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

Abstract

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).