Total Progeny in Killed Branching Random Walk

We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. 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 [26, 31]. In this paper, we confirm a conjecture of Aldous [3, 4] that E [Z] < ∞ while E [Z logZ] = ∞. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.