The Bergström-Grigelionis asymptotic expansions

Abstract

We consider a triangular array of independent identically distributed discrete random variables. We assume that the probability distribution of sums satisfies the necessary and sufficient conditions for the weak convergence to the compound Poisson distribution. The first known result (the case where random variables take only integer values) is due to B. Grigelionis, who estimated the convergence rate to the compound Poisson distribution. We extend the summation of random variables by including the variables taking discrete values and by using the Grigelionis ideas to obtain “lengthy” asymptotic expansions. These expansions are based on the well-known Bergström identity [H. Bergström, On asymptotic expansions of probability functions, Scand. Actuarial J., 34(1):1–33, 1951].