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2018 Biggins’ martingale convergence for branching Lévy processes
Jean Bertoin, Bastien Mallein
Electron. Commun. Probab. 23: 1-12 (2018). DOI: 10.1214/18-ECP185


A branching Lévy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet $(\sigma ^2,a,\Lambda )$, where the branching Lévy measure $\Lambda $ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins’ theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet $(\sigma ^2,a,\Lambda )$ for additive martingales to have a non-degenerate limit.


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Jean Bertoin. Bastien Mallein. "Biggins’ martingale convergence for branching Lévy processes." Electron. Commun. Probab. 23 1 - 12, 2018.


Received: 13 December 2017; Accepted: 17 October 2018; Published: 2018
First available in Project Euclid: 25 October 2018

zbMATH: 1402.60051
Digital Object Identifier: 10.1214/18-ECP185

Primary: 60G44 , 60J80

Keywords: additive martingale , branching Lévy process , spinal decomposition , uniform integrability


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