# 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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