Limit theorems of probability theory: Sequences of independent random variables, by
Limit theorems of probability theory: Sequences of independent random variables, by
复制标题
DOI：
10.2307/2291698
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
V. V. Petrov-V.
中科院分区：
文献类型：
--
作者：
V. V. Petrov-V.
统计独立性的简单概念是概率论中许多重要内容的核心。首先是De Moivre和拉普拉斯的经典中心极限定理（CLT），在Paul Lévy的最终形式中，它说n个独立同分布（i.i.d.）具有有限二阶矩的随机变量Xj（1 ≤ j ≤ n）当n→∞时渐近高斯或正态。然后是稳定的法律。如果不假设二阶矩的有限性，在标度序列an和bn > 0下an+bnSn的可能（非退化）极限定律是什么？限制不需要存在。但如果它是，比如说Q，那么它显然应该是“稳定的”：无论n ≥ 1，如果Xj是独立同分布的。对于分布Q，则存在an和bn > 0，使得an+ bnSn具有分布Q。对称化稳定定律的特征函数或傅里叶变换的形式为exp{-B|不|}，其中B > 0，α是稳定律的“指数”，0 0和一个有限的绝对三阶矩ρ3，即Sn/n 1/2的分布函数Fn（x）之间的差
The simple notion of statistical independence lies at the core of much that is important in probability theory. First there was the classical central limit theorem (CLT) of De Moivre and Laplace which, in its final form due to Paul Lévy, says that the sum Sn of n independent and identically distributed (i.i.d.) random variables Xj (1 ≤ j ≤ n) having finite second moments is asymptotically Gaussian, or normal, as n→∞. Then came the stable laws. If one did not assume finiteness of second moments, what could be the possible (nondegenerate) limit laws of an+bnSn under scaling sequences an and bn > 0? The limit need not exist. But if it does, say Q, then it should clearly be “stable”: whatever be n ≥ 1, if Xj are i.i.d. with distribution Q, then there exist an and bn > 0 such that an+ bnSn has distribution Q. The characteristic function, or the Fourier transform, of a symmetrized stable law is of the form exp{−b|t|}, where b > 0, and α is the “index” of the stable law, 0 0 and a finite absolute third moment ρ3, the difference between the distribution function Fn(x) of Sn/n 1/2