Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift. (English) Zbl 1403.60038
Summary: 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 \([0, \infty)\), 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.
MSC:
| 60G51 | Processes with independent increments; Lévy processes |
| 60K25 | Queueing theory (aspects of probability theory) |
Keywords:
compound Poisson process; \(M/G/1\) queue; heavy traffic; large deviations; uniform asymptotics; first passage time; supremumReferences:
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