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
Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip
Download PDF
Download PDF
  • Published: 11 August 2007

Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip

  • Ilya Ya. Goldsheid1 

Probability Theory and Related Fields volume 141, pages 471–511 (2008)Cite this article

  • 172 Accesses

  • Metrics details

Abstract

The main goal of this work is to study the asymptotic behaviour of hitting times of a random walk (RW) in a quenched random environment (RE) on a strip. We introduce enlarged random environments in which the traditional hitting time can be presented as a sum of independent random variables whose distribution functions form a stationary random sequence. This allows us to obtain conditions (stated in terms of properties of random environments) for a linear growth of hitting times of relevant random walks. In some important cases (e.g. independent random environments) these conditions are also necessary for this type of behaviour. We also prove the quenched Central Limit Theorem (CLT) for hitting times in the general ergodic setting. A particular feature of these (ballistic) laws in random environment is that, whenever they hold under standard normalization, the convergence is a convergence with a speed. The latter is due to certain properties of moments of hitting times which are also studied in this paper. The asymptotic properties of the position of the walk are stated but are not proved in this work since this has been done in Goldhseid (Probab. Theory Relat. Fields 139(1):41–64, 2007).

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Atkinson F.V. (1964). Discrete and Continuous Boundary Problems. Academic, New York

    MATH  Google Scholar 

  3. Bolthausen E. and Goldsheid I. (2000). Recurrence and transience of random walks in random environments on a strip. Commun. Math. Phys. 214: 429–447

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  5. Brémont J. (2002). On some random walks on \({\mathbb{Z}}\) in random medium Ann. Probab. 30: 1266–1312

    Article  MathSciNet  MATH  Google Scholar 

  6. Brémont J. (2004). Random walks on \({\mathbb{Z}}\) in random medium and Lyapunov spectrum Ann. Inst. H. Poincare Prob/Stat 40: 309–336

    MATH  Google Scholar 

  7. Derriennic Y. (1999). Sur la récurrence des marches aléatoires unidimensionnelles en environement aléatoire. C. R. Acad. Sci. Paris Sér. I Math. 329(1): 65–70

    MathSciNet  MATH  Google Scholar 

  8. Furstenberg H. and Kesten H. (1960). Products of random matrices. Ann. Math. Stat. 31: 457–469

    Article  MathSciNet  MATH  Google Scholar 

  9. Goldhseid I. (2007). Simple transient random walks in one-dimensional random environment. Probab. Theory Relat. Fields 139(1): 41–64

    Article  Google Scholar 

  10. Guivarc’h, Y., Le Page, E.: Simplicité de spectres de Lyapunov et propriété d’isolation spectrale pour une famille d’opérateurs sur l’espace projectif. In: Kaimanovitch, V. (ed.) Random walks and Geometry, pp. 181–259, De Gruyter (2004)

  11. Kesten H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131: 207–248

    Article  MathSciNet  MATH  Google Scholar 

  12. Kesten H., Kozlov M.V. and Spitzer F. (1975). Limit law for random walk in a random environment. Compos. Math. 30: 145–168

    MathSciNet  MATH  Google Scholar 

  13. Key E. (1984). Recurrence and transience criteria for a random walk in random environment. Ann. Probab. 12: 529–560

    Article  MathSciNet  MATH  Google Scholar 

  14. Kifer Y. (1996). Perron-Frobenius theorem, large deviations and random pertubations in random environments. Math. Zeitschrift 222: 677–698

    MathSciNet  MATH  Google Scholar 

  15. Kifer Y. (1998). Limit theorems for random transformations and processes in random environments. Trans. Am. Math. Soc. 348: 2003–2038

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  17. Ledrappier, F.: Quelques propriétés des exposants caractéristiques, Ecole d‘Eté de Saint-Flour 1982, Lecture Notes in Mathematics, vol. 1097, pp. 305–396. Springer, Berlin (1984)

  18. Letchikov A.V. (1993). A criterion for linear drift and the central limit theorem for one-dimensional random walks in a random environment. Russ. Acad. Sci. Sb. Math. 79: 73–92

    Article  MathSciNet  Google Scholar 

  19. Mayer-Wolf E., Roitershtein A. and Zeitouni O. (2004). Limit theorems for one-dimensional random walks in Markov random environments. Ann. Inst. H. Poincare Prob/Stat 40: 635–659

    Article  MathSciNet  MATH  Google Scholar 

  20. Molchanov, S.A.: Lectures on random media. Ecole d’Eté de Prbabilités de St. Flour XXII-1992. Lecture Notes in Mathematics, vol. 1581. Springer, Berlin (1994)

  21. Roitershtein, A.: Tranzient random walks on a strip in a random environment. Preprint (2006)

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

    Article  MathSciNet  Google Scholar 

  23. Sinai Ya.G. (1999). Simple random walk on tori. J. Stat. Phys. 94(3–4): 695–708

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  25. Sznitman A.-S. and Zerner M.P.W. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27(4): 1851–1869

    Article  MathSciNet  MATH  Google Scholar 

  26. Sznitman A.-S. (2000). Slowdown and central limit theorem for random walks in random environment. J. Eur. Math. Soc. 2: 93–143

    Article  MathSciNet  MATH  Google Scholar 

  27. Sznitman A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Relat. Fields 122(4): 509–544

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

  29. 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 (2004)

Download references

Author information

Authors and Affiliations

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

    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. Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip. Probab. Theory Relat. Fields 141, 471–511 (2008). https://doi.org/10.1007/s00440-007-0091-0

Download citation

  • Received: 04 September 2006

  • Revised: 11 June 2007

  • Published: 11 August 2007

  • Issue Date: July 2008

  • DOI: https://doi.org/10.1007/s00440-007-0091-0

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
  • Random walks on a strip
  • Quenched random environments
  • Central limit theorem

Mathematics Subject Classification (2000)

  • Primary: 60K37
  • 60F05
  • Secondary: 60J05
  • 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