A LOCAL LIMIT THEOREM FOR RANDOM WALKS IN RANDOM SCENERY AND ON RANDOMLY ORIENTED LATTICES

摘要

Random walks in random scenery are processes defined by Z_n := Σ_(k=1)~n ξ_(X_1+···+X_k), where (X_k, 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) 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~δx」), when x ∈ R is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.