On the small-time behavior of subordinators

Abstract

We prove several results on the behavior near t = 0 of Y-t(-1) for certain (0, infinity)-valued stochastic processes (Y-t)(t>0). In particular, we show for Levy subordinators that the Pareto law on [1, infinity) 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 -> 0 for a decreasing function L that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.