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

Abstract

For non-negative 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, 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.

Convergence of trimmed Lévy processes to trimmed stable random variables at 0

“…In the special case when the trimmed processes, with appropriate centering and norming, converge to a normal or degenerate distribution, it is shown in [7] that the original Lévy process, after being centered and normed with the same functions, will converge to the same normal or degenerate law as t ↓ 0. This implies that light trimming, i.e.…”

“…In the case of asymptotic degeneracy or normality, by Fan [7], the limit distribution of the trimmed process in (1.4) or (1.5) is the same. In general the limit distribution in (1.3) is different from that of (1.4) or (1.5).…”

“…It has been proved in [7] that if (1.4) or (1.5) holds with limit a normal or degenerate limit, S t also converges to the same law. This is derived in [7] by first showing that the tightness of (r,s) S t or (r )  S t implies the tightness of S t (see Theorem 1.1 in [7]). By eliminating the degenerate distribution, (1.4) or (1.5) implies that the untrimmed process X t is in the Feller class (refer to Maller and Mason [15] for more details on properties of Feller class) at 0, i.e.…”

“…Therefore, by adding a finite number of tight families, we can easily derive that S t is tight at 0. We can write X (r,±) t D = Π ±,← (Γ r /t) for each r ∈ N, where Γ r is distributed as a gamma random variable with parameter (r, 1) (see [2] or [7]) and Π ±,← denote the inverse functions. When f : (0, ∞)  → [0, ∞) is a nonincreasing function, its right-continuous inverse is…”

“…Now we need to introduce the distributional representations from [2,7]. By Theorem 2.1 in [2] and Section 2 in [7], an r, s-trimmed process has the following representation.…”