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This paper addresses heavy-tailed large deviation estimates for thedistribution tail of functionals of a class of spectrally one-sided L\'evyprocess. Our contribution is to show that these estimates remain valid in anear-critical regime. This complements recent similar results that have beenobtained for the all-time supremum of such processes. Specifically, we considerlocal asymptotics of the all-time supremum, the supremum of the process untilexiting $[0,\infty)$, the maximum jump until that time, and the time it takesuntil exiting $[0,\infty)$. The proofs rely, among other things, on propertiesof scale functions.

Uniform Asymptotics for Compound Poisson Processes with Regularly Varying Jumps and Vanishing Drift