Convergence of weighted sums of independent random variables

Abstract

1. Introduction. Let {X k : k ≥ 1} be a sequence of independent, but not necessarily identically distributed, random variables. Suppose that the random variables X n are uniformly bounded by a random variable X in the sense that (1) P(|X n | ≥x) ≤ P(|X| ≥ x)for all x > 0. Write q n (x) = P (|X n | ≥ x) and q (x) = P (|X| ≥ x).