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January 1997 Seneta-Heyde norming in the branching random walk
J. D. Biggins, A. E. Kyprianou
Ann. Probab. 25(1): 337-360 (January 1997). DOI: 10.1214/aop/1024404291


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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J. D. Biggins. A. E. Kyprianou. "Seneta-Heyde norming in the branching random walk." Ann. Probab. 25 (1) 337 - 360, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0873.60062
MathSciNet: MR1428512
Digital Object Identifier: 10.1214/aop/1024404291

Primary: 60J80

Keywords: Branching random walk , Functional equations , Martingales , Seneta-Heyde norming

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 1 • January 1997
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