Limit Theorems for Random Fields

Home , Random field, Stationary sequence

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Department of Mathematical Sciences College of Arts and Sciences University of Cincinnati, July 2019

Author: Na Zhang Chair: Magda Peligrad, Ph.D. Degrees: B.S. Mathematics, 2011, Committee: Wlodzimierz Bryc, Ph.D. Huaibei Normal University Yizao Wang, Ph.D. M.S. Financial Mathematics, 2014, Soochow University Abstract

The focus of this dissertation is on the dependent structure and limit theorems of high dimensional probability theory. In this dissertation, we investigate two related topics: the Central Limit Theorem (CLT) for stationary random fields (multi-indexed random variables) and for Fourier transform of stationary random fields.

We show that the CLT for stationary random processes under a sharp projective condition introduced by Maxwell and Woodroofe in 2000 [MW00] can be extended to random fields. To prove this result, new theorems are established herein for triangular arrays of martingale differences which have interest in themselves. Later on, to exploit the richness of martingale techniques, we establish the necessary and sufficient conditions for martingale approximation of random fields, which extend to random fields many corresponding results for random sequences (e.g. [DMV07]). Besides, a stronger form of convergence, the quenched convergence, is investigated and a quenched CLT is obtained under some projective criteria.

The discrete Fourier transform of random fields, (Xk)k∈Zd (d ≥ 2), where Z is the set of integers, is defined as the rotated sum of the random fields. Being one of the important tools to prove the CLT for Fourier transform of random fields, the law of large numbers

(LLN) is obtained for discrete Fourier transform of random sequences under a very mild regularity condition. Then the central limit theorem is studied for Fourier transform of random fields, where the dependence structure is general and no restriction isassumed on the rate of convergence to zero of the covariances.

First of all, I would like to thank my thesis advisor, Professor Magda Peligrad, for her excellent guidance, constant help and support since 2015. Without her great patience, stimulating discussions and thorough feedback, this dissertation would not have been possible. She has been a wonderful advisor to work with during my research career.

Besides, I am indebted to Professor Yizao Wang for his help, instruction in the probability theory course, and many insightful discussions on research. I would also like to thank Professor Magda Peligrad, Professor Yizao Wang and Professor Wlodzimierz Bryc for serving on my dissertation committee and for their time and effort to read and to give feedback on this dissertation.

In addition, I would like to extend my gratitude to the Department of Mathematical

Sciences at University of Cincinnati for its generous support for my graduate studies and the Charles Phelps Taft Research Center for funding the Dissertation Fellowship to me during my last year of study. Moreover, I would like to thank the faculty members, staff and my colleagues for making me feel at home in the department.

Finally, I am eternally grateful to my parents for their unconditional love, support and understanding. If it were not for them, I would not be where I am today. Also, I am grateful to my husband, Zheng Zhang, who stands by me all the time.

The research in this dissertation was supported in part by my advisor’s research grants

DMS-1512936 and DMS-1811373 from the National Science Foundation, and the author would like to thank the NSF for their generous contribution.

iv Contents

Abstract ii

Acknowledgments iv

1 Introduction 1

1.1 Limit theorems for random fields ...... 2

1.2 Limit theorems for Fourier transform of random fields ...... 6

2 Background 10

2.1 Stochastic processes ...... 11

2.1.1 Stationary processes and measure preserving transformations . . . . . 11

2.1.2 Shift transformations and canonical representation of stochastic pro-

cesses ...... 13

2.1.3 Markov chains ...... 14

2.1.4 Martingales ...... 16

2.2 Dunford-Schwartz operators and the Ergodic theorem ...... 17

2.2.1 Positive Dunford-Schwartz operators ...... 17

v 2.2.2 The Ergodic theorem for positive Dunford-Schwartz operators . . . . . 18

2.2.3 Multiparameter ergodic theorem for Dunford-Schwartz operators . . . 20

2.3 Preliminaries for Fourier transform ...... 21

2.3.1 Weak stationarity and spectral density ...... 23

2.3.2 Relation with discrete Fourier transform ...... 26

3 Central limit theorem for random fields via martingale methods 30

3.1 Results ...... 31

3.2 Proofs ...... 37

3.3 Auxiliary results ...... 46

4 Martingale approximations for random fields 49

4.1 Results ...... 50

4.2 Proofs ...... 53

4.3 Multidimensional index sets ...... 55

4.4 Examples ...... 58

5 Quenched central limit for random fields 61

5.1 Assumptions and Results ...... 62

5.2 Proofs ...... 66

5.3 Random fields with multi-dimensional index sets ...... 76

5.4 Examples ...... 81

5.5 Appendix ...... 84

6 Law of large numbers for discrete Fourier transform 85

6.1 Law of large numbers for discrete Fourier transform ...... 86

6.2 Rate of convergence in the strong law of large numbers ...... 89

vi 7 Central limit theorem for discrete Fourier transform 97

7.1 Preliminaries ...... 99

7.1.1 Spectral density and limiting variance ...... 99

7.1.2 Stationary random fields and stationary filtrations ...... 101

7.2 Results and proofs ...... 101

7.3 Random fields with multi-dimensional index set ...... 112

7.4 Applications ...... 114

7.4.1 Independent copies of a stationary sequence with “nonparallel” past

and future ...... 115

7.4.2 Functions of i.i.d...... 116

7.5 Supplementary results ...... 117

Bibliography 120

vii Chapter 1

Introduction

As is well-known, the central limit theorem (CLT) is considered to be one of the most fundamental and important theorems in both statistics and probability. For a sequence of random variables (Xn)n∈Z on a probability space (Ω, F,P ), with Z the set of integers, we say that a central limit theorem holds for (Xn)n∈Z if

n 1 X √ (X − EX ) ⇒ N (0, σ2) n n n k=1 where ⇒ denotes the convergence in distribution. The central limit theorem is often used to help measure the accuracy of many statistics and to get confidence intervals.

A stronger form of convergence, with many practical applications, is the central limit theorem for processes conditioned to start from a point or from a fixed past trajectory. This type of convergence is also known as the quenched central limit theorem.

Convergence in the quenched sense implies convergence in the annealed sense, but the reciprocal is not true. See for instance Volný and Woodroofe [VW10] and Ouchti and

Volný [OV08]. The limit theorems started at a fixed point or fixed past trajectory are often encountered in evolutions in random media and are useful for Markov chain Monte

1 Carlo procedures, which evolve from a seed. So they are of considerable importance for accurately applying statistical tools and predicting future outcomes.

This dissertation is mainly focused on central limit theorems for stationary random fields (multi-indexed random variables) and Fourier transform of stationary random fields.

In the field of limit theorems for stationary random fields, the central limit theorem is studied in Chapter 3 while the quenched convergence is investigated in Chapter 5 under some projective conditions, which are easy to verify in applications. Moreover, in

Chapter 4, necessary and sufficient conditions for martingale approximations of random fields are established.

In the limit theorems for Fourier transform of stationary random fields, the law of large numbers for discrete Fourier transform is investigated in Chapter 6. Later on, in

Chapter 7 of this dissertation, we point out a spectral density representation in terms of projections and show that the limiting distribution of the real and imaginary parts of the Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field’s spectral density.

The overviews of chapters of this dissertation are as follows.

1.1 Limit theorems for random fields

In the context of sequences of random variables (one-dimensional processes), it is well- known that martingale methods are crucial for establishing limit theorems. The theory of martingale approximation, initiated by Gordin [Gor69], was perfected in many subsequent papers. A random field consists of multi-indexed random variables (Xu)u∈Zd , d ≥ 2. The main difficulty when analyzing the asymptotic properties of random fields isthe

d lack of natural ordering of points in Z (d ≥ 2), that is, the future and the past do not have a unique interpretation. As a result, most technical tools available for sequences of

2 random variables do not fully extend to random fields. Nevertheless, it is still possible to try to exploit the richness of the martingale techniques. The main problem consists of the construction of meaningful filtrations. In order to overcome this difficulty mathematicians either used the lexicographic order or introduced the notion of commuting filtration. The lexicographic order appears in early papers, such as in Rosenblatt [Ros72], who pioneered the field of martingale approximation in the context of random fields.An important result was obtained by Dedecker [Ded98] who pointed out an interesting projective criteria for random fields, also based on the lexicographic order. The lexicographic order leads to normal approximation under projective conditions with respect to rather large, half-plane indexed σ-algebras. In order to reduce the size of the filtration used in projective conditions, mathematicians introduced the so-called commuting filtrations.

The traditional way for constructing commuting filtrations is to consider random fields which are functions of independent random variables. For literatures in this direction, see e.g. El Machkouri et al. [EVW13], Wang and Woodroofe [WW13], Volný and Wang

[VW14], Cuny et al. [CDV15].

Central limit theorem for random fields

One of the problems this dissertation investigates is the CLT for random fields.

Namely, that given a stationary random field (Xn,m)n,m∈Z, what kind of conditions will guarantee

n m 1 X X √ (X − EX ) ⇒ N (0, σ2)? (1.1.1) nm u,v u,v u=1 v=1 This problem has been extensively studied under mixing conditions, see, e.g., Bradley and Tone [BT17]. Such conditions are difficult to check for many stochastic processes.

Here, we would like to establish (1.1.1) for more general random fields under some easily verifiable projective-type conditions. For one-dimensional stochastic processes, these conditions have been used by many researchers for studying the central limit theorems

3 and invariance principles. See for example, Maxwell and Woodroofe [MW00], Peligrad and Utev [PU05], Wang and Woodroofe [WW13], among others. The extension of these results to random fields is not trivial, as the martingale approximation technique ismuch more difficult and complicated to apply in the multiparameter setting. Indeed, toprove a central limit theorem for stationary random fields, we will establish new theorems for triangular arrays of martingales differences in Chapter 3, which have interest in themselves.

Martingale approximations for random fields

In the one-dimensional case, as mentioned before, martingale approximation is a useful tool for establishing limit theorems for sequences of random variables. Besides, it is known that a stationary ergodic sequence of martingales differences satisfies the

CLT. However, this is not true for higher dimensional case. That means the ergodicity

d of Z (d ≥ 2) action is not enough to guarantee the CLT. A simple example is that

Di,j = XiYj, where (Xi)i∈Z, (Yj)j∈Z are mutually independent i.i.d. standard normal random variables. Clearly, (Di,j)i,j∈Z is an ergodic field of ortho-martingale differences, Pn Pm √ but the limiting distribution of i=1 j=1 Di,j/ nm fails to be normal (See the example in [Vol15] for details). However, Volný [Vol15] showed that the ergodicity in one direction is a sufficient condition for a CLT for stationary martingale differences fields.

Therefore, the CLT can be deduced for large classes of random fields from the martingale case. The theory of martingale approximation was started by Rosenblatt [Ros72] and its development is still in progress. Recently, the interest is in the approximations with ortho-martingales, introduced by Cairoli [Cai69]. New central limit theorems for stationary ortho-martingales can be found in [WW13] and Volný [Vol15]. There are also other forms of multiparameter martingales and corresponding CLT and functional CLT

(see e.g. Basu and Dorea [BD79] and Nahapetian and Petrosian [NP92]).

4 In order to exploit the richness of martingale approximations, in Chapter 4, we investigate the necessary and sufficient conditions for martingale approximation, generalizing known results by Dedecker et al. [DMV07], Zhao and Woodroofe [ZW08], and Peligrad

[Pel10].

Quenched central limit theorem for random fields

Quenched central limit theorem has been a subject of intense research for the last few decades. In the one-dimensional case, this notion informally can be introduced in the following way: Let (ξn)n∈Z be a real-valued stationary sequence in a probability space RZ, and f be a real-valued measurable function defined on the RN with the product sigma algebra, where N is the set of positive integers. Now we define a stationary stochastic processes (Xn)n∈Z by

Xn = f(ξj, j ≤ n), ∀n ∈ Z.

2 2 In addition, for simplicity, assume that E(X0) = 0 and E(X0 ) = σ .

If ω0 := (··· , ξ−1, ξ0) ∈ RN is fixed, consider the new stochastic process given by

Xω0,n = f(ω0, ξ1, ··· , ξn), n ∈ N.

A natural problem to ask is that if known

n 1 X √ X ⇒ N (0, σ2), n k k=1 can one obtain that n 1 X √ X ⇒ N (0, σ2)? n ω0,k k=1

We say that a quenched CLT holds for (Xn)n∈Z if the above CLT holds for almost every ω0 ∈ RN. The formal notion of quenched CLT for random fields will be introduced in Chapter 5. Such results are of considerable importance in various areas, especially in

5 Bayesian statistics, where a starting point or initial conditions are assumed most of the time.

While the problem of quenched convergence for stationary random sequences has been intensively explored, it has been rarely studied for random fields. The paper by Peligrad and Volný [PV18] is a remarkable step in this direction of research. As far as we know, that is the only paper on quenched convergence for random fields available in the literature. It was shown there, that a quenched functional central limit theorem holds for ortho-martingales and for random fields via co-boundary decomposition. In addition,

2 they showed that the moment condition E(X0,0 log(1 + |X0,0|)) < ∞ is optimal for the validity of this type of result.

Motived by their result, in Chapter 5, sufficient conditions for the quenched central limit theorem will be established, generalizing the quenched central limit theorem for ortho-martingales to more general random fields. The proof will be based on the ortho- martingale approximation, projective decomposition and ergodic theorems for Dunford-

Schwartz operators. The definition and preliminary results on Dunford-Schwartz operators will be provided in Chapter 2.

Remark 1.1.1. The main results in Chapters 3, 4 and 5 could also be found in [PZ18b],

[PZ18a] and [ZRP19], respectively, the first two of which have been published while the latter has been submitted to peer-reviewed journals.

1.2 Limit theorems for Fourier transform of random fields

The discrete Fourier transform, defined as

n X ikt Sn(t) = e Xk , (1.2.1) k=1 √ where i = −1 is the imaginary unit, plays an essential role in the study of stationary time series (Xj)j∈Z of centered random variables with finite second moment, defined

6 on a probability space (Ω, K,P ). Preliminary results on Fourier transform of random variables can be found in Section 2.3 of Chapter 2.

The periodogram, introduced as a tool by Schuster [Sch98], is essential for the esti- mation of the spectral density of the stationary processes. It is defined by

1 I (t) = |S (t)|2, t ∈ [−π, π). (1.2.2) n 2πn n

There is a vast literature concerning these statistics. They are often used to determine hidden periodicities. The asymptotic behavior of the discrete Fourier transform and the periodogram was studied under various dependence condition. We mention Rosen- blatt [Ros85] who considered mixing processes; Brockwell and Davis [BD91] and Walker

[Wal65] who discussed linear processes; Shao and Wu [SW07] who treated functions of i.i.d.

In this dissertation, we will analyze the asymptotics of Fourier transform of stationary random fields. For a stationary random field (Xu,v)u,v∈Z of centered random variables with finite second moment, the discrete Fourier transform is defined by

n m X X i(ut1+vt2) Sn,m(t1, t2) = e Xu,v u=1 v=1

where t1, t2 ∈ [−π, π).

Law of large numbers for discrete Fourier transform

It is known that the law of large numbers (LLN) is one of the fundamental tools to prove the CLT. The LLN holds for pairwise independent and identically distributed random variables is due to Etemadi [Ete81]. The rate of convergence of this result was provided by Baum and Katz [BK65] when the variables are i.i.d. with finite r- th moment (with 1 < r < 2) and by Stoica [Sto11] in the martingale difference case.

A careful examination of the proof in Stoica [Sto11] yields the same result for centered

7 pairwise independent random variables. In Chapter 6, we shall establish for a larger class of random variables a similar result. That is, the LLN for discrete Fourier transform of random variables is provided under a very weak condition (which will be used to prove the central limit theorem in Chapter 7). In addition, the speed of convergence in the strong law of large numbers will be given.

Central limit theorem for Fourier transform and periodogram of random fields

Denote by λ the Lebesgue measure on the real line. In Peligrad and Wu [PW10], it was proved that, under ergodicity and a very mild regularity condition, for λ-almost √ √ all frequencies t, the random variables Re Sn(t)/ n and Im Sn(t)/ n are asymptotically independent identically distributed random variables with normal distribution, mean 0 and variance πf(t). Here f is the spectral density of (Xj)j∈Z. This result implies that for 2 2 λ-almost all t, the periodogram In(t) converges in distribution to f(t)χ , where χ has a chi-square distribution with 2 degrees of freedom, even in the case of processes with long memory. The proof of this result is based on the celebrated Carleson theorem (Theorem

2.3.3 in Chapter 2) about almost sure convergence of Fourier transforms, combined with martingale approximations and Fourier analysis.

Motived by the spectral density of random evolutions, in Chapter 7 of this dissertation, similar results are established for random fields. Namely, in Chapter 7, itis

2 proved that, under certain regularity conditions, for almost all (t1, t2) ∈ [−π, π) , the √ real and imaginary part of Sn,m(t1, t2)/ nm converge to independent normal variables, whose variance is, up to a multiplicative constant, the spectral density of the random field (Xu,v)u,v∈Z. The results can be easily applied to derive the asymptotic behavior of the periodogram associated to random fields. The difficult part of the aforementioned generalization is that many important spectral analysis results relevant to the proofs do not fully extend to double indexed sequences, including the celebrated Fejér-Lebesgue

8 Theorem (cf Bary [Bar64], p. 139), the Carleson theorem (Theorem 2.3.3 in Chapter 2)

(see Fefferman [Fef71b]) or the Hunt and Young maximal inequality (Theorem 2.3.4in

Chapter 2). Due to the limited level of knowledge in harmonic analysis on the summability of multi-dimensional trigonometric Fourier series, our results are formulated for general random fields by using only summation over cubes and unrestricted rectangles.

The ingredients of the proof for the CLT for Fourier transform of random fields include:

(a) Martingale approximation for random fields, validated via a new, interesting representation for the spectral density in terms of projection operators; (b) CLT for Fourier transform of martingales; (c) LLN for Fourier sums, which has interest in itself.

Remark 1.2.1. The main results in Chapters 6 and 7 have been published in peer- reviewed journals ([Zha17] and [PZ19]).

9 Chapter 2

Background

In this chapter we review the preliminaries and literature relevant to our work. This chapter is organized as follows. In Section 2.1, we introduce important definitions of

(stationary) stochastic processes, measure preserving transformations, shift transformations and canonical representation of stochastic processes. These notions will be used a lot in the forthcoming chapters. In addition, two special kind of stochastic processes—

Markov chains and Martingales and their important properties will be discussed. In

Section 2.2, we will present some results necessary for the proofs of main results in

Chapter 5. Specifically, the notion of (positive) Dunford-Schwartz operators (Definition

2.2.1 and Definition 2.2.3) associated with a measure preserving transformation and their corresponding ergodic theorems (especially, Theorem 2.2.12 and Theorem 2.2.9) will be given, which are crucial in the proof of martingale approximation in Chapter 5.

Section 2.3 settles the ground for the subsequent discussion on the central limit theorem for Fourier transform. Particularly, the maximal inequalities for double Fourier series

(Theorem 2.3.5) is essential for the proof of spectral density representation for random fields in Chapter 7. At the end, we describe two Féjer-Lebesgue Theorems fordouble

Fourier series giving the almost everywhere (a.e.) convergence of the average of the

10 double Fourier series. These two theorems will be of great importance in validating the martingale approximation for the Fourier transform of random fields in Chapter 7.

2.1 Stochastic processes

Stochastic processes deal with the dynamics in probability theory. The concept of stochastic processes enlarge the random variable concept to include time.

Definition 2.1.1 (Stochastic processes). A stochastic process is simply a collection of random variables (Xt, t ∈ T ) on a probability space (Ω, F,P ) where T ⊆ R, R is the set of real numbers.

The set of values that Xt(w) assumes is called the state space or phase space of the process. The points of T are thought of as representing time. A stochastic process can be classified by the nature of the time parameter and the nature of the state space.If T is the set of integers, the stochastic process is called discrete time process, or else T is an interval of the line, the stochastic process is called continuous time process. However,

T can be quite arbitrary in general. Similarly, according to the nature of the state space, a stochastic process can be either a continuous state process or discrete state process.

A discrete time stochastic process is also called a random sequence, usually denoted by

(Xk, k ∈ Z) or (Xk)k∈Z where Z is the set of integers. But throughout this dissertation, we only consider discrete time stochastic processes.

2.1.1 Stationary processes and measure preserving transformations

Let us now recall the definition for stationary random processes.

Definition 2.1.2 (Strictly stationary random sequences). A sequence (Xk)k∈Z of random variables is said to be strictly stationary if for each integer k and each pos-

n itive integer n with t1, t2, ··· , tn ∈ Z, the distribution on R of the random vector

(Xt1 ,Xt2 , ··· ,Xtn ) is the same as that of the random vector (Xt1+k,Xt2+k, ··· ,Xtn+k).

11 Definition 2.1.3 (Measure preserving transformation). Given a probability space (Ω, F,P ), we say a map T :Ω → Ω is measure preserving transformation if it is measurable F/F and P (T −1A) = P (A) for all A ∈ F.

Remark 2.1.4. The map T induces an operator U on functions defined on a probability space (Ω, F,P ), namely, Uf(ω) = f(T ω). As T is measure preserving, it is easy to see that U is an isometry on Lp(Ω, F,P ) for any 1 ≤ p < ∞. In fact, by a change of variables, we have

Z Z E|Uf|p = |f(T ω)|pP (dω) = |f(ω)|pPT −1(dω) = E|f|p. Ω Ω

Moreover, U is an unitary operator on L2(Ω, F,P ) with inner product < ·, · >, since for any f, g ∈ (Ω, F,P ),

Z Z hUf, Ugi = f(T ω)g(T ω)P (dω) = f(ω)g(ω)P (dω) = hf, gi. Ω Ω

j For each j ∈ Z, the powers of this shift operator U can be formulated in the following obvious way: ∀ω ∈ Ω, (U jf)(ω) = f(T jω).

Then U i+j = U iU j for each pair of integers i and j.

Definition 2.1.5 (Trivial σ-algebras). A σ-algebra A is said to be trivial if P (A) = 0 or 1 for every A ∈ A.

