ Hω for any ω. For any ω ∈ Ω consider the set Yω of all complex continuous linear functionals x ˆ ∈ Bω∗ which have the form x ˆ(g) = g(x) for some x ∈ Eω . Consider the space Y=

[

{ω} × Yω .

ω∈Ω

Identifying x ˆ and x, we endow Y with the σ-algebra P on E induced by F × B. Then we are in the situation of Example 4.1.4 in Chapter 4 and so we can extend P to a σ-algebra G defined on B ∗ and with this extension a measurable family in the sense of Chapter 4 is just a measurable function g : E → C so that g(ω, ·) ∈ Bω for each ω and a measurable family of elements of (Bω )∗ is just a family of continuous linear functionals µω ∈ (Bω )∗ so that the map ω → µω (g(ω, ·)) is measureable for any measurable function g : E → C with g(ω, ·) ∈ Bω for any ω. Next, for any function ψ : E → C consider the random functions Snω ψ : Eω → C, ω ∈ Ω, n ∈ N given by Snω ψ =

n−1 X

ψθi ω ◦ Tωi

i=0

where ψω = ψ(ω, ·). Let the random variable Qω (H) be defined by Qω (H) =

∞ X

Hθ−j ω (γθ−j ω,j )−α .

(5.1.11)

j=1

Note that Qω is bounded when Hω is a bounded random variable and γω ≥ 1 + δ, P -a.s. for some δ > 0. The following simple distortion property is a direct consequence of (5.1.8). Lemma 5.1.4. Let ∈ Hα,ξ (H). For any ω and n ≥ 1, let x, x0 ∈ Eθn ω with ρ(x, x0 ) < ξ and y1 , ..., yk and y10 , ..., yk0 satisfy (5.1.7) and (5.1.8). Then for any 1 ≤ i ≤ k, |Snω ψ(yi ) − Snω ψ(yi0 )| ≤ ρα (x, x0 )

n−1 X j=0

Hθj ω (γθj ω,n−j )−α ≤ ρα (x, x0 )Qθn ω (H).

(5.1.12)

page 200

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Application to random locally distance expanding covering maps

5.2

LNcov

201

Transfer operators

Let φ, u : E → R be measurable functions. Assumption 5.2.1. For any ω ∈ Ω, the functions φω , uω : Eω → R given by φω (·) = φ(ω, ·) and uω (·) = u(ω, ·) are continuous and satisfy (5.1.12). By Lemma 5.1.4 this assumption holds true when φω , uω ∈ Hωα,ξ (H). Still, Assumption 5.2.1 holds true in more general situations which will make the results from this section applicable for the random transfer operators introduced in the proof of the nonconventional LLT from Chapter 2 in Part 1. Let z ∈ C and consider the transfer operators Lω z , ω ∈ Ω which map functions from Eω to functions X from Eθω by the formula eφω (y)+zuω (y) g(y). (5.2.1) Lω z g(x) = y∈Tω−1 {x}

Note that under Assumption 5.1.2 the operators Lω z , z ∈ C are well defined and when also Assumption 5.2.1 holds true they map a continuous function on Eω to a continuous function on Eθω . For any n ∈ N and z ∈ C consider the n-th step iterates Lω,n of the transfer operator given by z n−1 ω Lω,n = Lθz ω ◦ · · · ◦ Lθω (5.2.2) z z ◦ Lz . Then X ω ω Lω,n eSn φ(y)+zSn u(y) g(y) (5.2.3) z g(x) = y∈(Tωn )−1 {x}

is the transfer operator generated by the map Tωn and the + zSnω u. Next, consider the (global) transfer operator Lz acting on functions g : E → C by X the formula −1 Lz g(s) = eφθ−1 ω (y)+zuθ−1 ω (y) g(s0 ) = Lθz ω gθ−1 ω (x), s = (ω, x),

namely Lω,n z function Snω φ

s0 ∈T −1 {s}

(5.2.4) namely Lz is generated by the skew product map T and the function φ+zu. For any ψ : E → C and n ∈ N set n−1 X Sn ψ = ◦ T i. i=0

Then, Sn ψ(ω,X y) = Snω ψ(y) and the iterates of the operator Lz satisfy 0 0 −n n Lz g(s) = eSn φ(s )+zSn u(s ) g(s0 ) = Lθz ω,n gθ−n ω (x), s = (ω, x). s0 ∈T −n {s}

Assumption 5.2.2. The transfer operators Lz , z ∈ C map measurable functions on E to measurable functions on E.

page 201

March 13, 2018 15:46

202

5.3

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Real and complex cones

In this section we will describe the cones for which we apply the result from Chapter 4. Set Qω = Qω (2H) = 2Qω (H)

(5.3.1)

and let ω be so that Qω < ∞. Following [53], let s > 1 and consider the real proper convex cones (see Appendix A), Cω,R = Cω,R,s = {g : Eω → [0, ∞) : g(x) ≤ e

sQω ρα (x,x0 )

0

(5.3.2)

0

g(x ) if ρ(x, x ) < ξ}.

Next, let the functional lω : CEω → C be given by the formula lω (g) =

Lω X

g(xω,j )

(5.3.3)

j=1

where the xω,j ’s are the points in Eω satisfying (5.1.9). It is clear that lω is measurable in the sense described in Section 5.1 with the σ-algebra G on B ∗ defined there. Note that klω kα,ξ ≤ klω k∞ = lω (1) = Lω and therefore klω kα,ξ = Lω α,ξ where the (operator) norm klω kα,ξ of lω is either with respect to Hω,R or with respect to Hωα,ξ . In particular, klω kα,ξ is measurable in ω.

Theorem 5.3.1. Let ω ∈ Ω be such that Qω < ∞. α,ξ (i) The cone Cω,R is a closed subsets of the real Banach space Hω,R . Let Cω be the canonical complexification of Cω,R (see Appendix A). Then the complex cone Cω is linearly convex. Moreover, when Qω is P -a.s. finite then the first part of Assumption 4.1.1 holds true with the family bω ≡ 1, where 1 is the function√which takes the constant value 1. α α (ii) Set Kω = 2 2Lω (sQω esQω ξ + 1)esQω ξ and Mω = 8(1 − α e−sQω ξ )−2 . Then kxkα,ξ klω kα,ξ ≤ Kω |lω (x)| for any x ∈ Cω

(5.3.4)

kµkα,ξ ≤ Mω |µ(1)| for any µ ∈ Cω∗ .

(5.3.5)

and

Therefore when Qω is P -a.s. finite the cones Cω and Cω∗ satisfy Assumption 4.1.1 with the above Kω , Mω , the functional lω defined in (5.5.11) and the functional κω ∈ (Bω∗ )∗ given by κω µ = µ1 which is G-measurable, where G was defined in Section 5.1.

page 202

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Application to random locally distance expanding covering maps

LNcov

203

(iii) For any f ∈ Bω there exist f1 , ..., f8 ∈ Cω so that f = f1 +f2 +...+f8 and kf1 kα,ξ + kf2 kα,ξ + ... + kf8 kα,ξ ≤ rω kf kα,ξ

(5.3.6)

where rω = 4(1 + sQ2 ω ). Hence Assumption 4.1.3 holds true with kω = 8 and the above rω when Qω is P -a.s. finite.

