Self-normalized Moderate Deviations for Random Walk in Random Scenery

Let $$\{S_k:k\ge 0\}$$ { S k : k ≥ 0 } be a symmetric and aperiodic random walk on $$\mathbb {Z}^d$$ Z d , $$d\ge 3$$ d ≥ 3 , and $$\{\xi (z),z\in \mathbb {Z}^d\}$$ { ξ ( z ) , z ∈ Z d } a collection of independent and identically distributed random variables. Consider a random walk in random scenery defined by $$T_n=\sum _{k=0}^n\xi (S_k)=\sum _{z\in \mathbb {Z}^d}l_n(z)\xi (z)$$ T n = ∑ k = 0 n ξ ( S k ) = ∑ z ∈ Z d l n ( z ) ξ ( z ) , where $$l_n(z)=\sum _{k=0}^nI{\{S_k=z\}}$$ l n ( z ) = ∑ k = 0 n I { S k = z } is the local time of the random walk at the site z. Using $$(\sum _{z\in \mathbb {Z}^d}l_n(z)|\xi (z)|^p)^{1/p}$$ ( ∑ z ∈ Z d l n ( z ) | ξ ( z ) | p ) 1 / p , $$p\ge 2$$ 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.