Random Walk: A Modern Introduction
by Gregory F. Lawler , Vlada LimicFormat: Hardcover
Pub. Date: 2010-07-26
Publisher(s): Cambridge University Press
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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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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