## The Annals of Probability

- Ann. Probab.
- Volume 10, Number 4 (1982), 968-981.

### Invariance Principles for Mixing Sequences of Random Variables

#### Abstract

In this note we prove weak invariance principles for some classes of mixing sequences of $L_2$-integrable random variables under the condition that the variance of the sum of $n$ random variables is asymptotic to $\sigma^2n$ where $\sigma^2 > 0$. One of the results is simultaneously an extension to nonstationary case of a theorem of Ibragimov and an improvement of the $\varphi$-mixing rate used by McLeish in his invariance principle for nonstationary $\varphi$-mixing sequences.

#### Article information

**Source**

Ann. Probab. Volume 10, Number 4 (1982), 968-981.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176993718

**Digital Object Identifier**

doi:10.1214/aop/1176993718

**Mathematical Reviews number (MathSciNet)**

MR672297

**Zentralblatt MATH identifier**

0503.60044

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60B10: Convergence of probability measures

**Keywords**

Invariance principles mixing sequences of random variables

#### Citation

Peligrad, Magda. Invariance Principles for Mixing Sequences of Random Variables. Ann. Probab. 10 (1982), no. 4, 968--981. doi:10.1214/aop/1176993718. http://projecteuclid.org/euclid.aop/1176993718.