Cramér's estimate for the reflected process revisited

The reflected process of a random walk or Lévy process arises in many areas of applied probability, and a question of particular interest is how the tail of the distribution of the heights of the excursions away from zero behaves asymptotically. The Lévy analogue of this is the tail behaviour of the characteristic measure of the height of an excursion. Apparently, the only case where this is known is when Cramér's condition hold. Here, we establish the asymptotic behaviour for a large class of Lévy processes, which have exponential moments but do not satisfy Cramér's condition. Our proof also applies in the Cramér case, and corrects a proof of this given in Doney and Maller [Ann. Appl. Probab. 15 (2005) 1445-1450].