A local limit theorem for random walks in random scenery and on randomly oriented lattices

Abstract

Random walks in random scenery are processes defined by Z_n := ∑_k=1ⁿ ξ_X1+⋯+Xk, where (X_k, k ≥ 1) and (ξ_y, 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 Z_n, for some suitable δ > 0 depending on α and β. Here, we are interested in the convergence, as n → ∞, of n^δℙ(Z_n = ⌊n^δ x⌋), when x ∈ ℝ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.