Definition 2.1.6 (Invariant sets and sigma algebras, ergodic transformation). Given a probability space (Ω, F,P ), a set A ∈ F is invariant under T if T −1A = A. The class of invariant sets I is called invariant σ-algebra. A map T :Ω → Ω is ergodic if T is measure preserving and there are in F no “nontrivial” T-invariant sets.

12 2.1.2 Shift transformations and canonical representation of stochastic processes

Sometimes it is convenient to work on RZ itself as a probability space, where RZ can be

Z viewed as a product space Πn∈ZR. An element in R will be denoted by x = (xk)k∈Z. Consider the measurable space (RZ, RZ) where RZ is the σ-algebra of RZ generated by cylinders.

Definition 2.1.7 (The shift operator). Let x ∈ RZ, the shift operator T : RZ → RZ is defined by

∀k ∈ Z, (T x)k = xk+1.

That is, T (x) = (xk+1)k∈Z. j For each j ∈ Z, the power T is formulated in the following way:

j ∀k ∈ Z, (T x)k = xk+j.

Canonical representation of stochastic processes

Z As before, we denote the element in R by x = (xk)k∈Z. Let X = (Xn)n∈Z be a real- valued stochastic process on the probability space (Ω, F,P ), then X can be considered as a random variable (Ω, F) → (RZ, RZ) where RZ is the σ-algebra of RZ generated by cylinders. The distribution (or law) of X is the probability PX−1. For distinct t1, t2, ··· , tn ∈ Z, the distribution of (Xt1 ,Xt2 , ··· ,Xtn ) is called the n-dimensional marginal distribution of X corresponding to (t1, ··· , tn).

For all n ∈ Z, consider the n-th coordinate map Zn : RZ → R defined by Zn(x) = xn.

These coordinate maps introduce a process Z = (Zn)n∈Z on a new probability space 0 0 −1 n (RZ, RZ,P ) with P = PX . Clearly, for any A ∈ R and distinct t1, ··· , tn ∈ Z, we

13 have

−1 Z X x ∈ R : Zt1 (x),Zt2 (x), ··· ,Ztn (x) ∈ A

= ω ∈ Ω: Zt1 (X(ω)),Zt2 (X(ω)), ··· ,Ztn (X(ω)) ∈ A

Thus

−1 Z PX x ∈ R : Zt1 (x),Zt2 (x), ··· ,Ztn (x) ∈ A

= P ω ∈ Ω: Zt1 (X(ω)),Zt2 (X(ω)), ··· ,Ztn (X(ω)) ∈ A , which implies that, under P , the distribution of X is the same as the distribution of Z under under PX−1, by the fact that the distribution of a stochastic process is determined by its finite dimensional marginals. Therefore (Zn)n∈Z is called the canonical version

Z Z −1 of (Xn)n∈Z. The probability space (R , R ,PX ) is called the canonical space of

(Xn)n∈Z.

2.1.3 Markov chains

A special class of stochastic processes, with many applications, is the Markov chain.

Here we restrict ourselves to the real-valued case.

Definition 2.1.8 (Markov chains). A stochastic process (Xn)n∈N is a Markov chain with values in R if for n ≥ 1 and A ∈ R,

P (Xn+1 ∈ A|X1, ··· ,Xn) = P (Xn+1 ∈ A|Xn). (2.1.1)

Thus we can see, for Markov chains, their dependence on the past is only through the previous state. That is, given the present state, the future is independent of the past.

This property is referred to as the Markov property.

14 As part of the definition for time-homogeneous Markov chains, we recall the notion of a transition probability function.

Definition 2.1.9 (Transition probability function). Let (S, S) be a measurable space.

A function Q : S × S → [0, 1] is said to be a transition probability function if

• For every x ∈ S, Q(x, ·) is a probability measure on (S, S);

• For every A ∈ S, Q(·,A) is measurable with respect to S.

Equivalent to Definition 2.1.8, we say the stochastic process (Xn, n ∈ N) is a Markov chain with transition probability function Q if

P (Xn+1 ∈ A|X1, ··· ,Xn) = Q(Xn,A), for all A ∈ R.

The properties of Markov processes are entirely determined by the transition probability function and the initial distribution (the law of X1). In fact, by the Markov property and the chain rule for conditional probability, the finite-dimensional distribution of (Xn)n∈N can be computed as follows: for all A1,A2, ··· ,An ∈ R,

P (X1 ∈ A1,X2 ∈ A2, ··· ,Xn ∈ An)

= P (X1 ∈ A1)P (X2 ∈ A2|X1 ∈ A1)P (X3 ∈ A3|X2 ∈ A2) ··· P (Xn ∈ An|Xn−1 ∈ An−1).

The following remark gives one way to introduce stationary processes.

Remark 2.1.10. Any stationary sequence is a function of a Markov chain.

15 Proof. Let (Xn)n∈Z be a stationary process. Now we define a Markov chain (ξn)n∈Z by ξn = (Xi, i ≤ n). To prove this remark, let A ∈ RN,

P (ξn+1 ∈ A|ξ0, ··· , ξn) = P (ξn+1 ∈ A|(Xi, i ≤ 0), (Xi, i ≤ 1), ··· , (Xi, i ≤ n))

= P (ξn+1 ∈ A|(Xi, i ≤ n))

= P (ξn+1 ∈ A|ξn).

Thus each Xn can be viewed as a function of Markov chain ξn, that is, Xn = f(ξn). Stationary processes could also be introduced as a dynamical system.

Lemma 2.1.11. Let T :Ω → Ω be a bijective, bi-measurable, measure preserving transformation on (Ω, F,P ) and let F0 ⊂ F be a sub-σ-algebra of F satisfying F0 ⊂

−1 T F0. In addition assume that X0 is F0-measurable. Then the process (Xk)k∈Z defined k by Xk = X0 ◦ T is stationary.

2.1.4 Martingales

In this section, we discuss the notion of filtrations, adapted processes and martingales for the one-dimensional case. The corresponding notions for higher dimensional case will be given in Chapter 3. The setting of adapted processes will be one of the fundamental assumptions along the main results in this dissertation.

Definition 2.1.12 (Filtrations, adapted processes). Given a probability space (Ω, F,P ), for each k ∈ Z, Fk is a sub-σ-algebra of F. In addition assume that Xk is Fk-measurable and Fk ⊂ Fk+1 ⊂ F. Then

• (Fn)n∈Z is called a filtration;

• The process (Xn)n∈Z is adapted to the filtration (Fn)n∈Z.

Definition 2.1.13 (Martingales, submartingales). Given a probability space (Ω, F,P ), assume that (Xn)n∈Z is adapted to filtration (Fn)n∈Z and Xn is integrable. Then

16 • (Xn)n∈Z is a martingale adapted to (Fn)n∈Z if E(Xn+1|Fn) = Xn P -a.s.;

• (Xn)n∈Z is a martingale difference adapted to (Fn)n∈Z if E(Xn+1|Fn) = 0 P -a.s.;

• (Xn)n∈Z is a submartingale adapted to (Fn)n∈Z if E(Xn+1|Fn) ≥ Xn P -a.s.

One classical inequality for martingales can be found in the book by Hall and Heyde

[HH80] (Theorem 2.11, p. 23) (see also Theorem 6.6.7 Ch. 6, p. 322, de la Peña and

Giné [PG99]), says that

Pn Theorem 2.1.14 (Rosenthal’s Inequality). Let p ≥ 2. Let Mn = k=1 Xk where

{Mn, Fn} is a martingale with martingale differences (Xn)n≥1. Then there are constants

0 < cp,Cp < ∞ such that

n n p/2 X p X 2 cp E|Xk| + E E(Xk |Fk−1) k=1 k=1 n p/2 n p X 2 X p ≤ kMnkp ≤ Cp E E(Xk |Fk−1) + E|Xk| . k=1 k=1

2.2 Dunford-Schwartz operators and the Ergodic theorem

In this section, we present some maximal inequalities and the ergodic theorems for the

Dunford-Schwartz operators, which will justify the validity of the martingale approximation for random fields in the quenched sense in Chaper 5.

2.2.1 Positive Dunford-Schwartz operators

We will begin with the definition of Dunford-Schwartz operators.

17 Definition 2.2.1. Given a measure space (Ω, F, µ), an operator T : L1 → L1 is called a Dunford-Schwartz operator if for every f ∈ L1 ∩ L∞,

kT fk1 ≤ kfk1 and kT fk∞ ≤ kfk∞.

By Theorem 8.23 in Eisner et al. [Eis+15], we have the following property for Dunford-

Schwartz operators:

Proposition 2.2.2. Let T : L1 → L1 be a Dunford-Schwartz operator on (Ω, F, µ), then

p ∞ kT fkp ≤ kfkp for any f ∈ L ∩ L , p ≥ 1

Definition 2.2.3 (Positive operators). Let T : L1 → L1 be a bounded (linear) operator on (Ω, F, µ), then T is called positive if for all 0 ≤ f ∈ L1, T f ≥ 0.

2.2.2 The Ergodic theorem for positive Dunford-Schwartz operators

The following theorem is a combination of Definition 11.3, Theorem 11.4 and Theorem

11.6 in Eisner et al. [Eis+15].

Theorem 2.2.4 (Ergodic theorem for positive Dunford-Schwartz operators). Let (Ω, F, µ) be a finite measure space (that is, µ(Ω) < ∞) and T : L1 → L1 be a positive Dunford- n−1 1 X Schwartz operator on Ω. Then for every p ≥ 1 and every f ∈ Lp(Ω), lim T kf n→∞ n k=0 exists µ-a.e. and in Lp.

Remark 2.2.5 (Properties of the limit). Denote the limit in the conclusion of Theorem

2.2.4 by fˆ, then

1. fˆ is T −invariant: T fˆ = fˆ µ-a.e.; R ˆ R 2. Ω f = Ω f if T is measure preserving.

18 Proof. Let us denote the Cesàro-mean by Anf:

n−1 1 X A f := T kf. n n k=0

p As limn→∞ Anf = fˆ µ-a.e. and in L , by the continuity of T ,

T fˆ = T lim Anf = lim TAnf µ-a.e. n→∞ n→∞

n n Moreover, T f/n converge to zero µ−a.e. when n → ∞ (T f = (n + 1)An+1 − nAn), therefore

1 n T fˆ− fˆ = lim (TAnf − Anf) = lim (T f − f) = 0, µ-a.e. n→∞ n→∞ n

Thus the limit fˆ is T -invariant. R R To prove property 2, first notice that for the measure preserving T , Ω Anf = Ω f, and we have µ-a.e. convergence, so the result will follow if Anf is uniformly integrable.

The uniform integrability is a direct consequence of an inequality related to maximal operators which will be given later (see Theorem 2.2.6). Now we give estimates for the maximal operators, which is defined in the following way:

( n−1 ) 1 X k M∞f := sup T f (2.2.1) n≥0 n k=0

Theorem 2.2.6 ([Kre85], Theorem 6.3, p. 52). Let T be a positive Dunford-Schwartz operator in (Ω, F, µ), then for f ≥ 0,

p • kM∞fkp ≤ p−1 kfkp (1 < p < ∞)

e R + • kM∞fk1 ≤ e−1 µ(Ω) + Ω f log fdµ

19   + f(log f) , for f ≥ 0 where log is the natural logarithm and f log+ f = .  0, for f = 0

m Let L log L denote the class of all measurable functions f : (Ω, F, µ) → (R, R, λ) m such that |f| log+ |f| is integrable. Then we have :

Theorem 2.2.7 ([Kre85], Theorem 6.4, p. 54 ). Let T be a positive Dunford-Schwartz

1 m m−1 operator in L (Ω) with µ(Ω) < ∞. Then f ∈ L log L implies that M∞|f| ∈ L log L

(m ≥ 1).

Remark 2.2.8. If µ(Ω) < ∞, T is ergodic, measure preserving and one-to-one, then

1 M∞|f| ∈ L implies |f| ∈ L log L.

Proof. See Ornstein [Orn71]∗ for details. This remark shows the reverse of Theorem 2.2.7, that is, for ergodic, measuring preserving and one-to-one T and µ(Ω) < ∞, |f| ∈ L log L is also necessary for the integrability of M∞|f|.

2.2.3 Multiparameter ergodic theorem for Dunford-Schwartz operators

d Let d ≥ 2 be a fixed integer. For u = (u1, u2, ··· , ud) and v = (v1, v2, ··· , vd) ∈ Z , d by u ≤ v, we mean ui ≤ vi, i = 1, 2, ··· , d. For n = (n1, n2, ··· , nd) ∈ Z , let Qd d |n| = j=1 nj. For n ∈ Z and operators T1,T2, ··· ,Td, we shall use the notation

n n1 n2 nd 1 Pn k T = T1 T2 ··· Td , and Anf = |n| k=1 T f. The notation n → ∞ means that min1≤i≤d ni → ∞.

Ergodic theorem over rectangles

Krengel [Kre85] gives unrestricted almost everywhere convergence for the Cesàro-mean over rectangles.

∗ Ornstein [Orn71] also shows that if T is ergodic, µ(Ω) = ∞, M∞|f| cannot be integrable for f 6= 0.

20 1 1 Theorem 2.2.9 ([Kre85], Theorem 1.1, Ch. 6, p. 196). Let T1,T2, ··· ,Td : L → L be Dunford-Schwartz operators on a finite measure space (Ω, F, µ). Then for every d−1 h ∈ L log L, limn→∞ Anh exists µ-a.e. and

lim Anh = A∞(T1)A∞(T2) ··· A∞(Td)h n→∞

A (T )f = lim A (T )f = lim 1 Pni T kf where ∞ i ni→∞ ni i ni→∞ ni k=1 i .

Theorem 2.2.10 ( [Kre85], Theorem 1.2, Ch. 6, p. 196). Let T1,T2, ··· ,Td be positive Dunford-Schwartz operators in Lp (1 < p < ∞) of a σ-finite measure space (Ω, F, µ).

p d Then for every h ∈ L , Anh converges µ-a.e. and ksup{Anh : n ∈ Z }kp ≤ khkp (1 < p < ∞).

Remark 2.2.11. The integrability condition of h ∈ L logd−1 L cannot be weaken for the the unrestricted a.e. convergence. In fact, it was shown by Smythe [Smy73] that even for Pn the i.i.d. random variables (Xu)u∈Zd , the unrestricted a.e. convergence of u=1 Xu/|n| d−1 implies X0 ∈ L log L.

Ergodic theorem over squares

A direct application of Theorem 2.8 in Chapter 6 of [Kre85] gives the restricted a.e. convergence of the Cesàro-mean over squares.

Theorem 2.2.12 (Ergodic theorem for Dunford-Schwartz operators over squares). Let

1 T1,T2, ··· ,Td be positive Dunford-Schwartz operators in L of a measure space (Ω, F, µ).

p Then for every f ∈ L , 1 ≤ p < ∞, Anf converges µ-a.e. with n = (n, n, ··· , n).

2.3 Preliminaries for Fourier transform

Let us start this section by introducing the notion of Fourier transform of integrable functions.

21 Fourier transform of integrable functions

Definition 2.3.1 (Fourier transform). For an integrable function f on ([−π, π), B, λ), the Fourier transform of f, denoted by fˆ : R → C, is defined by

Z π fˆ(x) = f(t)e−itxλ(dt). −π

Definition 2.3.2 (Partial Fourier sum). Given an integrable function f on ([−π, π), B, λ) and a positive integer n, the partial Fourier sum of order n of f at frequency t ∈ [−π, π) is defined by X ikt Sn(f, t) = fˆ(k)e . |k|≤n

With this definition, we can state the celebrated Carleson’s Theorem proved by

Lennard Carleson [Car66], which establishes the almost sure convergence of the partial sum of the Fourier series of square integrable functions. ∗

Theorem 2.3.3 (Carleson’s Theorem). If f ∈ L2([−π, π), B, λ), then for almost all t ∈ [−π, π), 1 lim Sn(f, t) = f(t). n→∞ 2π

That is, there exists a set If ⊂ [−π, π) with λ(If ) = 2π such that for all t ∈ If , Sn(f, t) converges to f(t) when n → ∞.

Therefore, we have the following maximal inequality, which will be referred as, throughout the dissertation, the Hunt and Young’s maximal inequality.

∗The result also holds for functions in Lp for p > 1. See Lacey [Lac04].

22 Theorem 2.3.4 (A maximal inequality). For a square integrable function f on ([−π, π), B, λ), there exists a constant C such that

Z π 2 X 2 sup|Sn(f, t)| λ(dt) ≤ C |fˆ(n)| −π n∈ N n∈Z where fˆ is the transform of f (Definition 2.3.1) and Sn(f, t) is the partial sum of the

Fourier series of f (Definition 2.3.2).

However, these results do not fully extend to the d-dimensional setting (d ≥ 2). For the sake of discussion, we only consider the case d = 2. Let f ∈ L2([−π, π)2, B, λ2), then

2 the partial sum of the Fourier series of f at t = (t1, t2) ∈ [−π, π) is

Z X i(kt1+`t2) −i(kt1+`t2) Sn,m(f, t) = fˆ(k, `)e , with fˆ(k, `) = f(t1, t2)e dt. 2 |k|≤n,|`|≤m [−π,π) (2.3.1)

It was shown by Fefferman [Fef71b] that there exists a continuous function f on

2 2 [−π, π) such that limn∧m→∞ Sn,m(f, t)/(2π) = f holds nowhere. But later, Fefferman

[Fef71a] established the Carleson’s Theorem (Theorem 2.3.3) and the Hunt and Young’s maximal inequality (Theorem 2.3.4) for Sn,m(f, t), under some restriction. A particular version of the maximal equality given in Fefferman [Fef71a], sufficient for our purpose, is

Theorem 2.3.5. For f ∈ L2([−π, π)2, B, λ2), there exists a constant C such that

Z 2 X 2 sup|Sn,n(f, t)| dt ≤ C |fˆ(u)| . [−π,π)2 n∈N u∈Z2

2.3.1 Weak stationarity and spectral density

In many situations, the correlations between variables could be viewed as a measure of dependence and, in the Gaussian setting, they determine the distribution. The con-

23 densed information about the correlation structure is contained in the so called the

“process spectral measure” and, when it exists, in its density called the “spectral density function” of a stochastic process. Then, the covariances between variables are obtained as the Fourier coefficients of this function. Because the spectral density function en- capsulates all the information about covariances of a stochastic process, its study has a central role in their theory. We start by recalling the closely related notion of weak stationarity.

Weak Stationarity

Definition 2.3.6 (Weak stationarity or L2-stationarity). A sequence of complex-valued random variables (Xn)n∈Z is said to be weakly stationary (or second order stationary) if

• E(Xn) = c, for all n ∈ Z.

2 • E(|X0| ) < ∞.

• There exists γ(k) ∈ C, k ∈ Z, such that for all j, k ∈ Z, cov(Xj,Xk) = γ(j − k).

Remark 2.3.7. For a weakly stationary complex-valued random sequences (Xn)n∈Z, 2 γ(k) = γ(−k), for each k ∈ Z and γ(0) = E|X0 − c| .

Fact 2.3.8. For Gaussian sequences, the concept of weak stationarity and strict stationarity coincide.

Spectral density

The non-negative definite functions play an important role in the study of spectral properties of stationary processes due to the following definition and theorem.

Definition 2.3.9. A complex-valued function γ(·) defined on integers is non-negative definite if n X aiγ(i − j)aj ≥ 0, i,j=1

24 n for all positive integers n and all vectors a = (a1, a2, ··· , an) ∈ C .

Remark 2.3.10. The autocovariance function γ of a weakly stationary process (Xn)n∈Z is non-negative definite.

Proof. For each i, j ∈ Z, γ(i − j) = cov(Xi,Xj) = E(Xi − EXi)(Xj − EXj). Let n a = (a1, a2, ··· , an) ∈ C , X = (X1 − EX1,X2 − EX2, ··· ,Xn − EXn). Then

n  n  X X 2 aiγ(i − j)aj = E  ai(Xi − EXi)(Xj − EXj)aj = E|a · X| ≥ 0. i,j=1 i,j=1

The existence of spectral measure for weakly stationary random variables (Xn)n∈Z is a consequence of Herglotz’s Theorem (See Theorem 4.3.1 in Brockwell and Davis [BD06]).

Theorem 2.3.11 (Herglotz’s Theorem). A complex-valued function γ(·) defined on integers is non-negative definite if and only if there exists a right-continuous, non-decreasing, bounded function on [−π, π) satisfying F (−π) = 0, called the spectral distribution function, such that Z π ikt γ(k) = e F (dt), for each k ∈ Z. −π

If F is absolutely continuous with respect to the Lebesgue measure λ on [−π, π), then the Radon–Nikodym derivative f of F with respect to the Lebesgue measure is called the spectral density of γ(·), that is, F (dt) = f(t)dt and we have

Z π ikt γ(k) = e f(t)dt, for each k ∈ Z. −π

Combining Herglotz’s theorem (Theorem 2.3.11) and Remark 2.3.10 together with the fact that the function eikt is periodic with period 2π, we can get the following theorem:

Theorem 2.3.12 (Existence of spectral measure). Given a weakly stationary complex- valued random processes (Xn)n∈Z (Definition 2.3.6), there exists a unique positive mea-

25 sure ν, called the spectral measure, on [−π, π) such that

Z itk γ(k) = cov(Xk,X0) = e ν(dt), for all k ∈ Z. [−π,π)

Actually, Theorem 2.3.12 can be extended to weakly stationary random fields.