5.4

RPF triplets

Theorem 5.4.1. Suppose that Assumption 5.2.1 holds true and that Qω is P -a.s. finite. (i) The random operators Lω z , z ∈ C, map Bω to Bθω , they are continuous with respect to the k · kα,ξ -norms and are analytic in z. As a consequence, under Assumption 5.2.2 the family Aω z , ω ∈ Ω is G-measurable in the sense of Section 4.1 where G is defined in Section 5.1. (ii) Suppose, in addition, that there exists δ > 0 so that γω ≥ 1 + δ, P -a.s. and that the random variables Hω , nω , kφω k∞ , kuω k∞ and Dω,nω

(5.4.1)

are bounded. Then Qω is a bounded random variable and Assumption 4.1.2 ω holds true with Aω z = Lz in some deterministic neighborhood U of 0. Corollary 5.4.2. Suppose that all the assumptions of Theorem 5.4.1 hold true. Let U be the neighborhood of 0 from Theorem 5.4.1 (ii). Then P -a.s. for any z ∈ U there exists a unique triplet λω (z), hω (z) and νω (z) consisting of a nonzero complex number λω (z), a complex function hω (z) ∈ Cω and a complex continuous linear functional νω (z) ∈ Cω∗ such that ∗ Aω hω (z) = λω (z)hθω (z), (Aω z ) νθω (z) = λω (z)νω (z) and

(5.4.2)

νω (z)hω (z) = νω (z)1 = 1. For any z ∈ U the maps ω → λω (z) and (ω, x) → hω (z)(x), (ω, x) ∈ E are measurable and the family νω (z) is measurable in ω. When z = t ∈ R then λω (t) > 0, the function hω (t) is strictly positive, νω (t) is a probability measure and the equality νθω (t) Lω t g) = λω (t)νω (t)(g) holds true for any bounded Borel function g : Eω → C and when the maps Tω satisfy the assumptions from [53] then the triplet (λω (t), hω (t), νω (t)) coincides with the one constructed there with the function φ + tu.

page 203

March 13, 2018 15:46

ws-book9x6

204

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Moreover, when Lω is bounded then this triplet is analytic and uniformly bounded around 0. Namely, the maps
∗ λω (·) : U → C, hω (·) : U → Hωα,ξ and νω (·) : U → Hωα,ξ are analytic, where (Hωα,ξ )∗ is the dual space of Hωα,ξ , and there exists a deterministic neighborhood U0 ⊂ U of 0 such that for any k ≥ 0 there is a constant Ck > 0 so that   (k) (k) max sup |λ(k) (z)|, sup kh (z)k , sup kν (z)k ≤ Ck , P -a.s. α,ξ α,ξ ω ω ω z∈U0

z∈U0

z∈U0

(5.4.3) where g (k) stands for the k-th derivative of a function on the complex plane which takes values in some Banach space and kνkα,ξ is the operator norm of a linear functional ν : Hωα,ξ → C. Furthermore, the exponential convergences (4.2.2) and (4.2.3) hold true ω with Aω z = Lz and all the random variables k1 (ω), Cω , Dω and cω described in Theorem 4.2.2 can be replaced by constants when z ∈ U0 . Theorems 5.3.1 and 5.4.1 will be proved in the remaining part of this chapter. Before proving them we will describe their application to the situation of random transfer operators arising in the proof of the nonconventional LLT in Chapter 2. Let (X , ρ) be the metric space and T : X → X be the locally distance expanding map described in Section 2.5. Consider the measure preserving system (ΩΘ , FΘ , PΘ , θ), θ = ϑ−1 and the random ` family of transfer operators Lω z defined in Section 2.7.1 using the map T ` and the H¨ older continuous functions f : X → R and F : X → R. Let E = Ω × X , namely we consider the case when Eω = X for any ω. Consider ϑ−1 ω ω = Lθω the random transfer operators Lω z . Then for z given by Lz = Lz any n ∈ N, Lθz

−n

ω,n

n−1

ϑ ϑω = Lω z ◦ Lz ◦ · · · ◦ Lz

ω

and therefore Theorems 2.7.1 and 2.7.2 will follow if Corollary 5.4.2 holds true for the random transfer operators Lω z presented above. For any ω ∈ ΩΘ , set `

Tω = T , φ(ω, x) =

`−1 X

f (T k x) and u(ω, x) = Fω (x) = F (p0 (θ−1 ω), x),

k=0

where p0 is defined in Section 2.7.1. We claim that the assumptions of Corollary 5.4.2 hold true for this choice of Tω , φ and u. Indeed, φ and u are measurable since f and F are continuous and the random variables kφω k∞ and kuω k∞ are bounded since f and F are bounded. Moreover, the transfer

page 204

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

205

Application to random locally distance expanding covering maps

operators Lω z are measurable in the sense required in Assumption 5.2.2 since Tω does not depend on ω and φ and u are measurable. Furthermore, it follows by the definition of the map T introduced in Section 2.5.1 together with the pairing property discussed in Section 2.5.2 that the map Tω = T ` satisfies the assumptions from Section 5.1 with constant nω , γω and Dω . Finally, Assumption 5.2.1 holds true with a constant Hω since f ∈ Hα,ξ (X ) and F satisfies (2.5.18) and we can guarantee that Lω would not depend on ω since Eω = X . We conclude that Corollary 5.4.2 holds true with the θω random transfer operators Lω z = Lz and the system (ΩΘ , FΘ , PΘ , θ). 5.5

Properties of cones: proof of Theorem 5.3.1

Let ω ∈ Ω be such that Qω < ∞. We begin with the following lemma. Lemma 5.5.1. For any g ∈ Cω,R , Lω kgkα,ξ ≤ Kω,R lω (g)

(5.5.1)

where α

α

Kω,R = Lω (sQω esQω ξ + 1)esQω ξ . In particular Cω,R ⊂

(5.5.2)

α,ξ Hω,R .

This lemma means that the aperture of the real cone Cω,R (when considered α,ξ as a subset of Hω,R ) does not exceed Kω,R . The proof of the lemma goes exactly as the proof of Lemma 9.7 from [53] and is given here for readers’ convenience. Proof. Let g ∈ Cω,R . We first show that α

kgk∞ ≤ lω (g)esQω ξ .

(5.5.3)

Indeed, for any x ∈ Eω there exists an index 1 ≤ j ≤ Lω so that x ∈ B(xω,j , ξ). Therefore α

g(x) ≤ g(xω,j )esQω ξ ≤ lω (g)esQω ξ

α

and (5.5.3) follows by taking supremum over all x ∈ Eω . Next, let x, x0 ∈ Eω be such that ρ(x, x0 ) < ξ and g(x) ≥ g(x0 ). Then
α 0 |g(x) − g(x0 )| = g(x) − g(x0 ) ≤ esQω ρ (x,x ) − 1 g(x0 ) and so by the mean value theorem, |g(x) − g(x0 )| ≤ sQω esQω ρ

α

(x,x0 ) α

ρ (x, x0 )kgk∞ .

page 205

March 13, 2018 15:46

206

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Reversing the roles of x and x0 , we conclude that this inequality holds true for any x and x0 such that ρ(x, x0 ) < ξ. Dividing by ρα (x, x0 ) and taking supremum over all possible choices of x and x0 we conclude that α

vα,ξ (g) ≤ sQω esQω ξ kgk∞ which implies that α

kgkα,ξ ≤ (sQω esQω ξ + 1)kgk∞ and (5.5.1) follows now from (5.5.3). Next, by Lemma 5.5.1 the cone Cω,R is a closed subset of the real Banach α,ξ space Hω,R , namely it is closed with respect to the norm k·kα,ξ . We conclude that the cone Cω,R is a real proper closed convex cone (see Appendix A) α,ξ when considered as a subset of Hω,R . Let Cω ⊂ Hωα,ξ be the canonical complexification of Cω,R (see Appendix A) which according to (A.2.2) can also be written in the form
Cω = C0 Cω,R + iCω,R = C0 {x + iy : x ± y ∈ Cω,R }, C0 = C \ {0}. The following result is a corollary of Lemma 5.5.1 above and Lemma 5.3 in [58]. Corollary 5.5.2. For any g ∈ Cω , klω kα,ξ kgkα,ξ ≤ Kω |lω (g)| where

(5.5.4)

√ Kω = 2 2Kω,R

and therefore the aperture of the cone Cω (as a subset of exceed Kω .

