ABELIAN THEOREM FOR THE REGULARLY VARYING MEASURE AND ITS DENSITY IN ORTHANT

摘要

The paper is concerned with a sigma-finite measure U concentrated in the positive orthant R-+(n) = [0, infinity)(n) such that there exists the Laplace transform (U) over tilde(lambda) for lambda is an element of int R-+(n). Let functions R(t) > 0 and b(t) = (b(1)(t), ..., b(n)(t)) is an element of intR(+)(n) for t >= 0 be such that R(t) -> infinity, b(i)(t) -> infinity for any i = 1, ..., n. Under certain assumptions on these functions, the weak convergence of the measures U(b(t) .)/R(t) to Phi(.) as t -> infinity is shown to imply the convergence (U) over tilde (lambda/b(t)) -> (Phi) over tilde (lambda) < infinity for any lambda is an element of int R-+(n) (t -> infinity) (the multiplication and division of vectors are defined componentwise). A function f : R-+(n) -> R+ is said to be regularly varying at infinity in R-+(n) along b(t) if f(b(t)x(t))/f(b(t)) -> phi(x) is an element of (0,infinity) as t -> infinity for all x, x(t) is an element of R-+(n) {0} such that x(t) -> x. Sufficient conditions are given for such functions to give (f) over tilde (lambda/b(t)) equivalent to (U) over tilde (lambda/b(t)) -> (phi) over cap (lambda) equivalent to (Phi) over tilde (lambda) < infinity for any lambda is an element of int R-+(n) (t -> infinity) for U(dx) = f(x) dx, Phi(dx) = phi(x) dx. The Abelian theorem obtained here is applied at the end of the paper to investigate the limit behavior of multiple power series distributions.