Find out how to access previewonly content
Date:
22 Nov 2006
Simple transient random walks in onedimensional random environment: the central limit theorem
 Ilya Ya. Goldsheid
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the central limit theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.
 Alili, S: Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36, 334349 (1999) CrossRef
 Bolthausen, E., Sznitman, A.: Ten lectures on Random Media, DMVLectures, vol. 32, Birkhäuser, Basel (2002)
 Goldsheid, I.Ya.: Linear and sublinear growth and the CLT for hitting times of a random walk in random environment on a Strip (in preparation)
 Kesten, H., Kozlov, M.V., Spitzer, F.: Limit law for random walk in a random environment. Composito Mathematica 30, 145–168 (1975)
 Kozlov, M.V.: A random walk on a line with stochastic structure (in Russian). Probab. Theory Appl. 18, 406408 (1973)
 MayerWolf, E., Roitershtein, A., Zeitouni, O.: Limit theorems for onedimensional random walks in Markov random environments, Arxiv preprint math.0308154, 2003—arxiv.org
 Sinai, Ya.G.: The limiting behavior of a onedimensional random walk in a random medium. Theory Probab. Appl. 27, 256–268 (1982) CrossRef
 Solomon, F.: Random walks in a random environment. Ann. Probab. 3, 1–31 (1975)
 Sznitman, A.S.: Topics in random walks in random environment. In: School and Conference on Probability Theory, ICTP Lecture Notes Series, Trieste, vol. 17, pp. 203–266 (2004)
 Zeitouni, O.: Random walks in random environment, XXXI Summer school in Probability, St. Flour (2001). Lecture Notes in Mathematics, vol. 1837, pp. 193–312. Springer, Berlin Heidelberg New York (2004)
 Title
 Simple transient random walks in onedimensional random environment: the central limit theorem
 Journal

Probability Theory and Related Fields
Volume 139, Issue 12 , pp 4164
 Cover Date
 20070901
 DOI
 10.1007/s004400060038x
 Print ISSN
 01788051
 Online ISSN
 14322064
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 RWRE
 Simple random walks
 Quenched random environments
 Central limit theorem
 Primary 60K37
 Primary 60F05
 Secondary 60J05
 Secondary 82C44
 Industry Sectors
 Authors

 Ilya Ya. Goldsheid ^{(1)}
 Author Affiliations

 1. School of Mathematical Sciences, Queen Mary, University of London, London, E1 4NS, Great Britain