(5.5.5) Hωα,ξ )

does not

Another corollary of Lemma 5.5.1 goes as follows. Corollary 5.5.3. The cone Cω is linearly convex, namely for any g 6∈ Cω there exist µ ∈ Cω∗ such that µ(g) = 0. Proof. Since Qω is finite it follows from (5.5.1) that lω does not vanish on 0 Cω,R and so the canonical complexification Cω of Cω,R is linearly convex by Lemma 4.1 from [19]. We prove next the following. Lemma 5.5.4. The statement of Theorem 5.3.1 (iii) holds true.

page 206

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Application to random locally distance expanding covering maps

207

This lemma is a consequence of Lemma 3.13 from [53], and for readers’ convenience we provide the proof here. Proof. Let g ∈ Hωα,ξ be a nonnegative function. Consider the function q=g+

vα,ξ (g) sQω

(5.5.6)

which satisfies vα,ξ (g) = vα,ξ (q) and kqkα,ξ ≤ 1 +

1
kgkα,ξ . sQω

As in Lemma 3.13 from [53], this function is a member of Cω,R,s . Indeed, let x, x0 ∈ Eω be such that ρ(x, x0 ) < ξ. We have to show that q(x) ≤ q(x0 )esQω ρ

α

(x,x0 )

.

(5.5.7)

If q(x) ≤ q(x0 ) then this inequality trivially holds true, while when q(x) > q(x0 ) we have 0≤

|q(x) − q(x0 )| vα,ξ (g)ρα (x, x0 ) q(x) − 1 = ≤ ≤ sQω ρα (x, x0 ) q(x0 ) q(x0 ) q(x0 ) v

(g)

where we have used that q ≥ α,ξ sQω . Inequality (5.5.7) follows now from the inequality 1 + t ≤ et which holds true for any t ≥ 0. The constant function v (g) g2 ≡ − α,ξ sQω is a member of the complex cone Cω , and we conclude that any nonnegative member of Hωα,ξ can be represented as a sum of two members q = g1 and g2 of Cω which satisfy kg1 kα,ξ + kg2 kα,ξ ≤ 1 +

2
kgkα,ξ . sQω

Finally, any complex valued function g ∈ Hωα,ξ can be written as a linear combination of four nonnegative members of Hωα,ξ whose k · kα,ξ norms do not exceed kgkα,ξ , and thus can be written as a sum of eight members g1 , ...., g8 of the cone Cω,s which satisfy 8 X

kgi kα,ξ ≤ rω kgkα,ξ

i=1

where rω = 4(1 +

2 sQω ).

(5.5.8)

page 207

March 13, 2018 15:46

208

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Next, we will show that dual cone Cω∗ has bounded aperture. More precisely, we prove the following. Lemma 5.5.5. For any µ ∈ Cω∗ , kµkα,ξ ≤ Mω |µ(1)| = Mω |κω (µ)|

(5.5.9)

where α

Mω = 8(1 − e−sQω ξ )−2

(5.5.10)

and κω is the linear functional given by κω (µ) = µ(1) which satisfies kκω kα,ξ = 1. When Qω is P -a.s. finite this lemma means that (4.1.5) in Assumption 4.1.1 holds true with the above κω and Mω . We remark that in this case κω is clearly measurable in ω in the sense of the definition at the beginning of Section 4.1 with the σ-algebra G introduced in Section 5.1. Before proving this lemma we need the following notations. Let x, x0 ∈ Eω be such that 0 < ρ(x, x0 ) < ξ and consider the linear functional lω,x,x0 : CEω → C given by the formula lω,x,x0 (g) = g(x) − e−sQω ρ

α

(x,x0 )

g(x0 )

(5.5.11)

where CEω is the space of all functions g : Eω → C. Denote by Λω the set of linear functionals λ ∈ (Hωα,ξ )∗ which either have the form λ(g) = g(x) for some x ∈ Eω or the form λ = lω,x,x0 for some distinct x, x0 ∈ Eω such that ρ(x, x0 ) < ξ. Then we can write α,ξ Cω,R = {g ∈ Hω,R : λ(g) ≥ 0 ∀ λ ∈ Λω },

(5.5.12)

namely, the family Λω generates cone Cω,R . Therefore, by (A.2.3) we can write ¯ 1 (g)λ2 (g)) ≥ 0 ∀ λ1 , λ2 ∈ Λω } Cω = {g ∈ Hωα,ξ : 0 and κ ∈ (0, 1] such that P -a.s. for all xi , zi ∈ R℘ , i = 1, ..., `, |Fω (x) − Fω (z)| ≤ K[1 +

` ` X X (|xi |ι + |zi |ι )] |xj − zj |κ i=1

(7.5.5)

i=1

and |Fω (x)| ≤ K[1 +

` X

|xi |ι ]

(7.5.6)

i=1

where x = (x1 , ..., x` ) and z = (z1 , ..., z` ). For any N ≥ 1 set ω SN =

N X

ω ω Fθn ω (ξnω , ξ2n , ..., ξ`n ).

n=1

The main goal of this section is to prove a central limit theorem (CLT) for the normalized sums
1 ω ω − cN,ω ZN = N − 2 SN

page 258

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Limit theorems for processes in random environment

LNcov

259

for P -a.a. ω, where cN,ω are appropriate centralizing constants. For each θ > 0 and ω, set θ γθ,ω = sup kξnω kθLθ = sup EPω |ξnω |θ . n≥0

(7.5.7)

n≥0

Our results will rely on the following assumptions. Assumption 7.5.3. There exist b ≥ 2, q ≥ 1 and m > 0 such that ι κ 1 > + b m q and P -a.s. γm,ω < ∞ and γιb,ω < ∞. We will also need Assumption 7.5.4. There exist d ≥ 1 and θ > 2 such that P -a.s. for any n ∈ N, ω κ −θ φω n + (βq,n ) ≤ dn

(7.5.8)

where q is specified in Assumption 7.5.3. Theorem 7.5.5. Suppose that Assumptions 7.5.1, 7.5.3 and 7.5.4 hold true. Then P -a.s. the limit N

X
2 1 ω EPω SN − F¯θn ω N →∞ N n=1

Dω2 = lim exists, where F¯ω =

Z Fω (x1 , x2 , ..., x` )dµ(x1 )dµ(x2 )...dµ(x` )

and µ is the measure specified in Assumption 7.5.1. Moreover, for P -almost all ω the sequence of random variables 1

ω N − 2 SN −

N X


F¯θn ω , N ≥ 1

n=1

converges in distribution as N → ∞ towards a centered normal random variable with variance Dω2 .

page 259

March 13, 2018 15:46

ws-book9x6

260

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

In contrast to the case when ξ ω and Fω do not depend on ω (considered in Chapter 1), the proof that the limit Dω2 exists does not rely only on approximations of expectations such as the ones obtained in Lemma 3.3.2 and Corollary 1.3.14, and we explain here how to prove its existence in our situation. After this is proved, the above theorem will follow by Stein’s method. We begin with the following similar to [45] decomposition (see also Chapter 3). Let µ be the measure specified in Assumption 7.5.1 and write F

