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

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