Random Walk: A Modern Introduction

by Gregory F. Lawler , Vlada Limic

• ISBN13: 9780521519182
• ISBN10: 0521519187
• Format: Hardcover
• Copyright: 2010-07-26
• Publisher: Cambridge University Press

Summary

Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.

Table of Contents

Preface	p. ix
Introduction	p. 1
Basic definitions	p. l
Continuous-time random walk	p. 6
Other lattices	p. 7
Other walks	p. 11
Generator	p. 11
Filtrations and strong Markov property	p. 14
A word about constants	p. 17
Local central limit theorem	p. 21
Introduction	p. 21
Characteristic functions and LCLT	p. 25
Characteristic functions of random variables in Rd	p. 25
Characteristic functions of random variables in Zd	p. 27
LCLT - characteristic function approach	p. 28
Exponential moments	p. 46
Some corollaries of the LCLT	p. 51
LCLT - combinatorial approach	p. 58
Stirling's formula and one-dimensional walks	p. 58
LCLT for Poisson and continuous-time walks	p. 64
Approximation by Brownian motion	p. 72
Introduction	p. 72
Construction of Brownian motion	p. 74
Skorokhod embedding	p. 79
Higher dimensions	p. 82
An alternative formulation	p. 84
The Green's function	p. 87
Recurrence and transience	p. 87
The Green's generating function	p. 88
The Green's function, transient case	p. 95
Asymptotics under weaker assumptions	p. 99
Potential kernel	p. 101
Two dimensions	p. 101
Asymptotics under weaker assumptions	p. 107
One dimension	p. 109
Fundamental solutions	p. 113
The Green's function for a set	p. 114
One-dimensional walks	p. 123
Gambler's ruin estimate	p. 123
General case	p. 127
One-dimensional killed walks	p. 135
Hitting a half-line	p. 138
Potential theory	p. 144
Introduction	p. 144
Dirichlet problem	p. 146
Difference estimates and Harnack inequality	p. 152
Further estimates	p. 160
Capacity, transient case	p. 166
Capacity in two dimensions	p. 176
Neumann problem	p. 186
Beurling estimate	p. 189
Eigenvalue of a set	p. 194
Dyadic coupling	p. 205
Introduction	p. 205
Some estimates	p. 207
Quantile coupling	p. 210
The dyadic coupling	p. 213
Proof of Theorem 7.1.1	p. 216
Higher dimensions	p. 218
Coupling the exit distributions	p. 219
Additional topics on simple random walk	p. 225
Poisson kernel	p. 225
Half space	p. 226
Cube	p. 229
Strips and quadrants in Z2	p. 235
Eigenvalues for rectangles	p. 238
Approximating continuous harmonic functions	p. 239
Estimates for the ball	p. 241
Loop measures	p. 247
Introduction	p. 247
Definitions and notations	p. 247
Simple random walk on a graph	p. 251
Generating functions and loop measures	p. 252
Loop soup	p. 257
Loop erasure	p. 259
Boundary excursions	p. 261
Wilson's algorithm and spanning trees	p. 268
Examples	p. 271
Complete graph	p. 271
Hypercube	p. 272
Sierpinski graphs	p. 275
Spanning trees of subsets of Z2	p. 277
Gaussian free field	p. 289
Intersection probabilities for random walks	p. 297
Long-range estimate	p. 297
Short-range estimate	p. 302
One-sided exponent	p. 305
Loop-erased random walk	p. 307
h-processes	p. 307
Loop-erased random walk	p. 311
LERW in Zd	p. 313
d≥3	p. 314
d=2	p. 315
Rate of growth	p. 319
Short-range intersections	p. 323
Appendix	p. 326
Some expansions	p. 326
Riemann sums	p. 326
Logarithm	p. 327
Martingales	p. 331
Optional sampling theorem	p. 332
Maximal inequality	p. 334
Continuous martingales	p. 336
Joint normal distributions	p. 337
Markov chains	p. 339
Chains restricted to subsets	p. 342
Maximal coupling of Markov chains	p. 346
Some Tauberian theory	p. 351
Second moment method	p. 353
Subadditivity	p. 354
Bibliography	p. 360
Index of Symbols	p. 361
Index	p. 363
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