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).
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