This paper addresses heavy-tailed large-deviation estimates for the distribution tail of functionals of a class of spectrally one-sided Lévy processes. Our contribution is to show that these estimates remain valid in a near-critical regime. This complements recent similar results that have been obtained for the all-time supremum of such processes. Specifically, we consider local asymptotics of the all-time supremum, the supremum of the process until exiting [0,∞), the maximum jump until that time, and the time it takes until exiting [0,∞). The proofs rely, among other things, on properties of scale functions.

Compound Poisson process, First passage time, Heavy traffic, Large deviations, M/G/1 queue, Supremum, Uniform asymptotics
Stochastic Processes and their Applications
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Kamphorst, B, & Zwart, A.P. (2018). Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift. Stochastic Processes and their Applications. doi:10.1016/