Abstract

Fix a probability measure on the space of isometries of Euclidean space Rd. Let Y0 = 0, Y1, Y2,... ∈ Rd be a sequence of random points such that Yl+1 is the image of Yl under a random isometry of the pre-viously fixed probability law, which is independent of Yl. We prove a local limit theorem for Yl under necessary non-degeneracy conditions. Moreover, under more restrictive but still general conditions we give a quantitative estimate which describes the behavior of the law of Yl on scales e−cl1/4 < r < l1/2. 1