Theorem 2.3.13 (Existence of spectral measure for random fields). Given a weakly stationary random field (Xn)n∈Z2 on (Ω, F,P ), there exists a unique positive measure F on [−π, π)2, called the spectral measure, such that

Z it·u 2 γ(u) = cov(Xu,X0) = e F (dt), for all u ∈ Z , [−π,π)2 where u·t is the inner product. If F is absolutely continuous with respect to the Lebesgue measure on [−π, π)2 (F (dt) = f(t)dt), then the Radon–Nikodym derivative f of F with respect to the Lebesgue measure is called the spectral density, and we have

Z iu·t 2 γ(u) = e f(t)dt, for all u ∈ Z . [−π,π)2

2.3.2 Relation with discrete Fourier transform

Let us start by recalling the Féjer-Lebesgue Theorem in the one-dimensional setting which can be found in Bary [Bar64]:

Theorem 2.3.14 ([Bar64], Féjer-Lebesgue Theorem). Given an integrable function on ([−π, π), B, λ), denote the Fourier transform of f by fˆ (Definition 2.3.1), vn := 1 Pn itk ˆ 2π k=−n e f(k). Then for almost all t ∈ [−π, π)

n 1 X v → f as n → ∞. n m m=1

In the higher dimensional setting, Marcinkiewicz and Zygmund ([MZ39]) showed that

26 Theorem 2.3.15 ([MZ39], Féjer-Lebesgue Theorem). Given f ∈ L1([−π, π)2, B, λ2),

ˆ 1 Pn Pm i(kt1+`t2) ˆ denote the Fourier transform of f by f, set σn,m := (2π)2 k=−n `=−m e f(k, `). 2 Then for almost all t = (t1, t2) ∈ [−π, π)

n m 1 X X σ → f, nm u,v u=1 v=1 when n, m tends to ∞ such that m/n ≤ a and n/m ≤ a for some positive number a.

The other 2-dimensional result can be found in Jessen et al. [JMZ35], which gives the corresponding Féjer-Lebesgue when the summation is taken over unrestricted rectangles of size n × m:

Theorem 2.3.16 ([JMZ35], Féjer-Lebesgue Theorem). Under condition of Theorem

2.3.15 and if the function f log+ f is, in addition, integrable. Then for almost all

2 t = (t1, t2) ∈ [−π, π) n m 1 X X lim σu,v = f. n∧m→∞ nm u=1 v=1 These two theorems are crucial in the proof of the limiting distribution of Fourier transform of random fields in Chapter 7, since they give the limiting distribution ofthe variance of partial sum of Fourier transform of random fields. Let (Xn)n≥1 be a sequence of real-valued random variables on a probability space (Ω, F,P ), for t ∈ [−π, π). We define the discrete Fourier transform of (Xn)n≥1 at t in the following way:

Definition 2.3.17 (Discrete Fourier transform).

n X ikt Sn(t) = e Xk (2.3.2) k=1

When t = 0, Sn(0) = Sn. Note that Sn(t) is also a function of ω, as the random variable

Xk is a function of ω.

27 Peligrad and Wu [PW10] proved the CLT and its functional form for Sn(t) for almost every t ∈ [−π, π) under a certain regularity condition (which will be discussed in Chapter

7). One of the main purpose of this dissertation is to extend their results to random fields, the precise description of which can be found in Chapter 7.

For a stationary random field (Xn,m)n,m≥1, the discrete Fourier transform of (Xn,m)n,m≥1 is defined to be

n m X X i(ut1+vt2) Sn,m(t1, t2) = e Xu,v, where t1, t2 ∈ [−π, π). u=1 v=1 Then the variance of partial sums of Fourier transform of the stationary random field

(Xn,m)n,m≥1 is

2 X it·(u−v) E|Sn(t)| = cov(Xu,Xv)e 1≤u,v≤n X = γ(u − v)eit·(u−v), 1≤u,v≤n where t · u is the inner product and u ≤ v means u1 ≤ v1, u2 ≤ v2.

R ix·(u−v) Considering the fact that γ(u − v) = [−π,π)2 e f(x)dx (Theorem 2.3.13), with f being the spectral density of (Xn,m)n,m≥1, we have

Z 2 X i(x+t)·(u−v) E|Sn(t)| = e f(x)dx 2 1≤u,v≤n [−π,π) X Z = eix·(u−v)f(x − t)dx. 2 1≤u,v≤n [−π,π)

By applying Theorem 2.3.15 and Theorem 2.3.16, we will get the following limiting distribution of the variance of partial sums of Fourier transform of stationary (Xn,m)n,m≥1

: for almost all t ∈ [−π, π)2,

1 2 2 lim E|Sn,n(t)| = (2π) f(t), n→∞ n2

28 and if in addition f(u) ln+ f(u) is integrable,

1 2 2 lim E|Sn1,n2 (t)| = (2π) f(t), n1≥n2→∞ n1n2

which will be needed along the proof of the main result of Chapter 7.

29 Chapter 3

Central limit theorem for random fields via martingale methods

Most of the results in this chapter have been published in Peligrad and Zhang [PZ18b]. In this chapter, our focus in on establishing the central limit theorem for stationary random fields, for the situation when the variables satisfy a sharp projective condition, introduced by Maxwell and Woodroofe in the setting of random sequences. This is an interesting projective condition which defines a class of random variables satisfying the central limit theorem and its invariance principle, even in its quenched form. This condition is in some sense minimal for this type of behavior as shown in Peligrad and Utev [PU05]. Its importance was pointed out, for example, in papers by Maxwell and Woodroofe [MW00], who obtained a central limit theorem (CLT); Peligrad and Utev [PU05] obtained a maximal inequality and the functional form of the CLT; Cuny and Merlevède [CM14] obtained the quenched form of this invariance principle. The reinforced Maxwell-Woodroofe condition for random fields was formulated in Wang and Woodroofe [WW13], who also pointed out a variance inequality in the context of Bernoulli random fields (random fields induced by i.i.d. random fields).

30 Compared to the main result in Wang and Woodroofe [WW13], our results have double scope. First, to provide a central limit theorem under generalized Maxwell-Woodroofe condition that extends the original result of Maxwell and Woodroofe [MW00] to random fields. Second, to use more general random fields than Bernoulli fields. Our resultsare relevant for analyzing some statistics based on repeated independent samples from a stationary process.

We will apply martingale approximation approach. The tools for proving these results will consist of new theorems for triangular arrays of martingales differences which have interest in themselves. We begin in Section 3.1 with the main result. Moreover, we present applications of our result to linear and nonlinear random fields, which provide new limit theorems for these structures. In Section 3.2, we gather the proofs. Some auxiliary results needed along the proofs of the results are given in Section 3.3.

3.1 Results

Everywhere in this chapter we shall denote by k·k the norm in L2. By ⇒ we denote the convergence in distribution. In the sequel [x] denotes the integer part of x. As usual, a ∧ b stands for the minimum of a and b.

Maxwell and Woodroofe [MW00] introduced the following condition for a stationary processes (Xi)i∈Z, adapted to a stationary filtration (Fi)i∈Z :

X 1 Xk kE(Sk|F1)k < ∞,Sk = Xi, (3.1.1) k≥1 k3/2 i=1

√ and proved a central limit theorem for Sn/ n. In this chapter we extend this result to random fields.

For the sake of clarity we shall explain first the extension to random fields with double indexes and, at the end, we shall formulate the results for general random fields. We shall introduce a stationary random field adapted to a stationary filtration. In orderto

31 construct a flexible filtration, it is customary to start with a stationary real valuedran- dom field (ξn,m)n,m∈Z defined on a probability space (Ω, K,P ) and to introduce another stationary random field (Xn,m)n,m∈Z defined by

Xn,m = f(ξi,j, i ≤ n, j ≤ m), (3.1.2)

2 where f is a measurable function defined on RZ . Now we define the filtrations

Fn,m = σ(ξi,j, i ≤ n, j ≤ m). (3.1.3)

Note that Xn,m is adapted to the filtration Fn,m. Without restricting the generality we

2 Z shall define (ξu)u∈Z2 in a canonical way on the probability space Ω = R , endowed with the σ-field, B, generated by cylinders. As a matter of fact

n m Xn,m = f(T S (ξa,b)a≤0,b≤0) (3.1.4) where T is the vertical shift operator

T ((xu,v)(u,v)∈Z2 ) = (xu+1,v)(u,v)∈Z2 (3.1.5) and S is the horizontal shift operator

S((xu,v)(u,v)∈Z2 ) = (xu,v+1)(u,v)∈Z2 . (3.1.6)

We raise the question of normal approximation for stationary random fields under projection conditions with respect to the filtration (Fn,m)n,m∈Z. In several previous results involving various types of projective conditions, the methods take advantage of

32 the existence of commuting filtrations, i.e.

E(E(X|Fa,b)|Fu,v) = E(X|Fa∧u,b∧v). (3.1.7)

This type of filtration is induced by an initial random field (ξn,m)n,m∈Z of independent random variables, or, more generally can be induced by stationary random fields

(ξn,m)n,m∈Z where only the columns are independent, i.e. η¯m = (ξn,m)n∈Z are independent. This model often appears in statistical applications when one deals with repeated realizations of a stationary sequence. We prove this property in Lemma 3.3.5 in Section

3.3.

It is interesting to point out that commuting filtrations can be described by the equivalent formulation: for a ≥ u we have

E(E(X|Fa,b)|Fu,v) = E(X|Fu,b∧v).

This follows from this Markovian-type property, see for instance Problem 34.11 in

Billingsley [Bil95].

Our main result is the following theorem which is an extension of the CLT in Maxwell and Woodroofe [MW00] to random fields. Below we use the notation

Xk,j Sk,j = Xu,v. u,v=1

Theorem 3.1.1. Define (Xn,m)n,m∈Z by (3.1.2) and assume that (3.1.7) holds. Assume that the following projective condition is satisfied

X 1 kE(Sj,k|F1,1)k < ∞. (3.1.8) j,k≥1 j3/2k3/2

33 In addition assume that the vertical shift T is ergodic. Then there is a constant c such that 1 2 2 E(Sn1,n2 ) → c as min(n1, n2) → ∞ n1n2 and 1 2 √ Sn1,n2 ⇒ N (0, c ) as min(n1, n2) → ∞. (3.1.9) n1n2

By simple calculations involving the properties of conditional expectation we obtain the following corollary.

Corollary 3.1.2. Assume the following projective condition is satisfied

X 1 kE(X |F )k < ∞, (3.1.10) j1/2k1/2 j,k 1,1 j,k≥1 and T is ergodic. Then there is a constant c such that the CLT in (3.1.9) holds.

(X ) d The results are easy to extend to general random fields u u∈Z introduced in the following way. We start with a stationary random field (ξn)n∈Zd , defined on the canonical d Z probability space R and introduce another stationary random field (Xn)n∈Zd defined by Xk = f(ξj, j ≤ k), where f is a measurable function and j ≤ k denotes ji ≤ ki for all i. Note that Xk is adapted to the filtration Fk = σ(ξu, u ≤ k). As a matter of fact

k1 kd Xk = f(T1 ◦ ... ◦ Td (ξu)u≤0) (3.1.11)

where Ti’s are the coordinate-wise translation operators.

In the next theorem we shall consider commuting filtrations in the sense that for

d−1 a ≥ u ∈ R, b, v ∈ R we have

E(E(X|Fa,b)|Fu,v) = E(X|Fu,b∧v).

34 For example, this kind of filtration is induced by stationary random fields (ξn,m)n∈Z,m∈Zd d−1 such that the variables ηm = (ξn,m)n∈Z are independent, m ∈ Z . All the results extend in this context via mathematical induction. Below, |n| = n1 · ... · nd.

(X ) d (F ) d Theorem 3.1.3. Assume that u u∈Z and u u∈Z are as above and assume that the following projective condition is satisfied

X 1 kE(Su|F1)k < ∞. u≥1 |u|3/2

In addition assume that T1is ergodic. Then there is a constant c such that

1 E(S2 ) → c2 as min(n , ..., n ) → ∞ |n| n 1 d and 1 2 Sn ⇒ N (0, c ) as min(n1, ..., nd) → ∞. (3.1.12) p|n|

Corollary 3.1.4. Assume that

X 1 kE(Xu|F1)k < ∞ (3.1.13) u≥1 |u|1/2

and T1 is ergodic. Then the CLT in (3.1.12) holds.

Remark 3.1.5. Corollary 3.1.4 above shows that Theorem 1.1 in Wang and Woodroofe

[WW13] holds for functions of random fields which are not necessarily functions of i.i.d.

We shall give examples providing new results for linear and Volterra random fields.

For simplicity, they are formulated in the context of functions of i.i.d.

Example 3.1.6 (Linear field). Let (ξn)n∈Zd be a random field of independent, identically distributed random variables which are centered and have finite second moment. Define

X Xk = ajξk−j. j≥0

35 P 2 Assume that j≥0 aj < ∞ and

j X |bj| X X < ∞ where b2 = ( a )2. (3.1.14) |j|3/2 j u+i j≥1 i≥0 u=1

Then the CLT in (3.1.12) holds.

Let us mention how this example differs from other results available in the literature. Example 1 in El Machkouri et al. [EVW13] contains a CLT under the condition P d |a | < ∞. u u∈Z u If we take for instance for i positive integers

d Y 1 a = (−1)ui √ , u1,u2,...,ud u log u i=1 i i

P d |a | = ∞. then u∈Z u Furthermore, condition (3.1.13), which was used in this context by Wang and Woodroofe [WW13], is not satisfied but condition (3.1.14) holds.

Another class of nonlinear random fields are the Volterra processes, which plays an important role in the nonlinear system theory.

Example 3.1.7 (Volterra field). Let (ξn)n∈Zd be a random field of independent random variables identically distributed centered and with finite second moment. Define

X Xk = au,vξk−uξk−v, (u,v)≥(0,0)

P 2 where au,v are real coefficients with au,u = 0 and u,v≥0 au,v < ∞. Denote

j X cu,v(j) = ak+u,k+v k=1 and assume

X |bj| X < ∞ where b2 = (c2 (j) + c (j)c (j)). |j|3/2 j u,v u,v v,u j≥1 u≥0,v≥0,u6=v

36 Then the CLT in (3.1.12) holds.

Remark 3.1.8. In examples 3.1.6 and 3.1.7 the fields are Bernoulli. However, we can take as innovations the random field (ξn,m)n.m∈Z having as columns independent copies of a stationary and ergodic martingale differences sequence.

3.2 Proofs

Now we gather the proofs. They are based on a new result for a random field consisting of triangular arrays of row-wise stationary martingale differences, which allows us to find its asymptotic behavior by analyzing the limiting distribution of its columns.

Theorem 3.2.1. Assume that for each n fixed (Dn,k)k∈Z forms a stationary martingale difference sequence adapted to the stationary nested filtration (Fn,k)k∈Z and the 2 family (Dn,1)n≥1 is uniformly integrable. In addition assume that for all m ≥ 1 fixed,

(Dn,1, ..., Dn,m)n≥1 converges in distribution to (L1,L2, ..., Lm), and

m 1 X L2 → c2 in L1 as m → ∞. (3.2.1) m j j=1

Then 1 Xn √ Dn,k ⇒ cZ as n → ∞, n k=1 where Z is a standard normal variable.

√ Proof of Theorem 3.2.1. For the triangular array (Dn,k/ n)k≥1, we shall verify the conditions of Theorem 3.3.1, given for convenience in Section 3.3. Note that for

ε > 0 we have 1 2 2 2 √ E( max Dn,k) ≤ ε + E(Dn,1I(|Dn,1| > ε n)) (3.2.2) n 1≤k≤n

37 2 and, by the uniformly integrability of (Dn,1)n≥1, we obtain:

2 √ lim E(D I(|Dn,1| > ε n)) = 0. n→∞ n,1

Therefore, by passing to the limit in inequality (3.2.2), first with n → ∞ and then with

ε → 0, the first condition of Theorem 3.3.1 is satisfied. The result will follow from

Theorem 3.3.1 if we can show that

n 1 X D2 → c2 in L1 as n → ∞. n n,j j=1

2 To prove it, we shall apply the following lemma to the sequence (Dn,k)k∈Z after notic- 2 2 ing that, under our assumptions, for all m ≥ 1 fixed, (Dn,1, ..., Dn,m)n≥1 converges in 2 2 2 distribution to (L1,L2, ..., Lm).

Lemma 3.2.2. Assume that the triangular array of random variables (Xn,k)k∈Z is row- wise stationary and (Xn,1)n≥1 is a uniformly integrable family. For all m ≥ 1 fixed,

(Xn,1, ..., Xn,m)n≥1 converges in distribution to (X1,X2, ..., Xm) and

m 1 X X → c in L1 as m → ∞. (3.2.3) m u u=1

Then n 1 X X → c in L1 as n → ∞. n n,u u=1 Proof of Lemma 3.2.2. Let m ≥ 1 be a fixed integer and define consecutive blocks of indexes of size m, Ij(m) = {(j − 1)m + 1, ..., mj)}. In the set of integers from 1 to n we have kn = kn(m) = [n/m] such blocks of integers and a last one containing less than m indexes. Practically, by the stationarity of the rows and by the triangle inequality,

38 we write

1 Xn E| (Xn,u − c)| ≤ (3.2.4) n u=1 1 Xkn X 1 Xn E| (Xn,k − c)| + E| (Xn,u − c)| n j=1 k∈Ij (m) n u=knm+1 1 Xm m ≤ E| (Xn,u − c)| + E|Xn,1 − c|. m u=1 n

Note that, by the uniform integrability of (Xn,1)n≥1, we have

m m lim sup E|Xn,1 − c| ≤ lim sup (E|Xn,1| + |c|) = 0. n→∞ n n→∞ n

Now, by the continuous function theorem and by our conditions, for m fixed, we have the following convergence in distribution:

1 Xm 1 Xm (Xn,u − c) ⇒ (Xu − c). m u=1 m u=1

In addition, by the uniform integrability of (Xn,k)n and by the convergence of moments theorem associated to convergence in distribution, we have

1 Xm 1 Xm lim E| (Xn,u − c)| = E| (Xu − c)|, n→∞ m u=1 m u=1 and by assumption (3.2.3) we obtain

1 Xm E| (Xu − c)| → 0 as m → ∞. m u=1

The result follows by passing to the limit in (3.2.4), letting first n → ∞ followed by m → ∞.

When we have additional information about the type of the limiting distribution for the columns the result simplifies.

39 Corollary 3.2.3. If in Theorem 3.2.1 the limiting vector (L1,L2, ..., Lm) is stationary

Gaussian, then condition (3.2.1) holds and

1 Xn √ Dn,k ⇒ cZ as n → ∞, n k=1 where Z is a standard normal variable and c can be identified by

c2 = lim E(D2 ). n→∞ n,1

Proof. We shall verify the conditions of Theorem 3.2.1. Note that, by the martingale property, we have that cov(Dn,1,Dn,k) = 0. Next, by the condition of uniform integrability, by passing to the limit we obtain cov(L1,Lk) = 0. Therefore, the sequence (Lm)m is an i.i.d. Gaussian sequence of random variables and condition (3.2.1) holds.

In order to prove Theorem 3.1.1 we start by pointing out an upper bound for the variance given in Corollary 7.2 in Wang and Woodroofe [WW13]. It should be noticed that to prove it, the assumption that the random field is Bernoulli is not needed.

Lemma 3.2.4. Define (Xn,m)n,m∈Z by (3.1.2) and assume that (3.1.7) holds. Then, there is a universal constant C such that

1 X 1 √ kSn,mk ≤ C kE(Sj,i|F1,1)k. nm i,j≥1 (ji)3/2

By applying the triangle inequality, the contractivity property of the conditional expectation and changing the order of summations we easily obtain the following corollary.

Corollary 3.2.5. Under the conditions of Lemma 3.2.4 there is a universal constant C such that 1 X 1 √ kSn,mk ≤ C kE(Xj,i|F1,1)k . nm i,j≥1 (ji)1/2

40 Proof of Theorem 3.1.1. We shall develop the “small martingale method” in the context of random fields. To construct a row-wise stationary martingale approximation we shall introduce a parameter. Let ` be a fixed positive integer and denote k = [n2/`].

We start the proof by dividing the variables in each line in blocks of size ` and making sums in each block. Define

i` (`) 1 X X = X , i ≥ 1. j,i `1/2 j,u u=(i−1)`+1

Then, for each line j we construct the stationary sequence of martingale differences (`) (Yj,i )i∈Z defined by (`) (`) (`) (`) Yj,i = Xj,i − E(Xj,i |Fj,i−1),

(`) where Fj,k = Fj,k`. Also, we consider the triangular array of martingale differences (D(`) ) n1,i i≥1 defined by (`) 1 Xn1 (`) D = Y . n1,i √ j,i n1 j=1 √ (Pk D(`) / k) min(n , k) → ∞, In order to find the limiting distribution of i=1 n1,i k when 1 we shall apply Corollary 3.2.3. It is enough to show that

(D(`) , ..., D(`) ) ⇒ (L , ..., L ), n1,1 n1,N 1 N

(L , ..., L ) [(D(`) )2] where 1 N is stationary Gaussian and n1,1 n1 is uniformly integrable. Both these conditions will be satisfied if we are able to verify the conditions of Theorem 3.3.2, (`) (`) in Section 3.3, for the sequence (a1Yn,1 + ... + aN Yn,N )n, where a1, ..., aN are arbitrary, fixed real numbers. We have to show that, for ` fixed

k X 1 X (`) (`) (`) k E(a1Y + ... + aN Y |F )k < ∞. (3.2.5) k≥1 k3/2 j,1 j,N 1,N j=1

41 By the triangle inequality it is enough to treat each sum separately and to show that for all 1 ≤ v ≤ N we have

k X 1 X (`) (`) k E(Y |F )k < ∞. k≥1 k3/2 j,v 1,N j=1

(`) (`) (`) (`) By (3.1.7) we have that E(Yj,v |F1,N ) = E(Yj,v |F1,v). Therefore, by stationarity, the latter condition is satisfied if we can prove that

k X 1 X (`) (`) k E(Y |F )k < ∞. k≥1 k3/2 j,1 1,1 j=1

Now, by using once again (3.1.7), we deduce

(`) (`) (`) (`) (`) (`) E(Yj,1 |F1,1) = E(Xj,1 − E(Xj,1 |Fj,0 )|F1,1) = (`) (`) (`) (`) E(Xj,1 |F1,1) − E(Xj,1 |F1,0).

