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

Yuguang Fan

doi:10.1016/j.spa.2015.04.005

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

Yuguang Fan^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

Let ^(r,s)X_t be the Lévy process X_t with r largest jumps and s smallest jumps up till time t deleted and let ^(r)X˜_t be X_t with r largest jumps in modulus up till time t deleted. We show that (^(r,s)X_t-a_t)/b_t or (^(r)X˜_t-a_t)/b_t converges to a proper nondegenerate nonnormal limit distribution as t ↓ 0 if and only if (X_t-a_t)/b_t converges as t ↓ 0 to an α-stable random variable, with 0 < α < 2, where s_t and b_t > 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.

Original language	English
Pages (from-to)	3691-3724
Number of pages	34
Journal	Stochastic Processes and their Applications
Volume	125
Issue number	10
DOIs	https://doi.org/10.1016/j.spa.2015.04.005
Publication status	Published - 5 Dec 2015
Externally published	Yes

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title = "Convergence of trimmed L{\'e}vy processes to trimmed stable random variables at 0",

abstract = "Let (r,s)Xt be the L{\'e}vy 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 st 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 L{\'e}vy processes at 0.",

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AB - 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 st 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.

KW - Domain of attraction

KW - Lévy process

KW - Small times

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JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

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

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