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

Abstract

Let G be an infinite connected graph with vertex set V. Let {Sn:n∈N0} be the simple random walk on G and let {ξ(v):v∈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 Tn= ∑k=0nξ(Sk), and the normalization variables Vn=(∑k=0nξ2(Sk))1∕2 and Ln,2=(∑v∈Vln2(v))1∕2. For G=Zd and G=Td, the d-ary tree, we provide large deviations results for the self-normalized process Tnn∕(L n,2Vn) under only finite moment assumptions on the scenery.