## The Annals of Probability

### Strong approximation theorems for geometrically weighted random series and their applications

Li-Xin Zhang

#### Abstract

Let ${X_n;n\geq 0}$ be a sequence of random variables. We consider its geometrically weighted series $\xi(\beta)=\sum_{n=0}^\infty \betaX_n$ for $0<\beta < 1$. This paper proves that $\xi (\beta)$ can be approximated by $\sum_{n=0}^\infty \beta^n Y_n$ under some suitable conditions, where ${Y_n; n \geq 0}$ is a sequence of independent normal random variables. Applications to the law of the iterated logarithm for $\xi(\beta)$ are also discussed.

#### Article information

Source
Ann. Probab., Volume 25, Number 4 (1997), 1621-1635.

Dates
First available in Project Euclid: 7 June 2002

https://projecteuclid.org/euclid.aop/1023481105

Digital Object Identifier
doi:10.1214/aop/1023481105

Mathematical Reviews number (MathSciNet)
MR1487430

Zentralblatt MATH identifier
0903.60017

#### Citation

Zhang, Li-Xin. Strong approximation theorems for geometrically weighted random series and their applications. Ann. Probab. 25 (1997), no. 4, 1621--1635. doi:10.1214/aop/1023481105. https://projecteuclid.org/euclid.aop/1023481105

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