θn ω

(x1 , ...x` ) − F¯θn ω =

` X

Fθn ω,i (x1 , ..., xi )

i=1

where Z F

θ n ω,`

(x1 , ..., x` ) = F

θn ω

(x1 , ..., x` ) −

Fθn ω (x1 , ..., x` )dµ(x` )

and for all j = ` − 1, ` − 2, ..., 1, Z Fθn ω,j (ˆ xj ) = Fθn ω (x1 , ..., x` )dµ(x` )dµ(x`−1 ) · · · dµ(xj+1 ) Z − Fθn ω (x1 , ..., x` )dµ(x` ) · · · dµ(xj ), where x ˆj = (x1 , ..., xj ). Then relying on Assumptions 7.5.3 and 7.5.4, the arguments in [45] imply that Dω2 exists when the following limits X Z 1 Fθn ω,i (ˆ xi )Fθm ω,j (ˆ yj )dµω xi , yˆj ) lim i,j,u (ˆ N →∞ N 1≤n,m≤N in−jm=u

exist P -a.s. for any 1 ≤ i, j ≤ ` and u ∈ Z which is divisible by νi,j = gcd(i, j), where the measure µω i,j,u is constructed as follows. Let (sk , tk ), k = 1, ..., r be all the pairs (s, t) of natural numbers so that 1 ≤ s ≤ i, 1 ≤ t ≤ j and it = js and consider the measures µω k specified in Assumption 7.5.1. Then dµω xi , yˆj ) i,j,u (ˆ r Y Y Y = dµωusk (xsk , ytk ) dµ(xv ) dµ(yv0 ) k=1

i

v6=sk ∀k

(7.5.9)

v 0 6=tk ∀k

where note that usi k = sk n − tk m, k = 1, ..., r. Notice that the variables |ˆ xi |ι and |ˆ yj |ι are square integrable under the law µω i,j,u . Next, write ν = 0 0 νi,j = gcd(i, j), u = νu0 , i = i ν and j = j ν. When u = 0 then the equality in = jm means that n has the form n = kj 0 and then m = ki0 .

page 260

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Limit theorems for processes in random environment

LNcov

261

Suppose that u 6= 0 and let (n0 , m0 ) satisfy i0 n0 − j 0 m0 = u0 . Such n0 and m0 exist since gcd(i0 , j 0 ) = 1. Then any solution (n, m) of the equation in − jm = u = νu0 can be written in the form n = n0 + n ˜ and m = m0 + m ˜ where i˜ n = jn ˜ , namely (˜ n, m) ˜ = k(j 0 , i0 ) is a solution of the homogeneous 0 0 equation. Therefore, we can write θn = θn0 ◦ θkj and θm = θm0 ◦ θki , and so the above limits exist if the limits N Z 1 X yj )dµω xi , yˆj ) xi )Hθki0 ω,j (ˆ Gθkj0 ω,i (ˆ lim i,j,u (ˆ N →∞ N k=1

exist, P -a.s., where Gω,i = Fθn0 ω,i and Hω,j = Fθm0 ω,j . Note that the random functions Gω,i and Hω,j also satisfy (7.5.5) and (7.5.6), perhaps with some other constants. Changing the order of summation and integration, using the latter uniform (over ω) continuity in x ˆi and yˆj and then the dominated convergence theorem we see that such limits exist when almost sure limits of the form N 1 X lim gθj0 k ω hθi0 k ω N →∞ N k=1

exist, where g and h are bounded functions. Existence of such limits follows from the double recurrence theorem in [9], and we conclude that the limit Dω2 exists P -a.s. Next, suppose that Assumptions 7.5.2, 7.5.3 and 7.5.4 hold true. For any ω and n set Z n 2n `n F¯ω,n = Fθn ω (x1 , x2 , ..., x` )dµθ ω (x1 )dµθ ω (x2 )...dµθ ω (x` ). As in Theorem 7.5.5, when the limit N

Dω2

X
2 1 ω = lim E SN − F¯ω,n N →∞ N n=1

exists P -a.s. then using Stein’s method we obtain that the sequence 1

ω − N − 2 SN

N X


F¯ω,n , N ≥ 1

n=1

converges in distribution for P -a.a. ω as N → ∞ towards a centered normal random variable with variance Dω2 . In order to avoid a tedious presentation we will not formulate here a theorem stating that Dω2 exists under certain conditions, but instead we will explain how to prove its existence in several particular situations which will be easier to describe after the following.

page 261

March 13, 2018 15:46

ws-book9x6

262

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Consider the functions Fω,j,n , 1 ≤ j ≤ ` given by Z `n n Fω,`,n (x1 , ..., x` ) = Fθ ω (x1 , ..., x` ) − F (x1 , ..., x` )dµθ ω (x` ) and for j = ` − 1, ` − 2, ..., 1, Fω,j,n (x1 , ..., xj ) Z =

`n

(`−1)n

(j+1)n

ω ω Fθn ω (x1 , ..., x` )dµθ ω (x` )dµθ (x`−1 ) · · · dµθ (xj+1 ) Z `n (`−1)n jn ω − Fθn ω (x1 , ..., x` )dµθ ω (x` )dµθ (x`−1 ) · · · dµθ ω (xj )

where the measures µθ write F

θn ω

m

ω

are specified in Assumption 7.5.2. Then we can

(x1 , ..., x` ) − F¯ω,n =

` X

Fω,j,n (x1 , ..., xj ).

j=1

For any 1 ≤ i, j ≤ ` consider again the pairs (n, m) satisfying in − jm = u for some fixed u which is divisible by ν = gcd(i, j). Relying on Assumptions 7.5.3 and 7.5.4 the proof that Dω2 exists will proceed similarly to the situation of Theorem 7.5.5 assuming that limits X Z 1 Fω,i,n (ˆ xi )Fω,j,m (ˆ yj )dµω,n,m xi , yˆj ) lim i,j,u (ˆ N →∞ N 1≤n,m≤N in−jm=u

exist P -a.s, where the family of laws (θ n ω,θ 2 ω,...,θ `ω ,θ m ω,θ 2m ω,...,θ `m ω)

µω,n,m i,j,u = µi,j,u is given by

dµω,n,m xi , yˆj ) i,j,u (ˆ =

r Y k=1

sk n

dµθusk ω (xsk , ytk ) i

Y v6=sk ∀k

dµθ

vn

ω

(xv )

Y

dµθ

v0 m

ω

(yv0 ),

v 0 6=tk ∀k b

ω the pairs (sk , tk ), k = 1, ..., r are described before (7.5.9) and µθa−b ,a 0 for P -a.a. ω, |F (ω, x) − F (ω, y)| ≤ CF ρ`,∞ (x, y))α

(7.5.10)

for all x = (x1 , ..., x` ) and y = (y1 , ..., y` ) in X ` such that ρ`,∞ (x, y) := max ρ(xi , yi ) < ξ. 1≤i≤`

We assume in addition that Z ¯ Fω := Fω (x1 , x2 , ..., x` )dµ(x1 )dµ(x2 )...dµ(x` ) = 0, P -a.s.

page 263

March 13, 2018 15:46

264

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

where Fω (·) = F (ω, ·) for each ω. This is not really a restriction since we can always replace Fω with Fω − F¯ω . For any N ∈ N consider the random variable N X ω SN = Fθn ω (ζn , ζ2n , ..., ζ`n ). n=1

In this section we will explain how to prove an LLT for the sequences of ω random variables SN , N ≥ 1 for P -a.a. ω. By Theorem 2.2.3, once a (nondegenerate) CLT is established the appropriate LLT follows in both lattice and non-lattice cases when Assumptions 2.2.1 and 2.2.2 hold true, and we will focus on showing that the CLT holds true and providing conditions for the latter assumptions to be satisfied. Let Z0 be a random element of X which is distributed according to µ and set Zm = T m Z0 for any m ∈ N. Let Fm,n , −∞ ≤ m ≤ n ≤ ∞ be the family of σ-algebras generated by cylinder sets with coordinates indexed by max(0, m), ..., max(0, n). Then, in our circumstances (see [10]) there exist constants d > 0 and c ∈ (0, 1) so that for any n, r ≥ 0, φn ≤ dcn

