The Annals of Probability

Central Limit Theorems for Mixing Sequences of Random Variables Under Minimal Conditions

Herold Dehling, Manfred Denker, and Walter Philipp

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Abstract

Let $\{X_j, j \geq 1\}$ be a strictly stationary sequence of random variables with mean zero, finite variance, and satisfying a strong mixing condition. Denote by $S_n$ the $n$th partial sum and suppose that $\operatorname{Var} S_n$ is regularly varying of order 1. We prove that if $S_n (\operatorname{Var} S_n)^{-1/2}$ does not converge to zero in $L^1$, then $\{X_j, j \geq 1\}$ is in the domain of partial attraction of a Gaussian law. If, however, no subsequence of $\{S_n(\operatorname{Var} S_n)^{-1/2}, n \geq 1\}$ converges to zero in $L^1$ and if $E|S_n|$ is regularly varying of order $\frac{1}{2}$, then $\{X_j, j \geq 1\}$ is in the domain of attraction to a Gaussian law. In each case the norming constant can be chosen as $E|S_n|$.

Article information

Source
Ann. Probab., Volume 14, Number 4 (1986), 1359-1370.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992376

Digital Object Identifier
doi:10.1214/aop/1176992376

Mathematical Reviews number (MathSciNet)
MR866356

Zentralblatt MATH identifier
0605.60027

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems

Keywords
Central limit theorem normal distribution strong mixing

Citation

Dehling, Herold; Denker, Manfred; Philipp, Walter. Central Limit Theorems for Mixing Sequences of Random Variables Under Minimal Conditions. Ann. Probab. 14 (1986), no. 4, 1359--1370. doi:10.1214/aop/1176992376. https://projecteuclid.org/euclid.aop/1176992376


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