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 $\tau = t_2-t_1$.


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

  • 3
    $\begingroup$ I'll counter-ask: What's the physical interpretation of the autocorrelation? $\endgroup$ Nov 19, 2018 at 20:19
  • $\begingroup$ 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? $\endgroup$
    – A_A
    Nov 19, 2018 at 21:41
  • $\begingroup$ @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. $\endgroup$ Nov 19, 2018 at 22:33
  • $\begingroup$ @CarlosDanger PDF $\equiv$ moment generating function ($\equiv$ characteristic function) hence IMHO, in general you cannot compute higher order moments from lower order ones. $\endgroup$
    – AlexTP
    Nov 19, 2018 at 22:43
  • $\begingroup$ 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. $\endgroup$
    – Robert L.
    Nov 19, 2018 at 22:52

1 Answer 1


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


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.


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