Mathematics > Probability
[Submitted on 19 Oct 2014 (v1), last revised 19 Nov 2015 (this version, v3)]
Title:Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times
View PDFAbstract:Let $^{(r,s)}X_t$ be the Lévy 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 $t \downarrow 0$.
Submission history
From: Yuguang Fan [view email][v1] Sun, 19 Oct 2014 05:12:42 UTC (22 KB)
[v2] Thu, 8 Jan 2015 05:51:24 UTC (23 KB)
[v3] Thu, 19 Nov 2015 22:31:00 UTC (27 KB)
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