The Annals of Probability

Stationary Strongly Mixing Sequences Not Satisfying the Central Limit Theorem

Norbert Herrndorf

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Abstract

For every sequence $(\varepsilon_n)_{n \in N}$ in (0, 1) there exists a strictly stationary orthonormal sequence $(X_n)_{n \in N}$ of random variables with $|P(A \cap B) - P(A)P(B)| \leq \varepsilon_n$ for all $A \in \sigma(X_1, \cdots, X_k), B \in \sigma(X_{k+n}, X_{k+n+1}, \cdots), k \in \mathbb{N}, n \in \mathbb{N}$, such that the distribution of $n^{-1/2} \sum^n_{i=1} X_i$ is not weakly convergent to the standard normal distribution.

Article information

Source
Ann. Probab., Volume 11, Number 3 (1983), 809-813.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993529

Digital Object Identifier
doi:10.1214/aop/1176993529

Mathematical Reviews number (MathSciNet)
MR704571

Zentralblatt MATH identifier
0513.60033

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes

Keywords
Central limit theorem strongly mixing strictly stationary sequences

Citation

Herrndorf, Norbert. Stationary Strongly Mixing Sequences Not Satisfying the Central Limit Theorem. Ann. Probab. 11 (1983), no. 3, 809--813. doi:10.1214/aop/1176993529. https://projecteuclid.org/euclid.aop/1176993529


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