The Seneta–Heyde scaling for the branching random walk

By Elie Aidékon, Zhan Shi. 2011

97 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d4bf02b866da028bb4db401a7e04ad095691c90d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Branching random walks (2015, background): “They are collected or adapted from the appendices of [8], [16] and [17], but many of them can probably be found elsewhere in the literature in one form or another.”
Maximum of a log-correlated Gaussian field (2013, background): “Appendix B: On the one dimensional Brownian motion B We refer to [4] and [2] for the proof of the following lemmas.”
Gaussian multiplicative chaos and applications: A review (2013, methodology): “It is a random measure over the interval [0, 1].”
Uniqueness of critical Gaussian chaos (2017, background): “4) was proven by Aidekon and Shi in the important work [2] in the case of Mandelbrot chaos measures that can be thought of as a discrete analogue of continuous chaos.”