I am not very familiar with random vectors beyond the covariance analysis. Here is my question. For a scalar random variable $X$ it is straight forward to define for any $n\in \mathbb{R}$ \begin{align} E[X^n] \end{align}
However, what about random vectors. For example, generalization of $E[X^2]$ is the autocovarience matrix given by \begin{align} E[XX^T] \end{align}
My question is are generalizations to any other moments? I guess the generalization where $n$ is even is $E[ (XX^T)^n]$ since $(XX^T)$ is a symmetric matrix. However, what about $n=1.2$?
Thanks for any help. Also, please let me know how I can improve my question.