Convergence of Trimmed L\'evy Processes to Trimmed Stable Random Variables at $0$

TL;DR

This paper investigates the convergence behavior of trimmed Lévy processes at zero, establishing conditions under which they converge to stable or normal distributions, thereby completing the domain of attraction analysis for such processes.

Contribution

It characterizes the convergence of trimmed Lévy processes to stable or normal limits at zero, extending the understanding of their asymptotic behavior.

Findings

Convergence to stable laws occurs if the original process converges to an α-stable distribution.

Trimmed processes with large jumps removed also converge to the same stable law.

The work completes the domain of attraction results for trimmed Lévy processes at zero.

Abstract

Let $^{(r,s)}X_t$ be the L\'evy process $X_t$ with the $r$ largest jumps and $s$ smallest jumps up till time $t$ deleted and let $^{(r)}\tilde X_t$ be $X_t$ with the $r$ largest jumps in modulus up till time $t$ deleted. We show that $({}^{(r,s)}X_t - a_t)/b_t$ or $({}^{(r)}\tilde X_t - a_t)/b_t$ converges to a proper nondegenerate nonnormal limit distribution as $t \downarrow 0$ if and only if $(X_t-a_t)/b_t $ converges as $t \downarrow 0$ to an $\alpha$-stable random variable, with $ 0 <\alpha<2 $, where $a_t$ and $b_t>0$ are non stochastic functions in $t$. Together with the asymptotic normality case treated in \cite{fan2014an}, this completes the domain of attraction problem for trimmed L\'evy processes at $0$.