Random Tensors and their Normal Distributions

The main purpose of this paper is to introduce the random tensor with normal distribution, which promotes the matrix normal distribution to a higher order case. Some basic knowledge on tensors are introduced before we focus on the random tensors whose entries follow normal distribution. The random tensor with standard normal distribution(SND) is introduced as an extension of random normal matrices. As a random multi-array deduced from an affine transformation on a SND tensor, the general normal random tensor is initialised in the paper. We then investigate some equivalent definitions of a normal tensor and present the description of the density function, characteristic function, moments, and some other functions related to a random matrix. A general form of an even-order multi-variance tensor is also introduced to tackle a random tensor. Finally some equivalent definitions for the tensor normal distribution are described.