On the small-time behavior of subordinators

Abstract
We prove several results on the behavior near t=0 of for certain -valued stochastic processes . In particular, we show for L\'{e}vy subordinators that the Pareto law on is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of as for a decreasing function that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.
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