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April, 1987 Nonuniform Estimates in the Conditional Central Limit Theorem
Dieter Landers, Lothar Rogge
Ann. Probab. 15(2): 776-782 (April, 1987). DOI: 10.1214/aop/1176992171

Abstract

Let $X_n, n \in \mathbb{N}$, be i.i.d. with mean 0, variance 1, and $E(|X_1|^r) < \infty$ for some $r > 3$. Let $B$ be a measurable set such that its distances from the $\sigma$ fields $\sigma (X_1,\ldots, X_n)$ are of order $O(n^{-1/2} (\log n)^{-r/2})$. We prove that for such $B$ the conditional probabilities $P(n^{-1/2}\sum^n_{i=1} X_i \leq t\mid B)$ can be approximated by the standard normal distribution $\Phi (t)$ up to the classical nonuniform bound $(1 + |t|^r)^{-1} n^{-1/2}$. An example shows that this is not true any more if the distances of $B$ from $\sigma (X_1,\ldots, X_n)$ are only of order $O(n^{-1/2}(\log n)^{-r/2+\varepsilon})$ for some $\varepsilon > 0$. For the case $r = 3$ one can obtain the corresponding assertion only under a strengthened assumption.

Citation

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Dieter Landers. Lothar Rogge. "Nonuniform Estimates in the Conditional Central Limit Theorem." Ann. Probab. 15 (2) 776 - 782, April, 1987. https://doi.org/10.1214/aop/1176992171

Information

Published: April, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0622.60025
MathSciNet: MR885143
Digital Object Identifier: 10.1214/aop/1176992171

Subjects:
Primary: 60E15
Secondary: 60F05

Keywords: conditional central limit theorem , Independent and identically distributed random variables , Nonuniform bounds

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 2 • April, 1987
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