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 .