Uniform Asymptotic Normality Of Several Matrix-variate Distribution

we are familiar with the nature of some one-dimensional and multivariate distributions. In the multivariate analysis, the sample data matrix is a matrix form when we make statistical inference, which requires us to promote the definition of the distribution from vector theory to the matrix case. This is an important issue of the multivariate analysis.As the density function of the matrix-variate distribution is complicate, we want to find the conditions under which a matrix-variate distribution will approach uniformly and asymptotically a normal distribution mainly for computational ease. Currently, the recent monograph only proved the uniform asymptotic normality of the Wishart distribution and matrix-variate Gamma distributions. For other matrix-variate distributions,we have not discussed. This article is mainly to describe as the following aspects, to describe the relationship between matrix-variate Gamma functions and matrix-variate Beta functions, and the conditions of uniform asymptotic normality of matrix-variate Beta distribution.In the first chapter, we introduce the main study results of the matrix-variate distributions, and also introduce the premise of this study and some important lemmas.In the second chapter, we present the multivariate normal distribution and multivariate Beta distributions, introduce the Kullback-Leiber distance between the two density functions, in order to obtain the conditions of the uniform asymptotic normality of multivariate t-distribution.In the third chapter, after giving the definition of the matrix-variate Gamma function and matrix-variate Beta function, analogy to the nature of the multivariate function, we obtain the relationship between them.In the fourth chapter, we first introduce the definition and the the conditions of the uniform asymptotic normality of matrix-variate Gamma distributions,analogy to the matrix-variate Gamma distributions, according to the Kullback-Leiber distance between the two density functions,we give the definition of the matrix-variate Beta distributions. Finally,we give the conditions under which a Beta matrix-variate distribution will approach uniformly and asymptotically a normal distribution.