So, by the triangle inequality and the monotonicity of the L2-norm of the conditional expectation with respect to increasing random fields, we obtain

k k X (`) (`) X (`) (`) 1 k E(Y |F )k ≤ 2k E(X |F )k = 2 kE(S |F )k. j,1 1,1 j,1 1,1 `1/2 k,` 1,` j=1 j=1

Furthermore, since the filtration is commuting, by the triangle inequality we obtain

k ` k X X X kE(Sk,`|F1,`)k = k E(Xu,v|F1,v)k ≤ `k E(Xu,1|F1,1)k. u=1 v=1 u=1

By taking into account condition (3.1.8), it follows that we have

k X 1 X (`) (`) 1/2 X 1 k E(Y |F )k ≤ 2` kE(S |F1,1)k < ∞, k≥1 k3/2 j,v 1,N k≥1 k3/2 k,1 j=1

42 showing that condition (3.2.5) is satisfied, which implies that the conditions of Corollary

3.2.3 are satisfied. The conclusion is that

1 Xn1 Xk (`) 2 √ Yj,i ⇒ N (0, σ` ) as min(n1, k) → ∞, n1k j=1 i=1

2 where σ` is defined, in accordance with Theorem 3.3.2, by

1 Xn (`)2 σ2 = lim E Y . ` n→∞ n j=1 j,1

According to Theorem 3.2 in Billingsley [Bil99], in order to prove convergence and to √ find the limiting distribution of Sn1,n2 / n1n2 we have to show that

1 1 Xn1 Xk (`) lim lim supk√ Sn ,n − √ Y k = 0 (3.2.6) 1 2 j=1 i=1 j,i `→∞ n1,k→∞ n1n2 n1k

2 2 and N (0, σ` ) ⇒ N (0, σ ), which is equivalent to

2 2 σ` → σ as ` → ∞. (3.2.7)

√ 2 The conclusion will be that Sn1,n2 / n1n2 ⇒ N (0, σ ) as min(n1, n2) → ∞. Let us first prove (3.2.6). By the triangle inequality we shall decompose the difference in (3.2.6) into two parts. Relation (3.2.6) will be established if we show both

1 Xn1 Xk (`) (`) lim lim sup √ k E(X |F )k = 0. (3.2.8) j=1 i=1 j,i j,i−1 `→∞ n1,k→∞ n1k and 1 1 lim k√ Sn1,n2 − √ Sn1,k`k = 0. (3.2.9) n1,k→∞ n1n2 n1k`

In order to compute the standard deviation of the double sum involved, before taking the limit in (3.2.8), we shall apply Lemma 3.2.4 and a multivariate version of Remark

43 3.3.3 in Section 3.3. This expression is dominated by a universal constant times

X 1 Xj Xi (`) (`) k E(E(X(`) |F )|F )k. i,j≥1 (ij)3/2 u=1 v=1 u,v u,v−1 1,0

Now, Xj Xi (`) (`) (`) 1 E(E(X |F )|F ) = E(Sj,i`|F1,0). u=1 v=1 u,v u,v−1 1,0 `1/2

So, the quantity in (3.2.8) is bounded above by a universal constant times

1 X 1 kE(Sj,i`|F1,0)k, `1/2 i,j≥1 (ij)3/2 which converges to 0 as ` → ∞ under our condition (3.1.8), by Lemmas 2.7 and 2.8 in

Peligrad and Utev [PU05], applied in the second coordinate.

As far as the limit (3.2.9) is concerned, since by Lemma 3.2.4 and condition (3.1.8)

Pn1 Pn2 √ 2 the array j=1 i=1 Xj,i/ n1 n2 is bounded in L , it is enough to show that, for k` < n2 < (k + 1)`, we have

1 Xn1 Xn2 lim k√ Xj,ik = 0. n1,n2→∞ n1n2 j=1 i=k`+1

We just have to note that, again by Lemma 3.2.4, condition (3.1.8) and stationarity, there is a constant K such that

Xn1 Xn2 p k Xj,ik ≤ K n1` j=1 i=k`+1

and `/n2 → 0 as n2 → ∞.

We turn now to prove (3.2.7). By (3.2.6) and the orthogonality of martingale differences, 1 1 Xn1 (`) lim lim sup | k√ Sn1,n2 k − k√ Yj,0 k | = 0. `→∞ n1,n2→∞ n1n2 n1 j=1

44 So 1 lim lim sup | k√ Sn1,n2 k − σ` | = 0. `→∞ n1,n2→∞ n1n2

By the triangle inequality, this shows that σ` is a Cauchy sequence, therefore converges to a constant σ and also 1 lim k√ Sn1,n2 k = σ. n1,n2→∞ n1n2

The proof is now complete. d Proof of Theorem 3.1.3. The extensions to random fields indexed by Z , for d > 2, are straightforward following the same lines of proofs as for a two-indexed random field.

We shall point out the differences. To extend Lemma 3.2.4, we first apply a resultof

Peligrad and Utev [PU05] (see Theorem 3.3.2 in Section 3.3) to the stationary sequence Pm d−1 Yj(m) = i=1 Xj,i with j ∈ Z and then we apply induction. In order to prove Theorem 3.1.3, we partition the variables according to the last index.

Let ` be a fixed positive integer, denote k = [nd/`] and define

i` (`) 1 X X = X , i ≥ 1. j,i `1/2 j,u u=(i−1)`+1

(`) Then, for each j we construct the stationary sequence of martingale differences (Yj,i )i∈Z (`) (`) (`) (`) defined by Yj,i = Xj,i − E(Xj,i |Fj,i−1) and

0 (`) 1 Xn (`) D 0 = Y . n ,i p|n0| j=1 j,i

(`) (`) For showing that (Dn0,1, ..., Dn0,N ) ⇒ (L1, ..., LN ), we apply the induction hypothesis.

45 Proof of Example 3.1.6. Let us note first that the variables are square integrable and well defined. Note that

X X E(Su|F0) = ak−jξj 1≤k≤u j≤0 and therefore 2 X X 2 2 E(E (Su|F0)) = ( ak+i) E(ξ1). i≥0 1≤k≤u

The result follows by applying Theorem 3.1.3 (see Remark 3.3.3 and consider a multivariate analog of it). Proof of Example 3.1.7. Note that

j X X E(Sj|F0) = au,vξk−uξk−v k=1 (u,v)≥(k,k) j X X X = ak+u,k+vξ−uξ−v = cu,v(j)ξ−uξ−v. (u,v)≥(0,0) k=1 (u,v)≥(0,0)

Since by our conditions cu,u = 0 we obtain

2 X 2 2 E(E (Sj|F0)) = (cu,v(j) + cu,v(j)cv,u(j))E(ξuξv) . u≥0,v≥0,u6=v

3.3 Auxiliary results

For convenience we mention a classical result of McLeish which can be found on pp.

237-238 Gänssler and Häusler [GH79].

46 Theorem 3.3.1. Assume (Dn,i)1≤i≤n is an array of square integrable martingale differences adapted to an array (Fn,i)1≤i≤n of nested sigma fields. Suppose that

2 max |Dn,j| → 0 in L as n → ∞. 1≤j≤n and n X 2 2 Dn,j → c in probability as n → ∞. j=1

Pn 2 Then j=1 Dn,j converges in distribution to N (0, c ).

The following is a Corollary of Theorem 1.1 in Peligrad and Utev [PU05]. This central limit theorem was obtained by Maxwell and Woodroofe [MW00].

Theorem 3.3.2. Assume that (Xi)i∈Z is a stationary sequence adapted to a stationary filtration (Fi)i∈Z. Then there is a universal constant C1 such that

1/2 X∞ 1 kSnk ≤ C1n kE(Sk|F1)k. k=1 k3/2

If X∞ 1 kE(Sk|F1)k < ∞, k=1 k3/2

2 then (Sn/n)n is uniformly integrable and and there is a positive constant c such that

1 E (S )2 → c2 as n → ∞. n n

If in addition the sequence is ergodic we have

1 √ S ⇒ cN (0, 1) as n → ∞. n n

47 Remark 3.3.3. Note that we have the following equivalence:

X∞ 1 X∞ 1 kE(Sk|F1)k < ∞ if and only if kE(Sk|F0)k < ∞. k=1 k3/2 k=1 k3/2

Remark 3.3.4. The condition (3.1.1) is implied by

X∞ 1 kE(Xk|F1)k < ∞. k=1 k1/2

Lemma 3.3.5. Assume that X,Y,Z are integrable random variables such that (X,Y ) and Z are independent. Assume that g(X,Y ) is integrable. Then

E(g(X,Y )|σ(Y,Z)) = E(g(X,Y )|Y ) a.s. and

E(g(Z,Y )|σ(X,Y )) = E(g(Z,Y )|Y ) a.s.

Proof. Since (X,Y ) and Z are independent, it is easy to see that X and Z are conditionally independent given Y . The result follows from this observation by Problem

34.11 in [Bil95].

48 Chapter 4

Martingale approximations for random fields

Most results in this chapter has been published in Peligrad and Zhang [PZ18a]. In setting of sequences of random variables, martingale approximation is a fruitful approach to obtain limit theorems. The idea dates back to Gordin [Gor69]. Regarding random fields, formed by multi-indexed random variables, the theory of martingale approximation was started by Rosenblatt [Ros72] and its development is in progress. In recent years, the interest is in the approximation with ortho-martingales, which were introduced by Cairoli

[Cai69]. Different from the random sequences, ergodicity is not enough to guarantee the central limit theorem for stationary ortho-martingales. However, Volný [Vol15] investigated this problem and showed that the ergodicity in one direction of the stationary martingale differences field is a sufficient condition for the central limit theorem.

In order to exploit the richness of the martingale techniques, in this chapter we present necessary and sufficient conditions for an ortho-martingale approximation in mean square. These approximations extend to random fields the corresponding results obtained for sequences of random variables by Dedecker et al. [DMV07], Zhao and

49 Woodroofe [ZW08] and Peligrad [Pel10]. The tools for proving these results consist of projection decomposition.

We would like to mention several remarkable recent contributions, which provide interesting sufficient conditions for ortho-martingale approximations, by Gordin [Gor09], El

Machkouri et al. [EVW13], Volný and Wang [VW14], Cuny et al. [CDV15], Peligrad and

Zhang [PZ18b], and Giraudo [Gir18]. A special type of ortho-martingale approximation, so called co-boundary decomposition, was studied by El Machkouri and Giraudo [EG16] and Volný [Vol18]. Other recent results involve interesting mixingale-type conditions in

Wang and Woodroofe [WW13], and mixing conditions in Bradley and Tone [BT17].

This chapter is organized as follows. We state our assumptions and main results in

Section 4.1. In Section 4.2, we gather the proofs. The multidimensional indexed results are provided in Section 4.3. Applications to linear and nonlinear random fields are given in the last section.

4.1 Results

Due to the complicated notation, we shall explain first the results for double indexed random fields and, at the end, we shall formulate the results for general random fields.No technical difficulties arise when the double indexed random field is replaced by amultiple indexed one. To construct a stationary random field adapted to a stationary filtration, we use the same framework as in Section 3.1 of Chapter 3: we start with a stationary real

2 Z valued random field (ξn,m)n,m∈Z defined on a probability space (R , B,P ), and consider the stationary random field (Xn,m)n,m∈Z is defined by (3.1.2) and the filtration Fn,m defined by (3.1.3) is commuting in the sense of (3.1.7).

Below we use the notations

Xk,j Sk,j = Xu,v and E(X|Fa,b) = Ea,b(X). u,v=1

50 2 For an integrable random variable X and (u, v) ∈ Z , we introduce the projection operators defined by

Pu,v˜ (X) = (Eu,v − Eu−1,v)(X) (4.1.1)

Pu,v˜(X) = (Eu,v − Eu,v−1)(X). (4.1.2)

Note that, by the property of commuting filtration (See (3.1.7) in Chapter 3), we have

Pu,v(X) := Pu,v˜ ◦ Pu,v˜(X) = Pu,v˜ ◦ Pu,v˜ (X) and by an easy computation we have that

Pu,v(X) = Eu,v(X) − Eu,v−1(X) − Eu−1,v(X) + Eu−1,v−1(X). (4.1.3)

We shall introduce the definition of an ortho-martingale, which will be referred toas a martingale with multiple indexes or simply martingale.

Definition 4.1.1. Let d be a function and define

Dn,m = d(ξi,j, i ≤ n, j ≤ m), (4.1.4)

Assume integrability. We say that (Dn,m)n,m∈Z is a martingale differences field if

Ea,b(Dn,m) = 0 for either a < n or b < m.

Set Xk,j Mk,j = Du,v. u,v=1

In the sequel we shall denote by k·k the norm in L2. As before, ⇒ denotes the convergence in distribution and a ∧ b stands for the minimum of a and b.

Definition 4.1.2. We say that a random field (Xn,m)n,m∈Z defined by (3.1.2) admits a martingale approximation if there is a sequence of martingale differences (Dn,m)n,m∈Z

51 defined by (4.1.4) such that

1 2 lim kSn,m − Mn,mk = 0. (4.1.5) n∧m→∞ nm

Theorem 4.1.3. Assume that (3.1.7) holds. The random field (Xn,m)n,m∈Z defined by (3.1.2) admits a martingale approximation if and only if

n m 1 X X kP (S ) − D k2 → 0 when n ∧ m → ∞ (4.1.6) nm 1,1 j,k 1,1 j=1 k=1 and both

1 1 kE (S )k2 → 0 and kE (S )k2 → 0 when n ∧ m → ∞. (4.1.7) nm 0,m n,m nm n,0 n,m

Remark 4.1.4. Condition (4.1.7) in Theorem 4.1.3 can be replaced by

1 kS k2 → kD k2. (4.1.8) nm n,m 1,1

Theorem 4.1.5. Assume that (3.1.7) holds. The random field (Xn,m)n,m∈Z defined by (3.1.2) admits a martingale approximation if and only if

n m 1 X X P (S ) converges in L2 to D when n ∧ m → ∞ (4.1.9) nm 1,1 j,k 1,1 j=1 k=1 and the condition (4.1.8) holds.

Corollary 4.1.6. Assume that the vertical shift T (or horizontal shift S) is ergodic and either the conditions of Theorem 4.1.3 or Theorem 4.1.5 hold. Then

1 2 √ Sn1,n2 ⇒ N (0, c ) when n1 ∧ n2 → ∞, (4.1.10) n1n2

2 2 where c = kD0,0k .

52 4.2 Proofs

Proof of Theorem 4.1.3. We start from the following orthogonal representation

n m X X Sn,m = Pi,j(Sn,m) + Rn,m, (4.2.1) i=1 j=1 with

Rn,m = En,0(Sn,m) + E0,m(Sn,m) − E0,0(Sn,m).

Note that for all 1 ≤ a ≤ i − 1, 1 ≤ b ≤ j − 1 we have Pi,j(Xa,b) = 0; for all 1 ≤ b ≤ j − 1 we have Pi,j(Xi,b) = 0 and for all 1 ≤ a ≤ i − 1, Pi,j(Xa,j) = 0. Whence,

n m X X Pi,j(Sn,m) = Pi,j( Xu,v). u=i v=j

This shows that for any martingale differences sequence defined by (4.1.4), by orthogonality, we obtain

n m n m 2 X X X X 2 2 E(Sn,m − Mn,m) = kPi,j( Xa,b) − Di,jk + kRn,mk (4.2.2) i=1 j=1 a=i b=j n m n−i+1 m−j+1 X X X X 2 2 = kP1,1( Xa,b) − D1,1k + kRn,mk i=1 j=1 a=1 b=1 n m X X 2 2 = kP1,1(Si,j) − D1,1k + kRn,mk . i=1 j=1

A first observation is that we have a martingale approximation if and only if both(4.1.6)

2 is satisfied and kRn,mk /nm → 0 as n ∧ m → ∞.

Computation, involving the fact that the filtration is commuting, shows that

2 2 2 2 kRn,mk = kEn,0(Sn,m)k + kE0,m(Sn,m)k − kE0,0(Sn,m)k , (4.2.3)

53 2 2 and since kE0,0(Sn,m)k ≤ kE0,m(Sn,m)k we have that kRn,mk /nm → 0 as n ∧ m → ∞ if and only if (4.1.7) holds.

Proof of Theorem 4.1.5. Let us first note that D1,1 defined by (4.1.9) is a martingale difference. By using the translation operators we then define the sequence of martingale differences (Du,v)u,v∈Z and the sum of martingale differences (Mu,v)u,v∈Z. This time we evaluate 2 2 2 E(Sn,m − Mn,m) = E(Sn,m) + E(Mn,m) − 2E(Sn,mMn,m).

By using the martingale property, stationarity and simple algebra we obtain

n m n m n m X X X X X X E(Sn,mMn,m) = E(Du,vXi,j) = E(D1,1Su,v). u=1 v=1 i≥u j≥v u=1 v=1

A simple computation involving the properties of conditional expectation and the martingale property shows that

E(D1,1Su,v) = E(D1,1P1,1(Su,v)).

By (4.1.9) this identity gives that

1 2 lim E(Sn,mMn,m) = E(D ). n∧m→∞ nm 1,1

From the above considerations

1 2 1 2 2 lim E(Sn,m − Mn,m) = lim E(S ) − E(D ), n∧m→∞ nm n∧m→∞ nm n,m 1,1 whence the martingale approximation holds by (4.1.8).

Let us assume now that we have a martingale approximation. According to Theorem

4.1.3 condition (4.1.6) is satisfied. In order to show that (4.1.6) implies (4.1.9) weapply

54 the Cauchy-Schwarz inequality twice:

n m n m 1 X X 1 X X k (P (S ) − D )k2 ≤ k (P (S ) − D )k2 nm 1,1 i,j 1,1 nm2 1,1 i,j 1,1 i=1 j=1 i=1 j=1 n m 1 X X ≤ kP (S ) − D )k2. nm 1,1 i,j 1,1 i=1 j=1

Also, by the triangular inequality

1 1 √ kSn,mk − kD1,1k ≤ √ kSn,m − Mn,mk → 0 as n ∧ m → ∞, nm nm and (4.1.8) follows. Proof of Remark 4.1.4. If we have a martingale decomposition, then by Theorem

4.1.3 we have (4.1.6) and by Theorem 4.1.5 we have (4.1.8). Now, in the opposite direction, just note that (4.1.6) implies (4.1.9) and then apply Theorem 4.1.5. Proof of Corollary 4.1.6. This Corollary follows as a combination of Theorem 4.1.3

(or Theorem 4.1.5) with the main result in Volnỳ [Vol15] via Theorem 25.4 in Billingsley

[Bil95].

4.3 Multidimensional index sets

d The extensions to random fields indexed by Z , for d > 2, are straightforward following the same lines of proofs as for a double indexed random field. We use the same setting as in Chapter 3. Recall that by u ≤ n, we understand u = (u1, ..., ud), n = (n1, ..., nd) and 1 ≤ u1 ≤ n1,..., 1 ≤ ud ≤ nd.

The filtration (Fu)u∈Zd and stationary random fields (Xm)u∈Zd are defined by

Fu = σ(ξj : j ≤ u) (4.3.1)

55 and

k1 kd Xk = f(T1 ◦ ... ◦ Td (ξu)u≤0), (4.3.2)

where (ξu)u∈Zd is a stationary random field defined on the canonical probability space d RZ and Ti’s are the coordinate-wise translations. d−1 We say that the filtration is commuting if for a ≥ u ∈ R, b, v ∈ R we have

E(E(X|Fa,b)|Fu,v) = E(X|Fu,b∧v),

where the minimum is taken coordinate-wise and we used notation Eu(X) = E(X|Fu).

Let d be a function and define

Xk Xk Dm = d((ξj)j≤m) and set Mk = Du,Sk = Xu. u=1 u=1

Definition 4.3.1. Assume integrability. We say that (Dm)m∈Zd is a martingale differences field if Ea(Dm) = 0 is at least one coordinate of a is strictly smaller than the corresponding coordinate of m.

We have to introduce the d-dimensional projection operators. By using the fact that the filtration is commuting, it is convenient to define

Pu(X) := Pu(1) ◦ Pu(2) ◦ ... ◦ Pu(d)(X), (4.3.3) where

Pu(j)(Y ) := E(Y |Fu) − E(Y |Fu(j)).

Above, we used the notation: u(j) = u0 where u0 has all the coordinates of u with the exception of the j-th coordinate, which is uj − 1. For instance when d = 3,Pu(2)(Y ) =

E(Y |Fu1,u2,u3 ) − E(Y |Fu1,u2−1,u3 ).

56 We say that a random field (Xn)n∈Zd admits a martingale approximation if there is a sequence of martingale differences (Dm)m∈Zd such that

1 2 kSn − Mnk → 0 when min ni → ∞, (4.3.4) |n| 1≤i≤d

where |n| = n1 · ... · nd. Let us introduce the following regularity condition

1 2 2 kSnk → E(D1) when min ni → ∞. (4.3.5) |n| 1≤i≤d

Theorem 4.3.2. Assume that the filtration is commuting. The following statements are equivalent:

(a) The random field (Xn)n∈Zd admits a martingale approximation. (b) The random field satisfies (4.3.5) and

1 Xn 2 kP1(Sj) − D1k → 0 when min ni → ∞. (4.3.6) |n| j≥1 1≤i≤d

1 2 kEn (Sn)k → 0 when min ni → ∞, |n| j 1≤i≤d

d where nj ∈ Z has the j-th coordinate 0 and the other coordinates equal to the coordinates of n.

(d) The random field satisfies (4.3.5) and

n 1 X 2 P1(Sj) converges in L to D1 when min ni → ∞. (4.3.7) |n| 1≤i≤d j=1

57 Corollary 4.3.3. Assume that one of the shifts (Ti)1≤i≤d is ergodic and either one of the conditions of Theorem 4.3.2 holds. Then

1 2 Sn ⇒ N (0, c ) when min ni → ∞, p|n| 1≤i≤d

2 2 where c = kD0k .

4.4 Examples

Let us apply these results to linear and nonlinear random fields with independent innovations.

Example 4.4.1 (Linear field). Let (ξn)n∈Zd be a random field of independent, identically distributed random variables which are centered and have finite second moment, σ2 =

2 E(ξ0). For k ≥ 0 define X Xk = ajξk−j. j≥0

P 2 Pj−1 Assume that j≥0 aj < ∞ and denote bj = k=0 ak. Also assume that

n 1 X bj → c when min ni → ∞ (4.4.1) |n| 1≤i≤d j=1 and 2 E(Sn) 2 2 → c σ when min ni → ∞. |n| 1≤i≤d

Then the martingale approximation holds.

Proof of Example 4.4.1. The result follows by simple computations and by applying

Theorem 4.3.2 (d).

Example 4.4.2 (Volterra field). Let (ξn)n∈Zd be a random field of independent random 2 2 variables identically distributed centered and with finite second moment, σ = E(ξ0).

