## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 6 (1980), 1003-1036.

### Almost Sure Invariance Principles for Partial Sums of Mixing $B$-Valued Random Variables

#### Abstract

The approximation of partial sums of $\phi$-mixing random variables with values in a Banach space $B$ by a $B$-valued Brownian motion is obtained. This result yields the compact as well as the functional law of the iterated logarithm for these sums. As an application we strengthen a uniform law of the iterated logarithm for classes of functions recently obtained by Kaufman and Philipp (1978). As byproducts we obtain necessary and sufficient conditions for an almost sure invariance principle for independent identically distributed $B$-valued random variables and an almost sure invariance principle for sums of $d$-dimensional random vectors satisfying a strong mixing condition.

#### Article information

**Source**

Ann. Probab., Volume 8, Number 6 (1980), 1003-1036.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994565

**Digital Object Identifier**

doi:10.1214/aop/1176994565

**Mathematical Reviews number (MathSciNet)**

MR602377

**Zentralblatt MATH identifier**

0451.60008

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G17: Sample path properties 60B05: Probability measures on topological spaces 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60B10: Convergence of probability measures

**Keywords**

Brownian motion in a Banach space mixing random variables almost sure invariance principles law of the iterated logarithm approximation of partial sums of Banach space valued variables

#### Citation

Kuelbs, J.; Philipp, Walter. Almost Sure Invariance Principles for Partial Sums of Mixing $B$-Valued Random Variables. Ann. Probab. 8 (1980), no. 6, 1003--1036. doi:10.1214/aop/1176994565. https://projecteuclid.org/euclid.aop/1176994565