Acta mathematica scientia,Series B ›› 2013, Vol. 33 ›› Issue (6): 1736-1748.doi: 10.1016/S0252-9602(13)60119-8

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RENEWAL THEOREM FOR (L, 1)-RANDOM WALK IN RANDOM ENVIRONMENT

 HONG Wen-Ming, SUN Hong-Yan*   

  1. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
  • Received:2012-06-19 Revised:2012-11-06 Online:2013-11-20 Published:2013-11-20
  • Contact: SUN Hong-Yan,sunhy@mail.bnu.edu.cn E-mail:wmhong@bnu.edu.cn; sunhy@mail.bnu.edu.cn
  • Supported by:

    The project is supported by NSFC(11131003), 985-Project.

Abstract:

We consider a random walk on Z in random environment with possible jumps {−L, · · · , −1, 1}, in the case that the environment {ωi : i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of “the environment viewed from the particle” in both annealed probability and quenched probability, which generalize partially the results of Kesten (1977) and Lalley (1986) for the nearest random walk in random environment on Z, respectively. Our method is based on the intrinsic branching structure within the (L, 1)-RWRE formulated in Hong and Wang (2013).

Key words: random walk in random environment, renewal theorem, multitype branching process in random environment, coupling

CLC Number: 

  • 60K37
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