Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

K. M. Kosiński (1, 2) , O. J. Boxma (2, 3) , B. Zwart (4)

1 Korteweg-de Vries Institute for Mathematics
2 EURANDOM - European Institute for Statistics, Probability, Stochastic Operations Research and its Applications
3 Department of mathematics and computing science [Eindhoven]
4 CWI

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each >0, let be a sequence of independent and identically distributed random variables and be a Lévy process such that , and as ↓0. Let . Then, under some mild assumptions, , for some random variable and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.

Mots clés: Lévy processes; Heavy traffic; Functional limit theorems; Mittag-Leffler distribution; 60G50; 60G51; 60K25; 60F17

Domaines: Réseaux et télécommunications [cs.NI]

K. M. Kosiński, O. J. Boxma, B. Zwart. Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime. Queueing Systems, 2011, pp.295-304. ⟨10.1007/s11134-011-9215-4⟩. ⟨hal-00798710⟩