Weak convergence of the extremes of branching Lévy processes with regularly varying tails

Yan Xia Ren; Renming Song; Rui Zhang

doi:10.1017/jpr.2023.103

Weak convergence of the extremes of branching Lévy processes with regularly varying tails

Yan Xia Ren
, Renming Song
, Rui Zhang

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

Abstract

We study the weak convergence of the extremes of supercritical branching Lévy processes whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, converges weakly. As a consequence, we obtain a limit theorem for the order statistics of.

Original language	English (US)
Pages (from-to)	622-643
Number of pages	22
Journal	Journal of Applied Probability
Volume	61
Issue number	2
Early online date	Dec 6 2023
DOIs	https://doi.org/10.1017/jpr.2023.103
State	Published - Jun 6 2024

Keywords

Branching Lévy process
extremal process
regularly varying
rightmost position

ASJC Scopus subject areas

Statistics and Probability
General Mathematics
Statistics, Probability and Uncertainty

Online availability

10.1017/jpr.2023.103

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title = "Weak convergence of the extremes of branching L{\'e}vy processes with regularly varying tails",

abstract = "We study the weak convergence of the extremes of supercritical branching L{\'e}vy processes whose spatial motions are L{\'e}vy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, converges weakly. As a consequence, we obtain a limit theorem for the order statistics of.",

keywords = "Branching L{\'e}vy process, extremal process, regularly varying, rightmost position",

author = "Ren, \{Yan Xia\} and Renming Song and Rui Zhang",

note = "The research of Y.-X. Ren is supported by the National Key R\&D Program of China (No. 2020YFA0712900) and NSFC (Grant Nos. 12071011 and 11731009). The research of R. Song is supported by a grant from the Simons Foundation (\#960480, Renming Song). Part of the research for this paper was done while R. Song was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China (11931004, Yingchao Xie). The research of R. Zhang is supported by NSFC (Grant Nos. 11601354, 12271374, and 12371143), Beijing Municipal Natural Science Foundation (Grant No. 1202004), and the Academy for Multidisciplinary Studies, Capital Normal University.",

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doi = "10.1017/jpr.2023.103",

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AU - Zhang, Rui

N1 - The research of Y.-X. Ren is supported by the National Key R&D Program of China (No. 2020YFA0712900) and NSFC (Grant Nos. 12071011 and 11731009). The research of R. Song is supported by a grant from the Simons Foundation (#960480, Renming Song). Part of the research for this paper was done while R. Song was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China (11931004, Yingchao Xie). The research of R. Zhang is supported by NSFC (Grant Nos. 11601354, 12271374, and 12371143), Beijing Municipal Natural Science Foundation (Grant No. 1202004), and the Academy for Multidisciplinary Studies, Capital Normal University.

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N2 - We study the weak convergence of the extremes of supercritical branching Lévy processes whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, converges weakly. As a consequence, we obtain a limit theorem for the order statistics of.

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KW - extremal process

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KW - rightmost position

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Weak convergence of the extremes of branching Lévy processes with regularly varying tails

Abstract

Keywords

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