Total progeny in killed branching random walk
Louigi Addario-Berry, Nicolas Broutin

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 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.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Markov Chains and Monte Carlo Methods · Bayesian Methods and Mixture Models
