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 ) d L ( s ) for t ∈ R , where f is a deterministic function and L is a Lévy process whose increments, represented by L ( 1 ) , are subexponential and in the maximum domain of attraction of the Gumbel distribution. We give necessary and sufficient conditions for Y 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 f cause excesses of Y and thus properly chosen discrete-time points are sufficient for specifying the extremal behavior of the continuous-time process Y . We describe the extremal behavior of Y completely as a weak limit of marked point processes. A complementary result guarantees the convergence of running maxima of Y to the Gumbel distribution.