Bartlett分解定理的简便证明  

A Concise Proof of the Bartlett Decomposition Theorem

作　　者：朱道元[1] 赵桂芹[1] 

机构地区：[1]东南大学应用数学系,南京210096

出　　处：《东南大学学报（自然科学版）》2001年第5期125-127,共3页Journal of Southeast University：Natural Science Edition

摘　　要：对服从Wishart分布的随机矩阵W～Wp(n ,Ι)已有著名的Bartlett分解定理 ,结果非常完美 ,但证明过程既繁又长 ,本文用特征函数方法证明 2个服从n -i+ 1维标准正态分布、且相互独立的随机向量的内积应同分布于一个服从 χn -i + 1分布的随机变量与一个与其独立且服从N( 0 ,1 )分布的随机变量的乘积 .从而简单而直观地证明该定理 ,虽结论稍减弱为W =d T′T 。For a random matrix that represents the Whishart distribution, there exists the well-known Bartlett decomposition theorem. The result is perfect, but the proof is tedious. This article uses the characteristic function to prove the following result: the inner product of two independent (n-i+1) -dimensional random vectors subject to standard normal distribution has the same distribution of the product of two independent random variables in which one is subject to χ n-i+1 distribution and the other to N(0,1) distribution. Thus the theorem can be proved briefly and directly. Though the conclusion is reduced to W= d T′T , it does not affect the application of Bartlett decomposition theorem in most cases.

关 键 词：Bartlett分解定理 特征函数 WISHART分布 证明方法 随机变量 随机矩阵 多元统计分析 

分 类 号：O212.1[理学—概率论与数理统计] O211.3[理学—数学]