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Total progeny in killed branching random walk
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  • Published: 21 May 2010

Total progeny in killed branching random walk

  • L. Addario-Berry1 &
  • N. Broutin2 

Probability Theory and Related Fields volume 151, pages 265–295 (2011)Cite this article

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  • 14 Citations

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Abstract

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 12:805–814, 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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montreal, Canada

    L. Addario-Berry

  2. Projet algorithms, INRIA Rocquencourt, 78153, Le Chesnay, France

    N. Broutin

Authors
  1. L. Addario-Berry
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  2. N. Broutin
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Correspondence to L. Addario-Berry.

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Addario-Berry, L., Broutin, N. Total progeny in killed branching random walk. Probab. Theory Relat. Fields 151, 265–295 (2011). https://doi.org/10.1007/s00440-010-0299-2

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  • Received: 07 August 2009

  • Revised: 12 April 2010

  • Published: 21 May 2010

  • Issue Date: October 2011

  • DOI: https://doi.org/10.1007/s00440-010-0299-2

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Mathematics Subject Classification (2000)

  • 60J80
  • 60G50
  • 60G17
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