The Pennsylvania State University The Graduate School Eberly College of Science STUDIES ON THE LOCAL TIMES OF DISCRETE-TIME STOCHASTIC PROCESSES A Dissertation in Mathematics by Xiaofei Zheng c 2017 Xiaofei Zheng Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2017 The dissertation of Xiaofei Zheng was reviewed and approved∗ by the following: Manfred Denker Professor of Mathematics Dissertation Adviser Chair of Committee Alexei Novikov Professor of Mathematics Anna Mazzucato Professor of Mathematics Zhibiao Zhao Associate Professor of Statistics Svetlana Katok Professor of Mathematics Director of Graduate Studies ∗Signatures are on file in the Graduate School. Abstract This dissertation investigates the limit behaviors of the local times `(n; x) of the n Pn 1 partial sum fSng of stationary processes fφ ◦ T g: `(n; x) = i=1 fSi=xg. Under the conditional local limit theorem assumption: n kn BnP (Sn = knjT (·) = !) ! g(κ) if ! κ, P − a:s:; Bn we show that the limiting distribution of the local time is the Mittag-Leffler distri- bution when the state space of the stationary process is Z. The method is from the infinite ergodic theory of dynamic systems. We also prove that the discrete-time fractional Brownian motion (dfBm) admits a conditional local limit theorem and the local time of dfBm is closely related to but different from the Mittag-Leffler dis- tribution. We also prove that the local time of certain stationary processes satisfies an almost sure central limit theorem (ASCLT) under the additional assumption that the characteristic operator has a spectral gap. iii Table of Contents Acknowledgments vi Chapter 1 Introduction and Overview 1 1.0.1 Brownian local time . 1 1.0.2 Local time of discrete-time stochastic processes . 4 1.0.3 Connection between the Brownian local time and the local times of discrete-time processes . 9 Chapter 2 Local Limit Theorems 11 2.1 Motivation . 11 2.2 Local limit theorems for independent and identically distributed random variables . 12 2.3 Local limit theorems for Markov chains . 13 2.4 Conditional local limit theorems for stationary processes . 14 2.5 Conditional local limit theorem for discrete-time fractional Brown- ian motion . 19 2.5.1 Proof of the conditional local limit theorem . 20 2.5.2 Estimate of the variance . 24 2.5.3 Estimate of the mean . 31 Chapter 3 Limiting Distributions of Local Times 33 3.1 Local times of random walks . 33 3.2 Occupation times of Markov chains . 38 3.3 Ergodic sums of infinite measure preserving transformation . 44 iv 3.4 Asymptotic distribution of the local times `n of stationary processes with conditional local limit theorems . 47 3.5 Limit theorems of local times of discrete-time fractional Brownian motion . 59 3.5.1 Occupation times of discrete-time fractional Brownian motions 60 3.5.2 Occupation times of continuous fractional Brownian motions 63 Chapter 4 Almost Sure Central Limit Theorems 65 4.1 Almost sure central limit theorems for local times of random walks 65 4.2 Almost sure central limit theorem for stationary processes . 67 4.3 Proof of almost sure central limit theorem (ASCLT) . 69 4.3.1 Proof of Theorem 4.2.2 . 69 4.3.2 Proof of Proposition 4.3.1 . 70 4.4 Transfer operators . 74 4.5 Bounds of local times of stationary processes . 78 Chapter 5 Conclusion and Open Questions 80 Bibliography 82 v Acknowledgments Over the past five years I have received support and encouragement from a great number of individuals. I must first thank my adviser, Professor Manfred Denker, for his continuous guidance, endless encouragement and generous help during my graduate study and research. This dissertation could not have been finished without his advices and support. His guidance and friendship have made my graduate study a thoughtful and rewarding journey. I would also like to thank my dissertation committee members: Alexei Novikov, Anna Mazzucato and Zhibao Zhao for generously offering their time, insightful comments and support. I learned fractional Brownian motion from Professor Novikov and benefited a lot from his precious guidance and endless patience. I own my thanks to Professor Mazzucato for her invaluable advices as my mentor when I first came to Penn State. I am grateful to Professor Zhao for his valuable time. I offer my thanks to Professor Svetlana Katok for providing me the oppor- tunity to study in the Ph.D. program. I also thank the staffs