Diophantine distributions and local limit theorem on Rd

• 成果类型: Article
• 署名作者: Breuillard, E
• 署名单位: Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0388-1
• 发表日期: 2005
• 页码: 39-73
• 关键词: renewal theory; large deviations

摘要

We study the speed of convergence of n(d/2) integral fd mu(*n) in the local limit theorem on R-d under very general conditions upon the function f and the distribution mu. 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 Holder. We finally give an application to equidistribution of random walks.