Limit Theorems for Random Fields 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)k2Zd (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. ii © 2019 by Na Zhang. All rights reserved. Acknowledgments 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 prob- ability 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 Copyright iii 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)n2Z on a probability space (Ω; F;P ), with Z the set of integers, we say that a central limit theorem holds for (Xn)n2Z if n 1 X p (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 trajec- tory. 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 subse- quent papers. A random field consists of multi-indexed random variables (Xu)u2Zd ; 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 math- ematicians 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 pro- jective 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.