## The Annals of Probability

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

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

Herold Dehling and Walter Philipp

#### 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

**Subjects**

Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

**Keywords**

Almost sure invariances principles mixing and absolutely regular sequences of random variables Hilbert space Banach space Brownian motion

#### 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