(7.5.11)

and β∞ (r) := sup kρ(Zn , Zn,r )kL∞ ≤ dcr

(7.5.12)

n≥0

where Zn,r is an Fn−r,n+r -measurable random element of X whose coordinates with indexes n − r ≤ i ≤ n + r coincide with the coordinates of Zn with these indexes. Relying on (7.5.11) and (7.5.12) instead of (7.5.4) and taking into account (2.5.11), we obtain similarly to Theorem 7.5.5 that the limit 1 ω 2 E(SN ) Dω2 = lim N →∞ N 1

exists P -a.s. and N − 2 Snω converges in distribution as N → ∞ towards a centered normal random variable with variance Dω2 . Next, relying on (7.5.11) and (7.5.12) and the arguments in Sections 2.3 and 2.7, in order to show that Assumptions 2.2.1 and 2.2.2 are satisfied it is sufficient to approximate for P -a.a. ω the expectations

EΘ

N Y

(θ n ω,Θn )

∞

Lit

(7.5.13)

n=M +1 (ω,¯ x)

` where M = M` (N ) = N − 2[ N −N ], Lit g = L` (eitFω (¯x,·) g) and the 2 process Θ = {Θn : n ≥ 0} is defined in Section 2.7.1. Let M (Θ) =

page 264

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Limit theorems for processes in random environment

LNcov

265

(ΩΘ , BΘ , PΘ , ϑ) be the invertible MPS corresponding to the process Θ (see Section 2.7.1) and consider the product system τ = θ × ϑ. Then the expression in (7.5.13) can be written in the form Z N

Y τ n (ω,¯ ω)

dPΘ (¯ Lit ω) (7.5.14) ∞ n=M +1 (ω,¯ ω)

where Lz g = L` (gezFω (p0 (¯ω),·) ) for any complex z and the map p0 : `−1 ΩΘ → X is defined in Section 2.7.1. 2 Let Dω,` be the limit defined similarly to Dω2 but with the function Fω,` (defined after Theorem 7.5.5) in place of Fω (whose existence follows similarly to Dω2 ’s). 2 Proposition 7.5.6. Suppose that Dω,` > 0. Then, for P -almost all ω the ω sequence of random variables SN , N ≥ 1 satisfies Assumption 2.2.2.

We sketch here the proof of the above proposition. Observe that we can apply in these circumstances the RPF theorem with the map τ −1 and τ −1 (ω,¯ ω) . The main modification of the the random transfer operators Lz arguments in Section 2.8 is in the proof of an appropriate version of (2.8.16). In our situation we first consider the random variables Vk : Ω×ΩΘ → R, k ≥ 1 given by Vk (ω, ω ¯ ) = Varµ

k−1 X

Fτ j (ω,¯ω),` (ζ`j ) = Eµ

j=0

k−1 X


2 Fτ j (ω,¯ω),` (ζ`j )

j=0

where F(ω,¯ω),` (·) = Fω,` (p0 (¯ ω ), ·). (i) {ζn

Let ζ (i) = : n ≥ 0}, i = 1, 2, ..., ` be independent copies of ζ and for each j ≥ 0 set
˜ j = ζ (1) , ζ (2) , ..., ζ (`) . Θ j

2j

`j

Then using arguments similar to the ones proceeding Theorem 7.5.5 (or by Theorem 2.3 in [38]) the limit k−1 X
˜ j) 2 ˜ `2 = lim 1 E ˜ Fθj ω,` (Θ D Θ k→∞ k j=0

exists P -a.s. and it does not depend on ω. Using the arguments in Section ˜ 2 > 0 if and only if D2 > 0. We conclude that 4 of [30] we see that D ` ω,` Z 1 ˜ `2 > 0. Vk (ω, ω ¯ )dPθ (¯ ω) = D lim k→∞ k

page 265

March 13, 2018 15:46

266

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Therefore, there exists a constant b1 > 0 so that for a sufficiently large k, Z  1 Vk (ω, ω ¯ )dPθ (¯ ω ) > b1 > 0. P ω: k Fixing a sufficiently large k and considering the jk-th hitting times, j = 1, 2, ... by θn , n ≥ 1 to the above set of positive probability, we can repeat the argument in the proof of (2.8.16) using the blocks generated by these hitting times instead of the blocks generated by the time intervals [ik + 1, (i + 1)k], i = 0, 1, 2, ..., taking into account that by the mean ergodic theorem the number of such hitting times until time n grows linearly in n. Next, we will provide conditions which guarantee that Assumption 2.2.1 holds true. Assumption 7.5.7. The space Ω is a topological space, F is the Borel σalgebra and P assigns positive mass to open sets. The map θ is continuous and there exist ω0 ∈ Ω and n0 ∈ N such that θn0 ω0 = ω0 . Moreover, the map ω → Fω is continuous at the points ω = θi ω0 , 0 ≤ i < n0 when considered as a map from Ω to the space Hα,ξ (X ` ). Let x ¯0 be a periodic point of Tˆ = T ×T 2 ×· · ·×T `−1 and let m0 be such m0 ˆ ¯0 = x ¯0 . When Assumption 7.5.7 holds true then we can always that T x assume that n0 = m0 since otherwise we can replace both with m0 n0 . Assumption 7.5.8. The functions Fθi ω0 (·), 0 ≤ i < m0 satisfy Assumption 2.6.1. Consider next the function Fω0 ,¯x0 ,m0 : X → R given by Fω0 ,¯x0 ,m0 (x) =

m 0 −1 X

¯0 , T `j x). Fθj ω0 (Tˆj x

j=0

We distinguish between lattice and non-lattice cases similarly to Section 2.6. Namely, we call the case a non-lattice one when the function Fω0 ,¯x0 ,m0 is non-arithmetic in the sense of [28], i.e. if for any t ∈ R \ {0} there exist no nonzero g ∈ Hα,ξ and λ ∈ C, |λ| = 1 such that eitFω0 ,¯x0 ,m0 g = λg ◦ T `m0 , µ-a.s.

(7.5.15)

We call the case a lattice one when the function Fω0 ,¯x0 ,m0 cannot be written in the form Fω0 ,¯x0 ,m0 = a + β − β ◦ T `m0 + h0 k, µ-a.s.

(7.5.16)

for some h0 > h, a ∈ R, β : X → R such that eiβ ∈ Hα,ξ and an integer valued function k : X → Z.

page 266

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Limit theorems for processes in random environment

LNcov

267

Proposition 7.5.9. Suppose that Assumptions 7.5.7 and 7.5.8 hold true. Then in the above lattice and non-lattice cases, for P -almost all ω the seω quence of random variables S˜N , N ≥ 1 satisfies Assumption 2.2.1. The main difference in the proof of this proposition in comparison to Section 2.9 is as follows. Similarly to Section 2.10, given a compact interval J ⊂ R not containing the origin, we consider the visiting sequence n1 < n2 < n3 < ... with respect to the map θ to a neighborhood of ω0 for which relations similar to the ones in (2.10.2) hold true (in lattice and non-lattice cases). Now, instead of counting the number of disjoint blocks belonging to some Bowen ball which are contained in the word Θ1 Θ2 ...Θn and estimate the probability that there is proportional to n amount of blocks (see Corollary 2.9.3 together with (2.5.11)), we count the number of blocks beginning with indexes j of the form j = ni . By the mean ergodic theorem ni grows linearly in i, and an appropriate version of Corollary 2.9.3 follows. Next, consider sums of the form WNω (x) =

N X

Fθn ω (T n x, T 2n x, ..., T `n x)

n=1

where x is distributed according to µ. Then the LLT under appropriate conditions follows essentially in the same way as in Section 2.11.3. The situation of a two sided subshift of finite type can also be considered, observing that an appropriate version of Lemma 2.11.2 follows essentially in the same way.

