Convergence of trimmed Lévy processes to trimmed stable random variables at [formula omitted

Abstract

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