2011/03/23 by Castell, Fabienne, Guillotin--Plantard, Nadine, Pène, Françoise
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1103.4453
Random walks in random scenery are processes defined by Zn:=∑k=1nξX1+...+Xk, where (Xk,k≥ 1) and (ξy,y∈\mathbb Zd) are two independent sequences of i.i.d. random variables with values in \mathbb Zd and \mathbb 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 \citeKestenSpitzer and a local limit theorem in \citeBFFN. 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 \citebolthausen and recently in \citeDU in the particular case where β=2 (respectively for α=d=2 and α=d=1).