page 267

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

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

PART 3

Appendix

269

LNcov

page 269

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

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Appendix A

Real and complex cones

In this appendix we describe the theory of complex projective Hilbert metrics associated with complex cones, and the contraction properties of linear maps between such cones with respect to the corresponding Hilbert metrics. We introduce the basic notations and tools which were developed first in [58] and then in [18]. Still, for readers’ convenience, we first recall the definition and basic properties of real cones and the real Hilbert metric associated with it (see [7], [49], [13], [18], [19] and [58]). A.1

Real cones and real Hilbert metrics

Let X be a real vector space. A subset CR ⊂ X is called a proper real convex cone (or, in short, a real cone) if C is convex, invariant under multiplication of nonnegative numbers and CR ∩ −CR = {0}. Next, assume that X is a Banach space and let CR ⊂ X be a closed real cone. For any nonzero elements f, g of CR set βCR (f, g) = inf{t > 0 : tf − g ∈ CR }

(A.1.1)

where we use the convention inf ∅ = ∞. We note that βCR (f, g) > 0 since otherwise −g lays in (the closure of) CR which together with the inclusion g ∈ CR implies that g = 0. The real Hilbert (projective) metric dCR : CR × CR → [0, ∞] associated with the cone is given by
dCR (f, g) = ln βCR (f, g)βCR (g, f ) (A.1.2) where we use the convention ln ∞ = ∞. Next, let X1 and X2 be two real Banach spaces and let Ci ⊂ Xi , i = 1, 2 be two closed real cones. Let A : X1 → X2 be a continuous linear transformation such that AC1 \ {0} ⊂ C2 \ {0} and set D=

sup

dC2 (Ax1 , Ax2 ).

x1 ,x2 ∈C1 \{0}

271

page 271

March 13, 2018 15:46

272

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

The following theorem is a particular case of Theorem 1.1 in [49] (see [7] for the case when C1 = C2 ). Theorem A.1.1. For any nonzero x, x0 ∈ C1 we have 1
dC2 (Ax, Ax0 ) ≤ tanh D dC1 (x, x0 ) 4 where tanh ∞ := 1 This lemma means that any linear map between two (punctured) real closed cones weakly contracts the corresponding Hilbert metrics, and this contraction is strong if the (Hilbert) diameter of the image is finite. A.2 A.2.1

Complex cones and complex Hilbert metrics Basic notions

Let Y be a complex Banach space. We say that C ⊂ Y is a complex cone if C0 C ⊂ C, where C0 = C \ {0}. The cone C is said to be proper if its closure C¯ does not contain any two dimensional complex subspaces. The dual cone C ∗ ⊂ Y ∗ is the set given by C ∗ = {µ ∈ Y ∗ : µ(c) 6= 0 ∀c ∈ C 0 } where C 0 = C \ {0}, and, as usual, Y ∗ is the space of all continuous linear functionals µ : Y → C equipped with the operator norm. It is clear that C ∗ is a complex cone. We will say that C is linearly convex if for any x 6∈ C there exists µ ∈ C ∗ such that µ(x) = 0, i.e. the complement of C is the union of the kernels Ker(µ), µ ∈ C ∗ . Then the complex cone C ∗ is linearly convex, since for any c ∈ C the corresponding evaluation map ν → ν(c) is a member of the dual cone (C ∗ )∗ of C ∗ . We introduce now the notation of the complex Hilbert projective metric δC of a proper complex cone defined in [18]. Let x, y ∈ C 0 and consider the set EC (x, y) given by EC (x, y) = {z ∈ C : zx − y 6∈ C}. Since C is proper (and C0 invariant) the set EC (x, y) is nonempty. When x and y are collinear set δC (x, y) = 0 and otherwise set b
∈ [0, ∞] δC (x, y) = ln a where a = inf |EC (x, y)| ∈ [0, ∞] and b = sup |EC (x, y)| ∈ [0, ∞]

page 272

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Real and complex cones

LNcov

273

are the “largest” and “smallest” modulus of the set EC (x, y), respectively. Observe that δC (x, y) = δC (c1 x, c2 y) for any c1 , c2 ∈ C0 = C \ {0}, i.e. δC is projective. When C is linearly convex then δC satisfies the triangle inequality and so it is a projective metric (see [18] and [19]). We remark that a different notion of a complex Hilbert metric dC was defined in [58], which was prior to the definition of δC . We refer the readers to Section 5 in [18] for relations between dC and δC , among them a certain equivalence between them for a wide class of complex cones, including canonical complexifications of real cones introduced in the next section. This means that for such cones it makes no difference whether we use δC or dC , and in this section we will only present results concerning δC . A.2.2

The canonical complexification of a real cone

Next, let X be a real Banach and let CR ⊂ X be a real cone. Let Y = XC = X + iX be its complexification (see Section 5 from [58]). Following [58], we define the canonical complexification of CR by
CC = {x ∈ XC : < µ ¯(x)ν(x) ≥ 0 ∀µ, ν ∈ CR∗ } (A.2.1) where CR∗ = {µ ∈ X ∗ : µ(c) ≥ 0 ∀c ∈ CR } and X ∗ is the space of all continuous linear functions µ : X → R equipped with the operator norm. Then CC is a proper complex cone (see Theorem 5.5 of [58]) and by [58] and [18] we have the following polarization identities
CC = C0 CR + iCR = C0 {x + iy : x ± y ∈ CR } (A.2.2) where we recall that C0 = C \ {0}. Moreover, when CR = {x ∈ X : µ(x) ≥ 0 ∀µ ∈ S} for some S ⊂ X ∗ , then
CC = {x ∈ XC : < µ ¯(x)ν(x) ≥ 0 ∀µ, ν ∈ S}

(A.2.3)

since S generates the dual cone CR∗ . We conclude this section with the following result which appears as Lemma 4.1 in [19]. Lemma A.2.1. A canonical complexification CC of a real cone CR is linearly convex if there exists a continuous linear functional which is strictly positive on CR0 = CR \ {0}.

page 273

March 13, 2018 15:46

274

A.2.3

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

Apertures and contraction properties

The aperture of a real cone CR in some real Banach space is defined (see [58]) by K(CR ) =

inf

sup

µ∈(CR∗ )0 x∈C 0 R

kµkkxk µ(x)

namely it is the infimum of K-values for which there exists a continuous linear functional µ ∈ CR∗ such that kxkkµk ≤ Kµ(x) for any x ∈ CR which (see again [58]) is also the infimum of K-values for which there exists a continuous nonzero functional µ ∈ CR∗ such that kxk ≤ µ(x) ≤ Kkxk for any x ∈ CR .

(A.2.4)

Now, following [58], the aperture of a complex cone KC in some complex Banach space is defined similarly by K(KC ) = inf ∗ sup

µ∈KC x∈K0 C

kµkkxk . |µ(x)|

Then for any K > 0 we have K(KC ) ≤ K if there exists a continuous linear functional µ ∈ KC∗ such that for any x ∈ KC , kxkkµk ≤ K|µ(x)|

(A.2.5)

and we note that K(KC ) < K implies that (A.2.5) holds true for some µ ∈ KC∗ . The following result appears in [58] as Lemma 5.3. Lemma A.2.2. Let X be a real Banach space and let Y be its complexification. Let CR ⊂ X be a real cone, and assume that (A.2.4) holds true with some µ and √ K. Then the complexification CC of CR satisfies (A.2.5) with µC and 2 2K, where µC is the unique extension of µ to the complexified space Y . Next, the following assertion is formulated as Theorem 3.1 in [19] and it summarizes some of the main results from [18].
Theorem A.2.3. Let X, k · k be a complex Banach spaces and C ⊂ X be a complex cone. (i) Suppose that the cone C is linearly convex and of bounded sectional aperture. Then (C 0 / ∼, δC ) is a complete metric space, where x ∼ y if and only if C0 x = C0 y.

page 274

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Real and complex cones

LNcov

275

(ii) Let K > 0 and µ ∈ C ∗ be such that (A.2.5) holds true for any x ∈ C. Then for any x, y ∈ C 0 ,

x K y

− δC (x, y).

