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 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...

We discuss rates of convergence for the distribution of normalized sample extremes to the appropriate limit distribution. We show that the rate of convergence depends on that of the corresponding dependence functions and that of the marginals. The univariate results are well known by now, so we...