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