Israel Journal of Mathematics

, Volume 142, Issue 1, pp 205–220 | Cite as

Random walks in random environment: What a single trajectory tells

  • Omer Adelman
  • Nathanaël Enriquez


We present a procedure that determines the law of a random walk in an iid random environment as a function of a single “typical” trajectory. We indicate when the trajectory characterizes the law of the environment, and we say how this law can be determined. We then show how independent trajectories having the distribution of the original walk can be generated as functions of the single observed trajectory.


Random Walk Random Environment Single Trajectory Independent Trajectory Random Scenery 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I. Benjamini and H. Kesten,Distinguishing sceneries by observing the scenery along a random walk path, Journal d’Analyse Mathématique69 (1996), 97–135.MathSciNetCrossRefGoogle Scholar
  2. [2]
    N. Enriquez and C. Sabot,Edge-oriented reinforced random walks and RWRE, Comptes Rendus de l’Académie des Sciences, Paris335 (2002), 941–946.MATHMathSciNetGoogle Scholar
  3. [3]
    M. Löwe and H. Matzinger,Scenery reconstruction in two dimensions with many colors, The Annals of Applied Probability12 (2002), 1322–1347.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    S. A. Kalikow,Generalized random walk in a random environment, The Annals of Probability9 (1981), 753–768.MathSciNetGoogle Scholar
  5. [5]
    M. Keane and S. Rolles,Tubular recurrence, Acta Mathematica Hungarica97 (2002), 207–221.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    H. Kesten, M. Kozlov and F. Spitzer,A limit law for random walk in a random environment, Compositio Mathematica30 (1975), 145–168.MATHMathSciNetGoogle Scholar
  7. [7]
    H. Matzinger,Reconstructing a three-color scenery by observing it along a simple random walk path, Random Structures & Algorithms15 (1999), 196–207.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    R. Pemantle,Phase transition in reinforced random walk and RWRE on trees, The Annals of Probability16 (1988), 1229–1241.MATHMathSciNetGoogle Scholar
  9. [9]
    S. Rolles,How edge-reinforced random walk arises naturally, Probability Theory and Related Fields126 (2003), 243–260.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Y. Sinai,The limit behavior of a one-dimensional random walk in a random environment, (Russian) Teoriya Veroyatnostei i Primeneniya27 (1982), 247–258.MathSciNetGoogle Scholar
  11. [11]
    F. Solomon,Random walks in a random environment, The Annals of Probability3 (1975), 1–31.MATHGoogle Scholar
  12. [12]
    S. Zabell,Characterizing Markov exchangeable sequences, Journal of Theoretical Probability8 (1995), 175–178.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2004

Authors and Affiliations

  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris 6Paris cedex 05France

Personalised recommendations