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.
Citation
Marjorie G. Hahn. Michael J. Klass. "Uniform Local Probability Approximations: Improvements on Berry-Esseen." Ann. Probab. 23 (1) 446 - 463, January, 1995. https://doi.org/10.1214/aop/1176988394
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