Random walks in Euclidean space

Péter Pál Varjú

Published 2012-05-15, updated 2012-09-12Version 2

Consider a sequence of independent random isometries of Euclidean space with a previously fixed probability law. Apply these isometries successively to the origin and consider the sequence of random points that we obtain this way. We prove a local limit theorem under a suitable moment condition and a necessary non-degeneracy condition. Under stronger hypothesis, we prove a limit theorem on a wide range of scales: between e^(-cl^(1/4)) and l^(1/2), where l is the number of steps.