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

TL;DR

This paper investigates the tail behavior of the heights of excursions in reflected Lévy processes, extending known results beyond Cramér's condition and correcting previous proofs in the field.

Contribution

It establishes asymptotic tail behavior for a broad class of Lévy processes with exponential moments, including cases not satisfying Cramér's condition.

Findings

Derived asymptotic tail behavior for reflected Lévy processes with exponential moments.

Extended results to processes not satisfying Cramér's condition.

Provided a corrected proof for the Cramér case from previous literature.

Abstract

The reflected process of a random walk or L\'evy 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 L\'evy 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 Cram\'er's condition hold. Here we establish the asymptotic behaviour for a large class of L\'evy processes which have exponential moments but do not satisfy Cram\'er's condition. Our proof also applies in the Cram\'er case, and corrects a proof of this given in Doney and Maller [5].