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Logarithmic speeds for one-dimensional perturbed random walks in random environments

Menshikov, M.V.; Wade, Andrew R.

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Abstract

We study the random walk in a random environment on Z+={0,1,2,…}, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai’s regime; (ii) a simple random walk with a random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order (logt)β, for β∈(1,∞), depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.

Citation

Menshikov, M., & Wade, A. R. (2008). Logarithmic speeds for one-dimensional perturbed random walks in random environments. Stochastic Processes and their Applications, 118(3), 389-416. https://doi.org/10.1016/j.spa.2007.04.011

Journal Article Type Article
Publication Date Mar 1, 2008
Deposit Date Mar 1, 2011
Publicly Available Date Jan 31, 2013
Journal Stochastic Processes and their Applications
Print ISSN 0304-4149
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 118
Issue 3
Pages 389-416
DOI https://doi.org/10.1016/j.spa.2007.04.011
Keywords Random walk in perturbed random environment, Logarithmic speeds, Almost sure behaviour, Slow transience.

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Copyright Statement
NOTICE: this is the author’s version of a work that was accepted for publication in Stochastic processes and their applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Stochastic processes and their applications, 118(3), 2008, 10.1016/j.spa.2007.04.011




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