# A note on quenched moderate deviations for Sinai’s random walk in random environment

ESAIM: Probability and Statistics (2004)

- Volume: 8, page 56-65
- ISSN: 1292-8100

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topComets, Francis, and Popov, Serguei. "A note on quenched moderate deviations for Sinai’s random walk in random environment." ESAIM: Probability and Statistics 8 (2004): 56-65. <http://eudml.org/doc/244669>.

@article{Comets2004,

abstract = {We consider the continuous time, one-dimensional random walk in random environment in Sinai’s regime. We show that the probability for the particle to be, at time $t$ and in a typical environment, at a distance larger than $t^a$ ($0<a<1$) from its initial position, is $\exp \lbrace -\{\rm Const\}\cdot t^a / [(1-a)\ln t] (1+o(1))\rbrace $.},

author = {Comets, Francis, Popov, Serguei},

journal = {ESAIM: Probability and Statistics},

keywords = {random walk in random environment; Sinai’s regime; $t$-stable point; moderate deviations; Random walk in random environment; Sinai's regime; -stable point},

language = {eng},

pages = {56-65},

publisher = {EDP-Sciences},

title = {A note on quenched moderate deviations for Sinai’s random walk in random environment},

url = {http://eudml.org/doc/244669},

volume = {8},

year = {2004},

}

TY - JOUR

AU - Comets, Francis

AU - Popov, Serguei

TI - A note on quenched moderate deviations for Sinai’s random walk in random environment

JO - ESAIM: Probability and Statistics

PY - 2004

PB - EDP-Sciences

VL - 8

SP - 56

EP - 65

AB - We consider the continuous time, one-dimensional random walk in random environment in Sinai’s regime. We show that the probability for the particle to be, at time $t$ and in a typical environment, at a distance larger than $t^a$ ($0<a<1$) from its initial position, is $\exp \lbrace -{\rm Const}\cdot t^a / [(1-a)\ln t] (1+o(1))\rbrace $.

LA - eng

KW - random walk in random environment; Sinai’s regime; $t$-stable point; moderate deviations; Random walk in random environment; Sinai's regime; -stable point

UR - http://eudml.org/doc/244669

ER -

## References

top- [1] F. Comets and S.Yu. Popov, Limit law for transition probabilities and moderate deviations for Sinai’s random walk in random environment. Probab. Theory Relat. Fields 126 (2003) 571-609. Zbl1027.60091
- [2] A. Greven and F. den Hollander, Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381-1428. Zbl0820.60054MR1303649
- [3] Y. Hu and Z. Shi, The limits of Sinai’s simple random walk in random environment. Ann. Probab. 26 (1998) 1477-1521. Zbl0936.60088
- [4] Y. Hu and Z. Shi, Moderate deviations for diffusions with Brownian potentials. (2003) Preprint PMA–792 available at http://www.proba.jussieu.fr/mathdoc/preprints/index.html#2003 Zbl1066.60096
- [5] B. Hughes, Random Walks and Random Environments. Vol. 2. Random Environments. The Clarendon Press, Oxford University Press, New York (1996). Zbl0925.60076MR1420619
- [6] Z. Shi, Sinai’s Walk via Stochastic Calculus, in Milieux Aléatoires, F. Comets and E. Pardoux Eds., Société Mathématique de France, Paris, Panoramas et Synthèses 12 (2001). Zbl1031.60088
- [7] A. Shiryaev, Probability. 2nd edn., Springer, New York (1989). Zbl0835.60002MR1368405
- [8] Ya.G. Sinai, The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27 (1982) 256-268. Zbl0505.60086MR657919
- [9] O. Zeitouni, Lecture Notes on Random Walks in Random Environment. (2003) Preliminary version at http://www-ee.technion.ac.il/~zeitouni/ps/notes1.ps MR2071631

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