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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 ofdoi:10.1214/18-aap1399 fatcat:yeea2f25qzcrzcvv5m74u6h6wy