Moderate deviations for the self-normalized random walk in random scenery

Abstract

Let G be an infinite connected graph with vertex set V. Let $\{S_n:n\in {\mathbb{N}}_0\}$ be the simple random walk on G and let $\{\mathit{\xi }(v):v\in V\}$ be a collection of i.i.d. random variables which are independent of the random walk. Define the random walk in random scenery as $T_n={\sum }_{k=0}^n\mathit{\xi }(S_k)$, and the normalization variables $V_n={({\sum }_{k=0}^n{\mathit{\xi }}^2(S_k))}^{1∕2}$ and $L_{n,2}={({\sum }_{v\in V}{\ell }_n^2(v))}^{1∕2}$. For $G={\mathbb{Z}}^d$ and $G={\mathbb{T}}_d$, the d-ary tree, we provide large deviations results for the self-normalized process $T_n\sqrt{n}∕(L_{n,2}{V}_n)$ under only finite moment assumptions on the scenery.