关键词: random walk in random environment, renewal theorem, multitype branching process in random environment, coupling

Abstract:

We consider a random walk on Z in random environment with possible jumps {−L, · · · , −1, 1}, in the case that the environment {ω_i : 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