Open Access
February, 1979 Approximation Thorems for Independent and Weakly Dependent Random Vectors
Istvan Berkes, Walter Philipp
Ann. Probab. 7(1): 29-54 (February, 1979). DOI: 10.1214/aop/1176995146


In this paper we prove approximation theorems of the following type. Let $\{X_k, k \geqslant 1\}$ be a sequence of random variables with values in $\mathbb{R}^{d_k}, d_k \geqslant 1$ and let $\{G_k, k \geqslant 1\}$ be a sequence of probability distributions on $\mathbb{R}^{d_k}$ with characteristic functions $g_k$ respectively. If for each $k \geqslant 1$ the conditional characteristic function of $X_k$ given $X_1, \cdots, X_{k - 1}$ is close to $g_k$ and if $G_k$ has small tails, then there exists a sequence of independent random variables $Y_k$ with distribution $G_k$ such that $|X_k - Y_k|$ is small with large probability. As an application we prove almost sure invariance principles for sums of independent identically distributed random variables with values in $\mathbb{R}^d$ and for sums of $\phi$-mixing random variables with a logarithmic mixing rate.


Download Citation

Istvan Berkes. Walter Philipp. "Approximation Thorems for Independent and Weakly Dependent Random Vectors." Ann. Probab. 7 (1) 29 - 54, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60024
MathSciNet: MR515811
Digital Object Identifier: 10.1214/aop/1176995146

Primary: 60F05
Secondary: 60B10

Keywords: almost sure invariance principles , Approximation of weakly dependent random variables by independent ones , independent random vectors , mixing random variables

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 1 • February, 1979
Back to Top