Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Asymptotics for the survival probability in a killed branching random walk

Nina Gantert, Yueyun Hu, and Zhan Shi

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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 γε, 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).

Résumé

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 γε, 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).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 1 (2011), 111-129.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1294170232

Digital Object Identifier
doi:10.1214/10-AIHP362

Mathematical Reviews number (MathSciNet)
MR2779399

Zentralblatt MATH identifier
1210.60093

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching random walk Survival probability Maximal displacement

Citation

Gantert, Nina; Hu, Yueyun; Shi, Zhan. Asymptotics for the survival probability in a killed branching random walk. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 111--129. doi:10.1214/10-AIHP362. https://projecteuclid.org/euclid.aihp/1294170232


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References

  • [1] D. J. Aldous. A Metropolis-type optimization algorithm on the infinite tree. Algorithmica 22 (1998) 388–412.
  • [2] J. D. Biggins. The first- and last-birth problems for a multitype age-dependent branching process. Adv. in Appl. Probab. 8 (1976) 446–459.
  • [3] J. D. Biggins and A. E. Kyprianou. Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 (2005) 609–631, Paper 17.
  • [4] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.
  • [5] É. Brunet and B. Derrida. Shift in the velocity of a front due to a cutoff. Phys. Rev. E 56 (1997) 2597–2604.
  • [6] B. Derrida and D. Simon. The survival probability of a branching random walk in presence of an absorbing wall. Europhys. Lett. 78 (2007), Paper 60006.
  • [7] B. Derrida and D. Simon. Quasi-stationary regime of a branching random walk in presence of an absorbing wall. J. Stat. Phys. 131 (2008) 203–233.
  • [8] J. M. Hammersley. Postulates for subadditive processes. Ann. Probab. 2 (1974) 652–680.
  • [9] Y. Hu and Z. Shi. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 (2009) 742–789.
  • [10] K. Itô and H. P. McKean Jr. Diffusion Processes and Their Sample Paths. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer, Berlin, 1974.
  • [11] B. Jaffuel. The critical barrier for the survival of the branching random walk with absorption, 2009. Available at ArXiv math.PR/0911.2227.
  • [12] J.-P. Kahane and J. Peyrière. Sur certaines martingales de Mandelbrot. Adv. Math. 22 (1976) 131–145.
  • [13] H. Kesten. Branching Brownian motion with absorption. Stochastic Process. Appl. 7 (1978) 9–47.
  • [14] J. F. C. Kingman. The first birth problem for an age-dependent branching process. Ann. Probab. 3 (1975) 790–801.
  • [15] R. Lyons. A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes 217–221. K. B. Athreya and P. Jagers (Eds). IMA Volumes in Mathematics and Its Applications 84. Springer, New York, 1997.
  • [16] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of LlogL criteria for mean behavior of branching processes. Ann. Probab. 23 (1995) 1125–1138.
  • [17] C. McDiarmid. Minimal positions in a branching random walk. Ann. Appl. Probab. 5 (1995) 128–139.
  • [18] A. A. Mogulskii. Small deviations in the space of trajectories. Theory Probab. Appl. 19 (1974) 726–736.
  • [19] R. Pemantle. Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19 (2009) 1273–1291.