Error Estimates for the Weak Convergence to Certain Infinitely Divisible Laws

Abstract

Let $F_n$ denote the distribution function of the $n$th row sum of a triangular array of infinitesimal, rowwise independent random variables, and let $F^\ast$ denote the limiting infinitely divisible distribution function. Bounds are obtained for $\sup_{-\infty < x < \infty} |F_n(x) - F^\ast(x)|$ in the case that the means are finite and also for the attraction to a stable law with exponent $\alpha \leqq 1$. Conditions for convergence of these bounds are given.