Total progeny in killed branching random walk

Abstract

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 (Gantert and Muller in Markov Process. Relat. Fields 12:805–814, 2006; Hu and Shi in Ann. Probab. 37(2):742–789, 2009). In this paper, we confirm a conjecture of Aldous (Algorithmica 22:388–412, 1998; and Power laws and killed branching random walks) that E [Z] < ∞ while \({{\mathbf E}\left[Z\,{\rm log}\, Z\right]=\infty}\) . The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.