Boundedness of discounted branching random walks via generic chaining

TL;DR

This paper characterizes when a supercritical discounted branching random walk remains bounded, linking it to the finiteness of a specific moment of the increment distribution, extending prior results and addressing an open problem.

Contribution

It provides a precise criterion for the boundedness of discounted branching random walks based on moment conditions, extending previous work and solving an open problem.

Findings

Boundedness characterized by finite moment of order 1/H

Extends classical results of Athreya (1985)

Partially answers Aldous--Bandyopadhyay's open problem

Abstract

Consider a discrete-time supercritical discounted branching random walk, in which increments at depth $k$ are independent and identically distributed with the same law as $m^{-kH}Y$, where $Y$ has a fixed law, $H>0$, and $m>1$ is the expected number of offspring at depth one. We provide a clean characterization of the boundedness of the discounted branching random walk: under mild conditions on the offspring distribution, the process is almost surely bounded if and only if $\mathbb{E}[|Y|^{1/H}]<\infty$. This extends results of Athreya (1985) and A\"id\'ekon--Hu--Shi (2024), and provides a partial answer to Open Problem 31 of Aldous--Bandyopadhyay (2005).