Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times

Abstract

For nonnegative integers r, s, let $^{(r,s)}X_t$ be the Lévy process $X_t$ with the r largest positive jumps and the s smallest negative jumps up till time t deleted, and let $^{(r)}\widetilde{X}_t$ be $X_t$ with the r largest jumps in modulus up till time t deleted. Let $a_t \in \mathbb {R}$ and $b_t>0$ be non-stochastic functions in t. We show that the tightness of $({}^{(r,s)}X_t - a_t)/b_t$ or $({}^{(r)}{\widetilde{X}}_t - a_t)/b_t$ as $t\downarrow 0$ implies the tightness of all normed ordered jumps, and hence the tightness of the untrimmed process $(X_t -a_t)/b_t$ at 0. We use this to deduce that the trimmed process $({}^{(r,s)}X_t - a_t)/b_t$ or $({}^{(r)}{\widetilde{X}}_t - a_t)/b_t$ converges to N(0, 1) or to a degenerate distribution as $t\downarrow 0$ if and only if $(X_t-a_t)/b_t$ converges to N(0, 1) or to the same degenerate distribution, as $t \downarrow 0$.