Extremes of subexponential Lévy driven moving average processes

In this paper we study the extremal behavior of a stationary continuous-time moving average process Y(t)=∫−∞∞f(t−s)dL(s) for t∈Rt∈R, where ff is a deterministic function and LL is a Lévy process whose increments, represented by L(1)L(1), are subexponential and in the maximum domain of attraction of the Gumbel distribution. We give necessary and sufficient conditions for YY to be a stationary, infinitely divisible process, whose stationary distribution is subexponential, and in this case we calculate its tail behavior. We show that large jumps of the Lévy process in combination with extremes of ff cause excesses of YY and thus properly chosen discrete-time points are sufficient for specifying the extremal behavior of the continuous-time process YY. We describe the extremal behavior of YY completely as a weak limit of marked point processes. A complementary result guarantees the convergence of running maxima of YY to the Gumbel distribution.