## The Annals of Probability

- Ann. Probab.
- Volume 7, Number 1 (1979), 29-54.

### Approximation Thorems for Independent and Weakly Dependent Random Vectors

Istvan Berkes and Walter Philipp

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 7, Number 1 (1979), 29-54.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995146

**Digital Object Identifier**

doi:10.1214/aop/1176995146

**Mathematical Reviews number (MathSciNet)**

MR515811

**Zentralblatt MATH identifier**

0392.60024

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60B10: Convergence of probability measures

**Keywords**

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

#### Citation

Berkes, Istvan; Philipp, Walter. Approximation Thorems for Independent and Weakly Dependent Random Vectors. Ann. Probab. 7 (1979), no. 1, 29--54. doi:10.1214/aop/1176995146. https://projecteuclid.org/euclid.aop/1176995146