58 For k ≥ 1, define X Xk = au,vξk−uξk−v, (u,v)≥(0,0)

P 2 where au,v are real coefficients with au,u = 0 and u,v≥0 au,v < ∞. Denote

n j 1 X X c = (a + a ), (4.4.2) n,u,v |n| k−u,k−v k−v,k−u j=1 k=1

Denote A = {u ≤ 1, there is 1 ≤ i ≤ d with ui = 1} and B = {u ≤ 1} and assume that

X 2 lim (cn,u,v − cm,u,v) = 0. (4.4.3) n>m→∞ (u,v)∈(A,B)

Also assume that 2 E(Sn) 4 2 → σ c when min ni → ∞, |n| 1≤i≤d where c2 is the limit of

1 X 2 cn,u,v when min ni → ∞. |n| 1≤i≤d (u,v)∈(A,B)

Then the martingale approximation holds.

Proof of Example 4.4.2. We have

X X P1(Xk) = au,vP1(ξk−uξk−v) = ak−u,k−vP1(ξuξv). (u,v)≥(0,0) (u,v)≥(k,k)

Note that P1(ξuξv) 6= 0 if and only if u ∈ A and v ∈ B or v ∈ A and u ∈ B. Therefore,

X P1(Xk) = (ak−u,k−v + ak−v,k−u)ξuξv (u,v)∈(A,B)

59 and n n j 1 X 1 X X X P (S ) = (a + a )ξ ξ . |n| 1 j |n| k−u,k−v k−v,k−u u v j=1 (u,v)∈(A,B) j=1 k=1

By independence, and with the notation (4.4.2) this convergence happens if (4.4.3) holds.

It remains to apply Theorem 4.3.2 (d).

60 Chapter 5

Quenched central limit for random fields

Most results in this chapter can be found in [ZRP19], which has been submitted to a peer-reviewed journal. The quenched central limit theorem studies the asymptotic behavior of partial sums of random variables (Xk)k∈Zd , d ≥ 1 conditioned to start from a point or fixed past trajectory. This problem is difficult, as the stationary processes started from a fixed past trajectory are no longer stationary. In the caseof d = 1, this type of convergence has been widely explored for the last few decades. One of the remarkable result in this direction is given by Derriennic and Lin [DL01](see page 520).

It was shown that the stationary and ergodic martingale differences sequence satisfies the quenched central limit theorem. Since then, there has been a considerable amount of new research on the quenched central limit theorem for random processes.

However, this problem was rarely investigated for stationary random fields (for the case d ≥ 2). To the best of our knowledge, so far, the only quenched invariance principle for random fields is due to Peligrad and Volný [PV18]. Their paper contains a quenched functional CLT for ortho-martingales and a quenched functional CLT for random fields via co-boundary decomposition. By constructing an example of an ortho-martingale

61 which satisfies the CLT but not its quenched form, Peligrad and Volný [PV18] showed that the finite second moment condition is not enough for the quenched CLT andthey

2 provided a minimal moment condition, that is, EX0,0 log(1+|X0,0|) < ∞, for the validity of this type of results.

Here, we aim to establish sufficient conditions in terms of projective criteria such that a quenched CLT holds. The tools for proving these results consist of ortho-martingale approximations, projective decompositions and ergodic theorems for Dunford-Schwartz operators.

The chapter is organized as follows. In the first section, we present the assumptions and main results for double-indexed random fields. In Section 5.2, we prove results for double-indexed random fields. Extensions to general indexed random fields and their proofs are given in Section 5.3. In Section 5.4, we apply our results to linear and Volterra random fields with independent innovations, which are often encountered in economics.

Our results could also be formulated in the language of dynamical systems, leading to new results in this field. For the convenience of the reader, in the Section 5.5, weprovide a well-known inequality for martingales and an important theorem in decoupling theory which will be of great importance for the proof of our main results.

5.1 Assumptions and Results

In this section, as in the previous chapters, we shall only talk about the double-indexed random fields. After obtaining results for double-indexed random fields, we will extend

d them to random fields indexed by Z , d > 2. Using the same framework as in Section 3.1 of Chapter 3, we introduce a stationary random field adapted to a stationary filtration.

In other words, for each n, m ∈ Z, we assume Xn,m and Fn,m are defined by (3.1.4) and (3.1.3), respectively. For all i, j ∈ Z, we also define the following sigma algebras generated by the union of sigma algebras: F∞,j = ∨n∈ZFn,j, Fi,∞ = ∨m∈ZFi,m and

F∞,∞ = ∨i,j∈ZFi,j.

62 In addition, X0,0 is assumed to centered and square integrable, and the filtration

(Fn,m)n,m∈Z is assumed to be commuting in the sense of (3.1.7). Let φ : [0, ∞) → [0, ∞) be a Young function, that is, a convex function satisfying

φ(x) φ(x) lim = 0 and lim = ∞. x→0 x x→∞ x

We shall define the Luxemburg norm associated with φ which will be needed in the sequel. For any measurable function f from Ω to R, the Luxemburg norm of f is define by (Krasnosel’skii and Rutitskii [KR61], relation 9.18 and 9.19, p. 79)

kfkφ = inf{k ∈ (0, ∞): Eφ(|f|/k) ≤ 1}. (5.1.1)

ω ω In the sequel, for any ω ∈ Ω, we shall denote P (·) = P (·|F0,0)(ω) and denote by E the expectation corresponding to P ω.

Set Xk,j Xk,j Sk,j = Xu,v and Mk,j = Du,v u,v=1 u,v=1 where (Du,v)u,v∈Z is a martingale differences (Definition 4.1.1 in Chapter 4).

Definition 5.1.1. We say that a random field (Xn,m)n,m∈Z defined by (3.1.4) admits a martingale approximation if there is a field of martingale differences (Dn,m)n,m∈Z defined by (4.1.4) such that

1 ω 2 lim E (Sn,m − Mn,m) = 0 for almost all ω ∈ Ω. (5.1.2) n∧m→∞ nm

Relevant to the construction of martingale differences are the projection operators.

2 Recall that for each (u, v) ∈ Z , the projection operators are defined by

Pu,v˜ (X) = (Eu,v − Eu−1,v)(X),

63 Pu,v˜(X) = (Eu,v − Eu,v−1)(X) and

Pu,v(·) := Pu,v˜ ◦ Pu,v˜(·) = Pu,v˜ ◦ Pu,v˜ (·) = (Eu,v − Eu,v−1 − Eu−1,v + Eu−1,v−1)(·).

Our main result is the following theorem, which is an extension of the quenched CLT for ortho-martingales in Peligrad and Volný [PV18] to stationary random fields satisfying the generalized Hannan condition [Han73].

Theorem 5.1.2. Assume that (Xn,m)n,m∈Z is defined by (3.1.4) and the filtrations defined by (3.1.3) are commuting. Also assume that T (or S) is ergodic and in addition

X kP0,0(Xu,v)k2 < ∞. (5.1.3) u,v≥0

Then, for almost all ω ∈ Ω,

1 (S − R ) ⇒ N (0, σ2) under P ω when n → ∞. n n,n n,n

where Rn,n = En,0(Sn,n) + E0,n(Sn,n) − E0,0(Sn,n).

It should be noted that, for a stationary ortho-martingale, the existence of finite second moment is not enough for the validity of a quenched CLT when the summation in taken on rectangles (see Peligrad and Volný [PV18]). In order to assure the validity of a martingale approximation with a suitable moment condition we shall reinforce condition

(5.1.3) when dealing with indexes n and m which converge independently to infinity.

Theorem 5.1.3. Assume now that (5.1.3) is reinforced to

X kP0,0(Xu,v)kφ < ∞, (5.1.4) u,v≥0

64 2 where φ(x) = x log(1 + |x|) and k·kφ is defined by (5.1.1). Then, for almost all ω ∈ Ω,

1 (S − R ) ⇒ N (0, σ2) under P ω when n ∧ m → ∞, (5.1.5) (nm)1/2 n,m n,m

where Rn,m = En,0(Sn,m) + E0,m(Sn,m) − E0,0(Sn,m).

The random centering is not needed if we impose two regularity conditions.

Corollary 5.1.4. Assume that the conditions of Theorem 5.1.3 hold. If

2 2 E0,0 E0,m(Sn,m) E0,0 En,0(Sn,m) → 0 a.s. and → 0 a.s. when n ∧ m → ∞, nm nm (5.1.6) then for almost all ω ∈ Ω,

1 S ⇒ N (0, σ2) under P ω when n ∧ m → ∞. (5.1.7) (nm)1/2 n,m

If the conditions of Theorem 5.1.2 hold and (5.1.6) holds with m = n, then for almost all ω ∈ Ω, 1 S ⇒ N (0, σ2) under P ω when n → ∞. (5.1.8) n n,n

For the sake of applications, we provide a sufficient condition which will take care of both (5.1.4) and also of regularity assumptions (5.1.6).

Corollary 5.1.5. Assume that (Xn,m)n,m∈Z is defined by (3.1.4) and the filtrations are commuting. Also assume that T (or S) is ergodic and in addition for δ ≥ 0

X kE1,1(Xu,v)k2+δ < ∞. (5.1.9) (uv)1/(2+δ) u,v≥1

(a) If δ = 0, then the quenched convergence (5.1.8) holds.

(b) If δ > 0, then the quenched convergence (5.1.7) holds.

65 Remark 5.1.6. Assume that (Xn,m)n,m∈Z is defined by (3.1.4) and the filtrations are commuting. Also assume that T (or S) is ergodic, (5.1.4) holds for φ(x) = x2 log(1+|x|) and (5.1.9) holds for δ = 0. Then, the quenched convergence (5.1.7) holds.

Remark 5.1.7. Corollary 5.1.5 can be viewed as an extension to the random fields of

Proposition 12 in Cuny and Peligrad [CP12]. As we shall see the proof for random fields is much more involved and requires several intermediary steps and new ideas.

5.2 Proofs

Let us point out the main idea of the proof. Since Peligrad and Volný (2018) [PV18] proved a quenched CLT for ortho-martingales, the proof can be reduced to prove the existence of an almost sure ortho-martingale approximation for the random field we consider. We start with the proof of Theorem 5.1.3, since the proof of Theorem 5.1.2 is similar with the exception that we use different ergodic theorems.

Let us denote by Tˆ and Sˆ the operators on L2 defined by Tˆ f = f ◦ T , Sfˆ = f ◦ S, where T and S are the shift transformations in Chapter 3, defined by (3.1.5) and (3.1.6), respectively.

Proof of Theorem 5.1.3. Starting from condition (5.1.4), by triangle inequality we have that X f0 := |P0,0(Xu,v)| < ∞ a.s. (5.2.1) u,v≥0 and X kf0kφ ≤ kP0,0(Xu,v)kφ < ∞, u,v≥0

2 which clearly implies that E(f0 log(1 + |f0|)) < ∞.

Note that by (5.2.1) P1,1(Sn,m) is convergent almost surely. Denote the pointwise limit by X D1,1 = lim P1,1(Sn,m) = P1,1(Xu,v). n∧m→∞ u,v≥1

66 Meanwhile, by the triangle inequality and (5.1.4), we obtain

Thus, by the dominated convergence theorem, P1,1(Sn,m) converges to D1,1 a.s. and in

L2(P ) as n ∧ m → ∞.

Since E0,1(P1,1(Sn,m)) = 0 a.s. and E1,0(P1,1(Sn,m)) = 0 a.s., by defining for every ˆi−1 ˆj−1 i, j ∈ Z, Di,j = T S D1,1, we conclude that (Di,j)i,j∈Z is a martingale differences field. By the expression of D1,1 above,

X Di,j = Pi,j(Xu,v). (u,v)≥(i,j)

Now we look at the decomposition of Sn,m:

n m n m X X X X Sn,m − Rn,m = Pi,j( Xu,v) i=1 j=1 u=i v=j where

Rn,m = En,0(Sn,m) + E0,m(Sn,m) − E0,0(Sn,m).

Therefore

n m n m Sn,m − Rn,m − Mn,m 1 X X X X √ = √ P ( X ) − D . nm nm i,j u,v i,j i=1 j=1 u=i v=j

By the orthogonality of the martingale differences field (Pi,j −Di,j)i,j∈Z and the assumption that the filtration is commuting, we have

67 n m n m 2 1 1 X X X X E (S − R − M )2 = E P ( X ) − D . nm 0,0 n,m n,m n,m nm 0,0 i,j u,v i,j i=1 j=1 u=i v=j

From the main results in Peligrad and Volný [PV18], we know that the quenched CLT √ holds for Mn,m/ nm. Therefore by Theorem 25.4 in Billingsley [Bil95], in order to prove the conclusion of this theorem, it is enough to show that

1 2 lim E0,0 (Sn,m − Rn,m − Mn,m) = 0 a.s. n∧m→∞ nm

Define the operators

Q1(f) = E0,∞(Tˆ f); Q2(f) = E∞,0(Sfˆ ).

Then we can write

2 i j 2 E0,0 (Pi,j(Xu,v)) = Q1Q2(P0,0(Xu−i,v−j)) .

By simple algebra we obtain

n m 2 X X E0,0 Pi,j( Xu,v) − Di,j u=i v=j ∞ m ∞ ∞ 2 X X X X = E0,0 Pi,j(Xu,v) + Pi,j(Xu,v) . u=n+1 v=j u=i v=m+1

Therefore, by elementary inequalities we have the following bound

n m n m 2 1 1 X X X X E (S − R − M )2 = E P ( X ) − D nm 0,0 n,m n,m n,m nm 0,0 i,j u,v i,j i=1 j=1 u=i v=j

≤ 2(In,m + IIn,m),

68 where we have used the notations

n m ∞ ∞ 2 1 X X X X I = Qi Qj |P (X )| n,m nm 1 2 0,0 u,v i=1 j=1 u=n+1−i v=0 and n m ∞ ∞ 2 1 X X X X II = Qi Qj |P (X )| . n,m nm 1 2 0,0 u,v i=1 j=1 u=0 v=m+1−j

The task is now to show the almost sure negligibility of each term. By symmetry we treat only one of them.

Let c be a fixed integer satisfying c < n. We decompose In,m into two parts

n−c m ∞ ∞ 2 1 X X j X X Qi Q |P (X )| := A (c) (5.2.2) nm 1 2 0,0 u,v n,m i=1 j=1 u=n+1−i v=0 and n m ∞ ∞ 2 1 X X j X X Qi Q |P (X )| := B (c). (5.2.3) nm 1 2 0,0 u,v n,m i=n−c+1 j=1 u=n+1−i v=0

Note that

n m 1 X X B (c) ≤ Qi Qj f 2 n,m nm 1 2 0 i=n−c+1 j=1 n m n−c m 1 X X 1 X X = Qi Qj f 2 − Qi Qj f 2, nm 1 2 0 nm 1 2 0 i=1 j=1 i=1 j=1 where f0 is given by (5.2.1).

By the ergodic theorem for Dunford-Schwartz operators (Theorem 2.2.9 in Chapter

2), for each c fixed

n−c m 1 X X j lim Qi Q f 2 = E f 2 a.s. (5.2.4) n∧m→∞ nm 1 2 0 0 i=1 j=1

69 Therefore, for all c > 0

lim Bn,m(c) = 0 a.s. n∧m→∞

In order to treat the first term in the decomposition of In,m, note that

n−c m ∞ ∞ 1 X X j X X A (c) ≤ Qi Q f 2(c) where f (c) = |P (X )|. n,m nm 1 2 0 0 0,0 u,v i=1 j=1 u=c v=0

Again, by the ergodic theorem for Dunford-Schwartz operators (Theorem 2.2.9 in Chap- ter 2) , for each c fixed

n−c m 1 X X j lim Qi Q f 2(c) = E f 2(c) a.s. (5.2.5) n∧m→∞ nm 1 2 0 0 i=1 j=1

In addition, by (5.2.1), we know that limc→∞ |f0(c)| = 0. So, by the dominated convergence theorem, we have

2 lim lim An,m(c) ≤ lim E(f (c)) = 0 a.s. c→∞ n∧m→∞ c→∞ 0

The proof of the theorem is now complete. The proof of Theorem 5.1.2 requires only a slight modification of the proof of Theorem

5.1.3. Indeed instead of Theorem 1.1 in Ch. 6 in Krengel [Kre85] (Theorem 2.2.9), we shall use Theorem 2.8 in Ch. 6 in the same book (Theorem 2.2.12).

Proof of Corollary 5.1.4. By Theorem 5.1.3 together with Theorem 25.4 in Billings- ley [Bil95], it suffices to show that (5.1.6) implies that

1 2 lim E0,0(R ) = 0 a.s. (5.2.6) n∧m→∞ nm n,m

Simple computations involving the fact that the filtration is commuting gives that

2 2 2 2 E0,0(Rn,m) = E0,0 En,0(Sn,m) + E0,0 E0,m(Sn,m) − E0,0(Sn,m) (5.2.7)

70 2 2 and since E0,0(Sn,m) ≤ E0,0 E0,m(Sn,m) , we have

1 2 lim E0,0(R ) = 0 a.s. by condition (5.1.6). n∧m→∞ nm n,m

Proof of Corollary 5.1.5. Throughout the proof, denote by Cδ > 0 a generic constant depending on δ which may take different values from line to line.

Before we prove the corollary, we shall first establish a preparatory fact, namely that

(5.1.9) implies

X 1 X kP ˜(X )k < ∞. (5.2.8) u1/(2+δ) 0,0 u,v 2+δ u≥1 v≥0

By the Hölder’s inequality and the Rosenthal inequality for martingales (see Theorem

2.1.14 in Chapter 2), we have

X X kP0,0˜(Xu,v)k2+δ = kP−u,−v˜(X0,0)k2+δ v≥1 v≥1 2n+1−1 1 2n+1−1 2+δ X n 1+δ X 2+δ X n 1+δ X ≤ (2 ) 2+δ kP−u,−v˜(X0,0)k ≤ Cδ (2 ) 2+δ P−u,−v˜(X0,0) 2+δ n≥0 v=2n n≥0 v=2n 2+δ

X n 1+δ ≤ 2Cδ (2 ) 2+δ kE−u,−2n (X0,0)k2+δ. n≥0

Since the sequence (kE−u,−n(X0,0)k2)n≥1 is non-increasing in n, it follows that

2n−1 n 1+δ X kE−u,−k(X0,0)k2+δ (2 ) 2+δ kE n (X )k ≤ 2 . −u,−2 0,0 2+δ k1/(2+δ) k=2n−1

So ∞ X X kE−u,−k(X0,0)k2+δ kP ˜(X )k ≤ C . (5.2.9) 0,0 u,v 2+δ δ k1/(2+δ) v=1 k≥1

71 Thus relation (5.2.8) holds by (5.1.9), (5.2.9) and the stationarity. In addition we also have for any u ≥ 0 ∞ X kP0,0˜(Xu,v)k2+δ < ∞. (5.2.10) v=1 By symmetric roles of m and n, we have for any v ≥ 0

∞ X kP0˜,0(Xu,v)k2+δ < ∞. (5.2.11) u=1

Now we will proceed to prove Corollary 5.1.5 in two steps:

• Step 1. Condition (5.1.9) implies

1 2 lim E0,0(R ) = 0 a.s. n∧m→∞ nm n,m

First we show that (5.1.9) implies that

2 E0,0(Sn,m) → 0 a.s. when n ∧ m → ∞. nm

We bound this term in the following way

Now, (5.1.9) implies that

∞ ∞ X X |E0,0(Xu,v)| √ < ∞ a.s. uv u=1 v=1

72 Therefore, |E0,0(Sn,m)| √ → 0 a.s. (5.2.12) nm by letting n → ∞ followed by c → ∞.

By (5.2.7) and the symmetric roles of m and n, the corollary will follow if we can show that 2 (E0,m(Sn,m)) E → 0 a.s. when n ∧ m → ∞. 0,0 nm

By (5.2.12) this is equivalent to showing that

1 E (E (S ) − E (S ))2 → 0 a.s. when n ∧ m → ∞. nm 0,0 0,m n,m 0,0 n,m

We start from the representation

m n m 2 2 X X X E0,0 (E0,m(Sn,m) − E0,0(Sn,m)) = E0,0 P0,˜j Xu,v j=1 u=1 v=j m n m−j 2 X j X X = E0,0 Sb P0,0˜( Xu,v) . j=1 u=1 v=0

So,

m n m−j 2 1 2 1 X j X X E0,0 (E0,m(Sn,m) − E0,0(Sn,m)) = E0,0 Sb P ˜(Xu,v) nm mn 0,0 j=1 u=1 v=0 m c m−j 2 2 X j X X ≤ E0,0 Sb |P ˜(Xu,v)| mn 0,0 j=1 u=1 v=0 m n m−j 2 2 X j X 1 X + E0,0 Sb √ |P ˜(Xu,v)| m u 0,0 j=1 u=c+1 v=0

= In,m,c + IIn,m,c.

73 Let us introduce the operator

Q0(f) = E0,0(Sfb ).

We treat first the term In,m,c. For c fixed

2 m ∞ 2 2c X j X In,m,c ≤ sup E0,0 Sb |P ˜(Xu,v)| mn 0,0 1≤u≤c j=1 v=0 m ∞ 2 2c2 X X = sup Qj |P (X )| . mn 0 0,0˜ u,v 1≤u≤c j=1 v=0

By (5.2.10), the function ∞ X g(u) = |P0,0˜(Xu,v)| v=0 is square integrable. By the ergodic theorem for Dunford-Schwartz operators (see The- orem 11.4 in Eisner et al. [Eis+15] or Corollary 3.8 in Ch. 3, Krengel [Kre85], which is

Theorem 2.2.4 in Chapter 2)

m 1 X j 2 2 Q g (u) → E(g (u)) a.s. m 0 j=1 and therefore, since c is fixed,

lim In,m,c = 0 a.s. n∧m→∞

In order to treat the second term, note that

m ∞ ∞ 2 2 X X 1 X II ≤ Qj √ |P (X )| . n,m,c m 0 u 0,0˜ u,v j=1 u=c v=0

Denote ∞ ∞ X 1 X h(c) = √ |P (X )|. u 0,0˜ u,v u=c v=0

74 By (5.2.8), we know that

∞ ∞ X 1 X √ kP ˜(X )k < ∞. (5.2.13) u 0,0 u,v 2+δ u=1 v=0

So, E(h2(c)) < ∞. Again, by the ergodic theorem for the Dunford-Schwartz operators

(see Theorem 11.4 in Eisner et al. [Eis+15] or Corollary 3.8 in Ch. 3, Krengel [Kre85])

m ∞ ∞ !2 1 X X 1 X Qj (h2(c)) → E h2(c) ≤ √ kP (X )k . m 0 u 0,1˜ u,v 2 j=1 u=c v=0

So, by (5.2.13) m 1 X j lim lim Q (h2(c)) = 0 a.s. c→∞ m→∞ m 0 j=1

• Step 2. Condition (5.1.9) implies

X kP0,0(Xu,v)k2+δ < ∞, (5.2.14) u,v≥0 which clearly implies (5.1.4).

