The Annals of Probability

Minima in branching random walks

Louigi Addario-Berry and Bruce Reed

Full-text: Open access

Abstract

Given a branching random walk, let Mn be the minimum position of any member of the nth generation. We calculate EMn to within O(1) and prove exponential tail bounds for P{|MnEMn|>x}, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89–108], our results fully characterize the possible behavior of EMn when the branching random walk has bounded branching and step size.

Article information

Source
Ann. Probab., Volume 37, Number 3 (2009), 1044-1079.

Dates
First available in Project Euclid: 19 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1245434028

Digital Object Identifier
doi:10.1214/08-AOP428

Mathematical Reviews number (MathSciNet)
MR2537549

Zentralblatt MATH identifier
1196.60142

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
Branching random walks branching processes random trees

Citation

Addario-Berry, Louigi; Reed, Bruce. Minima in branching random walks. Ann. Probab. 37 (2009), no. 3, 1044--1079. doi:10.1214/08-AOP428. https://projecteuclid.org/euclid.aop/1245434028


Export citation

References

  • [1] Alon, N. and Spencer, J. H. (2000). The Probabilistic Method, 2nd ed. Wiley, New York. With an appendix on the life and work of Paul Erdős.
  • [2] Andersen, E. S. (1953). On the fluctuations of sums of random variables. Math. Scand. 1 263–285.
  • [3] Bachmann, M. (1998). Limit theorems for the minimal position in a branching random walk with independent logconcave displacements. Ph.D. thesis, Purdue Univ.
  • [4] Bachmann, M. (2000). Limit theorems for the minimal position in a branching random walk with independent logconcave displacements. Adv. in Appl. Probab. 32 159–176.
  • [5] Bahadur, R. R. and Ranga Rao, R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31 1015–1027.
  • [6] Biggins, J. D. (1976). The first- and last-birth problems for a multitype age-dependent branching process. Adv. in Appl. Probab. 8 446–459.
  • [7] Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 531–581.
  • [8] Bramson, M. D. (1978). Minimal displacement of branching random walk. Z. Wahrsch. Verw. Gebiete 45 89–108.
  • [9] Bramson, M. D. and Zeitouni, O. (2009). Tightness for a family of recursive equations. Ann. Probab. 37 615–653.
  • [10] Bramson, M. and Zeitouni, O. (2007). Tightness for the minimal displacement of branching random walk. J. Stat. Mech. Theory Exp. 7 P07010 (electronic).
  • [11] Broutin, N. and Devroye, L. (2006). Large deviations for the weighted height of an extended class of trees. Algorithmica 46 271–297.
  • [12] Broutin, N., Devroye, L. and McLeish, E. (2008). Weighted height of random trees. Acta Inform. 45 237–277.
  • [13] Chauvin, B. and Drmota, M. (2006). The random multisection problem, travelling waves and the distribution of the height of m-ary search trees. Algorithmica 46 299–327.
  • [14] Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493–507.
  • [15] Chung, K. L. and Erdös, P., (1952). On the application of the Borel–Cantelli lemma. Trans. Amer. Math. Soc. 72 179–186.
  • [16] Dekking, F. M. and Host, B. (1991). Limit distributions for minimal displacement of branching random walks. Probab. Theory Related Fields 90 403–426.
  • [17] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones & Bartlett, Boston, MA.
  • [18] Devroye, L. (1998). Branching processes and their applications in the analysis of tree structures and tree algorithms. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms and Combinatorics 16 249–314. Springer, Berlin.
  • [19] Devroye, L. (1999). Universal limit laws for depths in random trees. SIAM J. Comput. 28 409–432 (electronic).
  • [20] Devroye, L. and Reed, B. (1995). On the variance of the height of random binary search trees. SIAM J. Comput. 24 1157–1162.
  • [21] Drmota, M. (2003). An analytic approach to the height of binary search trees. II. J. ACM 50 333–374 (electronic).
  • [22] Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Probab. 2 652–680.
  • [23] Harris, T. E. (1963). The Theory of Branching Processes. Die Grundlehren der Mathematischen Wissenschaften 119. Springer, Berlin.
  • [24] 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.
  • [25] Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Probab. 3 790–801.
  • [26] Lifshits, M. A. (2007). Some limit theorems on binary trees. In preparation.
  • [27] McDiarmid, C. (1995). Minimal positions in a branching random walk. Ann. Appl. Probab. 5 128–139.
  • [28] Reed, B. (2003). The height of a random binary search tree. J. ACM 50 306–332 (electronic).