Limit theorems for one and two-dimensional random walks in random scenery

الأصل · EN

Random walks in random scenery are processes defined by Zₙ:=∑ₖ₌₁ⁿξₓ₁₊...₊ₓₖ, where (Xₖ,k≥ 1) and (ξy,y∈Zᵈ) are two independent sequences of i.i.d. random variables with values in Zᵈ and R respectively. We suppose that the distributions of X₁ and ξ₀ belong to the normal basin of attraction of stable distribution of index α∈(0,2] and β∈(0,2]. When d=1 and α≠ 1, a functional limit theorem has been established in KestenSpitzer and a local limit theorem in BFFN. In this paper, we establish the convergence of the finite-dimensional distributions and a local limit theorem when α=d (i.e. α= d=1 or α=d=2) and β∈ (0,2]. Let us mention that functional limit theorems have been established in bolthausen and recently in DU in the particular case where β=2 (respectively for α=d=2 and α=d=1).