of the Department of Mathematics for their kindly assistance. I am most grateful and indebted to my parents and the rest of my family for their unconditional love. Lastly, I must thank Changguang Dong for his unwavering love, patience and support. There are certainly many others who deserve mentioning to whom I offer a simple message: Thank You! vi Chapter 1 Introduction and Overview The local time of a continuous-time process is a stochastic process associated with an underlying stochastic process such as a Brownian motion, a Markov process, a diffusion process and so on, that characterizes the amount of time a particle has spent at a given level. It provides a very fine description of the sample paths of the underlying process. While the local time of a discrete-time process measures how often a state is visited and it is a refinement of the notion of recurrence. The local times in these two cases show many similar properties [2] and are closely connected by the invariance principle. 1.0.1 Brownian local time The notion of local time of a Brownian motion was first introduced by Le´vyin 1948 [40]. His contribution to the deep properties of the local time of Brownian motions laid the foundation of the theory of local times of stochastic processes. And later the theory was further developed by Trotter, Knight, Ray, It^o,and McKean, etc. Let fW (s); s ≥ 0g be a 1-dimensional Brownian motion. The occupation time R t 1 of a set A ⊂ R at time t is defined to be µt(A; !) = 0 fW (s;!)2Agds; which is a random measure on (R; B(R)). Le´vy(1948, [40]) proved that for almost all !, for any t ≥ 0, µt is absolutely continuous with respect to the Lebesgue measure, so the Radon-Nikodym derivative Lt;! exists and is Lebesgue-almost everywhere unique: Z µt(A; !) = Lt;!(x)dx: A 2 Trotter (1958, [61]) proved that for almost all ! 2 Ω, there exists a function L(t; x; !) that is continuous in (t; x) 2 [0; 1) × R, such that Z µt(A; !) = L(t; x; !)dx: A From now on, we use L(t; x), the jointed continuous version of the Brownian local times and we make the following remarks on the notations: 1. fL(t; x)gt≥0;x2R is called the Brownian local time. 2. Fix x, fL(t; x)gt≥0 is called the Brownian local time at level x. 3. Fix t, fL(t; x)gx2R is the Brownian local time at time t. 4. Fix x and t, L(t; x) is a random variable. 5. When x = 0, we use L(t) to denote L(t; 0) for short. So L(t; x) can be studied as a function of t or of x. Interesting questions of L(t; x) such as the exact distribution, the limiting distribution as t goes to 1 when x is fixed, the fluctuation of the Brownian local time are well studied. We recall some striking results related to the problems we are to deal with in this dissertation. As a function of t, the distribution of the Brownian local time at the level 0 is given by the following theorem. Theorem 1.0.1 (L´evyidentity, 1948). The processes f(jW (t)j;L(t; 0)) : t ≥ 0g and f(M(t) − W (t);M(t)) : t ≥ 0g have the same distribution, where M(t) = maxfW (s): s 2 [0; t]g. L´evy(1948) proved the theorem by showing that fM(t) − B(t): t ≥ 0g is a reflected Brownian motion. In [46], the theorem is proved by first defining the Brownian local time through the number of downcrossings of a Brownian motion and then using the embedded random walks into Brownian motions. The local time from this definition can be proved to be the density of the occupation measure. The importance of the concept Brownian local time also lies in its deep con- nection with It^o'sformula. 3 Theorem 1.0.2 (Tanaka's formula). Z t + 1 W (t) = 1fW (s)>0gdW (s) + L(t) 0 2 and Z t jW (t)j = sgn(W (s))dW (s) + L(t); 0 where sgn denotes the sign function 8 <+1; x > 0 sgn(x) = :−1; x ≤ 0: It is still true when W (t) is replaced by a continuous semimartingale. Tanaka's formula is the explicit Doob-Meyer decomposition of the submartingale jW (t)j into the martingale part and a continuous increasing process (local time). Tanaka's formula can be generalized by Ito-Tanaka's formula, which is also an extension of It^o'sformula to convex functions. Theorem 1.0.3 (It^o-Tanaka's formula). If f is the difference of two convex func- tions, then Z t Z 0 1 00 f(W (t)) = f(W (0)) + f−(W (s))dW (s) + L(t; x)f (dx): 0 2 R Recall that if f is convex, its second derivative f 00 in the sense of distributions is a positive measure. It^o-Tanaka's formula holds for any continuous semimartingale.