On The Limit Theorems of Probability Theory
On The Limit Theorems of Probability Theory
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DOI：
10.1007/978-94-011-2260-3_16
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发表时间：
1992
期刊：
影响因子：
--
通讯作者：
A. Shiryayev
中科院分区：
文献类型：
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作者：
A. Shiryayev
众所周知，P.Laplace[1]和P.L.Chebyshev[2]的基本思想随后在A.A.马尔科夫[3]和A.M.Lyapunov[4]的论文中得到了发展，最终得出了一个关于大量小的独立随机变量和的概率分布极限的非常一般的陈述(所谓的Lyapunov定理)。马尔科夫[5]和S.N.Bernshtein[6]的进一步研究表明，在许多情况下，类似的陈述也适用于独立变量的和。这些推广对于应用目的特别重要，但原则上它们不会走得太远；在这些作者研究的所有情况下，只有彼此接近的和是强相关的，而如果和被分解成足够长的部分和，则后几个和将几乎是独立的。更重要的结果是将Lyapunov定理在二维(和多维)推广到随机向量和的情况，这是由Bernshtein[6]首先严格证明的。
As is well known, the fundamental ideas of P. Laplace [1] and P.L. Chebyshev [2] were subsequently developed in papers by A.A. Markov [3] and A.M. Lyapunov [4], culminating in a very general statement (the so-called Lyapunov theorem) on the limit of probability distributions for sums of a large number of small independent random variables. Further studies by Markov [5] and S.N. Bernshtein [6] have demonstrated that in many cases a similar statement also holds for sums of independent variables. These generalizations are of special importance for applied purposes, but in principle they do not go much further; in all cases studied by these authors only summands that are near to each other are strongly dependent, whereas if the sum is decomposed into sufficiently long partial sums, the later ones will be almost independent. Of much greater consequence is the two- (and multi-) dimensional generalization of Lyapunov’s theorem to the case of sums of random vectors, which was first rigorously proved by Bernshtein [6].