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Simple random walk on the line in random environment
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  • Published: June 1986

Simple random walk on the line in random environment

  • Paul Deheuvels1 &
  • Pál Révész2,3 

Probability Theory and Related Fields volume 72, pages 215–230 (1986)Cite this article

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  • 19 Citations

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Summary

We obtain strong limiting bounds for the maximal excursion and for the maximum reached by a random walk in a random environment. Our results derive from a simple proof of Pólya's theorem for the recurrence of the random walk on the line. As applications, we obtain bounds for the number of visits of the random walk at the origin.

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References

  • Breim, L.: On the tail behaviour of sums of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.9, 21–25 (1967)

    Google Scholar 

  • Csáki, E.: On the lower limits of maxima and minima of Wiener process and partial sums. Z. Wahrscheinlichkeitstheor. Verw. Geb.43, 205–222 (1978)

    Google Scholar 

  • Csörgö, M., Révész, P.: Strong approximations in probability and statistics. New York: Academic Press 1981

    Google Scholar 

  • Erdös, P.: On the law of the iterated logarithm. Ann. Math.43, 419–436 (1942)

    Google Scholar 

  • Golosov, A.O.: Localisation of random walk in one-dimensional random environment. Comm. Math. Phys.92, 411–506 (1984)

    Google Scholar 

  • Hirsch, W.: A strong law for the maximum cumulative sum of independent random variables. Comm. Pure Appl. Math.18, 109–217 (1965)

    Google Scholar 

  • Itô, K., McKean, H.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965

    Google Scholar 

  • Pólya, G.: Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann.84, 149–160 (1921)

    Google Scholar 

  • Синай, Я.Г (Sina i, Ja. G.): Предемное поведение однолерного σлучийного длужданий в случайной среде (Limit behaviour of one dimensional random walks in random environment) Геория Ъероятностий и Применение (Theory of Probability and its Applications)27, 247–258 (1982).

    Google Scholar 

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

    Google Scholar 

  • Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 211–226 (1964)

    Google Scholar 

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Author information

Authors and Affiliations

  1. LSTA, t.45-55, Université Paris VI, E3, 4 Place Jussieu, F-75230, Paris Cedex 05, France

    Paul Deheuvels

  2. Mathematical Institute of the Hungarian Academy of Sciences, Realtanoda, U.13-15, Budapest V., Hungary

    Pál Révész

  3. Technische Universität Wien, Wiedner Hauptstrasse 6-10, A-1040, Wien, Austria

    Pál Révész

Authors
  1. Paul Deheuvels
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  2. Pál Révész
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Deheuvels, P., Révész, P. Simple random walk on the line in random environment. Probab. Th. Rel. Fields 72, 215–230 (1986). https://doi.org/10.1007/BF00699104

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  • Received: 18 February 1985

  • Accepted: 29 September 1985

  • Issue Date: June 1986

  • DOI: https://doi.org/10.1007/BF00699104

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Keywords

  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Mathematical Biology
  • Simple Proof
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