Electronic Journal of Probability

A Central Limit Theorem for Random Walk in a Random Environment on a Marked Galton-Watson Tree.

Gabriel Faraud

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Models of random walks in a random environment were introduced at first by Chernoff in 1967 in order to study biological mechanisms. The original model has been intensively studied since then and is now well understood. In parallel, similar models of random processes in a random environment have been studied. In this article we focus on a model of random walk on random marked trees, following a model introduced by R. Lyons and R. Pemantle (1992). Our point of view is a bit different yet, as we consider a very general way of constructing random trees with random transition probabilities on them. We prove an analogue of R. Lyons and R. Pemantle's recurrence criterion in this setting, and we study precisely the asymptotic behavior, under restrictive assumptions. Our last result is a generalization of a result of Y. Peres and O. Zeitouni (2006) concerning biased random walks on Galton-Watson trees.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 6, 174-215.

Accepted: 12 January 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Random Walk random environment tree branching random walk central limit theorem

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Faraud, Gabriel. A Central Limit Theorem for Random Walk in a Random Environment on a Marked Galton-Watson Tree. Electron. J. Probab. 16 (2011), paper no. 6, 174--215. doi:10.1214/EJP.v16-851. https://projecteuclid.org/euclid.ejp/1464820175

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  • Aidékon, Elie. Transient random walks in random environment on a Galton-Watson tree. Probab. Theory Related Fields 142 (2008), no. 3-4, 525–559.
  • Bercu, Bernard; Touati, Abderrahmen. Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18 (2008), no. 5, 1848–1869.
  • Biggins, J. D. Martingale convergence in the branching random walk. J. Appl. Probability 14 (1977), no. 1, 25–37.
  • Biggins, J. D.; Kyprianou, A. E. Seneta-Heyde norming in the branching random walk. Ann. Probab. 25 (1997), no. 1, 337–360.
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
  • Chernov A.A. Replication of a multicomponent chain. Biophysics 12 (1967), no. 2, 336–341.
  • Faraud, G.; Hu, Y. Shi, ZAn almost sure convergence for stochastichally biased random walk on a Galton-Watson tree (2010). Arxiv
  • Hu, Yueyun; Shi, Zhan. A subdiffusive behaviour of recurrent random walk in random environment on a regular tree. Probab. Theory Related Fields 138 (2007), no. 3-4, 521–549.
  • Hu, Yueyun; Shi, Zhan. Slow movement of random walk in random environment on a regular tree. Ann. Probab. 35 (2007), no. 5, 1978–1997.
  • Hu, Yueyun; Shi, Zhan. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 (2009), no. 2, 742–789.
  • Kemeny, John G.; Snell, J. Laurie; Knapp, Anthony W. Denumerable Markov chains. Second edition. With a chapter on Markov random fields, by David Griffeath. Graduate Texts in Mathematics, No. 40. Springer-Verlag, New York-Heidelberg-Berlin, 1976. xii+484 pp.
  • Kipnis, C.; Varadhan, S. R. S. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986), no. 1, 1–19.
  • Liu, Quansheng. On generalized multiplicative cascades. Stochastic Process. Appl. 86 (2000), no. 2, 263–286.
  • Liu, Quansheng. Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl. 95 (2001), no. 1, 83–107.
  • Lyons, Russell. The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 (1989), no. 2, 337–353.
  • Lyons, Russell; Pemantle, Robin. Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 (1992), no. 1, 125–136.
  • Lyons, Russell. Probability on trees and networks. (2005) Book
  • Mandelbrot, Benoit. Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire: quelques extensions. C. R. Acad. Sci. Paris Sér. A 278 (1974), 355–358.
  • Menshikov, Mikhail; Petritis, Dimitri. On random walks in random environment on trees and their relationship with multiplicative chaos. Mathematics and computer science, II (Versailles, 2002), 415–422, Trends Math., Birkhäuser, Basel, 2002.
  • Neveu, J. Arbres et processus de Galton-Watson. (French) [Galton-Watson trees and processes] Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 2, 199–207.
  • Peres, Yuval; Zeitouni, Ofer. A central limit theorem for biased random walks on Galton-Watson trees. Probab. Theory Related Fields 140 (2008), no. 3-4, 595–629.
  • Petrov, V. V. Sums of independent random variables. Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, 1975. x+346 pp.
  • Petrov, Valentin V. Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability, 4. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. xii+292 pp. ISBN: 0-19-853499-X
  • Zeitouni, Ofer. Random walks in random environment. Lectures on probability theory and statistics, 189–312, Lecture Notes in Math., 1837, Springer, Berlin, 2004.