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On the small-time behavior of subordinators

Abstract

We prove several results on the behavior near t=0 of YttY_t^{-t} for certain (0,)(0,\infty)-valued stochastic processes (Yt)t>0(Y_t)_{t>0}. In particular, we show for L\'{e}vy subordinators that the Pareto law on [1,)[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(Yt)tL(Y_t) as t0t\to0 for a decreasing function LL that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.

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