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August, 1984 An Example on the Central Limit Theorem for Associated Sequences
Norbert Herrndorf
Ann. Probab. 12(3): 912-917 (August, 1984). DOI: 10.1214/aop/1176993241

Abstract

We construct a strictly stationary associated sequence $(X_n)_{n \in \mathbb{N}}$ with $EX_n = 0, 0 < EX^2_n < \infty$ such that $K(R) = \operatorname{Cov}(X_1, X_1) + \sum^R_{j=2} \operatorname{Cov}(X_1, X_j) \sim \log R$ as $R \rightarrow \infty$, but $\sum^n_{j=1} X_j/(nK(n))^{1/2}$ does not converge to $\mathscr{N}(0, 1)$ in distribution. This is a counterexample to a conjecture of Newman and Wright (1981).

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Norbert Herrndorf. "An Example on the Central Limit Theorem for Associated Sequences." Ann. Probab. 12 (3) 912 - 917, August, 1984. https://doi.org/10.1214/aop/1176993241

Information

Published: August, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0544.60033
MathSciNet: MR744247
Digital Object Identifier: 10.1214/aop/1176993241

Subjects:
Primary: 60F05

Keywords: central limit theorem , strictly stationary associated sequences

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 3 • August, 1984
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