The Annals of Probability

The Seneta–Heyde scaling for the branching random walk

Elie Aidekon and Zhan Shi

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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.

Article information

Source
Ann. Probab., Volume 42, Number 3 (2014), 959-993.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1395838121

Digital Object Identifier
doi:10.1214/12-AOP809

Mathematical Reviews number (MathSciNet)
MR3189063

Zentralblatt MATH identifier
1304.60092

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems

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

Citation

Aidekon, Elie; Shi, Zhan. The Seneta–Heyde scaling for the branching random walk. Ann. Probab. 42 (2014), no. 3, 959--993. doi:10.1214/12-AOP809. https://projecteuclid.org/euclid.aop/1395838121


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