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

For nonnegative integers r, s, let $$^{(r,s)}X_t$$ ( r , s ) X t be the Lévy process $$X_t$$ X t with the r largest positive jumps and the s smallest negative jumps up till time t deleted, and let $$^{(r)}\widetilde{X}_t$$ ( r ) X ~ t be $$X_t$$ X t with the r largest jumps in modulus up till time t deleted. Let $$a_t \in \mathbb {R}$$ a t ∈ R and $$b_t>0$$ b t > 0 be non-stochastic functions in t. We show that the tightness of $$({}^{(r,s)}X_t - a_t)/b_t$$ ( ( r , s ) X t - a t ) / b t or $$({}^{(r)}{\widetilde{X}}_t - a_t)/b_t$$ ( ( r ) X ~ t - a t ) / b t as $$t\downarrow 0$$ t ↓ 0 implies the tightness of all normed ordered jumps, and hence the tightness of the untrimmed process $$(X_t -a_t)/b_t$$ ( X t - a t ) / b t at 0. We use this to deduce that the trimmed process $$({}^{(r,s)}X_t - a_t)/b_t$$ ( ( r , s ) X t - a t ) / b t or $$({}^{(r)}{\widetilde{X}}_t - a_t)/b_t$$ ( ( r ) X ~ t - a t ) / b t converges to N(0, 1) or to a degenerate distribution as $$t\downarrow 0$$ t ↓ 0 if and only if $$(X_t-a_t)/b_t $$ ( X t - a t ) / b t converges to N(0, 1) or to the same degenerate distribution, as $$t \downarrow 0$$ t ↓ 0 .