≤

µ(x) µ(y) 2kµk (iii) Let C1 be a complex cone in some complex Banach space X1 , and A : X → X1 be a complex linear map such that AC 0 ⊂ C10 . Set ∆ = sup δC1 (Au, Av) u,v∈C 0

and assume that ∆ < ∞. Then for any x, y ∈ C 0 , ∆
δC (x, y). δC1 (Ax, Ay) ≤ tanh 4 Theorem A.2.3 (ii) means that any linear map between two (punctured) cones whose image has finite δC1 (Hilbert) diameter is a weak contraction with respect to the appropriate projective metrics and that this contraction is strong. A.2.4

Comparison of real and complex operators

Let X and Y be real Banach spaces and denote their complexifications by XC and YC . Let S ⊂ Y ∗ , consider the real cone CR = {y ∈ Y : s(y) ≥ 0 ∀s ∈ S} and denote its canonical complexification by CC . Theorem A.2.4. Let K ⊂ X be a real closed cone and denote its canonical complexification by KC . Let P : X → Y be a real linear transformation such that P (K0 ) ⊂ CR0 . Suppose that the real Hilbert diameter satisfies D = sup dCR (P u, P v) < ∞. u,v∈K0

Let A : XC → YC be a complex linear transformation and assume that there exists ε > 0 such that D δ = δ(ε, D) := 2ε(1 + cosh( )) < 1 2 and that for any s ∈ S and x ∈ K, |s(Ax) − s(P x)| ≤ εs(P x). Then

A(KC0 )

⊂

CC0

and for any y ∈ KC0 , δCC (Ay, P y) ≤ 3 ln

1 1−δ

and in particular sup δCC (Ax, Ay) ≤ D + 6 ln

0 x,y∈KC

1 1−δ

page 275

March 13, 2018 15:46

276

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

In the case when K = C this is Theorem 4.5 from [19] and the proof of Theorem A.2.4 goes exactly in the same way. Remark that a close result was first obtained in Proposition 6.3 from [58] with the aforementioned metric dCC . A.2.5

Further properties of complex dual cones

We begin with a result which is proved in Lemma 2.4 from [18]. Lemma A.2.5. Let C be a complex linearly convex proper cone in a complex Banach space. Then for any x, y ∈ C 0 , f (y)g(x) . δC (x, y) = sup ln f (x)g(y) f,g∈C ∗ Next, let C1 be a complex cone in some complex Banach space. A direct calculation shows that for f, g ∈ C1∗ we have n g(x) o EC1∗ (f, g) = : x ∈ C10 . f (x) This together with the definition of δC1∗ yields that f (x)g(y) f (x)g(y) . = sup ln δC1∗ (f, g) = ln sup 0 0 g(x)f (y) g(x)f (y) x,y∈C1

x,y∈C1

Combining this with Lemma A.2.5 we derive the following. Lemma A.2.6. Let X1 and X be complex Banach spaces and C1 ⊂ X1 and C ⊂ X be complex cones such that C is proper and linearly convex. Let A : X1 → X be a complex linear map so that AC10 ⊂ C 0 . Then the dual operator A∗ : X ∗ → X1∗ satisfies A∗ C ∗ ⊂ C1∗ (this is true in general, of course) and sup δC (Ax, Ay) = sup δC1∗ (A∗ µ, A∗ ν)

x,y∈C10

(A.2.6)

µ,ν∈C ∗

namely, both images have the same diameter with respect to the appropriate complex Hilbert metrics. We conclude this appendix with the following simple lemma.
Lemma A.2.7. Let X, k · k be a complex Banach space and let C ⊂ X 1 )= be a complex cone. Let x0 ∈ X and M ∈ (0, ∞) be such that B(x0 , M 1 ∗ {x ∈ X : kx − x0 k < M } ⊂ C. Then for any µ ∈ C , kµk ≤ M |µ(x0 )|

(A.2.7)

namely, the aperture of the dual cone does not exceed M as can be seen by considering the evaluation functional µ → µ(x0 ).

page 276

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Real and complex cones

LNcov

277

1 . Proof. Let µ ∈ C ∗ , λ ∈ C and h ∈ X be such that |λ| < 1 and khk ≤ M Then |µ(x0 ) + λµ(h)| > 0, since x0 + λh ∈ C. Suppose that |µ(h)| > |µ(x0 )| 0) and let λ = − µ(x µ(h) . Then |λ| < 1 and so we obtain that

|µ(x0 ) − µ(x0 )| = |µ(x0 ) + λµ(h)| > 0 which is a contradiction. Thus, |µ(y)| ≤ M |µ(x0 )| for any y ∈ X such that kyk ≤ 1 and (A.2.7) follows.

page 277

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

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Bibliography

[1] A.D. Barbour, Stein’s Method for Diffusion Approximations, Probab. Th. Rel. Fields 84 (1990), 297-322. [2] A.D. Barbour and S. Janson, A functional combinatorial central limit theorem, Electron. J. Probab. 14 (2009), 2352-2370. [3] A.D. Barbour and L.H.Y. Chen, An Introduction to Stein’s Method, Singapore University Press, Singapore, 2005. [4] J.R. Blum, D.L. Hanson and L.H. Koopmans, On the strong law of large numbers for a class of stochastic processes, Z. Wahrsch. verw. Geb 2 (1963), 1-11. [5] P. Billingsley, Convergence of Probability Measures, 2nd ed. Wiley, New York, 1999. [6] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. [7] G. Birkhoff, Extension of Jentzsch’s theorem, Trans. A.M.S. 85 (1957), 219227. [8] E. Bolthausen, Markov process large deviations in τ -topology, Stoch. Proc. Appl. (1987), 95-108. [9] J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math. 404 (1990), 140-161. [10] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, volume 470, Springer Verlag, 1975. [11] R.C. Bradley, Introduction to Strong Mixing Conditions, Volume 1, Kendrick Press, Heber City, 2007. [12] L. Breiman, Probability, SIAM, Philadelphia (1992). [13] P.J. Bushell, The Cayley-Hilbert metric and positive operators, Linear Alg. and Appl. 84 (1986), 271-281. [14] I.P. Cornfeld, S.V. Fomin, and Ya.G. Sinai, Ergodic Theory, Springer-Verlag, Berlin, 1982. [15] L.H.Y Chen and Q.M. Shao, Normal approximation under local dependence, Ann. Probab. 32 (2004), 1985-2028. [16] C. Castaing and M.Valadier, Convex analysis and measurable multifunctions, Lecture Notes Math., vol. 580, Springer, New York, 1977. [17] M.J. Kasprzak, A.B. Duncan and S.J. Vollmer, Note on A. Barbours paper

279

page 279

March 13, 2018 15:46

280

[18] [19] [20]

[21] [22] [23]

[24] [25]

[26] [27] [28]

