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On the concentration of Sinai's walk

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  • Andreoletti, Pierre

Abstract

We consider Sinai's random walk in a random environment. We prove that for an interval of time [1,n] Sinai's walk sojourns in a small neighborhood of the point of localization for the quasi-totality of this amount of time. Moreover the local time at the point of localization normalized by n converges in probability to a well defined random variable of the environment. From these results we get applications to the favorite sites of the walk and to the maximum of the local time.

Suggested Citation

  • Andreoletti, Pierre, 2006. "On the concentration of Sinai's walk," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1377-1408, October.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:10:p:1377-1408
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    References listed on IDEAS

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    1. Mathieu, Pierre, 1995. "Limit theorems for diffusions with a random potential," Stochastic Processes and their Applications, Elsevier, vol. 60(1), pages 103-111, November.
    2. Hu, Yueyun, 2000. "Tightness of localization and return time in random environment," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 81-101, March.
    3. Kesten, Harry, 1986. "The limit distribution of Sinai's random walk in random environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 138(1), pages 299-309.
    4. Shi, Zhan, 1998. "A local time curiosity in random environment," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 231-250, August.
    5. Gantert, Nina & Shi, Zhan, 2002. "Many visits to a single site by a transient random walk in random environment," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 159-176, June.
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    Cited by:

    1. Andreoletti, Pierre, 2007. "Almost sure estimates for the concentration neighborhood of Sinai's walk," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1473-1490, October.

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