On the small-time behavior of subordinators

Abstract

We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty )$-valued stochastic processes $(Y_t{)}_{t>0}$. In particular, we show for Lévy subordinators that the Pareto law on $[1,\infty )$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to 0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.

Published in at http://dx.doi.org/10.3150/11-BEJ363 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (1)

• Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples