Accuracy of Convergence of Sums of Dependent Random Variables with Variances Not Necessarily Finite

Abstract

Let $S_n = \sum^{k_n}_{k=1} X_{nk}$ and $X$ be random variables with distribution functions $F_n(x)$ and $F(x)$. No assumptions are made that the $(X_{nk})$ have finite means or variances. Also, no independence conditions are assumed. A bound is found for $M_n = \sup_{-\infty<x<\infty}|F_n(x) - F(x)|.$ This bound involves various truncated moments and conditional probabilities and expectations. A typical quantity involved is $\sum^{k_n}_{k=1} E|E (X_{n^k}|\sum^{k-1}_{j=1} X_{nj}) - E(X_{n^k})|$. Using this bound, particular conditions are found so that $S_n$ converges in distribution to $X$.