Self-normalized Moderate Deviations for Random Walk in Random Scenery

Let { S k : k ≥ 0 } be a symmetric and aperiodic random walk on Z d , d ≥ 3 , and { ξ ( z ) , z ∈ Z d } a collection of independent and identically distributed random variables. Consider a random walk in random scenery defined by T n = ∑ k = 0 n ξ ( S k ) = ∑ z ∈ Z d l n ( z ) ξ ( z ) , where l n ( z ) = ∑ k = 0 n I { S k = z } is the local time of the random walk at the site z . Using ( ∑ z ∈ Z d l n ( z ) | ξ ( z ) | p ) 1 / p , p ≥ 2 , as the normalizing constants, we establish self-normalized moderate deviations for random walk in random scenery under a much weaker condition than a finite moment-generating function of the scenery variables.