Open Access
May 2014 The Seneta–Heyde scaling for the branching random walk
Elie Aidekon, Zhan Shi
Ann. Probab. 42(3): 959-993 (May 2014). DOI: 10.1214/12-AOP809


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.


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Elie Aidekon. Zhan Shi. "The Seneta–Heyde scaling for the branching random walk." Ann. Probab. 42 (3) 959 - 993, May 2014.


Published: May 2014
First available in Project Euclid: 26 March 2014

zbMATH: 1304.60092
MathSciNet: MR3189063
Digital Object Identifier: 10.1214/12-AOP809

Primary: 60F05 , 60J80

Keywords: additive martingale , Branching random walk , derivative martingale , Seneta–Heyde norming

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.42 • No. 3 • May 2014
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