Biggins’ martingale convergence for branching Lévy processes

A branching Lévy process can be seen as the continuous-time version of a branching random walk; see [BM17]. It describes a particle system on the real line in which particles move and reproduce independently one of the others, in a Poissonian manner. Just as for Lévy processes, the law of a branchin...

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Bibliographic Details
Published inElectronic communications in probability Vol. 23; no. none
Main Authors Bertoin, Jean, Mallein, Bastien
Format Journal Article
LanguageEnglish
Published Institute of Mathematical Statistics (IMS) 01.01.2018
Subjects
Mathematics
Probability
Branching Lévy process
spinal decomposition AMS subject classifications: 60G44
spinal decomposition
additive martingale
uniform integrability
60J80
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Summary:A branching Lévy process can be seen as the continuous-time version of a branching random walk; see [BM17]. It describes a particle system on the real line in which particles move and reproduce independently one of the others, in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet (σ 2 , a, Λ), where the Lévy measure Λ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem [Big77] in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet (σ 2 , a, Λ) for additive martingales of branching Lévy processes to have a non-degenerate limit. The proof is adapted from Lyons [Lyo97].
ISSN:1083-589X
1083-589X
DOI:10.1214/18-ECP185