Abelian and Tauberian theorems for the Laplace transform of functions in several variables (Q1825398)

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.