Asymptotic analysis of random walks : heavy-tailed distributions / A.A. Borovkov, K.A. Borovkov ; translated by O.B. Borovkova.

Format:

Book

Author/Creator:

Borovkov, A. A. (Aleksandr Alekseevich), 1931- author.

Contributor:

Borovkov, K. A. (Konstantin Aleksandrovich), editor.

Borovkova, O. B., translator.

Series:

Encyclopedia of mathematics and its applications ; v. 118.

Encyclopedia of mathematics and its applications ; volume 118

Language:

English

Subjects (All):

Random walks (Mathematics).

Asymptotic expansions.

Physical Description:

1 online resource (xxix, 625 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2008.

Language Note:

English

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.

Contents:

Preliminaries

Random walks with jumps having no finite first moment

Random walks with jumps having finite mean and infinite variance

Random walks with jumps having finite variance

Random walks with semiexponential jump distributions

Large deviations on the boundary of and outside the Cramer zone for random walks with jump distributions decaying exponentially fast

Asymptotic properties of functions of regularly varying and semiexponential distributions. Asymptotics of the distributions of stopped sums and their maxima. An alternative approach to studying the asymptotics of P(S[subscript n] [is equal to or greater than] x)

On the asymptotics of the first hitting times

Integro-local and integral large deviation theorems for sums of random vectors

Large deviations in trajectory space

Large deviations of sums of random variables of two types

Random walks with non-identically distributed jumps in the triangular array scheme in the case of infinite second moment. Transient phenomena

Random walks with non-identically distributed jumps in the triangular array scheme in the case of finite variances

Random walks with dependent jumps

Extension of the results of Chapters 2-5 to continuous-time random processes with independent increments

Extension of the results of Chapters 3 and 4 to generalized renewal processes.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88345-3

1-107-38409-5

1-107-38760-4

0-511-82627-3

1-107-39052-4

0-511-72139-0

1-107-39893-2

1-107-39532-1

OCLC:

776965731