## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 5 (1978), 847-869.

### Extremes of Moving Averages of Stable Processes

#### Abstract

In this paper we study extremes of non-normal stable moving average processes, i.e., of stochastic processes of the form $X(t) = \Sigma a(\lambda - t)Z(\lambda)$ or $X(t) = \int a(\lambda - t) dZ(\lambda)$, where $Z(\lambda)$ is stable with index $\alpha < 2$. The extremes are described as a marked point process, consisting of the point process of (separated) exceedances of a level together with marks associated with the points, a mark being the normalized sample path of $X(t)$ around an exceedance. It is proved that this marked point process converges in distribution as the level increases to infinity. The limiting distribution is that of a Poisson process with independent marks which have random heights but otherwise are deterministic. As a byproduct of the proof for the continuous-time case, a result on sample path continuity of stable processes is obtained.

#### Article information

**Source**

Ann. Probab., Volume 6, Number 5 (1978), 847-869.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995432

**Digital Object Identifier**

doi:10.1214/aop/1176995432

**Mathematical Reviews number (MathSciNet)**

MR494450

**Zentralblatt MATH identifier**

0394.60025

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G17: Sample path properties

**Keywords**

Extreme values stable processes moving average sample path continuity

#### Citation

Rootzen, Holger. Extremes of Moving Averages of Stable Processes. Ann. Probab. 6 (1978), no. 5, 847--869. doi:10.1214/aop/1176995432. https://projecteuclid.org/euclid.aop/1176995432