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October, 1978 Extremes of Moving Averages of Stable Processes
Holger Rootzen
Ann. Probab. 6(5): 847-869 (October, 1978). DOI: 10.1214/aop/1176995432


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.


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Holger Rootzen. "Extremes of Moving Averages of Stable Processes." Ann. Probab. 6 (5) 847 - 869, October, 1978.


Published: October, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0394.60025
MathSciNet: MR494450
Digital Object Identifier: 10.1214/aop/1176995432

Primary: 60F05
Secondary: 60G17

Keywords: Extreme values , moving average , sample path continuity , Stable processes

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 5 • October, 1978
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