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

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  • Fan, Yuguang

Abstract

Let (r,s)Xt be the Lévy process Xt with r largest jumps and s smallest jumps up till time t deleted and let (r)X˜t be Xt with r largest jumps in modulus up till time t deleted. We show that ((r,s)Xt−at)/bt or ((r)X˜t−at)/bt converges to a proper nondegenerate nonnormal limit distribution as t↓0 if and only if (Xt−at)/bt converges as t↓0 to an α-stable random variable, with 0<α<2, where at and bt>0 are nonstochastic functions in t. Together with the asymptotic normality case treated in Fan (2015) [7], this completes the domain of attraction problem for trimmed Lévy processes at 0.

Suggested Citation

  • Fan, Yuguang, 2015. "Convergence of trimmed Lévy processes to trimmed stable random variables at 0," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3691-3724.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:10:p:3691-3724
    DOI: 10.1016/j.spa.2015.04.005
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    Cited by:

    1. Yuguang Fan, 2017. "Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times," Journal of Theoretical Probability, Springer, vol. 30(2), pages 675-699, June.

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