# Biggins’ martingale convergence for branching Lévy processes

Bertoin, Jean; Mallein, Bastien (2018). Biggins’ martingale convergence for branching Lévy processes. Electronic Communications in Probability, 23(83):1-12.

## Abstract

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 $(σ2,a,Λ)$, where the branching Lévy measure $Λ$ 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 $(σ2,a,Λ)$ for additive martingales to have a non-degenerate limit

## Abstract

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 $(σ2,a,Λ)$, where the branching Lévy measure $Λ$ 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 $(σ2,a,Λ)$ for additive martingales to have a non-degenerate limit

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## Additional indexing

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Statistics and Probability Social Sciences & Humanities > Statistics, Probability and Uncertainty Statistics, Probability and Uncertainty, Statistics and Probability English 25 October 2018 17 Jan 2019 10:12 26 Jan 2022 20:01 Institute of Mathematical Statistics 1083-589X Gold Publisher DOI. An embargo period may apply. https://doi.org/10.1214/18-ecp185 https://projecteuclid.org/euclid.ecp/1540433049

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