Using two kinds of multivariate regular variation we prove several Abel-Tauber theorems for the Laplace transform of functions in several variables. We generalize some power series results of Alpar and apply our results in multivariate renewal theory.

A theorem on regularly varying functions in 2 is proved and applied to domains of attraction of stable laws with index 1 [less-than-or-equals, slant] [alpha] [less-than-or-equals, slant] 2. We also present a theory of [Pi]-variation in 2. Unlike the situation in 1 the latter is not connected...

Let G = [Sigma][infinity]n=0pnF*n denote the probability measure subordinate to F with subordinator {Pn}. We investigate the asymptotic behaviour of (1 - G(x))-([Sigma] npn)(1 - F(x)) as x -- [infinity] if 1 - F is regularly varying with index [varrho], 0 = [varrho] = 1. Applications to random...

A general form for regular variation in IR2 is introduced and applied to domains of attraction of stable distribution in IR2 where the components have different indices. The situation in IRd with d > 2 is more complicated but not essentially different. For simplicity this paper is limited to IR2.

Let (X,Y),(X1,Y1),(X2,Y2),... denote independent positive random vectors with common distribution function F(x,y)=P(X[less-than-or-equals, slant]x,Y[less-than-or-equals, slant]y) with F(x,y)1 for all x,y. Based on the Xi and the Yj we construct the sum sequences and respectively. For a double...