The Annals of Applied Probability

Almost sure central limit theorem for branching random walks in random environment

Makoto Nakashima

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Abstract

We consider the branching random walks in d-dimensional integer lattice with time–space i.i.d. offspring distributions. Then the normalization of the total population is a nonnegative martingale and it almost surely converges to a certain random variable. When d≥3 and the fluctuation of environment satisfies a certain uniform square integrability then it is nondegenerate and we prove a central limit theorem for the density of the population in terms of almost sure convergence.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 1 (2011), 351-373.

Dates
First available in Project Euclid: 17 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1292598038

Digital Object Identifier
doi:10.1214/10-AAP699

Mathematical Reviews number (MathSciNet)
MR2759206

Zentralblatt MATH identifier
1210.60108

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Branching random walk random environment central limit theorem linear stochastic evolutions phase transition

Citation

Nakashima, Makoto. Almost sure central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 21 (2011), no. 1, 351--373. doi:10.1214/10-AAP699. https://projecteuclid.org/euclid.aoap/1292598038


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References

  • [1] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • [2] Biggins, J. D. (1990). The central limit theorem for the supercritical branching random walk, and related results. Stochastic Process. Appl. 34 255–274.
  • [3] Birkner, M., Geiger, J. and Kersting, G. (2005). Branching processes in random environment—a view on critical and subcritical cases. In Interacting Stochastic Systems 269–291. Springer, Berlin.
  • [4] Bolthausen, E. (1989). A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 529–534.
  • [5] Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 705–723.
  • [6] Comets, F., Shiga, T. and Yoshida, N. (2004). Probabilistic analysis of directed polymers in a random environment: A review. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 115–142. Math. Soc. Japan, Tokyo.
  • [7] Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746–1770.
  • [8] Constantine, G. M. and Savits, T. H. (1996). A multivariate Faà di Bruno formula with applications. Trans. Amer. Math. Soc. 348 503–520.
  • [9] Durrett, R. (2004). Probability: Theory and Examples, 3rd ed. Duxbury Press, Belmont, CA.
  • [10] Hu, Y. and Yoshida, N. (2009). Localization for branching random walks in random environment. Stochastic Process. Appl. 119 1632–1651.
  • [11] Liggett, T. M. (2005). Interacting Particle Systems. Classics in Mathematics. Springer, Berlin.
  • [12] Nagahata, Y. and Yoshida, N. (2009). Central limit theorem for a class of linear systems. Electron. J. Probab. 14 960–977.
  • [13] Nakashima, M. (2009). Central limit theorem for linear stochastic evolutions. J. Math. Kyoto Univ. 49 201–224.
  • [14] Petersen, L. C. (1982). On the relation between the multidimensional moment problem and the one-dimensional moment problem. Math. Scand. 51 361–366.
  • [15] Shiozawa, Y. (2009). Central limit theorem for branching Brownian motions in random environment. J. Stat. Phys. 136 145–163.
  • [16] Shiozawa, Y. (2009). Localization for branching Brownian motions in random environment. Tohoku Math. J. (2) 61 483–497.
  • [17] Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40 814–827.
  • [18] Song, R. and Zhou, X. Y. (1996). A remark on diffusion of directed polymers in random environments. J. Statist. Phys. 85 277–289.
  • [19] Spitzer, F. (1976). Principles of Random Walks, 2nd ed. Springer, New York.
  • [20] Yoshida, N. (2008). Central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 18 1619–1635.
  • [21] Yoshida, N. (2010). Localization for linear stochastic evolutions. J. Stat. Phys. 138 598–618.
  • [22] Yoshida, N. (2008). Phase transitions for the growth rate of linear stochastic evolutions. J. Stat. Phys. 133 1033–1058.