SOME LIMIT-THEOREMS IN LOG DENSITY

被引:60
作者
BERKES, I [1 ]
DEHLING, H [1 ]
机构
[1] UNIV GRONINGEN,DEPT MATH,9700 AV GRONINGEN,NETHERLANDS
关键词
PATHWISE CENTRAL LIMIT THEOREM; LOG-AVERAGING METHODS; STABLE CONVERGENCE; STRONG APPROXIMATION; LAW OF LARGE NUMBERS;
D O I
10.1214/aop/1176989135
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
Motivated by recent results on pathwise central limit theorems, we study in a systematic way log-average versions of classical limit theorems. For partial sums S(k) of independent r.v.'s we prove under mild technical conditions that (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)I{S(k)/a(k) is-an-element-of .} --> G(.) (a.s.) if and only if (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)P(S(k)/a(k) is-an-element-of .) --> G(.). A functional version of this result also holds. For partial sums of i.i.d. r.v.'s attracted to a stable law, we obtain a pathwise version of the stable limit theorem as well as a strong approximation by a stable process on log dense sets of integers. We also give necessary and sufficient conditions for the law of large numbers in log density.
引用
收藏
页码:1640 / 1670
页数:31
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