Total progeny in killed branching random walk

TL;DR

This paper studies a killed branching random walk where particles with negative displacement are removed, confirming a conjecture that the expected number of surviving particles is finite but their entropy is infinite.

Contribution

It proves a conjecture by Aldous that the expected size of the un-killed particles is finite, yet the expected value of Z log Z is infinite, using large deviations and ballot theorem techniques.

Findings

Expected number of un-killed particles is finite (Exp[Z] < infinity).

Expected Z log Z is infinite, indicating heavy tail behavior.

Uses large deviations and ballot theorem methods for analysis.

Abstract

We consider a branching random walk for which the maximum position of a particle in the n'th generation, M_n, has zero speed on the linear scale: M_n/n --> 0 as n --> infinity. We further remove ("kill") any particle whose displacement is negative, together with its entire descendence. The size $Z$ of the set of un-killed particles is almost surely finite. In this paper, we confirm a conjecture of Aldous that Exp[Z] < infinity while Exp[Z log Z]=infinity. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.