In fact, by applying twice the Rosenthal inequality for martingales (Theorem 2.1.14 in Chapter 2), for any integers a ≤ b and c ≤ d, we have

b d b d 2+δ X X 2+δ X X P−k,−k0 (X0,0) ≤ Cδ P−k,−k0 (X0,0) . (5.2.15) 2+δ k=a k0=c k=a k0=c 2+δ

In addition, note that for any integers a ≤ b and c ≤ d, we have

b d X X 2+δ 2+δ 2+δ k P−k,−k0 (X0,0)k2+δ ≤ 4 kE−a,−c(X0,0)k2+δ. (5.2.16) k=a k0=c

75 Then by the Hölder’s inequality together with (5.2.15) and (5.2.16), we obtain

2n+1−1 2m+1−1 1 2+δ X X n m 1+δ X X 2+δ 2+δ 0 kP−u,−v(X0,0)k2+δ ≤ (2 2 ) P−k,−k (X0,0) 2+δ u,v≥1 n,m≥0 k=2n k0=2m

1+δ X n m 2+δ ≤ 4Cδ (2 2 ) kE−2n,−2m (X0,0)k2+δ . n,m≥0

Since kE−2n,−2m (X0,0)k is non-increasing in n and m, it follows that

2n−1 2m−1 n m 1+δ X X kE−u,−v(X0,0)k2+δ (2 2 ) 2+δ kE n m (X )k ≤ 4 . −2 ,−2 0,0 2+δ (uv)1/(2+δ) u=2n−1 v=2m−1

Therefore, by the relations above, we have proved that (5.1.9) implies

X kP−u,−v(X0,0)k2+δ < ∞. u,v≥1

Similarly we have

∞ ∞ X X kP−u,0(X0,0)k2+δ < ∞ and kP0,−v(X0,0)k2+δ < ∞. u=1 v=1

Thus by stationarity (5.2.14) holds.

The proof of the corollary is now complete by a combination of Theorem 5.1.3 for

δ > 0 and Theorem 5.1.2 for δ = 0 via Theorem 25.4 in Billingsley [Bil95]. Proof of Remark 5.1.6. The remark follows by the proof of Step 1 of Corollary

5.1.5 and Theorem 5.1.3, via Theorem 25.4 in Billingsley [Bil95].

5.3 Random fields with multi-dimensional index sets

d In this section we extend our results to random fields indexed by Z , d > 2. We use d the same notation as in Section 4.3 of Chapter 4. As before, for each vector u ∈ Z , we d write u = (u1, ··· , ud) and u, n ∈ Z , u ≤ n stands for ui ≤ ni for i = 1, 2, ··· , d.

76 We say that a random field (Xn)n∈Zd admits a martingale approximation if there is a field of martingale differences (Dm)m∈Zd (Definition 4.3.1) such that for almost all ω ∈ Ω

1 ω 2 E (Sn − Mn) → 0 when min ni → ∞, (5.3.1) |n| 1≤i≤d

Pn Pn where Sn = u=1 Xu, Mn = u=1 Du, and |n| = n1 · ... · nd.

Let Rn be the remainder term of the decomposition of Sn such that

n X Sn = Pu(Sn) + Rn, u=1

where Pu(·) is the projection operator defined by (4.3.3) in Chapter 4.

In the context of Section 4.3 of Chapter 4, we have:

Theorem 5.3.1. Assume that (Xn)n∈Zd is defined by (4.3.2) and there is an integer i, 1 ≤ i ≤ d, such that Ti is ergodic, and the filtration (Fn)n∈Zd defined by (4.3.1) is commuting. In addition assume that

X kP0(Xu)k2 < ∞. (5.3.2) u≥0

Then, for almost all ω ∈ Ω,

d/2 2 ω (Sn,··· ,n − Rn,··· ,n)/n ⇒ N (0, σ ) under P when n → ∞.

Theorem 5.3.2. Furthermore, assume now condition (5.3.2) is reinforced to

X kP0(Xu)kϕ < ∞, (5.3.3) u≥0

77 2 d−1 where ϕ(x) = x log (1 + |x|) and k·kϕ is defined by (5.1.1).

Then, for almost all ω ∈ Ω,

1 2 ω (Sn − Rn) ⇒ N (0, σ ) under P when min ni → ∞. p|n| 1≤i≤d

Corollary 5.3.3. Assume that the conditions of Theorem 5.3.2 hold and for all j,

1 ≤ j ≤ d we have

1 2 E0 En (Sn) → 0 a.s. when min ni → ∞. (5.3.4) |n| j 1≤i≤d

d where nj ∈ Z has the j-th coordinate 0 and the other coordinates equal to the coordinates of n. Then, for almost all ω ∈ Ω,

p 2 ω Sn/ |n| ⇒ N (0, σ ) under P when min ni → ∞. (5.3.5) 1≤i≤d

If the conditions of Theorem 5.3.1 hold and (5.3.4) holds with n = (n, n, ··· , n), then for almost all ω ∈ Ω,

1 S ⇒ N (0, σ2) under P ω when n → ∞. (5.3.6) nd/2 n,··· ,n

Corollary 5.3.4. Assume that (Xn)n∈Zd is defined by (4.3.2) and the filtration (Fn)n∈Zd defined by (4.3.1) is commuting. Also assume that there is an integer i, 1 ≤ i ≤ d, such that Ti is ergodic and in addition for δ ≥ 0,

X kE1(Xu)k2+δ < ∞. 1/(2+δ) (5.3.7) u≥1 |u|

(a) If δ = 0, then the quenched CLT (5.3.6) holds.

(b) If δ > 0, then the quenched convergence (5.3.5) holds.

78 As for the case of random fields with two indexes, we start with the proof of Theorem

5.3.2, since the proof of Theorem 5.3.1 is similar with the exception that we use different ergodic theorems.

Proof of Theorem 5.3.2. The proof of this theorem is straightforward following the same lines of proofs as for a double-indexed random field. It is easy to see that, by using the commutativity property of the filtration, the martingale approximation

2 d argument in the proof of Theorem 5.1.3 remains unchanged if we replace Z with Z for d ≥ 3. The definition of the approximating martingale is also clear. Theonly difference in the proof is that for the validation of the limit in (5.2.4) and (5.2.5) when min1≤i≤d ni → ∞, in order to apply the ergodic theorem for Dunford-Schwartz operators, conform to Theorem 1.1 in Ch. 6 in Krengel [Kre85] (Theorem 2.2.9), we have to assume

2 d−1 E f0 log (1 + |f0|) < ∞, which is implied by (5.3.3).

More precisely, let us denote by Tˆi, 1 ≤ i ≤ d, the operators defined by Tˆif = f ◦ Ti.

d i d ik Then for i = (i1, ··· , id) ∈ Z , we define Q = Πk=1Qk where (Qi)1≤i≤d are operators associated with coordinate-wise translations (Ti)1≤i≤d defined as follows

Q1(f) = E0,∞,··· ,∞(Tˆ1f),Q2(f) = E∞,0,∞,··· ,∞(Tˆ2f), ··· ,Qd(f) = E∞,··· ,∞,0(Tˆdf).

Then, we bound the following quantity

1 E |S − R − M |2 |n| 0 n n n by the sum of d terms with the first term of them in the form

n ∞ 2 1 X X X I = Qi |P (X )| where v ∈ d−1. n |n| 0 u,v Z i=1 u=n1+1−i1 v≥0

79 By symmetry, we only need to deal with this one. Let c be a fixed integer satisfying c < n1, we decompose In into two parts:

0 n1−c n ∞ 2 1 X X i X X Q |P0(Xu,v)| := An(c) |n| 0 i1=1 i =1 u=n1+1−i1 v≥0 and 0 n1 n ∞ 2 1 X X i X X Q |P0(Xu,v)| := Bn(c) |n| 0 i1=n1−c+1 i =1 u=n1+1−i1 v≥0

0 0 with i = (i2, ··· , id) and n = (n2, ··· , nd). Afterwards, we just proceed by following step by step the proof for negligibility of An,m(c) and Bn,m (see (5.2.2) and (5.2.3) from the proof of Theorem 5.1.3).

The proof of Theorem 5.3.1 follows by similar arguments, just replacing Theorem 1.1 in Ch. 6 in Krengel (Theorem 2.2.9) by Theorem 2.8 in Ch.6 in the same book (Theorem

2.2.12).

Proof of Corollary 5.3.3. The negligibility of the reminder Rn can be shown exactly in the same way as the negligibility of the term Rn,m in the proof of Corollary 5.1.4. Proof of Corollary 5.3.4. As in the proof of (5.3.9) and (5.3.10) in Corollary 5.1.5, we can show that (5.3.7) implies the following facts:

X 1 X kP0(d)(Xu,v)k2+δ < ∞, (5.3.8) p|u| u≥1 v≥0

X kP0(Xu,v)k2+δ < ∞ (5.3.9) v≥0 and X 1 X √ kP (X )k < ∞, (5.3.10) u 0 u,v 2+δ u≥1 v≥0

d d−1 where 0 = (0, ··· , 0) ∈ Z , u, v ∈ Z and P0 = P0(2) ◦ P0(3) ◦ · · · ◦ P0(d) with P0(j) defined by (4.3.3).

80 To prove the corollary, we need to show that

1 2 E0 E (k) (Sn) → 0 a.s. when min ni → ∞, (5.3.11) |n| n 1≤i≤d

(k) d where n ∈ Z has k coordinates equal to the corresponding coordinates of n and the other n − k coordinates zero for all 0 ≤ k ≤ d − 1. We will proceed by induction.

First, we have to show that

2 E0(Sn) 1 2 → 0 a.s. and E0 E0,··· ,0,n (Sn) → 0 a.s. when min ni → ∞, |n| |n| d 1≤i≤d

which are easy to establish by similar arguments as in the proof of Corollary 5.1.5, by using (5.3.7) and (5.3.8). That is, (5.3.11) holds for k = 0 and k = 1. Now assume for k < d − 1 the result holds. We need to show the result for k = d − 1 which follows straightforward by using (5.3.9) and (5.3.10). The proof of this corollary is complete now.

5.4 Examples

We shall give examples providing new results for linear and Volterra random fields. Let d be an integer greater than 1 and δ ≥ 0. Throughout this section, as before by Cδ > 0, we denote a generic constant depending on δ which may be different from line to line.

Example 5.4.1 (Linear field). Let (ξn)n∈Zd be a random field of independent, identically 2+δ distributed random variables which are centered and E |ξ0| < ∞. For k ≥ 0 define

X Xk = ajξk−j . j≥0

Assume that 1 X 1 X 2 a2 < ∞. (5.4.1) |k|1/(2+δ) j k≥1 j≥k−1

81 Then the results of Corollary 5.3.4 hold.

Proof. Since X E1(Xk) = ajξk−j, j≥k−1 by the independence of ξn and the Rosenthal inequality (see Theorem 2.1.14, given in

Chapter 2), we obtain

2+δ X 2+δ kE1(Xk)k2+δ = k ajξk−jk2+δ j≥k−1

 2+δ  2 X 2 2 X 2+δ ≤ Cδ  ajE(ξk−j) + E|ajξk−j|  j≥k−1 j≥k−1

 2+δ  2 2+δ X 2 2 2 X 2+δ 2+δ ≤ Cδ  aj Eξ0 + |aj| E |ξ0|  . j≥k−1 j≥k−1

By the monotonicity of norms in `p, we have

1 1 2+δ 2 X 2+δ X 2 |aj| ≤ aj , j≥k−1 j≥k−1 therefore 1 2 X 2 kE1(Xk)k2+δ ≤ Cδ aj . j≥k−1

So condition (5.3.7) is implied by (5.4.1). Whence the results of Corollary 5.3.4 holds.

Example 5.4.2 (Volterra field). Let (ξn)n∈Zd be a random field of independent random 2+δ variables identically distributed centered and E |ξ0| < ∞. For k ≥ 0, define

X Xk = au,vξk−uξk−v. (u,v)≥(0,0)

82 P 2 where au,v are real coefficients with au,u = 0 and u,v≥0 au,v < ∞. In addition, assume that

1/2 X 1 X a2 < ∞. (5.4.2) |k|1/(2+δ) u,v k≥1 (u,v)≥(k−1,k−1) u6=v

Then the results of Corollary 5.3.4 hold.

Proof. Note that

X E1(Xk) = au,vξk−uξk−v. (u,v)≥(k−1,k−1)

0 00 Let (ξn)n∈Zd and (ξn)n∈Zd be two independent copies of (ξn)n∈Zd . By independence and ak,k = 0, applying the decoupling inequality together with the Rosenthal inequality (see Theorem 5.5.1 given in the Appendix and Theorem 2.1.14 from Chapter 2), we obtain

2+δ 2+δ 2+δ X X 0 00 kE1(Xk)k = au,vξk−uξk−v ≤ C2 au,vξ ξ 2+δ k−u k−v u,v≥k−1,k−1 2+δ u,v≥k−1 2+δ u6=v u6=v 2+δ 2 X 2 0 00 2 X 2+δ 0 00 2+δ ≤ Cδ au,vE(ξk−uξk−v) + |au,v| E |ξk−uξk−v| u,v≥k−1 u,v≥k−1 u6=v u6=v 2+δ 2 2 X 2 2 2+δ X 2+δ 2+δ ≤ Cδ au,v E(ξ0) + |au,v| E |ξ0| . u,v≥k−1 u,v≥k−1 u6=v u6=v

Above, the first inequality holds by Theorem 5.5.1 while the second one is implied by

Theorem 2.1.14.

Again by the monotonicity of norms in `p, we have

1 2 X 2 kE1(Xk)k2+δ ≤ Cδ au,v . u,v≥k−1

83 Thus the results of Corollary 5.3.4 hold.

5.5 Appendix

For convenience, we mention a decoupling result for U-statistics, which can be found on p.99, Theorem 3.1.1, de la Peña and Giné [PG99].

Theorem 5.5.1 (Decouping inequality). Let (Xi)1≤i≤n be n independent random vari-

k ables and let (Xi )1≤i≤n, k = 1, ··· , m, be m independent copies of this sequences. m m For each (i1, i2, ··· , im) ∈ In , let hi1,··· ,im : R → R be a measurable function such

E|hi1,··· ,im (Xi1 , ··· ,Xim )| < ∞. Let f : [0, ∞) → [0, ∞) be a convex non-decreasing m function such that Ef(|hi1,··· ,im (Xi1 , ··· ,Xim )|) < ∞ for all (i1, i2, ··· , im) ∈ In , where m In = {(i1, ··· , im): ij ∈ N, 1 ≤ ij ≤ n, ij 6= ik, if j 6= k}. Then there exists Cm > 0 such that

X X 1 m Ef(| hi1,··· ,im (Xi1 , ··· ,Xim )|) ≤ Ef(Cm| hi1,··· ,im (Xi1 , ··· ,Xim )|). m m In In

84 Chapter 6

Law of large numbers for discrete Fourier transform

The results of this chapter have been published in [Zha17]. In this chapter we shall establish the law of large numbers for the discrete Fourier transform of random variables with finite first moment under condition P (|Xn| > x) ≤ P (|X1| > x) for all x ≥ 0; for

1 < p < 2, we establish the Marcinkiewicz-Zygmund type rate of convergence for the discrete Fourier transform of random variables with finite p-th moment under condition 1 Pn n k=1 P (|Xk| > x) ≤ MP (|X1| > x) for all x ≥ 0 and some positive constant M. The law of large numbers is valid for pairwise independent identically distributed random variables, result due to Etemadi [Ete81]. This is a surprising result since a sequence of pairwise independent identically distributed random variables may not be ergodic. A way to look into the speed of convergence of this result when the variables have finite moments of order r, 1 < r < 2, is provided by Baum and Katz [BK65] in the i.i.d. case and by Stocia [Sto11] in the martingale difference case. By carefully examining the proof in Stoica [Sto11], we notice that the proof can be adapted to centered pairwise independent random variables and we can formulate the following result.

85 Proposition 6.0.1. Let (Xn)n≥1 be a pairwise independent sequence of random variables Pn with the same distribution and denote the partial sum by Sn = k=1 Xk. 1 (a) Assume Xn ∈ L . Then

Sn → EX ,P -a.s. as n → ∞. n 1

p (b) Assume Xn ∈ L , 1 < p < 2. Then for all 1 ≤ r ≤ p and > 0,

∞ X p/r−2 1/r n P |Sn| > n < ∞. n=1

The goal of this chapter is to study the law of large numbers for the discrete Fourier transform of a sequence of random variables with finite p-th moment for p = 1 and its

Marcinkiewicz-Zygmund type rate of convergence for 1 < p < 2 under weaker conditions than identical distribution as described in the abstract and to show that, from some point of view, the variables have similar properties as pairwise independent random variables with the same distribution.

Let (Xn)n≥1 denote a sequence of real-valued random variables on a probability space

(Ω, F,P ). No dependence between the variables is assumed. For −π ≤ t < π, define the discrete Fourier transform n X ikt Sn(t) = e Xk. (6.0.1) k=1

6.1 Law of large numbers for discrete Fourier transform

Theorem 6.1.1. Assume

P (|Xn| > x) ≤ P (|X1| > x), for all x ≥ 0, n ≥ 1. (UD)

86 If E|X1| < ∞, then for almost all t ∈ [−π, π),

Sn(t) lim = 0,P -a.s. n→∞ n

Proof of Theorem 6.1.1. Throughout this whole paper, C > 0 denotes a generic constant which may take different values from line to line.

The proof has two steps: First, we show that the proof can be reduced to truncated random variables. Then we prove the result of Theorem 6.1.1 for truncated random variables.

Truncation argument

Lemma 6.1.2. Assume (Xn)n≥1 satisfies condition (UD) and E|X1| < ∞. Let Yk = n ∗ X ikt XkI{|Xk| ≤ k} and Sn(t) = e Yk. Then for all t in [−π, π), k=1

1 1 ∗ lim Sn(t) − S (t) = 0,P -a.s. n→∞ n n n

Proof of Lemma 6.1.2. As P (|Xn| > x) ≤ P (|X1| > x), we obtain

∞ ∞ X X P (Xn 6= Yn) = P (|Xn| > n) n=1 n=1 ∞ X Z ∞ ≤ P (|X1| > n) ≤ P (|X1| > x)dx = E|X1| < ∞. n=1 0

By the Borel-Cantelli Lemma, we know P (Xn 6= Yn i.o.) = 0. That is, for almost all

ω ∈ Ω, Xn(ω) = Yn(ω), for all n sufficiently large, say for all n ≥ m(ω) := m. Thus for all t ∈ [−π, π),

87 n 1 1 ∗ 1 X ikt Sn(t) − S (t) = e (Xk − Yk) n n n n k=1 m m 1 X 1 X ≤ |eikt| · |X − Y | = |X − Y | → 0,P -a.s. n k k n k k k=1 k=1

That is,

1 1 ∗ lim Sn(t) − S (t) = 0,P -a.s. n→∞ n n n

By Lemma 6.1.2, the proof of Theorem 6.1.1 will be complete if we can show

1 lim S∗(t) = 0,P -a.s. (6.1.1) n→∞ n n

n ikt X e Yk In order to prove (6.1.1), we proceeds as following: First, let us show that k k=1 converges a.s. for almost all t ∈ [−π, π). ∞ X EY 2 From Durrett [Dur10] (page 64), we know that k ≤ 4E|X | < ∞. So, for k2 1 k=1 ∞ X Y 2(ω) almost all ω ∈ Ω, k < ∞. k2 k=1 n ikt X e Yk(ω) Then, by Carleson’s Theorem (Theorem 2.3.3), for almost all ω ∈ Ω, k k=1 converges almost everywhere in t. That is, for almost all ω ∈ Ω, there exists Iω ⊂ [−π, π) n ikt X e Yk(ω) with λ(I ) = 2π, such that for all t ∈ I , converges, where λ is the Lebesgue ω ω k k=1 measure on [−π, π). ∞ X Y 2(ω) Let Ω = {ω : k < ∞}, then P (Ω ) = 1. 0 k2 0 k=1 It is convenient to work on the product space [−π, π) × Ω with product measure

Pe := λ × P . ∞ ikt X e Yk(ω) Define A = {(t, ω): is convergent} ⊂ [−π, π) × Ω. k k=1

88 Using Fubini Theorem,

Z Z Z π Pe(A) = IA(t, ω)dPe = IA(t, ω) λ(dt)dP [−π,π)×Ω Ω −π Z Z π Z = IA(t, ω) λ(dt)dP = 2πdP Ω0 −π Ω0 = 2π Z π Z = IA(t, ω) dP λ(dt). −π Ω

n ikt X e Yk Thus, for almost all t ∈ [−π, π), converges a.s. k k=1 Now by applying Kronecker Lemma [Shi19] for complex numbers (If an ↑ ∞ and P∞ −1 Pn n=1 (xn/an) converges, then an m=1 xm → 0, which also holds for complex numbers), for almost all t ∈ [−π, π), we obtain

n 1 X ikt 1 ∗ lim e Yk = lim S (t) = 0,P -a.s. n→∞ n n→∞ n n k=1

6.2 Rate of convergence in the strong law of large numbers

The following theorem describes the rate of convergence in the strong law of large numbers.

