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A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

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Quan Shi

Résumé

A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching L\'evy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. 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 $(\sigma^2,a,\Lambda)$. 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 L\'evy processes conditioned to stay positive.
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Dates et versions

hal-03251327 , version 1 (26-01-2022)
hal-03251327 , version 2 (17-11-2022)

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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. 2022. ⟨hal-03251327v1⟩
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