The Annals of Probability

The precise tail behavior of the total progeny of a killed branching random walk

Elie Aïdékon, Yueyun Hu, and Olivier Zindy

Full-text: Open access

Abstract

Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases, which solves an open problem of Aldous [Power laws and killed branching random walks, http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html].

Article information

Source
Ann. Probab., Volume 41, Number 6 (2013), 3786-3878.

Dates
First available in Project Euclid: 20 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1384957777

Digital Object Identifier
doi:10.1214/13-AOP842

Mathematical Reviews number (MathSciNet)
MR3161464

Zentralblatt MATH identifier
1288.60105

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems

Keywords
Killed branching random walk total progeny spinal decomposition Yaglom-type theorem time reversed random walk

Citation

Aïdékon, Elie; Hu, Yueyun; Zindy, Olivier. The precise tail behavior of the total progeny of a killed branching random walk. Ann. Probab. 41 (2013), no. 6, 3786--3878. doi:10.1214/13-AOP842. https://projecteuclid.org/euclid.aop/1384957777


Export citation

References

  • [1] Addario-Berry, L. and Broutin, N. (2011). Total progeny in killed branching random walk. Probab. Theory Related Fields 151 265–295.
  • [2] Addario-Berry, L. and Reed, B. (2009). Minima in branching random walks. Ann. Probab. 37 1044–1079.
  • [3] Aïdékon, E. (2010). Tail asymptotics for the total progeny of the critical killed branching random walk. Electron. Commun. Probab. 15 522–533.
  • [4] Aldous, D. Power laws and killed branching random walks. Available at http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html.
  • [5] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2013). The genealogy of branching Brownian motion with absorption. Ann. Probab. 41 527–618.
  • [6] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Probab. 22 2152–2167.
  • [7] Biggins, J. D. (1976). The first- and last-birth problems for a multitype age-dependent branching process. Adv. in Appl. Probab. 8 446–459.
  • [8] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37.
  • [9] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544–581.
  • [10] Buraczewski, D. (2009). On tails of fixed points of the smoothing transform in the boundary case. Stochastic Process. Appl. 119 3955–3961.
  • [11] Chang, J. T. (1994). Inequalities for the overshoot. Ann. Appl. Probab. 4 1223–1233.
  • [12] Doney, R. A. (1980). Moments of ladder heights in random walks. J. Appl. Probab. 17 248–252.
  • [13] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [14] Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Probab. 2 652–680.
  • [15] Heyde, C. C. (1967). Asymptotic renewal results for a natural generalization of classical renewal theory. J. R. Stat. Soc. Ser. B Stat. Methodol. 29 141–150.
  • [16] Hu, Y. and Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 742–789.
  • [17] Iglehart, D. L. (1972). Extreme values in the $GI/G/1$ queue. Ann. Math. Statist. 43 627–635.
  • [18] Jaffuel, B. (2012). The critical barrier for the survival of branching random walk with absorption. Ann. Inst. Henri Poincaré Probab. Stat. 48 989–1009.
  • [19] Jagers, P. (1989). General branching processes as Markov fields. Stochastic Process. Appl. 32 183–212.
  • [20] Jelenković, P. R. and Olvera-Cravioto, M. (2012). Implicit renewal theory and power tails on trees. Adv. in Appl. Probab. 44 528–561.
  • [21] Kallenberg, O. (1976). Random Measures. Akademie-Verlag, Berlin.
  • [22] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
  • [23] Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Probab. 3 790–801.
  • [24] Liu, Q. (2000). On generalized multiplicative cascades. Stochastic Process. Appl. 86 263–286.
  • [25] Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41 520–527.
  • [26] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994). IMA Vol. Math. Appl. 84 217–221. Springer, New York.
  • [27] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • [28] Maillard, P. (2013). The number of absorbed individuals in branching Brownian motion with a barrier. Ann. Inst. Henri Poincaré, Probab. Stat. 49 428–455.
  • [29] Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrsch. Verw. Gebiete 57 365–395.
  • [30] Pemantle, R. (1999). Critical killed branching process tail probabilities. Unpublished manuscript.
  • [31] Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. The Clarendon Press Oxford Univ. Press, New York.
  • [32] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • [33] Tanaka, H. (1989). Time reversal of random walks in one-dimension. Tokyo J. Math. 12 159–174.
  • [34] Vatutin, V. A. and Wachtel, V. (2009). Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 177–217.