## The Annals of Probability

### Almost Sure Invariance Principles for Weakly Dependent Vector-Valued Random Variables

#### Abstract

We obtain the almost sure approximation of the partial sums of random variables with values in a separable Hilbert space and satisfying a strong mixing condition by a suitable Brownian motion. This is achieved by a modification of the proof of a similar result by Kuelbs and Philipp (1980) on $\phi$-mixing Banach space valued random variables. As by-products we get almost sure invariance principles for sums of absolutely regular sequences of random variables with values in a Banach space and necessary and sufficient conditions for the almost sure invariance principle for sums of independent, identically distributed random variables.

#### Article information

Source
Ann. Probab., Volume 10, Number 3 (1982), 689-701.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993777

Digital Object Identifier
doi:10.1214/aop/1176993777

Mathematical Reviews number (MathSciNet)
MR659538

Zentralblatt MATH identifier
0487.60006

JSTOR
links.jstor.org

#### Citation

Dehling, Herold; Philipp, Walter. Almost Sure Invariance Principles for Weakly Dependent Vector-Valued Random Variables. Ann. Probab. 10 (1982), no. 3, 689--701. doi:10.1214/aop/1176993777. https://projecteuclid.org/euclid.aop/1176993777