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On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Shai Covo
Shai Covo*
Affiliation:
Bar Ilan University
*
∗Postal address: Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, Israel. Email address: green355@netvision.net.il
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Abstract

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Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let Xε be the sum of jumps not exceeding ε, and let µ(ε)=E[Xε(1)]. We study the question of weak convergence of Xε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of Xε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

Keywords

SubordinatorLévy processsmall jumpsweak convergenceDickman distributiongamma process

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 732 - 755
Copyright
Copyright © Applied Probability Trust 2009 

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