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

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 )}X~ _t be X_t with the r largest jumps in modulus up till time t deleted. Let a_t∈ 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 )}X~ _t- a_t) / b_t as t↓ 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 )}X~ _t- a_t) / b_t converges to N(0, 1) or to a degenerate distribution as t↓ 0 if and only if (X_t- a_t) / b_t converges to N(0, 1) or to the same degenerate distribution, as t↓ 0.