The Annals of Probability

Stationary Strongly Mixing Sequences Not Satisfying the Central Limit Theorem

Norbert Herrndorf

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

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