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

Menshikov, M. V. and Wade, Andrew R. (2008) 'Logarithmic speeds for one-dimensional perturbed random walks in random environments.', Stochastic processes and their applications., 118 (3). pp. 389-416.


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.

Item Type:Article
Keywords:Random walk in perturbed random environment, Logarithmic speeds, Almost sure behaviour, Slow transience.
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Publisher 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/
Date accepted:No date available
Date deposited:31 January 2013
Date of first online publication:March 2008
Date first made open access:No date available

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