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

K.M. Kosinski, O.J. Boxma, B. Zwart

Research output: Contribution to journalArticleAcademicpeer-review

4 Citations (Scopus)
2 Downloads (Pure)

Abstract

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let {Y(a)n:n ³ 1}Yn(a):n1 be a sequence of independent and identically distributed random variables and {X(a)t:t ³ 0}Xt(a):t0 be a Lévy process such that X1(a)=dY1(a)Unknown control sequence '\stackrel', \mathbbEX1(a) <0EX1(a)0 and \mathbbEX1(a)­0EX1(a)0 as a¿0. Let S(a)n=åk=1n Y(a)kSn(a)=nk=1Yk(a). Then, under some mild assumptions, , for some random variable and some function ¿(·). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.
Original languageEnglish
Pages (from-to)295-304
JournalQueueing Systems: Theory and Applications
Volume67
Issue number4
DOIs
Publication statusPublished - 2011

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