Random walks in random scenery are processes defined by Zn := ∑k=1n ξX1+⋯+Xk, where (Xk, k ≥ 1) and (ξy, y ∈ ℤ) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2], respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when α ≠ 1 and as n → ∞, of n−δ Zn, for some suitable δ > 0 depending on α and β. Here, we are interested in the convergence, as n → ∞, of nδℙ(Zn = ⌊nδ x⌋), when x ∈ ℝ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.
"A local limit theorem for random walks in random scenery and on randomly oriented lattices." Ann. Probab. 39 (6) 2079 - 2118, November 2011. https://doi.org/10.1214/10-AOP606