Abelian and Tauberian theorems for the Laplace transform of functions in several variables
type: Item
instance of: scholarly article
title: Abelian and Tauberian theorems for the Laplace transform of functions in several variables
zbMATH Open document ID: 0684.44002
DOI: 10.1016/0047-259X(89)90041-9
author: Yanyan Li
published in: Journal of Multivariate Analysis
publication date: 1989-01-01T00:00:00Z
review text: The authors discuss Abelian- and Tauberian theorems for Laplace transforms of bivariate nondecreasing functions f using two types of regular variation. For the first kind of regular variation, here \(\lim_{t\to \infty}f(r(t)\cdot x,s(t)\cdot y)/h(t)\) exists for all \(x,y>0\) with certain positive functions r, s, h, the results were obtained by \textit{L. de Haan}, \textit{E. Omey} and \textit{S. Resnick} [Anal. 14, 17-33 (1984; Zbl 0536.60027)]. The main purpose of this paper are Abelian-Tauberian theorems for functions of weak regular variation, here \(\lim_{\min (s,t)\to \infty}f(sx,ty)/h(s,t)\) exists for all \(x,y>0\) with a suitable \(h: {\mathbb{R}}^ 2_+\to {\mathbb{R}}_+.\) The results are applied to questions related to bivariate power series as e.g. the asymptotics of convolutions of certain bivariate sequences. Furthermore they are used to obtain the asymptotic behaviour of a bivariate renewal function.
reviewed by: Ulrich Stadtmüller
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 44A20
Mathematics Subject Classification ID: 44A30
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 4120756
zbMATH Keywords: Abelian theorems
zbMATH Keywords: Tauberian theorems
zbMATH Keywords: Laplace transforms
zbMATH Keywords: functions of weak regular variation
zbMATH Keywords: bivariate power series
zbMATH Keywords: asymptotics of convolutions
zbMATH Keywords: bivariate sequences
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: bivariate renewal function
MaRDI profile type: Publication
cites work: Q3861880
cites work: Q3036015
cites work: Renewal Theory in the Plane
cites work: Q5512461
cites work: Derivatives of regularly varying functions in \(R^d\) and domains of attraction of stable distributions
cites work: Integrals and derivatives of regularly varying functions in \(R^ n\) and domains of attraction of stable distributions. II
cites work: Domains of attraction and regular variation in \({\mathbb{R}}^ d\)
cites work: Renewal theory in two dimensions: Basic results
cites work: Renewal theory in two dimensions: asymptotic results
cites work: Q3889534
description: scientific article; zbMATH DE number 4120756