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October, 1986 Central Limit Theorems for Mixing Sequences of Random Variables Under Minimal Conditions
Herold Dehling, Manfred Denker, Walter Philipp
Ann. Probab. 14(4): 1359-1370 (October, 1986). DOI: 10.1214/aop/1176992376

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 $E|S_n|$.

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Herold Dehling. Manfred Denker. Walter Philipp. "Central Limit Theorems for Mixing Sequences of Random Variables Under Minimal Conditions." Ann. Probab. 14 (4) 1359 - 1370, October, 1986. https://doi.org/10.1214/aop/1176992376

Information

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

zbMATH: 0605.60027
MathSciNet: MR866356
Digital Object Identifier: 10.1214/aop/1176992376

Subjects:
Primary: 60F05

Keywords: central limit theorem , normal distribution , Strong mixing

Rights: Copyright © 1986 Institute of Mathematical Statistics

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Vol.14 • No. 4 • October, 1986
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