Functionals of infinitely divisible stochastic processes with exponential tails

Abstract

We investigate the tail behavior of the distributions of subadditive functionals of the sample paths of infinitely divisible stochastic processes when the Lévy measure of the process has suitably defined exponentially decreasing tails. It is shown that the probability tails of such functionals are of the same order of magnitude as the tails of the same functionals with respect to the Lévy measure, and it turns out that the results of this kind cannot, in general, be improved. In certain situations we can further obtain both lower and upper bounds on the asymptotic ratio of the two tails. In the second part of the paper we consider the particular case of Lévy processes with exponentially decaying Lévy measures. Here we show that the tail of the maximum of the process is, up to a multiplicative constant, asymptotic to the tail of the Lévy measure. Most of the previously published work in the area considered heavier than exponential probability tails.