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.

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