# Seneta-Heyde norming in the branching random walk

### Reference:

Biggins, J. D. and Kyprianou, A. E., 1997. Seneta-Heyde norming in the branching random walk. *Annals of Probability*, 25 (1), pp. 337-360.

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### Official URL:

http://dx.doi.org/10.1214/aop/1024404291

### Abstract

In the discrete-time supercritical branching random walk, there is a Kesten-Stigum type result for the martingales formed by the Laplace transform of the $n$th generation positions. Roughly, this says that for suitable values of the argument of the Laplace transform the martingales converge in mean provided an "$X \log X$" condition holds. Here it is established that when this moment condition fails, so that the martingale ..converges to zero, it is possible to find a (Seneta-Heyde) renormalization of the martingale that converges in probability to a finite nonzero limit when the process survives. As part of the proof, a Seneta-Heyde renormalization of the general (Crump-Mode-Jagers) branching process is obtained; in this case the convergence holds almost surely. The results rely heavily on a detailed study of the functional equation that the Laplace transform of the limit must satisfy.

### Details

Item Type | Articles |

Creators | Biggins, J. D.and Kyprianou, A. E. |

DOI | 10.1214/aop/1024404291 |

Departments | Faculty of Science > Mathematical Sciences |

Refereed | Yes |

Status | Published |

ID Code | 24193 |

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