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

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 $0$.