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The reflected process of a random walk or L\'evy process arises in many areasof applied probability, and a question of particular interest is how the tailof the distribution of the heights of the excursions away from zero behavesasymptotically. The L\'evy analogue of this is the tail behaviour of thecharacteristic measure of the height of an excursion. Apparently the only casewhere this is known is when Cram\'er's condition hold. Here we establish theasymptotic behaviour for a large class of L\'evy processes which haveexponential moments but do not satisfy Cram\'er's condition. Our proof alsoapplies in the Cram\'er case, and corrects a proof of this given in Doney andMaller [5].

Cram\'er's Estimate for the Reflected Process Revisited