Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions

by A. A. Borovkov , K. A. Borovkov

• ISBN13: 9780521881173
• ISBN10: 052188117X
• Format: Hardcover
• Copyright: 2008-06-30
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Introduction

Preliminaries

Random walks with jumps having no finite first moment

Random walks with finite mean and infinite variance

Random walks with jumps having finite variance

Random walks with semiexponential jump distributions

Random walks with exponentially decaying distributions

Asymptotic properties of functions of distributions

On the asymptotics of the first hitting times

Large deviation theorems for sums of random vectors

Large deviations in the space of trajectories

Large deviations of sums of random variables of two types

Non-identically distributed jumps with infinite second moments

Non-identically distributed jumps with finite variances

Random walks with dependent jumps

Extension to processes with independent increments

Extensions to generalised renewal processes

Bibliographic notes

Index of notations

Bibliography

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