The Seneta--Heyde scaling for the branching random walk

Abstract

We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609--631] in a one-dimensional super-critical branching random walk, and study the additive martingale $(W_n)$. We prove that, upon the system's survival, $n^{1/2}{W}_n$ converges in probability, but not almost surely, to a positive limit. The limit is identified as a constant multiple of the almost sure limit, discovered by Biggins and Kyprianou [Adv. in Appl. Probab. 36 (2004) 544--581], of the derivative martingale.

Published in at http://dx.doi.org/10.1214/12-AOP809 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Convergence in law of the minimum of a branching random walk
• Minima in branching random walks
• Discretization methods for homogeneous fragmentations
• Traveling waves and homogeneous fragmentation

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