Physical interpretation of 4th-order correlations BACKGROUND:

Let's say we have samples of a random process $X(t)$ at two different times, $t_1$ and $t_2$, denoted $X(t_1), X(t_2)$.  The values of $X(t)$ represent some voltage-like quantity (i.e. a voltage, current, or field). Assume $X(t)$ is a zero-mean, real-valued wide-sense stationary (WSS) process.

For the second-order statistics,  we usually treat the quantity $\mathbb E[X^2(t)]$ as the average power.
The expectation $\mathbb E[X(t_1) X(t_2)]$ is just the autocorrelation function evaluated at lag $\tau = t_2-t_1$. 

QUESTION:

Is there some physical interpretation of the 4th-order correlations, i.e. $\mathbb E[X^3 (t_1)X(t_2)]$, $\mathbb E[X^2 (t_1)X^2(t_2)]$, and $\mathbb E[X (t_1) X^3(t_2)]$?
If you are willing to relax the WSS assumption, and consider cyclostationary signals, you might check out my theory of pure and impure sine-wave components of higher-order probabilistic parameters:

On pure and impure sine-waves, higher-order moments, and cyclic cumulants

and

Oddities concerning pure nth-order sine waves

I've thought about your question a lot, more so in the past than lately, and all I've been able to come up with is that higher-order moments contain characteristic sine-wave components (characteristic of the process, the order, and the number of conjugated factors), and higher-order cumulants characterize what is new in the moment that cannot be accounted for by products of lower-order moments. Unlike the easy physical interpretation of autocorrelation as power (think of the zero-lag autocorrelation), I don't have a simple physical unit to assign here, but maybe my hint will lead you to a more satisfying answer.