[29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

on Steins method for diffusion approximations, Electron. Commun. Probab. 22 (2017), 1-8. L. Dubois, Projective metrics and contraction principles for complex cones, J. London Math. Soc. 79 (2009), 719-737. L. Dubois, An explicit Berry-Esseen bound for uniformly expanding maps on the interval, Israel J. Math. 186 (2011), 221-250. M.D. Donsker and S.R.S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure. Appl. Math. (1987), 1-47. H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, NJ 1981. H. Furstenberg, Nonconventional ergodic averages, Proc. Symp. Pure Math. 50 (1990), 43-56. Y. Guivar´ch and J. Hardy, Th´eor`emes limites pour une classe de chaˆınes de Markov et applications aux diff´eomorphismes d’Anosov, Ann. Inst. H. Poincar´e Probab. Statist. 24 (1988), no. 1, 73-98. A.L. Gibbs and F.E. Su, On choosing and bounding probability metrics, Int. Statist. Rev. 70 (2002), 419-435. Y. Gutman, W. Huang, S. Shao and X. Ye, Almost sure convergence of the multiple ergodic average for certain weakly mixing systems, arXiv:1612.02873v2. Y. Hafouta, Stein’s method for nonconventional sums, arXiv:1704.01094. P.G. Hall and C.C. Hyde, Martingale central limit theory and its application, Academic Press, New York, 1980. H. Hennion and L. Herv´e, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics vol. 1766, Springer, Berlin, 2001. Y. Hafouta and Yu. Kifer, A nonconventional local limit theorem, J. Theoret. Probab. 29 (2016), 1524-1553. Y. Hafouta and Yu. Kifer, Berry-Esseen type inequalities for nonconventional sums, Stoch. Proc. Appl. 126 (2016), 2430-2464. Y. Hafouta and Yu. Kifer, Nonconventional polynomial CLT, Stochastics, 89 (2017), 550-591. W. Huang, S. Shao and X. Ye, Pointwise convergence of multiple ergodic averages and strictly ergodic models, arXiv:1406.5930v2. I.A. Ibragimov and Yu.V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971. R.M. de Jong, A strong law of large numbers for triangular mixingale arrays, Stat. & Probab. Lett. 27 (1996), 1-9. J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd. ed., Springer-Verlag, Berlin, 2003. T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Yu. Kifer, Perron-Frobenius theorem, large deviations, and random perturbations in random environments, Math. Z. 222(4) (1996), 677-698. Yu. Kifer, Limit theorems for random transformations and processes in

page 280

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

Bibliography

LNcov

281

random environments, Trans. Amer. Math. Soc. 350 (1998), 1481-1518. [39] Yu. Kifer, Optimal stopping and strong approximation theorems, Stochastics, 79 (2007), 253-273. [40] Yu. Kifer, Thermodynamic formalism for random transformations revisited, Stoch. Dyn. 8 (2008), 77-102. [41] Yu. Kifer, Nonconventional limit theorems, Probab. Th. Rel. Fields, 148 (2010), 71-106. [42] Yu. Kifer, A nonconventional strong law of large numbers and fractal dimensions of some multiple recurrence sets, Stoch. Dynam. 12 (2012), 1150023. [43] Yu. Kifer, Nonconventional Poisson limit theorems, Israel J. Math. 195 (2013), 373–392. [44] Yu. Kifer, Strong approximations for nonconventional sums and almost sure limit theorems, Stochastic Process. Appl. 123 (2013), 2286-2302. [45] Yu. Kifer and S.R.S. Varadhan. Nonconventional limit theorems in discrete and continuous time via martingales, Ann. Probab. 42 (2014), 649-688. [46] Yu. Kifer and S.R.S. Varadhan, Nonconventional large deviations theorems, Probab. Th. Rel. Fields 158 (2014), 197-224. [47] Yu. Kifer and A. Rapaport, Poisson and compound Poisson approximation in conventional and nonconventional setups, Probab. Th. Rel. Fields 160 (2014), 797-831. [48] Yu. Kifer, Ergodic theorems for nonconventional arrays and an extension of the Szemer´edi theorem, Discrete Cont. Dynam. Sys. 38 (2018). [49] C. Liverani, Decay of correlations, Ann. Math. 142 (1995), 239-301. [50] S.V. Nagaev, Some limit theorems for stationary Markov chains, Theory Probab. Appl. 2 (1957), 378-406. [51] S.V. Nagaev, More exact statements of limit theorems for homogeneous Markov chains, Theory Probab. Appl. 6 (1961), 62-81. [52] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974. [53] V. Mayer, B. Skorulski and M. Urba´ nski, Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry, Lecture Notes in Mathematics, vol. 2036 (2011), Springer. [54] M. Pollicott, A complex Ruelle-Perron-Frobenius theorem and two counterexamples, Ergodic Theory Dynam. Systems, 4(1) (1984), 135-146. [55] D. Pollard, Convergence of stochastic processes, Springer, New York, 1984. [56] Y. Rinott, On normal approximation rates for certain sums of dependent random variables, J. Comput. Appl. Math. 55 (1994), 135-143. [57] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987. [58] H.H. Rugh, Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory, Ann. Math. 171 (2010), 1707-1752. [59] B.A. Sevast’yanov, Poisson limit law for a scheme of sums of dependent random variables, Theory of Probab. Appl. 17 (1972), 695-699. [60] A.N. Shiryaev, Probability, Springer-Verlag, Berlin, 1995. [61] C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 (1972), 583-602. Univ. California Press, Berkeley.

page 281

March 13, 2018 15:46

282

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

[62] C. Stein, Approximation Computation of Expectations, IMS, Hayward, CA (1986). [63] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1975.

page 282

March 13, 2018 15:46

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Index

canonical complexification, 277 dual cone, 277 Hilbert diameter, 276 real Hilbert metric, 275

approximation coefficients, 12, 88, 142, 261, 268 asymptotic variance, 13, 89, 133 nonconventional arrays, 154 Berry-Esseen fiberwise, vi, vii, 255 nonconventional, 14

decay of correlations complex Gibbs functional, 231 Gibbs measure, 229 distance Kolmogorov, 4 Wasserstein, 4 distance expanding maps, 66 deterministic, 81 random covering maps, 201 Doeblin condition, vi, 65, 66, 74 random, 236

central limit theorem, v, 4 functional, 4 nonconventional, 89 nonconventional arrays, 151 concentration inequality, 6 cones comparison, 221, 243, 279 complex cone, 276 aperture, 178, 206, 237, 278 complex Hilbert diameter, 178 complex Hilbert metric, 276 dual cone, 178, 212, 239, 276 Hilbert diameter, 279 Hilbert image diameter, 224, 244, 279 linearly convex, 276 proper, 276 random, 178, 206, 210, 212, 237 reproducing, 179, 207, 237 real cone, 275 aperture, 278

ergodic theorems, v functional central limit theorem, 43, 52 nonconventional, 51 Gibbs complex functional, 230 invariance, 230 measure, v, 65, 228 decay of correlations, 85, 229 invariance, 85, 229 283

page 283

March 13, 2018 15:46

284

ws-book9x6

Nonconventional Limit Theorems and Random Dynamics

LNcov

Nonconventional Limit Theorems and Random Dynamics

integral operator, 235 complex, 236 large deviations, v, 113 largest characteristic exponent analytic, 194 random transfer operators, 232 local central limit theorem fiberwise, vii, 248, 267 Fourier approach, 67 general, 67 lattice case, 67 nonconventional, 92, 133 non-lattice case, 67 nonconventional, 91, 133 nonconventional, vi, 74, 91, 129, 267 local limit theorem fiberwise, 127 martingale approximation, vi, 159 mixing, v, 3, 4, 88, 268 coefficients, 8, 10, 12, 142, 261 nonconventional arrays, vii, 139 central limit theorem, 151 Poisson limit theorem, 161 strong law of large numbers, 141 sums, v Berry-Esseen, 14 central limit theorem, 261 functional CLT, 51 local limit theorem, 73, 91, 267

Poisson limit theorem, v nonconventional, 161 pressure analytic, 99, 197, 250 random dynamics fiberwise Berry-Esseen, vi, vii, 255 fiberwise local limit theorem, vii, 248, 267 nonconventional sums reduction, 97 RPF theorem exponential convergence random complex, 180 random complex integral operators, 238 random complex transfer operators, 207 real deterministic, 85 random complex, 179 integral operators, 238 nonconventional sums, 93 transfer operators, 207 real deterministic, 85 Stein equation, 5 identity, 34 method, vi, 5 operator, 5 subshift of finite type, v, 65, 84, 88, 121, 129, 267 transfer operator, 66 deterministic, 81 random, 92 complex, 205

page 284