Probabilistic applications of Tauberian theorems
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Bibliographic Information
Probabilistic applications of Tauberian theorems
(Modern probability and statistics)
VSP, c2005
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Note
Includes bibliographical references and index
Description and Table of Contents
Description
01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information.
The Tauberian theory has found a widespread application in probability theory. Despite the strong interest of probabilists in Tauberian theorems, no book specially devoted to this topic has been published yet. This monograph is intended to fill this gap.
In last three decades, much thought has been given to multidimensional Tauberian theory. This is primarily due to the fact that Tauberian theorems are finding ever-widening application in mathematical physics, the theory of differential equations, and probability theory.
By Abelian theorems are meant those assertions which allow to deduce from the asymptotic behaviour of sequences and functions the asymptotic properties of their generating functions and Laplace transforms (as well as other integral transforms). Theorems converse to Abelian are referred to as Tauberian. Usually, direct methods are used to prove Abelian theorems. It is much more difficult to prove the corresponding Tauberian theorems, and a wide spectrum of analytical techniques is involved.
This monograph places particular emphasis on the multidimensional studies. It contains Tauberian theorems and their applications to analyse the asymptotic behaviour of stochastic processes, record processes, random permutations, and infinitely divisible random variables.
Tauberian theorems are contained in the first chapter of the book. Chapters 2-5 cover probabilistic applications of Tauberian theorems.
Table of Contents
Introduction
1 Tauberian theorems
Regularly varying functions in a cone
Weakconvergence ofmeasures and functions
Multidimensional Tauberian theorems of Karamata type
Weakly oscillating functions
A multidimensional Tauberian comparison theorem
One-dimensional Tauberiantheorems
Tauberian theorems of Drozhzhinov-Zavyalov type
Three multidimensional Tauberian theorems
Remarks to Chapter 1
2 Applications to branching processes
Bounded below branching processes
Bellman-Harris branching processes
Convergence of finite-dimensional distributions
The number of long-living particles
3 Random A-permutations
The number of A-permutations
Auxiliary limit theorems
Fundamental limit theorems
Uniformly distributed sequences
Examples of sets A
Random sets A
4 Infinitely divisible distributions
Probabilities of large deviations
Asymptotic behaviour of adensity at infinity
Multidimensional case
5 Limit theorems in the record model
Intervals betweenstate change times in the record process
The asymptotic behaviour of the kth record times
Bibliography
Index
by "Nielsen BookData"