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High order tensor moments of random vectors

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.