The Gaussian Distribution Of The Random Tensors

Gaussian distribution is one of the important distributions in probability statistical.It is widely used in many fields.In multivariate statistical analysis,Gaussian distribution generally refers to Gaussian distribution of random vectors or random matrices.Traditionally,We usually matriculate or vectorize the data for the distribution of high-dimensional(order)random variables.However,obviously this approach breaks the original structure of the data.While the random tensor is a good representation to show the data structure and simplify data processing.In this paper,we study the random tensors Gaussian distribution.This paper includes four parts: Firstly,we define the random tensors Gaussian distribution based on several equivalent definitions of the random matrices Gaussian distribution,and derive the theorem that the matrices which are matriculate from the random tensors still follow Gaussian distribution.Secondly,according to the definition of the density function and the characteristic function of the random matrices,we define the density function and characteristic function of the random tensors.Then,we obtain the density function and characteristic function of standard Gaussian distribution of the random tensors on the basis of last theorem and the density function and the characteristic function of the random matrices.And since the fact that the random tensors which follow Gaussian distribution still follow Gaussian distribution after linear transformation,we derive the density function and characteristic function of the general random tensors Gaussian distribution.Thirdly,we define the blocks of the random tensors and consider the distributions and properties of the blocks that follow Gaussian distribution.Fourthly,the tensor form of the multivariate linear model is established.It is assumed that the error tensor follow the Gaussian distribution of the random tensors and its parameter tensor is estimated.