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January, 1986 Second-Order Approximation in the Conditional Central Limit Theorem
Dieter Landers, Lothar Rogge
Ann. Probab. 14(1): 313-325 (January, 1986). DOI: 10.1214/aop/1176992630


Let $X_n, n \in \mathbb{N}$ be i.i.d. with mean 0 and variance 1. Let $B \in \sigma(X_n: n \in \mathbb{N})$ be a set such that its distances from the $\sigma$-fields $\sigma(X_1,\cdots, X_n)$ are of order $O(1/n(lg n)^{2 + \varepsilon})$ for some $\varepsilon > 0$. We prove that for those $B$ the conditional probabilities $P((1/\sqrt n)\sum^n_{i=1} X_i \leq t\mid B)$ can be approximated by a modified Edgeworth expansion up to order $O(1/n)$. An example shows that this is not true any more if the distances of $B$ from $\sigma(X_1,\cdots, X_n)$ are only of order $O(1/n(lg n)^2)$.


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Dieter Landers. Lothar Rogge. "Second-Order Approximation in the Conditional Central Limit Theorem." Ann. Probab. 14 (1) 313 - 325, January, 1986.


Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0595.60024
MathSciNet: MR815973
Digital Object Identifier: 10.1214/aop/1176992630

Primary: 60F15
Secondary: 60G50

Keywords: asymptotic expansion , conditional central limit theorem

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 1 • January, 1986
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