Considérons une marche aléatoire branchante surcritique à temps discret. Nous nous intéressons à la probabilité qu'il existe un rayon infini du support de la marche aléatoire branchante, le long duquel elle croît plus vite qu'une fonction linéaire de pente γ - ε, où γ désigne la vitesse asymptotique de la position de la particule la plus à droite dans la marche aléatoire branchante. Sous des hypothèses générales peu restrictives, nous prouvons que, lorsque ε → 0, cette probabilité décroît comme exp{-(β+o(1)) / ε1/2}, où β est une constante strictement positive dont la valeur dépend de la loi de la marche aléatoire branchante. Dans le cas spécial où des variables aléatoires i.i.d. de Bernoulli(p) (avec 0 < p < ½) sont placées sur les arêtes d'un arbre binaire enraciné, ceci répond à une question ouverte de Robin Pemantle (Ann. Appl. Probab. 19 (2009) 1273-1291).
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).
Mots-clés : branching random walk, survival probability, maximal displacement
@article{AIHPB_2011__47_1_111_0,
author = {Gantert, Nina and Hu, Yueyun and Shi, Zhan},
title = {Asymptotics for the survival probability in a killed branching random walk},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
pages = {111--129},
publisher = {Gauthier-Villars},
volume = {47},
number = {1},
year = {2011},
doi = {10.1214/10-AIHP362},
mrnumber = {2779399},
zbl = {1210.60093},
language = {en},
url = {http://www.numdam.org/articles/10.1214/10-AIHP362/}
}
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 SP - 111 EP - 129 VL - 47 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/10-AIHP362/ DO - 10.1214/10-AIHP362 LA - en ID - AIHPB_2011__47_1_111_0 ER -
%0 Journal Article %A Gantert, Nina %A Hu, Yueyun %A Shi, Zhan %T Asymptotics for the survival probability in a killed branching random walk %J Annales de l'I.H.P. Probabilités et statistiques %D 2011 %P 111-129 %V 47 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/10-AIHP362/ %R 10.1214/10-AIHP362 %G en %F AIHPB_2011__47_1_111_0
Gantert, Nina; Hu, Yueyun; Shi, Zhan. Asymptotics for the survival probability in a killed branching random walk. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 111-129. doi : 10.1214/10-AIHP362. http://www.numdam.org/articles/10.1214/10-AIHP362/
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