Convergence of trimmed Lévy processes to trimmed stable random variables at 00

Let (r,s)Xt(r,s)Xt be the Lévy process XtXt with rr largest jumps and ss smallest jumps up till time tt deleted and let (r)X˜t be XtXt with rr largest jumps in modulus up till time tt deleted. We show that ((r,s)Xt−at)/bt((r,s)Xt−at)/bt or ((r)X˜t−at)/bt converges to a proper nondegenerate nonnormal limit distribution as t↓0t↓0 if and only if (Xt−at)/bt(Xt−at)/bt converges as t↓0t↓0 to an αα-stable random variable, with 0<α<20<α<2, where atat and bt>0bt>0 are nonstochastic functions in tt. Together with the asymptotic normality case treated in Fan (2015) [7], this completes the domain of attraction problem for trimmed Lévy processes at 00.