Electronic Communications in Probability

Biggins’ martingale convergence for branching Lévy processes

Jean Bertoin and Bastien Mallein

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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.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 83, 12 pp.

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

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Zentralblatt MATH identifier

Primary: 60G44: Martingales with continuous parameter 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

branching Lévy process additive martingale uniform integrability spinal decomposition

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Bertoin, Jean; Mallein, Bastien. Biggins’ martingale convergence for branching Lévy processes. Electron. Commun. Probab. 23 (2018), paper no. 83, 12 pp. doi:10.1214/18-ECP185. https://projecteuclid.org/euclid.ecp/1540433049

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