May 2024 Maximal displacement of spectrally negative branching Lévy processes
Christophe Profeta
Author Affiliations +
Bernoulli 30(2): 961-982 (May 2024). DOI: 10.3150/23-BEJ1620

Abstract

We consider a branching Markov process in continuous time in which the particles evolve independently as spectrally negative Lévy processes. When the branching mechanism is critical or subcritical, the process will eventually die and we may define its overall maximum, i.e. the maximum location ever reached by a particle. The purpose of this paper is to give asymptotic estimates for the survival function of this maximum. In particular, we show that in the critical case the asymptotics is polynomial when the underlying Lévy process oscillates or drifts towards +, and is exponential when it drifts towards .

Acknowledgments

The author would like to thank the anonymous referees for suggesting several improvements, in particular Remark 4.2.

Citation

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Christophe Profeta. "Maximal displacement of spectrally negative branching Lévy processes." Bernoulli 30 (2) 961 - 982, May 2024. https://doi.org/10.3150/23-BEJ1620

Information

Received: 1 November 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699541
Digital Object Identifier: 10.3150/23-BEJ1620

Keywords: branching process , Extreme values , spectrally negative Lévy process

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Vol.30 • No. 2 • May 2024
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