A revisited proof of the Seneta-Heyde norming for branching random walks under optimal assumptions

Pierre Boutaud,Pascal Maillard

doi:10.1214/19-ejp350

Abstract

We introduce a set of tools which simplify and streamline the proofs of limit theorems concerning near-critical particles in branching random walks under optimal assumptions. We exemplify our method by giving another proof of the Seneta-Heyde norming for the critical additive martingale, initially due to A\"id\'ekon and Shi. The method involves in particular the replacement of certain second moment estimates by truncated first moment bounds, and the replacement of ballot-type theorems for random walks by estimates coming from an explicit expression for the potential kernel of random walks killed below the origin. Of independent interest might be a short, self-contained proof of this expression, as well as a criterion for convergence in probability of non-negative random variables in terms of conditional Laplace transforms.

Highlights

In the theory of branching processes, many limit theorems hold under so-called L log L-type moment conditions which are both sufficient and necessary
W is the limit of the martingale Wn = m−nZn and another statement of the theorem says that the martingale (Wn)n≥0 is uniformly integrable if and only if E[L log L] < ∞
In the context of branching random walks, Lyons [19] has shown an analogous theorem for the so-called additive martingales arising naturally in this context

Summary

Introduction

In the theory of branching processes, many limit theorems hold under so-called L log L-type moment conditions which are both sufficient and necessary. Another important example is the so-called Seneta-Heyde norming of the additive martingale at critical parameter: it has been shown by Aïdékon and Shi [2] that this martingale, properly renormalized, converges in probability to the same limit as the derivative martingale, under Aïdékon’s condition Their proof has been adapted by He, Liu and Zhang [13] to cases where a certain variance σ2 (defined in Equation (1.2) below) is infinite and by Aru, Powell and Sepúlveda [3] to an analogous result for Gaussian multiplicative chaos. We give a new proof of the Seneta-Heyde norming for the critical additive martingale in branching random walks, valid under optimal assumptions We find this proof to be simpler and more streamlined than the original one by Aïdékon and Shi due to several technical improvements, amongst others:. Appendix C contains a certain Tauberian-type lemma involving truncated first moments of a non-negative random variable

Spinal decomposition and renewal functions

The renewal function R

A Random walks on the half-line

B Laplace transform criterion for convergence in probability

Findings

C A Tauberian-type lemma