Self-normalized Moderate Deviations for Random Walk in Random Scenery

Abstract

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