So I was thinking of generalization of notions in statistics, like auto-correlation or cross-correlation (auto-correlation is a specific example of cross-correlation where we take the same proccess).
The problem is how do I generalize for example the notion of WSS (wide-sense-stationary)?
Take for example
$$R_{X,Y,Z}[n_1,n_2,n_3] = \mathbb{E}(X[n_1]Y[n_2]Z[n_3])$$
How would I generalize the notion of WSS, such that the cross correlation is a specific example of this function.
My naive notion of WSS in this case may be:
R is WSS when:
$$ R_{X,Y,Z}[n_1,n_2,n_3] = R_{X,Y}[n_1 -n_2] + R_{X,Z}[n_1 -n_3] + R_{Y,Z} [n_2 - n_3] $$
But I don't see how do I get the WSS cross correlation case when $ Z=1 \ w.p\ 1$.
Has this been done before, or tried to generalize this notion in statistics?
When you have more than 2 r.v., things get more complicated. You don't have just one mixed moment anymore, but several: for example $E(X^3)$, $E(X^2 Y)$, $E(XYZ)$, etc. Thus `the' correlation depends on what power of your r.v.s you want to get involved: there are multiple covariances of mixed type to take into account. Also, in your proposed expression, you could argue that there may need to be an extra term $-R_{X,Y,Z}(n_1-n_2, n_1-n_3)$ for WSS processes, because the covariances are defined as integrations (similar to the sum formula in probability theory for multiple events). There have been extensions for the case of Gaussian r.v., where Isserlis's formula (Gaussian moment theorem) can be used, which splits the moment for every even number of r.v.s into a sum of products of pairs (covariances), and all odd numbers giving zero. For more general distributions, I do not think it can be done because you will need to account for the skewness of the PDF.