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

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 $({}^{(r,s)}{X}_t-a_t)\mathrm{/}{b}_t$ or $({}^{(r)}{\tilde{X}}_t-a_t)\mathrm{/}{b}_t$ converges to a proper nondegenerate nonnormal limit distribution as $t\downarrow 0$ if and only if $(X_t-a_t)\mathrm{/}{b}_t$ 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 $0$.