Heavy tails of a Lévy process and its maximum over a random time interval.

Let { X, t ⩾ 0} be a Lévy process with Lévy measure ν on (−∞,∞), and let τ be a nonnegative random variable independent of { X, t ⩾ 0}. We are interested in the tail probabilities of X and X = sup X. For various cases, under the assumption that either the Lévy measure ν or the random variable τ has a heavy right tail we prove that both Pr( X > x) and Pr( X > x) are asymptotic to E τν(( x, ∞))+Pr( τ > x/(0 ∨ E X)) as x → ∞, where Pr( τ > x/0) = 0 by convention. [ABSTRACT FROM AUTHOR]