On the small-time behavior of subordinators

Abstract

We prove several results on the behavior near $t = 0$ of $Y_t^{−t}$ for certain $(0, ∞)$-valued stochastic processes ($Y_t)_{t>0}$. In particular, we show for Lévy subordinators that the Pareto law on [1, ∞) 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.