Renewal Theorem for (L, 1)-Random Walk in Random Environment

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).