The Seneta-Heyde scaling for the branching random walk

E. Aidekon (1) , Z. Shi (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609-631] in a one-dimensional supercritical 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, of the derivative martingale.

Domaines: Probabilités [math.PR]

E. Aidekon, Z. Shi. The Seneta-Heyde scaling for the branching random walk. Annals of Probability, 2014, 42 (3), pp.959-993. ⟨10.1214/12-AOP809⟩. ⟨hal-00988171⟩