## The Annals of Probability

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

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

Herold Dehling, Manfred Denker, and Walter Philipp

#### 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