Overview

The simplest mathematical model of the Brownian motion of physics is the simple, symmetric random walk. This book collects and compares current results — mostly strong theorems which describe the properties of a random walk. The modern problems of the limit theorems of probability theory are treated in the simple case of coin tossing. Taking advantage of this simplicity, the reader is familiarized with limit theorems (especially strong ones) without the burden of technical tools and difficulties. An easy way of considering the Wiener process is also given, through the study of the random walk.Since the first and second editions were published in 1990 and 2005, a number of new results have appeared in the literature. The first two editions contained many unsolved problems and conjectures which have since been settled; this third, revised and enlarged edition includes those new results. In this edition, a completely new part is included concerning Simple Random Walks on Graphs. Properties of random walks on several concrete graphs have been studied in the last decade. Some of the obtained results are also presented.Contents: Simple Symmetric Random Walk in ℤ1 Introduction of Part IDistributionsRecurrence and the Zero-One LawFrom the Strong Law of Large Numbers to the Law of Iterated LogarithmLévy ClassesWiener Process and Invariance PrincipleIncrementsStrassen Type TheoremsDistribution of the Local TimeLocal Time and Invariance PrincipleStrong Theorems of the Local TimeExcursionsFrequently and Rarely Visited SitesAn Embedding TheoremA Few Further ResultsSummary of Part I Simple Symmetric Random Walk in ℤd The Recurrence TheoremWiener Process and Invariance PrincipleThe Law of Iterated LogarithmLocal TimeThe RangeHeavy Points and Heavy BallsCrossing and Self-crossingLarge Covered BallsLong ExcursionsSpeed of EscapeA Few Further Problems Random Walk in Random Environment: Introduction of Part IIIIn the First Six DaysAfter the Sixth DayWhat Can a Physicist Say About the Local Time ξ(0, n )?On the Favourite Value of the RWIREA Few Further Problems Random Walks in Graphs: Introduction of Part IVRandom Walk in CombRandom Walk in a Comb and in a Brush with CrossingsRandom Walk on a SpiderRandom Walk in Half-Plane-Half-CombReadership: Graduate students and researchers in probability theory and statistical physics.

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