Distributions diophantiennes et théorème limite local sur ℝd

Abstract

We study the speed of convergence of n d/2 ∫ fd μ *n in the local limit theorem on ℝd under very general conditions upon the function f and the distribution μ. We show that this speed is at least of order 1/n and we give a simple characterization (in diophantine terms) of those measures for which this speed (and the full local Edgeworth expansion) holds for smooth enough f. We then derive a uniform local limit theorem for moderate deviations under a mild moment assumption. This in turn yields other limit theorems when f is no longer assumed integrable but only bounded and Lipschitz or Hölder. We finally give an application to equidistribution of random walks. © Springer-Verlag 2004.