Tail estimates for one-dimensional non nearest-neighbor random walk in random environment

Suppose that the integers are assigned i.i.d. random variables {(βxg,…,βx1,αx} (each taking values in the unit interval and the sum of them being 1), which serve as an environment. This environment defines a random walk {Xx} (called RWRE) which, when at x, moves one step of length 1 to the right with probability αx and one step of length k to the left with probability βxk for 1 ⩽ k ⩽ g. For certain environment distributions, we determine the almost-sure asymptotic speed of the RWRE and show that the chance of the RWRE deviating below this speed has a polynomial rate of decay. This is the generalization of the results by Dembo, Peres and Zeitouni in 1996. In the proof we use a large deviation result for the product of random matrices and some tail estimates and moment estimates for the total population size in a multi-type branching process with random environment.