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

Part of: Stochastic processes Markov processes Limit theorems

Published online by Cambridge University Press:  06 December 2023

Yan-xia Ren ,
Renming Song Open the ORCID record for Renming Song [Opens in a new window]  and
Rui Zhang
Yan-xia Ren*
Affiliation:
Peking University
Renming Song*
Affiliation:
University of Illinois Urbana-Champaign
Rui Zhang*
Affiliation:
Capital Normal University
*
*Postal address: LMAM School of Mathematical Sciences and Center for Statistical Science, Peking University, Beijing, 100871, P. R. China. Email: yxren@math.pku.edu.cn
**Postal address: Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA. Email: rsong@illinois.edu
***Postal address: School of Mathematical Sciences & Academy for Multidisciplinary Studies, Capital Normal University, Beijing, 100048, P.R. China. (Corresponding Author). Email: zhangrui27@cnu.edu.cn
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Abstract

We study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ 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, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.

Keywords

Branching Lévy processextremal processregularly varyingrightmost position

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems
Secondary: 60G57: Random measures 60G70: Extreme value theory; extremal processes

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 2 , June 2024 , pp. 622 - 643
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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