Theorem 6.2.1. Assume

n 1 X P (|X | > x) ≤ MP (|X | > x), for all x ≥ 0, n ≥ 1. (MD) n k 1 k=1

p Let 1 < p < 2, 1 ≤ r ≤ p. If E|X1| < ∞, then for every > 0 and for almost all t ∈ [−π, π), ∞ X p/r−2 1/r n P [ max |Sk(t)| > n ] < ∞. (6.2.1) 1≤k≤n n=1

89 Before proceeding to prove Theorem 6.2.1, we shall establish a preparatory lemma.

p Lemma 6.2.2. Assume (Xn)n≥1 satisfies condition (MD) and E|X1| < ∞. Then

∞ n X p/r−1/r−2 X h 1/r i p n E |Xk|I(|Xk| > n ) ≤ CE|X1| < ∞; (6.2.2) n=1 k=1

∞ n X p/r−2/r−2 X h 2 1/r i p n E |Xk| I(|Xk| ≤ n ) ≤ CE|X1| < ∞. (6.2.3) n=1 k=1

Proof of Lemma 6.2.2.

We shall use the following facts: For X ≥ 0 a.s. and A > 0,

Z ∞ Z ∞ E(X) = P (X > x)dx; E(XI(X > A)) = AP (X > A)+ P (X > x)dx (6.2.4) 0 A

and

Z A Z A E(XI(X ≤ A)) = −AP (X > A) + P (X > x)dx ≤ P (X > x)dx. (6.2.5) 0 0

Therefore Z A E(X2I(X2 ≤ A)) ≤ P (X2 > x)dx. (6.2.6) 0

As (Xn)n≥1 satisfies condition (MD), by fact (6.2.4), weget

n X h 1/r i h 1/r i E |Xk|I(|Xk| > n ) ≤ MnE |X1|I(|X1| > n ) . k=1

So then

∞ n ∞ X p/r−1/r−2 X h 1/r i X p/r−1/r−1 h 1/r i n E |Xk|I(|Xk| > n ) ≤ M n E |X1|I(|X1| > n ) . n=1 k=1 n=1

By Stoica [Sto11] (page 912), we have

90 ∞ X p/r−1/r−1 h 1/r i p n E |X1|I(|X1| > n ) ≤ CE|X1| < ∞. n=1

Combining this result with our computation, we obtain (6.2.2).

By condition (MD) and (6.2.6), we get

n n Z n2/r X 2 1/r X 2 EXk I(|Xk| ≤ n ) ≤ P (Xk > x)dx k=1 k=1 0 Z n2/r 2 ≤ nM P (X1 > x)dx 0 n2/r X 2 ≤ Cn P (X1 > k). k=1

Now

∞ n2/r ∞ X p/r−2/r−2 X 2 X 2 X p/r−2/r−1 n n P (X1 > k) ≤ P (X1 > k) n n=1 k=1 k=1 n≥kr/2 ∞ X 2 p−2 p ≤ C P (X1 > k)k 2 ≤ CE|X1| , k=1

because Z ∞ p 2 p/2 p p−2 2 E|X1| = E[(X1 ) ] = x 2 P (X1 > x)dx. 2 0

Therefore we have (6.2.3). Proof of Theorem 6.2.1. Define the following random variables for k = 1, 2, ..., n :

0 itk 1/r 00 itk 1/r Xk = e XkI{|Xk| ≤ n },Xk = e XkI{|Xk| > n }.

n itk 0 00 0 00 0 X 0 Clearly, e Xk = Xk + Xk and Sn(t) = Sn(t) + Sn(t) where Sn(t) = Xk and k=1 00 Pn 00 Sn(t) = k=1 Xk .

91 By Markov’s Inequality,

" 2# 0 1/r 1 −2/r 0 P max |Sk(t)| > n ≤ n E max |Sk(t)| , (6.2.7) 1≤k≤n 2 1≤k≤n and by Hunt and Young’s maximal inequality (Theorem 2.3.4),

Z π n 0 2 X 0 2 max |Sk(t)| λ(dt) ≤ C |Xk| . (6.2.8) 1≤k≤n −π k=1

Using Fubini Theorem and properties (6.2.7) and (6.2.8), we obtain

Z π 0 1/r 0 1/r Pe max |Sk(t)| > n = P max |Sk(t)| > n λ(dt) 1≤k≤n −π 1≤k≤n Z π " 2# 1 −2/r 0 ≤ 2 n E max |Sk(t)| λ(dt) −π 1≤k≤n " n # C X 0 ≤ n−2/rE |X |2 2 k k=1 n C X h i = n−2/r E (X )2I(|X | ≤ n1/r) . 2 k k k=1

By Lemma 6.2.2, we get

∞ X p/r−2 0 1/r n Pe max |Sk(t)| > n < ∞. (6.2.9) 1≤k≤n n=1 By Markov’s inequality,

00 1/r 1 −1/r 00 P max |Sk (t)| > n ≤ n E max |Sk (t)| 1≤k≤n 1≤k≤n " n # 1 X 00 ≤ n−1/rE |X | k k=1 n 1 X h i = n−1/r E |X |I{|X | > n1/r} . k k k=1

92 Again, by Lemma 6.2.2, we obtain

∞ X p/r−2 00 1/r p n P max |Sk (t)| > n ≤ CE|X1| < ∞. (6.2.10) 1≤k≤n n=1

Using Fubini Theorem and relation (6.2.10),

∞ ∞ Z π X p/r−2 00 1/r X p/r−2 00 1/r n Pe max |Sk (t)| > n = n P max |Sk (t)| > n λ(dt) 1≤k≤n 1≤k≤n n=1 n=1 −π p ≤ CE|X1| < ∞.

(6.2.11)

Combining (6.2.9) and (6.2.11), we get

∞ X p/r−2 1/r n Pe[ max |Sk(t)| > n ] 1≤k≤n n=1 ∞ ∞ X p/r−2 0 1/r X p/r−2 00 1/r ≤ n Pe[ max |Sk(t)| > n ] + n Pe[ max |Sk (t)| > n ] < ∞. 1≤k≤n 2 1≤k≤n 2 n=1 n=1

By Fubini Theorem, we have

∞ X p/r−2 1/r n Pe[ max |Sk(t)| > n ] 1≤k≤n n=1 ∞ Z π Z X p/r−2 1/r = n I{ max |Sk(t)| > n }dP λ(dt) 1≤k≤n n=1 −π Ω Z π ∞ X p/r−2 1/r = n P max |Sk(t)| > n λ(dt) < ∞. 1≤k≤n −π n=1

Thus, for almost all t ∈ [−π, π),

∞ X p/r−2 1/r n P max |Sk(t)| > n < ∞. 1≤k≤n n=1

93 Remark 6.2.3. In fact, Gut [Gut92] gave an example to show that condition (MD) is strictly weaker than condition (UD). But in Theorem 6.1.1, it is an open question whether condition (UD) can be weakened to condition (MD).

Remark 6.2.4. Differently from the i.i.d. case, the reciprocal of Theorem 6.2.1 isno longer true. That is, (6.2.1) does not imply X1 have finite p-th moment.

Proof of Remark 6.2.4.

p It is well-known that E|X1| < ∞ is equivalent to (6.2.1) in the i.i.d. case, but this is no longer true in our setting. For instance, let

X1 = X2 = ··· = Xn = ··· .

Then k ikt X ijt 1 − e max |Sk(t)| = max e Xk = max · |X1|. 1≤k≤n 1≤k≤n 1≤k≤n 1 − eit j=1

Because ikt 1 − e 2 1 ≤ max ≤ , 1≤k≤n 1 − eit |1 − eit|

(6.2.1) is equivalent to

∞ X p/r−2 1/r n P (|X1| > Cn ) < ∞, n=1

p which is certainly not equivalent to E|X1| < ∞.

Corollary 6.2.5. Under the assumption of Theorem 6.2.1, for almost all t ∈ [−π, π),

Sn(t) lim = 0,P -a.s. n→∞ n1/p

Proof of Corollary 6.2.5. By Theorem 6.2.1, when 1 < r = p < 2, we have: for almost all t ∈ [−π, π),

94 ∞ X −1 1/p n P max |Sk(t)| > n < ∞, 1≤k≤n n=1 which is equivalent to

∞ X N/p P max |Sk(t)| > 2 < ∞. 1≤k≤2N N=1

This is because

∞ ∞ 2k+1−1 X 1 1/r X X 1 1/r P ( max |Sk(t)| > n ) = P ( max |Sk(t)| > n ) n 1≤k≤n n 1≤k≤n n=1 k=0 n=2k ∞ 2k+1−1 X X 1 k/r ≤ P ( max |Sk(t)| > 2 ) 2k 1≤k≤2k+1 k=0 n=2k ∞ X −1/r (k+1)/r = P ( max |Sk(t)| > (2 )2 ) 1≤k≤2k+1 k=0 ∞ X 0 k/r = P ( max |Sk(t)| > 2 ). 1≤k≤2k k=1

The same argument for the opposite direction

∞ ∞ 2k+1−1 X 1 1/r X X 1 1/r P ( max |Sk(t)| > n ) = P ( max |Sk(t)| > n ) n 1≤k≤n n 1≤k≤n n=1 k=0 n=2k ∞ 2k+1−1 X X 1 (k+1)/r ≥ P ( max |Sk(t)| > 2 ) 2k+1 1≤k≤2k k=0 n=2k ∞ 1 X 1/r k/r = P ( max |Sk(t)| > (2 )2 ) 2 1≤k≤2k k=0 ∞ 1 X 00 k/r = P ( max |Sk(t)| > 2 ). 2 1≤k≤2k k=1

Then, by the Borel-Cantelli Lemma, for almost all t ∈ [−π, π),

1/p Sn(t)/n → 0,P -a.s.

Remark 6.2.6. Both Theorem 6.1.1 and Theorem 6.2.1 hold if the variables have the same distribution.

96 Chapter 7

Central limit theorem for discrete Fourier transform

The results of this chapter have been published in Peligrad and Zhang [PZ19]. In this chapter, we analyze the asymptotic properties of the Fourier transform for random fields.

Let d be a positive integer. We start with a strictly stationary random field (Xu)u∈Zd of square integrable and centered random variables. We introduce the discrete Fourier transform for random fields by the rotated sum

X iu·t Sn(t) = e Xu, 1≤u≤n where we have 1 ≤ u ≤ n and t ∈ I = [−π, π)d. By u ≤ n we understand u =

(u1, ..., ud), n = (n1, ..., nd) and 1 ≤ u1 ≤ n1,..., 1 ≤ ud ≤ nd. Also u·t = u1t1+...+udtd. The main result of this chapter is a natural extension from sequences of random variables, indexed by integers, to random fields of the result of Peligrad and Wu [PW10].

Under certain regularity conditions we shall prove that, almost surely in t ∈ I, both the √ real and imaginary part of Sn(t)/ n1...nd converge to independent normal variables whose variance is, up to a multiplicative constant, the spectral density of the random

97 field, denoted by f(t). The ergodicity condition is imposed to only one of the directions of the random field. Our result implies that, for almost all frequencies t ∈ I, the limiting

2 2 distribution of In(t) is f(t)χ (2), where χ (2) is a chi-square distribution with two degrees of freedom and In(t) is the periodogram of the random field (Xu)u∈Zd

1 X iu·t In(t) = e Xu , t ∈ I. (2π)dn ...n 1 d 1≤u≤n

The proof is based on a new, interesting representation for the spectral density in terms of projection operators, which is the most important tool for establishing our result. The proof also involves a martingale approximation for random fields as well as laws of large numbers for Fourier sums, which have interest in themselves.

We consider two types of summations. The first result is for summations of the variables in a multi-dimensional cube. The reason we first restrict ourselves to summations indexed by the cubes is due to the relation between our results and optimal results available in harmonic analysis. For example, for d = 2, Theorem 1 in Marcinkiewicz and

Zygmund [MZ39] shows that the Fejér-Lebesgue theorem holds for spectral densities in

L1 when the summation is taken over rectangles of size m × n, provided that m, n → ∞ such that m/n ≤ a and n/m ≤ a for some positive number a (see Theorem 2.3.15 in

Chapter 2) . This result fails when the summation is taken over general rectangles.

However, if the summation is taken over the sets 1 ≤ u1 ≤ n, 1 ≤ u2 ≤ m where n ≥ m → ∞, one should assume the integrability of f(u) ln+ f(u) as a minimal condition for the validity of the Fejér-Lebesgue theorem (see Jessen et al. [JMZ35], which is Theorem 2.3.16 in Chapter 2). We shall also give a result in this context, where the summation is taken over unrestricted rectangles.

When dealing with random fields the notation can become rather complicated. This is the reason why, for presenting the material, we implemented the following strategy:

98 We treat first the case d = 2. Then, we mention the small differences for treating the general case of multi-dimensional index set by using the mathematical induction.

The presentation is organized as follows. We start in Section 7.1 with a primer on the limiting variance of the Fourier series, introduce the notions of stationary random fields and commuting filtrations. In Section 7.2 we obtain a representation ofthespec- tral density in terms of projection operators, which extends a recent result by Lifshitz and Peligrad [LP15] beyond the setting of Bernoulli shifts. We also state and prove our main results on the limiting distribution of double indexed, random Fourier sums.

The extension to general index set is given in Section 7.3. Section 7.4 is dedicated to examples, such as functions of Gaussian sequences, linear and nonlinear random fields with independent innovations. It is remarkable that the only condition required for the validity of our results for linear or Volterra random fields with independent innovations is equivalent to merely the existence of these fields. In a supplementary section we prove two laws of large numbers.

7.1 Preliminaries

7.1.1 Spectral density and limiting variance

We call the complex valued zero mean field of random variables (Xm)m∈Z2 defined on a probability space (Ω, K,P ), weakly stationary (or second order stationary), if there are

2 2 complex numbers γ(m), m ∈ Z , such that for all u, v ∈ Z ,

cov(Xu,Xv) = E(XuX¯v) = γ(u − v).

In the context of weakly stationary random fields it is known that there exists a unique measure on I = [−π, π)2, such that

Z iu·x 2 γ(u) = e F (dx), for all u ∈ Z , I

99 where u · x is the inner product. If F is absolutely continuous with respect to Lebesgue measure λ2 on I = [−π, π)2 then, the Radon-Nikodym derivative f of F with respect to the Lebesgue measure is called spectral density (F (dt) = f(t)dt), and we have

Z iu·x 2 γ(u) = e f(x)dx, for all u ∈ Z . I

The variance of partial sums on rectangles is

2 X it·(u−v) E|Sn(t)| = γ(u − v)e . 1≤u,v≤n

Well-known computations show that

Z 2 X it·(u−v) ix·(u−v) E|Sn(t)| = e e f(x)dx 1≤u,v≤n I Z X = eix·(u−v)f(x − t)dx. I 1≤u,v≤n

So, with the notation x = (x1, x2), one can rewrite

Z 1 2 E|Sn(t)| = Kn1 (x1)Kn2 (x2)f(x − t)dx, n1n2 I where Kn(x) is the Fejér Kernel

X |j| ijx Kn(x) = (1 − )e . |j|

Furthermore, by Theorem 1 in Marcinkiewicz and Zygmund [MZ39] (Theorem 2.3.15 in

Chapter 2), for λ2-almost all t in I, we obtain a limiting representation for the spectral density, namely 1 2 2 lim E|Sn,n(t)| = (2π) f(t). (7.1.1) n→∞ n2

100 If in addition f(u) ln+ f(u) is integrable (Theorem 2.3.16 in Chapter 2), then

1 2 2 lim E|Sn1,n2 (t)| = (2π) f(t). (7.1.2) n1≥n2→∞ n1n2

7.1.2 Stationary random fields and stationary filtrations

As in Chapter 3, to construct stationary filtrations, we shall start with a strictly stationary real valued random field ξ = (ξu)u∈Z2 , defined on a probability space (Ω, K,P ) and the filtration (Fk)k∈Z2 is defined by (3.1.3). We shall consider that the filtration is commuting in the sense of (3.1.7).

Now we introduce the stationary random field (Xm)m∈Z2 , in the following way. We define first

X0 = g((ξu)u∈Z2 ).

2 where g : RZ → C and 0 = (0, 0). Then define j k Xj,k = g(T S (ξu)u∈Z2 ), (7.1.3)

2 where T and S are shift operators on RZ as in Chapter 3, defined by (3.1.5) and (3.1.6), respectively.

7.2 Results and proofs

In this section we first find a useful representation of the spectral density for regular functions and commuting filtrations. It extends a result of Lifshitz and Peligrad [LP15] beyond the case of Bernoulli shifts. The proof follows the same lines as in Lifshitz and

Peligrad [LP15]. We shall point out the differences and give it here for completeness, clarification and equivalent definitions.

2 For each (u, v) ∈ Z , the projection operator Pu,v is defined by (4.1.3) as in Section 4.1 of Chapter 4.

101 Define F−∞,m = ∩u∈ZFu,m and Fm,−∞ = ∩v∈ZFm,v. 2 Now let X0 be defined as before, in L (Ω, F,P ). Then we have the following orthogonal representation X X0 = PuX0 + Rnm + Unm, u∈Jn,m where n and m are two positive integers, Jn,m = [−n, ..., n] × [−m, ..., m],

Rnm = E(X0|F−n−1,m) + E(X0|Fn,−m−1) − E(X0|F−n−1,−m−1), and

Unm = X0 − E(X0|Fn,m).

We assume the following two regularity conditions

E(X0|F−∞,0) = 0 a.s. and E(X0|F−∞,0) a.s. (7.2.1)

Note that

E(X0|F−n−1,−m−1) = E(E(X0|F0,−m−1)|F−n−1,0).

By passing to the limit and using the reverse martingale theorem and arguments similar to Theorem 34.2 (V) in Billingsley [Bil95], we obtain that

2 lim lim Rnm = 0 a.s. and in L . n→∞ m→∞

Since X0 is measurable with respect to ∨u∈Z2 Fu = F∞,∞, by the martingale convergence theorem, 2 lim lim Unm = 0 a.s. and in L . n→∞ m→∞

Therefore n m X X 2 X0 = lim lim P−j,−kX0 a.s. and in L . n→∞ m→∞ j=−n k=−m

102 We shall denote this limit by X X0 = PuX0. (7.2.2) u∈Z2

Note that for u 6= v and for all X and Y in L2(Ω, K,P ) we have

cov(PuX, PvY ) = 0. (7.2.3)

Observe also that, by taking into account (7.2.2), (7.2.3) and stationarity, we have

X 2 2 E|PuX0| = E|X0| < ∞. (7.2.4) u∈Z2

We would like now to define a random variable which will be used to characterize the spectral density of random fields. For random variables this was achieved in Peligrad and Wu [PW10] by using Carleson (Theorem 2.3.3) and also Hunt and Young theorems

(Theorem 2.3.4). For random fields these theorems do not hold in general. Wecould use instead a weaker form of them or, as an alternative, an iterated procedure.

In the sequel we shall use the notation kXk2 = E|X|2.

Martingale difference construction

We start from the identity (7.2.4) and note that this identity implies

X 2 |PuX0| < ∞ P -a.s. (7.2.5) u∈Z2

Let Ω0 ⊂ Ω with P (Ω0) = 1 be such that the convergence above holds for all ω ∈ Ω0.

By the main theorem in Fefferman [Fef71a] for convergence of double Fourier series, we obtain the almost sure convergence in the following sense: For ω ∈ Ω0 we have

X −ij·t X −i j·t 2 e P0Xj = lim e P0Xj λ -a.e., n→∞ u∈Z2 j∈In

103 where In = [−n, n] × [−n, n]. By Fubini Theorem, for almost all t ∈I, we also have that

X −i j·t X −i j·t e P0Xj = lim e P0Xj P -a.s. (7.2.6) n→∞ u∈Z2 j∈In

Furthermore, by relation (1) in Fefferman’s paper [Fef71a] and by (7.2.5), for a positive constant C, we have that

Z X −i j·t 2 X 2 sup | e P0Xj| dt ≤ C |PuX0| P -a.s. I n j∈In u∈Z2

Whence, by integrating and using (7.2.4), we obtain

Z X −i j·t 2 X 2 2 E sup | e P0Xj| dt ≤ CE |PuX0| ≤ CkX0k . I n j∈In u∈Z2

It follows that for almost all t ∈I

X −i j·t 2 E(sup | e P0Xj| ) < ∞. n j∈In

By the dominated convergence theorem, the convergence in (7.2.6) also holds in L2.

Let us denote by

X −i j·t 2 D0(t) = lim e P0Xj P -a.s. and in L . (7.2.7) n→∞ j∈In

In the next theorem we point out a representations for the spectral density by using definition (7.2.7).

(X ) 2 Theorem 7.2.1. Let k k∈Z be a stationary sequence defined by (7.1.3) and the filtration (Fk)k∈Z2 is commuting as in (3.1.7). Assume that the second moment is finite and the regularity condition (7.2.1) is satisfied. Then, the sequence (Xk)k∈Z2 has spectral

104 density which has the representation:

1 f(t) = E |D (t)|2, t ∈ I. (7.2.8) (2π)2 0

Proof. Let us compute the covariance of Xk and X0. By using the projection decomposition in (7.2.2), written for both Xk and X0, together with the orthogonality of the

2 projections in (7.2.3) and stationarity, we have for all k ∈ Z ,

X X cov(Xk,X0) = cov( PjXk, PuX0) (7.2.9) j∈Z2 u∈Z2 X X = cov(PjXk, PjX0) = cov(P0Xk+j, P0Xj). j∈Z2 j∈Z2

Let us analyze the function f(t) defined in (7.2.8). By Fubini theorem and (7.2.4) we have Z Z 1 X −i j·t 2 X 2 f(t)dt = 2 E | e P0Xj| dt = E |PjX0| < ∞. I (2π) I j∈Z2 j∈Z2

2 Now, let us compute the Fourier coefficients of f(t). For every k ∈ Z , by the definition of f(t) and Fubini theorem we have

Z Z i k·t 1 X i (k−j+u)·t e f(t) dt = 2 E e P0XjP0Xudt. I (2π) I u,j∈Z2

By using the orthogonality of the exponential functions, we obtain

Z i k·t X e f(t)dt = cov (P0Xj, P0Xu) 1{k−j+u=0} I j,u∈Z2 X = cov (P0Xu+k, P0Xu) . u∈Z2

Now, comparing this expression with (7.2.9) we see that f in formula (7.2.8) is the spectral density for (Xk)k∈Z2 .

