Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T21:14:00.620Z Has data issue: false hasContentIssue false
  • English
  • Français

The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment

Part of: Combinatorics Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Jean Bérard
Jean Bérard*
Affiliation:
Université Claude Bernard Lyon 1
*
∗ Postal address: Laboratoire de Probabilités, Combinatoire et Statistique, Université Claude Bernard Lyon 1, 50, avenue Tony Garnier, 69366 Lyon Cedex 07, France. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabei et al. for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for an arbitrary level of randomness.

Keywords

Random walkrandom environmentcentral limit theoremgenerating function

MSC classification

Primary: 60F05: Central limit and other weak theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K37: Processes in random environments 05A15: Exact enumeration problems, generating functions
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 83 - 92
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bernabei, M. S. (2001). Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media. Prob. Theory Relat. Fields 119, 410–432.Google Scholar
Bernabei, M. S., Boldrighini, C., Minlos, R. A., and Pellegrinotti, A. (1998). Almost-sure central limit theorem for a model of random walk in fluctuating random environment. Markov Process. Relat. Fields 4, 381–393.Google Scholar
Boldrighini, C., and Pellegrinotti, A. (2001). T−1/4 -noise for random walks in dynamic environment on ℤ. Moscow Math. J. 1, 365–380.Google Scholar
Boldrighini, C., Minlos, R. A., and Pellegrinotti, A. (2000). Random walk in a fluctuating random environment with Markov evolution. In On Dobrushin's Way. From Probability Theory to Statistical Physics (AMS Transl. Ser. 198), American Mathematical Society, Providence, RI, pp. 13–35.Google Scholar
Conlon, J. G., and Song, R. (1999). Gaussian limit theorems for diffusion processes and an application. Stoch. Process. Appl. 81, 103–128.CrossRefGoogle Scholar
Dembo, A., and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38), 2nd edn. Springer, New York.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Hughes, B. (1996). Random Walks and Random Environments, Vol. 2, Random Environments. Oxford University Press.Google Scholar