Higher Moments of a random vector

Question

Let $X = (X_1,X_2,\dots,X_n)$ a random vector. I know we can define the variance matrix of $X$ (supposing $\mathbb{E}(X)=0$ ) as $$\mathbb{E}(XX^{T}), $$ but I'm wondering what is the a natural way for define higher moments for $X$?

Answer 1

The $m$'th moments of the vector $X$ are $\mathbb E[X^k] = \mathbb E[X_1^{k_1} \ldots X_n^{k_n}]$ where $k_1, \ldots, k_n$ are nonnegative integers with $\sum_i k_i = m$. These are the components of a tensor: $\mathbb E[X \otimes \ldots \otimes X]$.