2017/08/08 by R. A. Doney, Doney, R. A., Philip S. Griffin +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F10 #60G51 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1708.02470
openalex publication_date 2017/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
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 [5].