Extremes of Regularly Varying Lévy Driven Mixed Moving Average Processes

In this paper we study the extremal behavior of stationary mixed moving average processes Y (t) = ∫ R+×R f(r, t − s) dΛ(r, s) for 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 modelled by marked point processes at 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 on a high level. We obtain also convergence of partial maxima to the Fréchet distribution. Our models and results cover short and long range dependence regimes. AMS 2000 Subject Classifications: primary: 60G70 secondary: 60F05, 60G10, 60G55