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February 2023 A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process
Bastien Mallein, Quan Shi
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Bernoulli 29(1): 597-624 (February 2023). DOI: 10.3150/22-BEJ1470

Abstract

A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet (σ2,a,Λ). We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of (σ2,a,Λ). This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred Lévy processes conditioned to stay positive.

Funding Statement

Part of this work was supported by a PEPS JCJC grant. BM is partially supported by the Projet ANR-16-CE93-0003 (ANR MALIN). QS was partially supported by the SNSF Grant P2ZHP2_171955.

Acknowledgments

The authors would like to thank Samuel Baguley, Loïc Chaumont, Leif Döring and Andreas Kyprianou for valuable inputs in the proof of Proposition 3.3. We also thank Haojie Hou and an anonymous referee for their helpful comments.

Citation

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Bastien Mallein. Quan Shi. "A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process." Bernoulli 29 (1) 597 - 624, February 2023. https://doi.org/10.3150/22-BEJ1470

Information

Received: 1 June 2021; Published: February 2023
First available in Project Euclid: 13 October 2022

Digital Object Identifier: 10.3150/22-BEJ1470

Keywords: branching Lévy process , derivative martingale , Lévy process , perpetual integral , spinal decomposition

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Vol.29 • No. 1 • February 2023
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