矩阵正态统计分布的一个性质

A CHARACTERIZATION OF MATRIX VARIATE NORMAL DISTRIBUTION

对于服从相同统计分布的两个 n维随机向量 X和 Y,若 X关于 Y的条件概论分布为多元正态分布Nn(ATy+ b,φ0 ) ,则由 X和 Y构成的 2 n维随机向量也服从多元正态分布 ,并且︳(A) <1;利用条件分布和特征函数的唯一性定理 ,证明了矩阵正态分布也存在类似结论 .

For the two random vectors X n×1 and Y n×1 having the identical statistical distribution,if there exist A∈R n×n ,b∈R n and Σ 0>0 such that the distribution of X given Y=y,y∈R n is the multivariate normal distribution N n(A Ty+b,Σ 0),then random vector consisting of X and Y is a 2 n variate normal random vector ,and ρ(A)<1;the similar result about the matrix variate normal distributions is proved based on the random vectors’ conditional distributions and the uniqueness of characteristic functions.