Journal of Statistical Physics

, Volume 160, Issue 2, pp 357–370 | Cite as

Application of Moderate Deviation Techniques to Prove Sinai Theorem on RWRE

  • Marcelo Ventura Freire


We apply the techniques developed in Comets and Popov (Probab Theory Relat Fields 126:571–609, 2003) to present a new proof to Sinai theorem Sinai (Theory Probab Appl 27:256–268, 1982) on one-dimensional random walk in random environment (RWRE), working in a scale-free way to avoid rescaling arguments and splitting the proof in two independent parts: a quenched one, related to the measure \(P_{\omega }\) conditioned on a fixed, typical realization \(\omega \) of the environment, and an annealed one, related to the product measure \(\mathbb {P}\) of the environment \(\omega \). The quenched part still holds even if we use another measure (possibly dependent) for the environment.


Random walk Random environment Sinai walk Moderate deviations Metastability \(t\)-Stable point 

Mathematics Subject Classification

60K37 60G50 



The author wishes to express his gratitude to both reviewers for their valuable suggestions.


  1. 1.
    Andreoletti, P.: Alternative proof for the localization of Sinai’s walk. J. Stat. Phys. 118(5–6), 883–933 (2005). doi: 10.1007/s10955-004-2122-x MATHMathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Andreoletti, P.: On the concentration of Sinai’s walk. Stoch. Process. Appl. 116(10), 1377–1408 (2006). doi: 10.1016/ MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Andreoletti, P.: Almost sure estimates for the concentration neighborhood of Sinais walk. Stoch. Process. Appl. 117(10), 1473–1490 (2007). doi: 10.1016/ MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Berkes, I., Liu, W., Wu, W.B.: Komlós-Major-Tusnády approximation under dependence. Ann. Probab. 42(2), 794–817 (2014). doi: 10.1214/13-AOP850 MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Brox, T.: A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14(4), 1206–1218 (1986)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Comets, F., Menshikov, M., Popov, S.: Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26(4), 1433–1445 (1998)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Comets, F., Popov, S.: Limit law for transition probabilities and moderated deviations for Sinai’s random walk in random environment. Probab. Theory Relat. Fields 126, 571–609 (2003)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Golosov, A.O.: Localization of random walks in one-dimensional random environment. Commun. Math. Phys. 92, 491–506 (1984)MATHMathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV’s and the sample DF. I. Zeit. Wahrsch. verw. Geb. 32, 111–131 (1975)MATHCrossRefGoogle Scholar
  10. 10.
    Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV’s and the sample DF. II. Zeit. Wahrsch. verw. Geb. 34, 33–58 (1976)MATHCrossRefGoogle Scholar
  11. 11.
    Mathieu, P.: Zero White Noise Limit through Dirichlet forms, with application to diffusions in random medium. Probab. Theory Relat. Fields 99, 549–580 (1994)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Sinai, Y.G.: The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27, 256–268 (1982)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Solomon, F.: Random walks in random environments. Ann. Probab. 3, 1–31 (1975)MATHCrossRefGoogle Scholar
  14. 14.
    Zeitouni, O.: Random walk in random environment. In: Picard, J. (ed.) Lectures on Probability Theory and Statistics. Ecole d’Eté de Probabilité de Saint-Flour XXXI, Lecture Notes in Mathematics, vol. 1837, pp. 190–312. Springer, Berlin (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Escola de Artes, Ciências e HumanidadesUniversidade de São Paulo (EACHUSP)São PauloBrazil

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