Asymptotic Analysis of Random Walks
Heavy-Tailed Distributions

A comprehensive monograph presenting a unified systematic exposition of the large deviations theory for heavy-tailed random walks.

A. A. Borovkov (Author), K. A. Borovkov (Author)

9780521881173, Cambridge University Press

Hardback, published 12 June 2008

656 pages, 5 b/w illus.
24.2 x 16.2 x 4.8 cm, 1.22 kg

'… an up-to-date, unified and systematic exposition of the field. Most of the results presented are appearing in a monograph for the first time and a good proportion of them were obtained by the authors. … The book presents some beautiful and useful mathematics that may attract a number of probabilists to the large deviations topic in probability.' EMS Newsletter

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.

Introduction
1. Preliminaries
2. Random walks with jumps having no finite first moment
3. Random walks with finite mean and infinite variance
4. Random walks with jumps having finite variance
5. Random walks with semiexponential jump distributions
6. Random walks with exponentially decaying distributions
7. Asymptotic properties of functions of distributions
8. On the asymptotics of the first hitting times
9. Large deviation theorems for sums of random vectors
10. Large deviations in the space of trajectories
11. Large deviations of sums of random variables of two types
12. Non-identically distributed jumps with infinite second moments
13. Non-identically distributed jumps with finite variances
14. Random walks with dependent jumps
15. Extension to processes with independent increments
16. Extensions to generalised renewal processes
Bibliographic notes
Index of notations
Bibliography.

Subject Areas: Probability & statistics [PBT], Calculus & mathematical analysis [PBK]