# 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 $$\mathbb E[X (t_1) X^3(t_2)]$$?

• I'll counter-ask: What's the physical interpretation of the autocorrelation? – Marcus Müller Nov 19 '18 at 20:19
• I wonder if this could be considered, conceptually, a duplicate of this question. This was the first thing that came to my mind when trying to infer the physical meaning from the units. In any case, autocorrelation is a product, between two time series of the same unit, so it is that unit squared. So, voltage squared gives you power. But $X^2(t)$ would already be "Power" and the autocorrelation of that would be Power squared...So, not very helpful..What motivated the question? – A_A Nov 19 '18 at 21:41
• @CarlosDanger that's a pretty different question that you're asking!! And the answer is: you can't. I can give you a signal that has pretty much zero autocorrelation, but high $\mathbb E\left[ X^2(t) X^2(t+\tau) \right]$. But then I can give you a signal with the same $\mathbb E\left[ X^2(t) X^2(t+\tau) \right]$, but very different autocorrelation. – Marcus Müller Nov 19 '18 at 22:33
• @CarlosDanger PDF $\equiv$ moment generating function ($\equiv$ characteristic function) hence IMHO, in general you cannot compute higher order moments from lower order ones. – AlexTP Nov 19 '18 at 22:43
• I think you guys are getting too caught up in my example "motivation", so I've deleted it. For now, I'm just looking for the answer to the literal question. – Robert L. Nov 19 '18 at 22:52