## The Annals of Probability

### Extremes of Moving Averages of Stable Processes

Holger Rootzen

#### 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