Advances in Applied Probability

Extremes of regularly varying Lévy-driven mixed moving average processes

Vicky Fasen

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In this paper, we study the extremal behavior of stationary mixed moving average processes of the form Y(t)=∫+×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.

Article information

Adv. in Appl. Probab. Volume 37, Number 4 (2005), 993-1014.

First available in Project Euclid: 14 December 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G55: Point processes

Continuous-time moving average process extreme-value theory independently scattered random measure long-range dependence marked point process point process regular variation shot noise process supOU process


Fasen, Vicky. Extremes of regularly varying Lévy-driven mixed moving average processes. Adv. in Appl. Probab. 37 (2005), no. 4, 993--1014. doi:10.1239/aap/1134587750.

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