Abstract.
We study the speed of convergence of nd/2∫fdμ*n in the local limit theorem on under very general conditions upon the function f and the distribution μ. We show that this speed is at least of order 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.
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Breuillard, E. Distributions diophantiennes et théorème limite local sur . Probab. Theory Relat. Fields 132, 13–38 (2005). https://doi.org/10.1007/s00440-004-0388-1
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DOI: https://doi.org/10.1007/s00440-004-0388-1