随机向量的高阶张量矩
High order tensor moments of random vectors
摘要：
随机向量$\bx\in \R^n$是其坐标都是随机变量的向量。如果一个随机向量服从高斯分布，则称之为高斯向量。这些术语也可以扩展到随机（高斯）矩阵和随机（高斯）张量。随机向量$\bx\in \R^n$的$k$阶矩（对于任何正整数$k$）的经典形式通常表示为由$\bx$的特征函数或矩母函数的$k$阶导数生成的大小为$n\times n^{k-1}$的矩阵形式，并且即使对于标准正态分布向量，$k$阶矩的表示也非常复杂。利用张量形式，我们可以简化与高阶矩有关的所有表达式。本文主要介绍张量形式的随机向量的高阶矩和标准正态分布向量的高阶矩。最后给出了服从高斯分布的随机向量的高阶矩的表达式。
英文摘要：
A random vector $\bx\in \R^n$ is a vector whose coordinates are all random variables. A random vector is called a Gaussian vector if it follows Gaussian distribution. These terminology can also be extended to a random (Gaussian) matrix and random (Gaussian) tensor. The classical form of an $k$-order moment (for any positive integer $k$) of a random vector $\bx\in \R^n$ is usually expressed in a matrix form of size $n\times n^{k-1}$ generated from the $k$th derivative of the characteristic function or the moment generating function of $\bx$ , and the expression of an $k$-order moment is very complicate even for a standard normal distributed vector. With the tensor form, we can simplify all the expressions related to high order moments. The main purpose of this paper is to introduce the high order moments of a random vector in tensor forms and the high order moments of a standard normal distributed vector. Finally we present an expression of high order moments of a random vector that follows a Gaussian distribution.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
