随机张量及其正态分布
Random Tensors and their Normal Distributions
摘要：
本文的主要目的是引入具有正态分布的随机张量，将矩阵正态分布推广到更高阶的情况。首先介绍了张量的一些基本知识，然后重点讨论了元服从正态分布的随机张量。引入了具有标准正态分布（SND）的随机张量作为随机正规矩阵的推广。作为对SND张量进行仿射变换得到的随机多重阵列，本文对一般正态随机张量进行了初始化。然后，我们研究了正规张量的一些等价定义，并给出了密度函数，特征函数，矩，和其他一些有关的随机矩阵函数的描述。为了处理随机张量，还引入了偶数阶多方差张量的一般形式。最后给出了张量正态分布的一些等价定义。
英文摘要：
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.
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