A Central Limit Theorem for Two-Dimensional Random Walks in Random Sceneries

Let Sₙ, n ∈ N, be a recurrent random walk on Z² (S₀ = 0) and ξ(α), α ∈ Z², be i.i.d. R-valued centered random variables. It is shown that ∑ⁿi = 1ξ(Sᵢ)/ √n log n satisfies a central limit theorem. A functional version is presented.