Uniform Local Probability Approximations: Improvements on Berry-Esseen

Abstract

Let $X_1, X_2,\ldots$ be independent, mean zero, uniformly bounded random variables with $S_n = X_1 + \cdots + X_n$. Optimal criteria are determined on the length and location of an interval $\Gamma$ so that $P(S_n \in \Gamma)$ is proportional to $(|\Gamma|/\sqrt{\operatorname{Var} S_n)} \wedge 1$. The proof makes an unusual use of support considerations.