A principle of subsequences in probability theory: The central limit theorem
A principle of subsequences in probability theory: The central limit theorem
复制标题
概率论中的子序列原理：中心极限定理
DOI：
10.1016/0001-8708(74)90064-4
复制
发表时间：
1974
影响因子：
1.7
通讯作者：
S. D. Chatterji
中科院分区：
文献类型：
--
作者：
S. D. Chatterji
本文的目的是详细地证明我们最近有机会提出的下列子序列的一般原理[5]，并在几个地方证明了它的适当性[2，41]。所讨论的一般原理可以表示如下：如果某一数量渐近性质n对于任何独立的、同分布的随机变量序列X是有效的，该序列属于由范数11*[IL]的有限性确定的某个可积类，则类似性质+将对任何概率空间上的函数序列F的适当子序列{f，}有效，使得sup{(]fIIL：f)<；co.此外，可以以这样的方式选择子序列，即任何进一步的子序列将具有相同的属性+。当7r是Kolmogorov强大数定律时，该原理的有效性对应于Komlbs[8]的一个显著定理，该定理指出，从L1中的任何范数有界序列F，可以提取F的任意子序列F，使得F的任意子序列f，将具有LIM，，，(f，+*.*+f，)/n的性质，并且等于函数01 E L1，该函数与所考虑的F的特定子序列Fn无关。(自然，01取决于对F，，)的选择。这里的属性在两个方面不同于v。首先，极限函数01不一定是常数，其次，不保证(fi+**.+f，)/n在L1中收敛到OL。然而，事实仍然是，Komlbs定理是刚才提到的后继原理的一个清楚的例证。我们已经在其他地方证明了[2，4]，如果r是科尔莫戈洛夫强大定律的Marcinkiewicz推广，则31
The purpose of this article is to establish in detail an important verification of the following general principle of subsequences which we have had occasion to formulate recently [5] and the appropriateness of which we have demonstrated at several places [2, 41. The general principle in question can be formulated as follows: if a certain quantitative asymptotic property n is valid for any sequence of independent, identically distributed random variables X, belonging to some integrability class determined by the finiteness of a norm 11*[IL, an analogous property+ will be valid for a suitable subsequence {f,} of any sequence F of functions on any probability space such that sup {(] fIIL: f Ef)< co. Moreover, the subsequence can be chosen in such a way that any further subsequence will have the same property+. When 7r is the Kolmogorov strong law of large number, the validity of the principle corresponds to a remarkable theorem of Komlbs [8] which states that from any norm-bounded sequence F in L1, a subsequence F, can be so extracted that any subsequence f, of F. will have the property that lim,,,(f,+*.*+ f,)/n exists ae and equals a function 01 E L1 which is independent of the particular subsequence fn of F,, under consideration.(Naturally, 01 depends on the choice of F,,). The property here differs from v in two respects. First, the limit function 01 is not necessarily a constant and second, the convergence of (fi+**.+ f,)/n to OL in L1 is not guaranteed. The fact remains, however, that Komlbs’ theorem is a clear illustration of the subsequence principle just mentioned. We have shown elsewhere [2, 4] that if r is taken to be the Marcinkiewicz generalization of Kolmogorov’s strong law, then 31