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

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 be a sequence of independent and identically distributed random variables and be a Lévy process such that , and as a ↓0. Let . Then, under some mi...

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Bibliographic Details
Published in:Queueing systems 2011-04, Vol.67 (4), p.295-304
Main Authors: Kosiński, K. M., Boxma, O. J., Zwart, B.
Format: Article
Language:English
Subjects:
Approximation
Business and Management
Computer Communication Networks
Computer Science
Control
Convergence
Mathematics
Networking and Internet Architecture
Operations Research/Decision Theory
Probability Theory and Stochastic Processes
Queues
Queuing
Random variables
Random walk
Studies
Supply Chain Management
Systems Theory
Theorems
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Summary: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 be a sequence of independent and identically distributed random variables and be a Lévy process such that , and as a ↓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.
ISSN:0257-0130
1572-9443
DOI:10.1007/s11134-011-9215-4