The Seneta--Heyde scaling for the branching random walk

2011/02/28 by Elie Aidekon, Zhan Shi · 1 citation
Mathematics · #math.PR

paper · pdf · doi:10.1214/12-aop809

published as Annals of Probability 2014, Vol. 42, No. 3, 959-993 · 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)

arxiv created 2014/04/04 · arxiv updated 2014/04/07

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^1/2W_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.

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