First I'm going to give some context on the question. The question arose in reading one Physics paper, but the question is itself just about Math involved.
In "The Quantum Theory of Optical Coherence" by Glauber, he defines what he calls $n$-th order correlation functions. In that context he has a set of $2n$ random variables $E^-(x_1),\dots ,E^-(x_n)$ and $E^+(x_{n+1}),\dots, E^+(x_{2n})$ and defines (here $\langle\rangle$ is a mean value) $$G^{(n)}(x_1,\dots x_n,x_{n+1},\dots, x_{2n})=\langle E^-(x_1)\cdots E^-(x_n)E^+(x_{n+1})\cdots E^+(x_{2n})\rangle\tag{1}$$
He calls this $G^{(n)}$ the $n$-th order correlation function.
Now, in the $n=1$ case we see that $G^{(1)}(x_1,x_2)$ is connected to the covariance of the two random variables: $$\operatorname{Cov}(E^-(x_1),E^+(x_2))=\langle E^-(x_1)E^+(x_2)\rangle -\langle E^-(x_1)\rangle \langle E^+(x_2)\rangle\tag{2}.$$
Therefore if we assume the means are set to zero, $G^{(1)}(x_1,x_2)$ is exactly the covariance of the two random variables $E^-(x_1)$ and $E^+(x_2)$.
Since the Pearson correlation coefficient is just the covariance normalized by dividing by the standard deviations, in this sense $G^{(1)}(x_1,x_2)$ codifies linear correlations between $E^-(x_1)$ and $E^+(x_2)$. So it is justified to call $G^{(1)}$ the first order correlation.
Now $G^{(n)}$ involves $2n$ random variables.
Is there some sense in which an object like $G^{(n)}$, a mean value of $2n$ random variables, encodes $n$-th order correlations?
Is this some extension of the Pearson correlation coefficients to quantify correlations more general than linear correlations?
Finally, correlations for me is a relation between two random variables. What it even means to talk about a correlation between $2n$ random variables?