Random walks in Euclidean space

Abstract

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.

62 pages, 1 figure, revision based on referee's report, proofs and results unchanged

References in corpus (1)

• Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts

Cited by in corpus (3)

• Random walks in the group of Euclidean isometries and self-similar measures
• The local limit theorem on nilpotent Lie groups
• Anticoncentration in Ramsey graphs and a proof of the Erdős-McKay conjecture