Extremes of Lévy Driven Mixed MA Processes with Convolution Equivalent Distributions

We investigate the extremal behavior of stationary mixed MA processes Y (t) = ∫ R+×R f(r, t− s) dΛ(r, s) for t ≥ 0, where f is a deterministic function with f(r, 0) = f+ and |f(r, s)| < f+ for s 6= 0 and r ∈ R+. The random measure Λ is infinitely divisible and independently scattered, whose finite dimensional distributions, represented by L(1) = Λ(R+ × [0, 1]), are in the class of convolution equivalent distributions and in the maximum domain of attraction of the Gumbel distribution. It is shown that the tail of the stationary distribution of Y decreases faster to 0 than the tail of f+L(1). In contrast to this the tail of the maximum of Y over a fixed time interval decreases of the same order of magnitude as the tail of f+L(1) and is linearly in the length of the interval. We divide the positive real line into properly chosen randomly intervals and denote the maxima of the process in these intervals by (Mk)k∈N. The extremal behavior of Y is completely described by a weak limit of marked point processes based on (Mk)k∈N. A complementary result guarantees the convergence of running maxima of Y to the Gumbel distribution. AMS 2000 Subject Classifications: primary: 60G70 secondary: 60F05, 60G10, 60G55