Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Simple transient random walks in one-dimensional random environment: the central limit theorem
Download PDF
Download PDF
  • Published: 22 November 2006

Simple transient random walks in one-dimensional random environment: the central limit theorem

  • Ilya Ya. Goldsheid1 

Probability Theory and Related Fields volume 139, pages 41–64 (2007)Cite this article

  • 203 Accesses

  • 27 Citations

  • Metrics details

Abstract

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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Alili, S: Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36, 334-349 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bolthausen, E., Sznitman, A.: Ten lectures on Random Media, DMV-Lectures, vol. 32, Birkhäuser, Basel (2002)

  3. Goldsheid, I.Ya.: Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a Strip (in preparation)

  4. Kesten, H., Kozlov, M.V., Spitzer, F.: Limit law for random walk in a random environment. Composito Mathematica 30, 145–168 (1975)

    MATH  MathSciNet  Google Scholar 

  5. Kozlov, M.V.: A random walk on a line with stochastic structure (in Russian). Probab. Theory Appl. 18, 406-408 (1973)

    Google Scholar 

  6. Mayer-Wolf, E., Roitershtein, A., Zeitouni, O.: Limit theorems for one-dimensional random walks in Markov random environments, Arxiv preprint math.0308154, 2003—arxiv.org

  7. Sinai, Ya.G.: The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27, 256–268 (1982)

    Article  MathSciNet  Google Scholar 

  8. Solomon, F.: Random walks in a random environment. Ann. Probab. 3, 1–31 (1975)

    MATH  Google Scholar 

  9. 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)

  10. 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)

Download references

Author information

Authors and Affiliations

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

    Ilya Ya. Goldsheid

Authors
  1. Ilya Ya. Goldsheid
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilya Ya. Goldsheid.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Goldsheid, I.Y. Simple transient random walks in one-dimensional random environment: the central limit theorem. Probab. Theory Relat. Fields 139, 41–64 (2007). https://doi.org/10.1007/s00440-006-0038-x

Download citation

  • Received: 31 May 2006

  • Revised: 27 September 2006

  • Published: 22 November 2006

  • Issue Date: September 2007

  • DOI: https://doi.org/10.1007/s00440-006-0038-x

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • RWRE
  • Simple random walks
  • Quenched random environments
  • Central limit theorem

Mathematics Subject Classification (2000)

  • Primary 60K37
  • Primary 60F05
  • Secondary 60J05
  • Secondary 82C44
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature