Asymptotics for the survival probability in a killed branching random walk

Nina Gantert; Yueyun Hu; Zhan Shi

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 1, page 111-129
  • ISSN: 0246-0203

Abstract

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Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope γ − ε, where γ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when ε → 0, this probability decays like exp{−(β+o(1)) / ε1/2}, where β is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli(p) random variables (with 0 < p < ½) assigned on a rooted binary tree, this answers an open question of Robin Pemantle (see Ann. Appl. Probab.19 (2009) 1273–1291).

How to cite

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Gantert, Nina, Hu, Yueyun, and Shi, Zhan. "Asymptotics for the survival probability in a killed branching random walk." Annales de l'I.H.P. Probabilités et statistiques 47.1 (2011): 111-129. <http://eudml.org/doc/243873>.

@article{Gantert2011,
abstract = {Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope γ − ε, where γ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when ε → 0, this probability decays like exp\{−(β+o(1)) / ε1/2\}, where β is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli(p) random variables (with 0 &lt; p &lt; ½) assigned on a rooted binary tree, this answers an open question of Robin Pemantle (see Ann. Appl. Probab.19 (2009) 1273–1291).},
author = {Gantert, Nina, Hu, Yueyun, Shi, Zhan},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {branching random walk; survival probability; maximal displacement},
language = {eng},
number = {1},
pages = {111-129},
publisher = {Gauthier-Villars},
title = {Asymptotics for the survival probability in a killed branching random walk},
url = {http://eudml.org/doc/243873},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Gantert, Nina
AU - Hu, Yueyun
AU - Shi, Zhan
TI - Asymptotics for the survival probability in a killed branching random walk
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 1
SP - 111
EP - 129
AB - Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope γ − ε, where γ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when ε → 0, this probability decays like exp{−(β+o(1)) / ε1/2}, where β is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli(p) random variables (with 0 &lt; p &lt; ½) assigned on a rooted binary tree, this answers an open question of Robin Pemantle (see Ann. Appl. Probab.19 (2009) 1273–1291).
LA - eng
KW - branching random walk; survival probability; maximal displacement
UR - http://eudml.org/doc/243873
ER -

References

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