Tail asymptotics for the total progeny of the critical killed branching random walk
Abstract
We consider a branching random walk on $R$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that $P(Z>n)$ is of order $(n\ln^2(n))^{-1}$, which confirms the prediction of Addario-Berry and Broutin [1].
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Pages: 522-533
Publication Date: November 2, 2010
DOI: 10.1214/ECP.v15-1583
References
- L. Addario-Berry and N. Broutin. Total progeny in killed branching random walk. To appear in Prob. Th. Rel. Fields. (2010).
- D. Aldous. Power laws and killed branching random walks. URL http://www.stat.berkeley.edu/aldous/Research/OP/brw.html.
- J. Berestycki, N. Berestycki, and J. Schweinsberg. The genealogy of branching Brownian motion with absorption. Arxiv:1001.2337 (2010).
- J.T. Chang. Inequalities for the overshoot. Ann. Appl. Probab. 4 (1994), 1223--1233. Math. Review 95k:60218
- J.W. Harris and S.C. Harris. Survival probabilities for branching Brownian motion with absorption. Electron. Comm. Probab. 12 (2007), 81--92. Math. Review 2008a:60212
- H. Kesten. Branching Brownian motion with absorption. Stoch. Proc. Appl. 7 (1978), 9--47. Math. Review 58 #13384
- P. Maillard. The number of absorbed individuals in branching Brownian motion with a barrier. Arxiv:1004.1426 (2010).
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