TY - JOUR

T1 - Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

AU - Kosinski, K.M.

AU - Boxma, O.J.

AU - Zwart, B.

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

U2 - 10.1007/s11134-011-9215-4

DO - 10.1007/s11134-011-9215-4

M3 - Article

VL - 67

SP - 295

EP - 304

JO - Queueing Systems: Theory and Applications

JF - Queueing Systems: Theory and Applications

SN - 0257-0130

IS - 4

ER -