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Random walk in random environment with asymptotically zero perturbation

Menshikov, Mikhail V. and Wade, Andrew (2006) Random walk in random environment with asymptotically zero perturbation. Journal of the European Mathematical Society, 8 (3). pp. 491-513. ISSN 1435-9855

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We give criteria for ergodicity, transience and null recurrence for the random walk in random environment on \Z+={0,1,2,…}, with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different to these previously studied cases. Our method is based on a martingale technique - the method of Lyapunov functions.