On the small-time behavior of subordinators

We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\mathrm{\infty })$-valued stochastic processes $(Y_t{)}_{t>0}$. In particular, we show for L'{e}vy subordinators that the Pareto law on $[1,\mathrm{\infty })$ 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.