Small Time Convergence of Subordinators with Regularly or Slowly Varying Canonical Measure

TL;DR

This paper studies the small-time behavior of subordinators with specific tail properties, establishing convergence results to stable and extremal processes, and extends classical results to new boundary cases and trimmed versions.

Contribution

It introduces new convergence results for subordinators with regularly and slowly varying measures, including boundary cases and trimmed processes, generalizing classical stable process findings.

Findings

Convergence of scaled tail measures to stable distributions as time approaches zero.

Identification of the limit distribution as a function of the stability index.

Extension of results to extremal and trimmed jump processes.

Abstract

We consider subordinators $X_\alpha=(X_\alpha(t))_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator $(S_\alpha(t))_{t\ge 0}$ (where $\alpha\in(0,1)$); thus, with the property that $\overline{\Pi}_\alpha$, the tail function of the canonical measure of $X_\alpha$, is regularly varying of index $-\alpha\in (-1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha=0$, when $\overline{\Pi}_\alpha$ is slowly varying at 0. When $\alpha\in(0,1)$, we show that $(t \overline{\Pi}_\alpha (X_\alpha(t)))^{-1}$ converges in distribution, as $t\downarrow 0$, to the random variable $(S_\alpha(1))^\alpha$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha\downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from a process. The $\alpha=0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.