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3080 results in Probability theory and stochastic processes

Page 95 of 154

Uniform Central Limit Theorems
  • R. M. Dudley
  • Published online:
    20 May 2010
    Print publication:
    28 July 1999
  • This book shows how the central limit theorem for independent, identically distributed random variables with values in general, multidimensional spaces, holds uniformly over some large classes of functions. The author, an acknowledged expert, gives a thorough treatment of the subject, including several topics not found in any previous book, such as the Fernique-Talagrand majorizing measure theorem for Gaussian processes, an extended treatment of Vapnik-Chervonenkis combinatorics, the Ossiander L2 bracketing central limit theorem, the Giné-Zinn bootstrap central limit theorem in probability, the Bronstein theorem on approximation of convex sets, and the Shor theorem on rates of convergence over lower layers. Other results of Talagrand and others are surveyed without proofs in separate sections. Problems are included at the end of each chapter so the book can be used as an advanced text. The book will interest mathematicians working in probability, mathematical statisticians and computer scientists working in computer learning theory.

Asymptotic Analysis of Random Walks
  • Heavy-Tailed Distributions
  • A. A. Borovkov, K. A. Borovkov
  • Published online:
    07 May 2010
    Print publication:
    12 June 2008
  • This book focuses on the asymptotic behaviour of the probabilities of large deviations of the trajectories of random walks with 'heavy-tailed' (in particular, regularly varying, sub- and semiexponential) jump distributions. Large deviation probabilities are of great interest in numerous applied areas, typical examples being ruin probabilities in risk theory, error probabilities in mathematical statistics, and buffer-overflow probabilities in queueing theory. The classical large deviation theory, developed for distributions decaying exponentially fast (or even faster) at infinity, mostly uses analytical methods. If the fast decay condition fails, which is the case in many important applied problems, then direct probabilistic methods usually prove to be efficient. This monograph presents a unified and systematic exposition of the large deviation theory for heavy-tailed random walks. Most of the results presented in the book are appearing in a monograph for the first time. Many of them were obtained by the authors.

Stochastic Analysis
  • Proceedings of the Durham Symposium on Stochastic Analysis, 1990
  • Edited by M. T. Barlow, N. H. Bingham
  • Published online:
    31 March 2010
    Print publication:
    25 October 1991
  • Durham Symposia traditionally constitute an excellent survey of recent developments in many areas of mathematics. The Symposium on stochastic analysis, which took place at the University of Durham in July 1990, was no exception. This volume is edited by the organizers of the Symposium, and contains papers contributed by leading specialists in diverse areas of probability theory and stochastic processes. Of particular note are the papers by David Aldous, Harry Kesten and Alain-Sol Sznitman, all of which are based upon short courses of invited lectures. Researchers into the varied facets of stochastic analysis will find that these proceedings are an essential purchase.

  • 9 - Intersections and self-intersections of Brownian paths

  • Peter Mörters, University of Bath, Yuval Peres
  • Book:
    Brownian Motion
    Published online:
    05 June 2012
    Print publication:
    25 March 2010, pp 255-289

  • Summary

    In this chapter we study multiple points of d-dimensional Brownian motion. We shall see, for example, in which dimensions the Brownian path has double points and explore how many double points there are. This chapter also contains some of the highlights of the book: a proof that planar Brownian motion has points of infinite multiplicity, the intersection equivalence of Brownian motion and percolation limit sets, and the surprising dimension-doubling theorem of Kaufman.

    Intersection of paths: Existence and Hausdorff dimension

    Existence of intersections

    Suppose that {B1 (t): t ≥ 0} and {B2(t): t ≥ 0} are two independent d-dimensional Brownian motions started in arbitrary points. The question we ask in this section is, in which dimensions the ranges, or paths, of the two motions have a nontrivial intersection, in other words whether there exist times t1, t1 > 0 such that B1(t1) = B2(t2). As this question is easy if d = 1 we assume d ≥ 2 throughout this section.

    We have developed the tools to decide this question in Chapter 4 and Chapter 8. Keeping the path {B1(t): t ≥ 0} fixed, we have to decide whether it is a polar set for the second Brownian motion. By Kakutani's theorem, Theorem 8.20, this question depends on its capacity with respect to the potential kernel. As the capacity is again related to Hausdorff measure and dimension, the results of Chapter 4 are crucial in the proof of the following result.

Page 95 of 154