On the small-time behavior of subordinators

TL;DR

This paper investigates the small-time behavior of subordinators, establishing conditions for weak convergence of transformed processes and identifying Pareto law as the unique limit for certain Lévy subordinators.

Contribution

It provides new results on the asymptotic behavior of subordinators near zero, including necessary and sufficient conditions for convergence and the characterization of the Pareto law as the limit.

Findings

Pareto law is the only weak limit for certain subordinators

Conditions for convergence of $tL(Y_t)$ as $t o 0$

Examples illustrating applicability of the theoretical results

Abstract

We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show for L\'{e}vy subordinators that the Pareto law on $[1,\infty)$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.