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Random Walk: a Modern Introduction

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Random Walk: a Modern Introduction Random Walk: A Modern Introduction Gregory F. Lawler and Vlada Limic Contents Preface page 6 1 Introduction 9 1.1 Basic definitions 9 1.2 Continuous-time random walk 12 1.3 Other lattices 14 1.4 Other walks 16 1.5 Generator 17 1.6 Filtrations and strong Markov property 19 1.7 A word about constants 21 2 Local Central Limit Theorem 24 2.1 Introduction 24 2.2 Characteristic Functions and LCLT 27 2.2.1 Characteristic functions of random variables in Rd 27 2.2.2 Characteristic functions of random variables in Zd 29 2.3 LCLT — characteristic function approach 29 2.3.1 Exponential moments 42 2.4 Some corollaries of the LCLT 47 2.5 LCLT — combinatorial approach 51 2.5.1 Stirling’s formula and 1-d walks 52 2.5.2 LCLT for Poisson and continuous-time walks 56 3 Approximation by Brownian motion 63 3.1 Introduction 63 3.2 Construction of Brownian motion 64 3.3 Skorokhod embedding 68 3.4 Higher dimensions 71 3.5 An alternative formulation 72 4 Green’s Function 75 4.1 Recurrence and transience 75 4.2 Green’s generating function 76 4.3 Green’s function, transient case 81 4.3.1 Asymptotics under weaker assumptions 84 3 4 Contents 4.4 Potential kernel 86 4.4.1 Two dimensions 86 4.4.2 Asymptotics under weaker assumptions 90 4.4.3 One dimension 92 4.5 Fundamental solutions 95 4.6 Green’s function for a set 96 5 One-dimensional walks 103 5.1 Gambler’s ruin estimate 103 5.1.1 General case 106 5.2 One-dimensional killed walks 112 5.3 Hitting a half-line 115 6 Potential Theory 119 6.1 Introduction 119 6.2 Dirichlet problem 121 6.3 Difference estimates and Harnack inequality 125 6.4 Further estimates 132 6.5 Capacity, transient case 136 6.6 Capacity in two dimensions 144 6.7 Neumann problem 152 6.8 Beurling estimate 154 6.9 Eigenvalue of a set 157 7 Dyadic coupling 166 7.1 Introduction 166 7.2 Some estimates 167 7.3 Quantile coupling 170 7.4 The dyadic coupling 172 7.5 Proof of Theorem 7.1.1 175 7.6 Higher dimensions 176 7.7 Coupling the exit distributions 177 8 Addtional topics on Simple Random Walk 181 8.1 Poisson kernel 181 8.1.1 Half space 181 8.1.2 Cube 184 8.1.3 Strips and quadrants in Z2 188 8.2 Eigenvalues for rectangles 191 8.3 Approximating continuous harmonic functions 192 8.4 Estimates for the ball 193 9 Loop Measures 198 9.1 Introduction 198 9.2 Definitions and notations 198 9.2.1 Simple random walk on a graph 201 9.3 Generating functions and loop measures 201 9.4 Loop soup 206 Contents 5 9.5 Loop erasure 207 9.6 Boundary excursions 209 9.7 Wilson’s algorithm and spanning trees 214 9.8 Examples 216 9.8.1 Complete graph 216 9.8.2 Hypercube 217 9.8.3 Sierpinski graphs 220 9.9 Spanning trees of subsets of Z2 221 9.10 Gaussian free field 230 10 Intersection Probabilities for Random Walks 237 10.1 Long range estimate 237 10.2 Short range estimate 240 10.3 One-sided exponent 243 11 Loop-erased random walk 245 11.1 h-processes 245 11.2 Loop-erased random walk 248 11.3 LERW in Zd 250 11.3.1 d 3 250 ≥ 11.3.2 d = 2 251 11.4 Rate of growth 254 11.5 Short-range intersections 257 12 Appendix 259 12.1 Some expansions 259 12.1.1 Riemann sums 259 12.1.2 Logarithm 260 12.2 Martingales 263 12.2.1 Optional Sampling Theorem 263 12.2.2 Maximal inequality 265 12.2.3 Continuous martingales 267 12.3 Joint normal distributions 267 12.4 Markov chains 269 12.4.1 Chains restricted to subsets 272 12.4.2 Maximal coupling of Markov chains 275 12.5 Some Tauberian theory 278 12.6 Second moment method 280 12.7 Subadditivity 281 References 285 Index of Symbols 286 Index 288 Preface Random walk – the stochastic process formed by successive summation of independent, identically distributed random variables – is one of the most basic and well-studied topics in probability theory. For random walks on the integer lattice Zd, the main reference is the classic book by Spitzer [16]. This text considers only a subset of such walks, namely those corresponding to increment distributions with zero mean and finite variance. In this case, one can summarize the main result very quickly: the central limit theorem implies that under appropriate rescaling the limiting distribution is normal, and the functional central limit theorem implies that the distribution of the corresponding path-valued process (after standard rescaling of time and space) approaches that of Brownian motion. Researchers who work with perturbations of random walks, or with particle systems and other models that use random walks as a basic ingredient, often need more precise information on random walk behavior than that provided by the central limit