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Extremes of regularly varying Lévy-driven mixed moving average processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Vicky Fasen
Vicky Fasen*
Affiliation:
Munich University of Technology
*
∗ Postal address: Graduate Program in Applied Algorithmic Mathematics, Centre for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany. Email address: [email protected]
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Abstract

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In this paper, we study the extremal behavior of stationary mixed moving average processes of the form Y(t)=∫ℝ+×ℝf(r,t-s) dΛ(r,s), t∈ℝ, where f is a deterministic function and Λ is an infinitely divisible, independently scattered random measure whose underlying driving Lévy process is regularly varying. We give sufficient conditions for the stationarity of Y and compute the tail behavior of certain functionals of Y. The extremal behavior is modeled by marked point processes on a discrete-time skeleton chosen properly by the jump times of the underlying driving Lévy process and the extremes of the kernel function. The sequences of marked point processes converge weakly to a cluster Poisson random measure and reflect extremes of Y at a high level. We also show convergence of the partial maxima to the Fréchet distribution. Our models and results cover short- and long-range dependence regimes.

Keywords

Continuous-time moving average processextreme-value theoryindependently scattered random measurelong-range dependencemarked point processpoint processregular variationshot noise processsupOU process

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 993 - 1014
Copyright
Copyright © Applied Probability Trust 2005 

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