Advances in Applied Probability

A comparison of random walks in dependent random environments

Werner R. W. Scheinhardt and Dirk P. Kroese

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We provide exact computations for the drift of random walks in dependent random environments, including k-dependent and moving average environments. We show how the drift can be characterized and evaluated using Perron-Frobenius theory. Comparing random walks in various dependent environments, we demonstrate that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment.

Article information

Adv. in Appl. Probab., Volume 48, Number 1 (2016), 199-214.

First available in Project Euclid: 8 March 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60G50: Sums of independent random variables; random walks
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Random walk dependent random environment drift Perron-Frobenius eigenvalue


Scheinhardt, Werner R. W.; Kroese, Dirk P. A comparison of random walks in dependent random environments. Adv. in Appl. Probab. 48 (2016), no. 1, 199--214.

Export citation


  • Alili, S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Prob. 36, 334–349.
  • Bean, N. G. et al. (1997). The quasi-stationary behavior of quasi-birth-and-death processes. Ann. Appl. Prob. 7, 134–155.
  • Brereton, T. et al. (2012). Efficient simulation of charge transport in deep-trap media. In Proc. 2012 Winter Simulation Conference (Berlin), IEEE, New York, pp. 1–12.
  • Chernov, A. A. (1962). Replication of multicomponent chain by the `lighting mechanism'. Biophysics 12, 336–341.
  • Dolgopyat, D., Keller, G. and Liverani, C. (2008). Random walk in Markovian environment. Ann. Prob. 36, 1676–1710.
  • Greven, A. and den Hollander, F. (1994). Large deviations for a random walk in random environment. Ann. Prob. 22, 1381–1428.
  • Hughes, B. D. (1996). Random Walks and Random Environments, Vol. 2. Oxford University Press.
  • Kesten, H., Kozlov, M. W. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compositio Math. 30, 145–168.
  • Kozlov, S. M. (1985). The method of averaging and walks in inhomogeneous enviroments. Russian Math. Surveys 40, 73–145.
  • Mayer-Wolf, E., Roitershtein, A. and Zeitouni, O. (2004). Limit theorems for one-dimensional transient random walks in Markov environments. Ann. Inst. H. Poincaré Prob. Statist. 40, 635–659.
  • Révész, P. (2013). Random Walk in Random and Non-Random Environments, 3rd edn. World Scientific, Hackensack, NJ.
  • Scheinhardt, W. R. W. and Kroese, D. P. (2014). Computing the drift of random walks in dependent random environments. Preprint. Available at
  • Sinai, Y. G. (1983). The limiting behavior of a one-dimensional random walk in a random medium. Theory Prob. Appl. 27, 256–268.
  • Solomon, F. (1975). Random walks in a random environment. Ann. Prob. 3, 1–31.
  • Stenzel, O. et al. (2014). A general framework for consistent estimation of charge transport properties via random walks in random environments. Multiscale Model. Simul. 12, 1108–1134.
  • Sznitman, A.-S. (2004). Topics in random walks in random environment. In School and Conference on Probability Theory (ICTP Lecture Notes XVII), Abdus Salem, Trieste, pp. 203–266.
  • Temkin, D. E. (1969). The theory of diffusionless crystal growth. J. Crystal Growth 5, 193–202.
  • Zeitouni, O. (2004). Part II: Random walks in random environment. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1837), Springer, Berlin, pp. 189–312.
  • Zeitouni, O. (2012). Random walks in random environment. In Computational Complexity, Springer, New York, pp. 2564–2577.