theorems. In particular, it is important to understand the size of the error resulting from the approximation of random walk by Brownian motion. For this reason, there is need for more detailed analysis. This book is an introduction to the random walk theory with an emphasis on the error estimates. Although “mean zero, finite variance” assumption is both necessary and sufficient for normal convergence, one typically needs to make stronger assumptions on the increments of the walk in order to get good bounds on the error terms. This project embarked with an idea of writing a book on the simple, nearest neighbor random walk. Symmetric, finite range random walks gradually became the central model of the text. This class of walks, while being rich enough to require analysis by general techniques, can be studied without much additional difficulty. In addition, for some of the results, in particular, the local central limit theorem and the Green’s function estimates, we have extended the discussion to include other mean zero, finite variance walks, while indicating the way in which moment conditions influence the form of the error. The first chapter is introductory and sets up the notation. In particular, there are three main d classes of irreducible walks in the integer lattice Z — d (symmetric, finite range), ′ (aperiodic, P Pd mean zero, finite second moment), and (aperiodic with no other assumptions). Symmetric Pd∗ random walks on other integer lattices such as the triangular lattice can also be considered by taking a linear transformation of the lattice onto Zd. The local central limit theorem (LCLT) is the topic for Chapter 2. Its proof, like the proof of the usual central limit theorem, is done by using Fourier analysis to express the probability of interest 6 Preface 7 in terms of an integral, and then estimating the integral. The error estimates depend strongly on the number of finite moments of the corresponding increment distribution. Some important corollaries are proved in Section 2.4; in particular, the fact that aperiodic random walks starting at different points can be coupled so that with probability 1 O(n 1/2) they agree for all times − − greater than n is true for any aperiodic walk, without any finite moment assumptions. The chapter ends by a more classical, combinatorial derivation of LCLT for simple random walk using Stirling’s formula, while again keeping track of error terms. Brownian motion is introduced in Chapter 3. Although we would expect a typical reader to already be familiar with Brownian motion, we give the construction via the dyadic splitting method. The estimates for the modulus of continuity are given as well. We then describe the Skorokhod method of coupling a random walk and a Brownian motion on the same probability space, and give error estimates. The dyadic construction of Brownian motion is also important for the dyadic coupling algorithm of Chapter 7. The Green’s function and its analog in the recurrent setting, the potential kernel, are studied in Chapter 4. One of the main tools in the potential theory of random walk is the analysis of martingales derived from these functions. Sharp asymptotics at infinity for the Green’s function are needed to take full advantage of the martingale technique. We use the sharp LCLT estimates of Chapter 2 to obtain the Green’s function estimates. We also discuss the number of finite moments needed for various error asymptotics. Chapter 5 may seem somewhat out of place. It concerns a well-known estimate for one-dimensional walks called the gambler’s ruin estimate. Our motivation for providing a complete self-contained argument is twofold. Firstly, in order to apply this result to all one-dimensional projections of a higher dimensional walk simultaneously, it is important to shiw that this estimate holds for non- lattice walks uniformly in few parameters of the distribution (variance, probability of making an order 1 positive step). In addition, the argument introduces the reader to a fairly general technique for obtaining the overshoot estimates. The final two sections of this chapter concern variations of one-dimensional walk that arise naturally in the arguments for estimating probabilities of hitting (or avoiding) some special sets, for example, the half-line. In Chapter 6, the classical potential theory of the random walk is covered in the spirit of [16] and [10] (and a number of other sources). The difference equations of our discrete space setting (that in turn become matrix equations on finite sets) are analogous to the standard linear partial differential equations of (continuous) potential theory.
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