The central limit theorem and Poincaré-type inequalities
The central limit theorem and Poincaré-type inequalities
复制标题
中心极限定理和庞加莱型不等式
DOI：
--
复制
发表时间：
1988
期刊：
影响因子：
--
通讯作者：
Louis H. Y. Chen
中科院分区：
文献类型：
--
作者：
Louis H. Y. Chen
文摘：利用Poincare型不等式证明了中心极限定理中Lindeberg条件的充分性和必要性。中心极限定理是概率统计中的一个基本定理。它指出，随着观测数量的增加，大量相互独立的小随机数值观测之和的概率分布接近正态分布。Lindeberg条件是中心极限定理成立的条件。这已被证明是必要的和充分的。庞加莱型不等式是一个将函数的平方积分与其导数的平方积分联系起来的不等式。本文利用Poincare型不等式证明Lindeberg条件的必要性和充分性，给出了中心极限定理的一个新的证明。
Abstract : We use Poincare type inequalities to prove the sufficiency and necessity of the Lindeberg condition in the central limit theorem. The central limit theorem is a fundamental theorem in probability and statistics. It states that the probability distribution of the sum of a large number of small and mutually independent random numerical observations approaches a normal distribution as the number of observations increases. The Lindeberg condition is a condition for which the central limit theorem holds. It has been proved to be both necessary and sufficient. A Poincare type inequality is an inequality which relates the integral of the square of a function to the integral of the square of its derivative. In this report we give a new proof of the central limit theorem by using Poincare type inequalities to prove both the necessity and sufficiency of the Lindeberg condition.