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, R _n , has zero speed on the linear scale: R _n /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 Müller in Markov Process. Relat. Fields 12:805?C814, 2006; Hu and Shi in Ann. Probab. 37(2):742?C789, 2009). In this paper, we confirm a conjecture of Aldous (Algorithmica 22:388?C412, 1998; and Power laws and killed branching random walks) that E Z]?<??? while ${{\mathbf E}\leftZ\,{\rm log}\, Z\right]=\infty}$ . The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.