L¹-norm of Lévy processes with exponential tails

17 Remark 2 We cannot characterize so far all L evy processes for which Theorem 3.2 holds. However, the present argument given for that theorem easily shows that its statement holds when, say, + is equivalent to the tail of a distribution in S(), and 0 is of a smaller order. Then, in particular, lim x!1 P R T 0 jX(t)jdt > x A + () T ~ + (x=T) = T : (3:30) The case when 0 is equivalent to the tail of a distribution in S(), and + is of a smaller order is, of course, similar. Acknowledgment Remarks of the anonymous referee helped the authors to improve the presentation of Proposition 2.1 and to simplify its proof.