Tightness and Convergence of Trimmed L\'evy Processes to Normality at Small Times

TL;DR

This paper investigates the conditions under which trimmed Lévy processes converge to normality at small times, establishing that tightness and convergence properties of trimmed processes imply similar properties for the original process.

Contribution

It provides new criteria linking the tightness and convergence of trimmed Lévy processes to those of the original process at small times.

Findings

Tightness of trimmed processes implies tightness of all jumps.

Convergence of trimmed processes to normality is equivalent to that of the original process.

Results apply to both normal and degenerate limit distributions.

Abstract

Let $^{(r,s)}X_t$ be the L\'evy process $X_t$ with the $r$ largest positive jumps and $s$ smallest negative jumps up till time $t$ deleted and let $^{(r)}\widetilde X_t$ be $X_t$ with the $r$ largest jumps in modulus up till time $t$ deleted. Let $a_t \in \mathbb{R}$ 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 $({}^{(r)}\widetilde X_t - a_t)/b_t$ at $0$ 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 $({}^{(r)}\widetilde X_t - a_t)/b_t$ converges to $N(0,1)$ or to a degenerate distribution if and only if $(X_t-a_t)/b_t $ converges to $N(0,1)$ or to the same degenerate distribution, as $t \downarrow 0$.