Tightness and Convergence of Trimmed L\'evy Processes to Normality at Small Times

Let ${}^{(r,s)}{X}_t$ be the L'evy process $X_t$ with the $r$ largest positive jumps and $s$ smallest negative 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. Let $a_t\in \mathbb{R}$ and $b_t>0$ be non-stochastic functions in $t$. We show that the tightness of $({}^{(r,s)}{X}_t-a_t)\mathrm{/}{b}_t$ or $({}^{(r)}{\tilde{X}}_t-a_t)\mathrm{/}{b}_t$ at $0$ implies the tightness of all normed ordered jumps, hence the tightness of the untrimmed process $(X_t-a_t)\mathrm{/}{b}_t$ at $0$. We use this to deduce that the trimmed process $({}^{(r,s)}{X}_t-a_t)\mathrm{/}{b}_t$ or $({}^{(r)}{\tilde{X}}_t-a_t)\mathrm{/}{b}_t$ converges to $N(0,1)$ or to a degenerate distribution if and only if $(X_t-a_t)\mathrm{/}{b}_t$ converges to $N(0,1)$ or to the same degenerate distribution, as $t\downarrow 0$.