The Seneta--Heyde scaling for the branching random walk

TL;DR

This paper investigates the boundary case of a one-dimensional super-critical branching random walk, showing that a scaled version of the additive martingale converges in probability to a positive limit, linked to the derivative martingale.

Contribution

It establishes the convergence in probability of the scaled additive martingale to a constant multiple of the derivative martingale's limit in the boundary case.

Findings

n^{1/2}W_n converges in probability to a positive limit

The limit is proportional to the derivative martingale's almost sure limit

Convergence is not almost sure, only in probability

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.