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Journal Articles Probability Theory and Related Fields Year : 2011

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, $M_n$, has zero speed on the linear scale: $M_n/n \to 0$ as $n\to\infty$. 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 $\mathbf E[Z]<\infty$ while $\mathbf E [Z\log Z]=\infty$. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.

Dates and versions

hal-01220798 , version 1 (27-10-2015)

Identifiers

Cite

L. Addario-Berry, Nicolas Broutin. Total progeny in killed branching random walk. Probability Theory and Related Fields, 2011, 151, pp.265-295. ⟨10.1007/s00440-010-0299-2⟩. ⟨hal-01220798⟩

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