Journal of Theoretical Probability

, Volume 30, Issue 2, pp 675–699 | Cite as

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

Article

Abstract

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 \(^{(r)}\widetilde{X}_t\) 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 \(({}^{(r)}{\widetilde{X}}_t - a_t)/b_t\) as \(t\downarrow 0\) 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 \(({}^{(r)}{\widetilde{X}}_t - a_t)/b_t\) converges to N(0, 1) or to a degenerate distribution as \(t\downarrow 0\) if and only if \((X_t-a_t)/b_t \) converges to N(0, 1) or to the same degenerate distribution, as \(t \downarrow 0\).

Keywords

Trimmed Lévy processes Domain of normal attraction  Small time convergence Tightness Extreme jumps of Lévy processes 

Mathematics Subject Classification (2010)

60G05 60G07 60G51 

Notes

Acknowledgments

The author is very grateful to Prof. Ross Maller and Dr. Boris Buchmann for many helpful discussions and for critically reading the manuscript.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.ARC Center of Excellence for Mathematical and Statistical Frontiers, School of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
  2. 2.Research School of Finance, Actuarial Studies & StatisticsThe Australian National UniversityCanberraAustralia

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