Random walks in Euclidean space

Abstract

Fix a probability measure on the space of isometries of Euclidean space Rd. Let Y0 = 0, Y1, Y2,. ε Rd be a sequence of random points such that Yl+1 is the image of Yl under a random isometry of the previously fixed probability law, which is independent of Yl. We prove a Local Limit Theorem for Yl under necessary nondegeneracy 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