The Annals of Probability

Seneta-Heyde norming in the branching random walk

J. D. Biggins and A. E. Kyprianou

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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.

Article information

Source
Ann. Probab., Volume 25, Number 1 (1997), 337-360.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404291

Digital Object Identifier
doi:10.1214/aop/1024404291

Mathematical Reviews number (MathSciNet)
MR1428512

Zentralblatt MATH identifier
0873.60062

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Martingales functional equations Seneta-Heyde norming branching random walk

Citation

Biggins, J. D.; Kyprianou, A. E. Seneta-Heyde norming in the branching random walk. Ann. Probab. 25 (1997), no. 1, 337--360. doi:10.1214/aop/1024404291. https://projecteuclid.org/euclid.aop/1024404291


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