## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 1 (1985), 179-195.

### Limit Theory for Moving Averages of Random Variables with Regularly Varying Tail Probabilities

Richard Davis and Sidney Resnick

#### Abstract

Let $\{Z_k, -\infty < k < \infty\}$ be iid where the $Z_k$'s have regularly varying tail probabilities. Under mild conditions on a real sequence $\{c_j, j \geq 0\}$ the stationary process $\{X_n: = \sum^\infty_{j=0} c_jZ_{n-j}, n \geq 1\}$ exists. A point process based on $\{X_n\}$ converges weakly and from this, a host of weak limit results for functionals of $\{X_n\}$ ensue. We study sums, extremes, excedences and first passages as well as behavior of sample covariance functions.

#### Article information

**Source**

Ann. Probab., Volume 13, Number 1 (1985), 179-195.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993074

**Digital Object Identifier**

doi:10.1214/aop/1176993074

**Mathematical Reviews number (MathSciNet)**

MR770636

**Zentralblatt MATH identifier**

0562.60026

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60F17: Functional limit theorems; invariance principles 60G55: Point processes 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

**Keywords**

Extreme values stable laws regular variation moving average point processes

#### Citation

Davis, Richard; Resnick, Sidney. Limit Theory for Moving Averages of Random Variables with Regularly Varying Tail Probabilities. Ann. Probab. 13 (1985), no. 1, 179--195. doi:10.1214/aop/1176993074. https://projecteuclid.org/euclid.aop/1176993074