Diophantine distributions and local limit theorem on Rd

被引:0
作者
Breuillard, E [1 ]
机构
[1] Ecole Normale Super, DMA, Paris, France
关键词
D O I
10.1007/s00440-004-0388-1
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
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.
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页码:39 / 73
页数:35
相关论文
共 22 条
[1]  
AMOSOVA NN, 1974, LIT MAT SB, V14, P401
[2]  
[Anonymous], 1968, Probability
[3]  
CARLSSON H, 1982, COMPOS MATH, V46, P227
[4]   REMAINDER TERM ESTIMATES OF THE RENEWAL FUNCTION [J].
CARLSSON, H .
ANNALS OF PROBABILITY, 1983, 11 (01) :143-157
[5]  
CRAMER H, 1962, ELEMENTS PROBABILITY
[6]  
Cramer H., 1938, Les Sommes et Les Fonctions de Variables Aleatoires
[7]  
Feller W., 1971, An introduction to probability theory and its applications, VII
[8]  
Gnedenko B., 1954, LIMIT DISTRIBUTIONS
[9]   REMARKS ON DIRECTLY RIEMANN INTEGRABLE FUNCTIONS [J].
HINDERER, K .
MATHEMATISCHE NACHRICHTEN, 1987, 130 :225-230
[10]  
HOGLUND T, 1988, B SCI MATH, V112, P111