Probabilistic Applications of Tauberian Theorems

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