105 Remark 7.2.2. For defining the spectral density iterated limits are also possible. By applying Carleson (Theorem 2.3.3) and Hunt and Young (Theorem 2.3.4) theorems twice, consecutively in each variable, one can show that the following limits exist: for λ2-almost

2 all t ∈ [−π, π) , we can define a random variable D˜0(t) in the following sense

n m X X −i u·t 2 D˜0(t) = lim lim P0(Xu ,u )e P -a.s. and in L . n→∞ m→∞ 1 2 u1=−n u2=−m

Similarly, we can also define the other iterated limit

n m X X −i u·t 2 Dˆ0(t) = lim lim P0(Xu ,u )e P -a.s. and in L . m→∞ n→∞ 1 2 u1=−n u2=−m

Also, we can obtain the following alternative definitions for the spectral density:

1 1 f(t) = E|Dˆ (t)|2 = E|D˜ (t)|2. (2π)2 0 (2π)2 0

Note that in all the characterizations of f(t) the limits commute with the integrals.

Remark 7.2.3. By (3.1.7) and its definition, D0(t) is a martingale difference in each coordinate

E0,−1D0(t) = 0 and E−1,0D0(t) = 0 P -a.s.

where we have used notation Ea,b(X) = E(X|Fa,b).

We are ready to state our main result.

(X ) 2 (F ) 2 Theorem 7.2.4. Assume that k k∈Z and k k∈Z are as in Theorem 7.2.1. In addition, assume that one of the shifts T or S is ergodic. Then, for λ2-almost all t ∈ I,

1 (Re S (t), Im S (t)) ⇒ (N ,N ) as n → ∞, n n,n n,n 1 2

106 where N1,N2 are i.i.d. normally distributed random variables with mean 0 and variance

2 2π f(t), f(t) (X ) 2 where is the spectral density of the sequence k k∈Z . Furthermore, if f(u) ln+ f(u) is integrable then

1 √ (Re Sn1,n2 (t), Im Sn1,n2 (t)) ⇒ (N1,N2) as n1 ∧ n2 → ∞ n1n2

with N1,N2 as above.

Proof of Theorem 7.2.4. This proof has several steps. Let us point out the idea of the proof. First we show that the proof can be reduced to random variables with continuous spectral density. Then, we construct a random field which is a martingale difference in each coordinate and has the same limiting distribution as the original sigma field. To validate this approximation we shall use the limiting variance given in(7.1.1) and (7.1.2) along with the representation of the spectral density given in Theorem 7.2.1.

The result will follow by obtaining the central limit theorem for the martingale random field. To fix the ideas let us assume that the shift S is ergodic.

Martingale approximation

Let us recall the definition of D0(t) given in (7.2.7) and introduce a new notation:

(`) X −i j·t X −i j·t 2 D0 (t) = P0(Xj)e → P0(Xj)e = D0(t) P -a.s. and in L . (7.2.10) 2 j∈I` j∈Z

(`) Note that D0(t) and D0 (t) are functions of (ξu)u∈Z2 . By using stationarity and (`) 2 translation operators T and S we define Dk (t) and Dk(t) for any k ∈ Z . Note that, by (`) Remark 7.2.3, both (Du,v(t)) and (Du,v(t)) are coordinate-wise martingale differences with respect to the filtrations (F∞,v)v and (Fu,∞)u respectively.

107 For almost all t ∈ I we shall approximate Sn(t) by the martingale

n X i j·t Mn(t) = e Dj(t). (7.2.11) j=1

To validate this approximation, we first consider the situation when n = (n, n). Define the martingale n (`) X i j·t (`) Mn (t) = e Dj (t) (7.2.12) j=1 and, for t0 fixed, the “proper” Fourier series in t,

n (`) 0 X i j·t (`) 0 Mn (t, t ) = e Dj (t ). j=1

Note that we can bound

2 (`) 0 2 |Sn(t) − Mn(t)| ≤ 3(|Sn(t) − Mn (t, t )|

(`) 0 (`) 2 (`) 2 +|Mn (t, t ) − Mn (t)| + |Mn (t) − Mn(t)| ).

By (7.1.1), for almost all t ∈ I

1 (`) 0 2 2 (`) 0 lim E|Sn(t) − Mn (t, t )| = (2π) f (t, t ), n→∞ n2

(`) 0 (`) 0 where f (t, t ) is the spectral density of (Xk − Dk (t ))k. By using the representation (7.2.8) given in Theorem 7.2.1, and taking into account

(`) 0 2 that P0Dj (t ) = 0 P -a.s. for j ∈ Z with j 6= 0 we obtain

1 X (`) 1 (`) f (`)(t, t0) = E | P X e−i j·t − D (t0)|2 = E |D (t) − D (t0)|2. (2π)2 0 j 0 (2π)2 0 0 j∈Z2

108 On the other hand, by the orthogonality of the projections,

1 (`) 0 (`) 2 (`) 0 (`) 2 E|Mn (t, t ) − Mn (t)| = E|D (t ) − D (t)| (7.2.13) n2 0 0 and 1 E|M (`)(t) − M (t)|2 = E|D(`)(t) − D (t)|2. n2 n n 0 0

So, overall, by the above considerations,

1 2 (`) 0 2 lim sup 2 E|Sn(t) − Mn(t)| ≤ 3(E |D0(t) − D0 (t )| n→∞ n (`) 0 (`) 2 (`) 2 +E|D0 (t ) − D0 (t)| + E|D0 (t) − D0(t)| ).

(`) 0 0 Note now that D0 (t ) is continuous in t and so, by the dominated convergence theorem, (`) 0 (`) 2 limt0→t D0 (t ) = D0 (t) in L . Therefore, by taking into account (7.2.10), and letting first t0 → t and then ` → ∞, we obtain for λ2-almost all t ∈I the approximation

1 2 lim E|Sn(t) − Mn(t)| = 0. n→∞ n2

+ Furthermore if n = (n1, n2) and f(u) ln f(u) is integrable, by replacing in the proof the limit given in (7.1.1) by (7.1.2), for λ2-almost all t ∈I we have

1 2 lim E|Sn(t) − Mn(t)| = 0. (7.2.14) n1>n2→∞ n1n2

By using Theorem 25.4 in Billingsley [Bil95], the limit (7.2.14) shows that, the proof of Theorem 7.2.4 is now reduced to prove the central limit theorem for Mn(t).

The central limit theorem for the martingale.

Proposition 7.2.5. Consider Mn(t) defined by (7.2.11) where n = (n1, n2). Then the real and imaginary part of Mn(t) converge to independent normal random variables with

2 variance E|D0,0(t)| /2 when n1 ∧ n2 → ∞.

109 Proof of Proposition 7.2.5.

To ease the notation we shall drop t and denote Dj,k = Dj,k(t),Mn = Mn(t).

We start by writing (t = (t1, t2))

1 1 n1 1 n2 X ijt1 X ikt2 √ Mn = √ e √ e Dj,k. n1n2 n1 j=1 n2 k=1

Note that, by construction and since the filtration (Fj,k ) is commuting, the sequence (D0 ) n2,k k defined by 1 n2 0 X ikt2 Dn2,j = √ e Dj,k n2 k=1 is a triangular array of complex martingale differences with respect to the filtration

(Fj,∞ )j.

For a and b real numbers let us find the limiting distribution of

1 1 1 Xn1 0 a√ Re Mn + b√ Im Mn = √ [(a cos jt1 + b sin jt1) Re Dn2,j (7.2.15) n1n2 n1n2 n1 j=1

0 1 Xn1 +(b cos jt1 − a sin jt1) Im Dn2,j)] = √ ∆n2,j. n1 j=1

−1/2 Pn1 In order to find the limiting distribution of n1 j=1 ∆n2,j we have to prove now a central limit theorem for the triangular array of martingale differences (∆n2,j)j≥1. According to a classical result, which can be found in Gänssler and Häusler [GH79], we have to establish that

1 L2 max √ |∆n2,j| → 0 as n1 ∧ n2 → ∞, 1≤j≤n1 n1 and to verify the Raikov type condition, namely

1 Xn1 2 2 E| (∆n2,j − E∆n2,j)| → 0 as n1 ∧ n2 → ∞. (7.2.16) n1 j=1

110 The first condition is easy to verify since, by the stationarity involved in themodel (|D0 |2) and the main result in Peligrad and Wu [PW10], the variables n2,j j are uniformly 2 integrable and therefore, (|∆n2,j| )j in (7.2.15) are also uniformly integrable. In order to verify (7.2.16), after using the well-known trigonometric formulas

2 cos2 x = 1 + cos 2x, 2 sin2 x = 1 − cos 2x

cos2 x − sin2 x = cos 2x, 2(cos x)(sin x) = sin 2x, by Lemma 7.5.1 in Section 7.5, it follows that for almost all t ∈ I, the terms involving cos 2jt1 or sin 2jt1 in (7.2.16) are negligible as n1 ∧ n2 → ∞.

After simple computations, proving (7.2.16) is reduced to show that

1 Xn1 0 2 0 2 E| (|Dn2,j| − E|Dn2,j| )| → 0. n1 j=1

We shall apply Lemma 7.5.2 below. Clearly it is enough to prove that

1 Xn1 2 0 2 0 E| (Re Dn2,j − E Re Dn2,j)| → 0 n1 j=1 and 1 Xn1 2 0 2 0 E| (Im Dn2,j − E Im Dn2,j)| → 0. n1 j=1

Their proofs are similar and we shall deal only with the first one involving the real part.

Let m be a positive integer. By using Cramèr theorem, trigonometric formulas, the main Theorem in Peligrad and Wu [PW10], ergodicity of S and Lemma 7.5.1 from Section 7.5, we can easily show that the vector valued sequence of martingales (Re D0 , ..., Re D0 ) (N , ..., N ) n2,1 n2,m n2 converges to a Gaussian vector 1 m with the covariance structure

cov(N1,Nj) = cov(Nk,Nk+j).

111 The computations are simple and left to the reader. Because of the martingale property, (Re D0 ) we have the orthogonality of n2,k k. In addition we also have uniform integrability (|D0 |2) of the family n2,1 n2 , provided by the results in Peligrad and Wu (2010). By applying the continuous mapping theorem and the convergence of moments theorem associated to the convergence in distribution (Theorem 25.12 in [Bil95]), we obtain

0 0 cov(N1,Nk) = lim cov(Re Dn2,0, Re Dn2,k) = 0. n2→∞

This shows that the Gaussian limit (Nk)k is a stationary and independent sequence. It m ∈ N, (Re2 D0 , ..., Re2 D0 ) follows that, for all n2,1 n2,m n2 converges to an independent 2 2 2 vector (N1 , ..., Nm) and (Nk )k is stationary and ergodic. Therefore, (7.2.16) holds by Lemma 7.5.2 in Section 7.5.

By all of the above considerations we obtain

1 1 2 2 2 a√ Re Mn(t) + b√ Im Mn(t) ⇒ (a + b )N (0,E|D0,0(t)| ), n1n2 n1n2 and the result follows.

7.3 Random fields with multi-dimensional index set

d In this section we discuss the differences which occur when the index setis Z , d > 2. The main difference is that we use some recent results on summability ofmulti- dimensional trigonometric Fourier series which are surveyed and further developed in

Weisz [Wei12]. Many summability results, needed for our proofs, have already been extended from dimension 1 to dimension d, but the results are very different depending

d on the summation type and on the shape of the regions in Z containing the indexes of summations. Our intention is to present a method rather than the most general results. The probabilistic tools are completely developed in our paper. However, the

112 statements are limited by the level of knowledge in harmonic analysis. In order to construct the approximating martingale we can always base ourselves on the summation on cubes, where the celebrated Carleson-Hunt theorem extends completely for square integrable functions (see Theorem 4.4. in [Wei12]) or we can use an iterative procedure.

However, the statements of the CLT and the conditions imposed to the spectral density, strongly depend on shape of the summation region and the extensions of the Fejér-

Lebesgue theorem, namely on the validity of (7.1.1). These regions of summation can be

d restricted by using conditions imposed to various norms on Z or the summations can be taken over nonrestricted rectangles. In the latter case, additional restrictions have to be imposed to the spectral density. This is an active field of research in harmonic analysis and our results can be reformulated whenever a progress is achieved. We shall formulate the general results by using only summations over cubes and nonrestricted rectangles.

To introduce the regularity conditions, as in the chapters 3, 4 and 5, we defined the filtrations Fu = σ(ξj : j ≤ u), where ξ = (ξu)u∈Zd is a strictly stationary real- d valued random field, defined on the canonical probability space RZ . The filtration is commuting if EuEaX = Eu∧aX, where the minimum is taken coordinate-wise. By taking intersections of sigma algebras or sigma algebra generated by unions of sigma algebras, we can consider the coordinates of u in Fu being valued in Z ∪ {−∞, ∞}. We

k1 kd define X0 = f((ξu)u∈Zd ) and also define Xk = f(T1 ◦ ... ◦ Td (ξu)u∈Zd ), where Ti’s the coordinate-wise translations. We call f regular if E(X0|Fu) = 0 a.s., when at least a coordinate of u is −∞.

Our general result is summarized in the following theorem:

(X ) d (F ) d f Theorem 7.3.1. Assume that k k∈Z and k k∈Z are as above and is regular. In d addition, assume that one of the shifts Ti is ergodic,1 ≤ i ≤ d. Then, for λ -almost all t ∈ [0, 2π)d, 1 (Re S (t), Im S (t)) ⇒ (N ,N ) as n → ∞, nd/2 n n 1 2

113 where n = (n, ..., n),N1,N2 are i.i.d. normally distributed random variables with mean

d−1 d 0 2 π f(t), f(t) (X ) d and variance and is the spectral density of the sequence k k∈Z . Furthermore, if f(u)(ln+ f(u))d−1 is integrable, then

1 √ (Re Sn(t), Im Sn(t)) ⇒ (N1,N2) as ∧1≤i≤d ni → ∞, n1n2...nd

with N1,N2 as above.

Proof of Theorem 7.3.1. The proof of this theorem follows the same lines as of

Theorem 7.2.4 with the following differences. In order to be able to obtain a characteriza- tion of the spectral density, we need to introduce the d-dimensional projection operators.

The projection operators are defined the same way as in Section 4.3 of Chapter 4,that

d is, for each u ∈ Z , the projection operator Pu is defined by (4.3.3). We can easily see that, by using the commutativity property, this definition is a generalization of the case d = 2. We note that, by using this definition of P0(X), the

2 d statement and the proof of Theorem 7.2.1 remain unchanged if we replace Z with Z . The definition of the approximating martingale is also clear as well as the proof. Wepoint out the following two differences in the proof. One difference is that instead of Theorem

1 in Marcinkiewicz and Zygmund [MZ39] we use Corollary 14.4 in Weisz [Wei12], which assures the validity of (7.1.1) for λd-almost all t in [0, 2π)d. Another difference in the proof is that Proposition 7.2.5 is proved by induction. More precisely, we use instead of the results in Peligrad and Wu [PW10] the induction hypothesis. For proving the second part of the Theorem 7.3.1, we use several results in [Wei12], namely Corollary

16.5 about unrestricted summability and the line above relation 15.2 on page 123.

7.4 Applications

In this section, we give examples of stationary random fields (ξu)u∈Z2 which generate commuting filtration and in addition both F0,−∞ and F−∞,0 are trivial.

114 7.4.1 Independent copies of a stationary sequence with “nonparallel” past and future

The ρ-mixing coefficient, also known as maximal coefficient of correlation isdefinedas

2 2 ρ(A, B) = sup{Cov(X,Y )/kXk2kY k2 : X ∈ L (A),Y ∈ L (B)}.

For the stationary sequence of random variables (ξk)k∈Z, denote by F0 the past σ–field n generated by ξk with indices k ≤ 0 and by F the future σ-field after n−steps generated by ξj with indices j ≥ n. The sequence of coefficients (ρn)n≥1 is then defined by

n ρn = ρ(F0, F ).

If ρn < 1 for some n > 1, then the tail sigma field F−∞ = ∩n∈Zσ((ξj)j≤n) is trivial; see n Proposition (5.6) in Bradley [Bra07]. In this case it is customary to say that F0 and F ¯ are not parallel. Now we take a random field with columns ξj = (ξk,j)k∈Z independent copies of a stationary sequence with ρn < 1 for some n > 1. Clearly, because the columns are independent the sigma field F0,−∞ is trivial. Furthermore, we shall argue that F−∞,0 is also trivial. To prove it, we apply Theorem 6.2 in Csáki and Fisher [CF63] (see also

Theorem 6.1 in [Bra07]). According to this theorem

ρ(σ((ξk,j)j∈Z,k≤0), σ((ξk,j)j∈Z,k≥n)

= sup ρ(σ((ξk,j)k≤0), σ((ξk,j)k≥n)) = ρn < 1. j

Therefore we also have that F−∞,0 is trivial and our theorem applies. For this case we obtain the following corollary:

Corollary 7.4.1. Assume that the random field (ξk,j)k,j∈Z consists of columns, which are independent copies of a stationary sequence (ξj)j∈Z having ρn < 1 for some n > 1.

115 Construct (Xn)n∈Z2 by (7.1.3) and assume that the variables are centered and square integrable. Then the results of Theorems 7.2.1 and 7.2.4 hold.

As a particular example we can take, as generator of the commuting sigma algebras, independent copies of a Gaussian sequence with a special type of spectral density. It is convenient to define the spectral density on the unit circle in the complex plane, denoted by T . Let µ denote normalized Lebesgue measure on T (normalized so that µ(T ) = 1).

For a given random sequence X := (Xk)k∈Z, a “spectral density function” (if one exists) can also be viewed as a real, nonnegative, Borel, integrable function f : T → [0, ∞) such that for every k ∈ Z

Z k cov(Xk,X0) = t f(t)µ(dt). t∈T

Let (ξj)j∈Z be a stationary Gaussian sequence and let n be a positive integer. The following two conditions are equivalent:

(a) ρn < 1.

(b) (ξj)j∈Z has a spectral density function f (on T ) of the form

f(t) = |p(t)|exp(u(t) +v ˜(t)), t ∈ T where p is a polynomial of degree at most n − 1 (constant if n = 1), u and v are real bounded Borel functions on T with kvk∞ < π/2, and v˜ is the conjugate function of v.

For n = 1, this equivalence is due to Helson and Szegö [HS60]. For general n ≥ 1, it is due to Helson and Sarason [HS67] (Theorem 6).

7.4.2 Functions of i.i.d.

Our results also hold for any random field which is Bernoulli, i.e. a function of i.i.d. random field. For instance, if (ξn)n∈Zd is a random field of independent, identically

116 distributed random variables and we define (Xk)k∈Zd and (Fk)k∈Zd as in Theorem 7.3.1. Then the filtration is commuting and the regularity conditions of Theorem 7.3.1 are satisfied provided the variables are centered. If in addition X0 is square integrable, then the result of Theorem 7.3.1 holds.

For the next two examples the only conditions imposed are equivalent to the existence of the fields involved.

Example 7.4.2. (Linear field) Let (ξn)n∈Zd be a random field of independent, identically distributed random variables which are centered and have finite second moment. Define

X Xk = ak−jξj. j∈Zd

P 2 d a < ∞ Assume that j∈Z j . Then the CLT in Theorem 7.3.1 holds.

Example 7.4.3. (Volterra field) Let (ξn)n∈Zd be a random field of independent random variables identically distributed centered and with finite second moment. Define

X Xk = au,vξk−uξk−v , u,v∈Zd

a a = 0 P a2 < ∞. where u,v are real coefficients with u,u and u,v∈Zd u,v Then the CLT in Theorem 7.3.1 holds.

7.5 Supplementary results

In this section we prove two auxiliary results. They are laws of large numbers which have interest in themselves.

The following lemma is an extension of a result in [Zha17] (Theorem 6.1.1 in Chapter

6).

117 Lemma 7.5.1. Assume that the triangular array (Xn2,k)k∈Z is row-wise stationary and

(Xn2,k)n2≥1 is uniformly integrable for any k fixed. In addition assume that Xn2,k ⇒ Xk, 1 where Xk’s have the same distribution and are in L . Then, for λ-almost all t ∈ [−π, π),

1 Xn1 ikt e Xn2,k → 0 a.s. when n1 ∧ n2 → ∞. n1 k=1

Proof. Let m ≥ 1 be a fixed integer and define consecutive blocks of indexes ofsize m, Ij(m) = {(j −1)m+1, ..., mj}. In the set of integers from 1 to n we have [n1/m] such blocks of integers and a last one containing less than m indexes. By the stationarity of

(Xn2,k)n2≥1 we have

Now, again by the uniform integrability of (Xn2,k)n2≥1 and the convergence of moments associated to the weak convergence (see Theorem 25.12 in Billingsley [Bil95]) we have

Since the Xk’s have the same distribution, by Theorem 6.1.1 in Chapter 6, for almost all t ∈ [−π, π)

1 Xm 1 Xm 1 Xk sin kt → 0 and Xk cos kt → 0 P -a.s. and in L . m k=1 m k=1

118

Lemma 7.5.2. Assume that the triangular array (Xn2,k)k∈Z is row-wise stationary, mean 0 and (Xn2,k)n2≥1 is uniformly integrable for any k fixed. In addition assume that the finite dimensional distributions of (Xn2,k)k converge in distribution to those of (Xk)k 1 as n2 → ∞, where (Xk)k is stationary and ergodic and in L . Then

1 Xn1 1 Xn2,k converges in L to 0 when n1 ∧ n2 → ∞. n1 k=1

Proof. As in the previous lemma, we make blocks of variables as before and use the inequality

1 Xn1 1 Xm E| Xn2,k| ≤ E| Xn2,k| + on1 (1) as n1 → ∞. n1 k=1 m k=1

Now, by the uniform integrability and the weak convergence of the sequence (Xn2,k)n2≥1, we obtain Xm Xm lim E| Xn2,k| = E| Xk|. n2→∞ k=1 k=1

Furthermore, note also that by the conditions of this lemma we also have E(Xk) = 0 for Pm 1 all k. By the ergodic theorem k=1 Xk/m → 0 a.s. and in L and therefore

1 Xm E| Xk| → 0 as m → ∞. m k=1

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