Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift

This paper addresses heavy-tailed asymptotics of functionals of a class of spectrally one-sided L\'evy process that 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 $[0,\infty)$, the maximum jump until that time, and the time it takes until exiting $[0,\infty)$. The proofs rely, among other things, on properties of scale functions. Keywords: compound Poisson process, M/G/1 queue, heavy traffic, large deviations, uniform asymptotics, first passage time, supremum