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I'm having trouble understanding how the ergodicity concept must be understood and how it is different od the WSS concept. If I understood correctly, a process is said to be ergodic in some parameter if the average on the ensemble is equal of the average on time for that parameter.

So, I'm focusing on the mean and autocorrelation ergodicity. What I found in most places is that for a process to be mean ergodic then it must hold that $\lim_{T \to \infty}<\mu_x(t)>_T = \mu_x(t)$. From this it follows that $\mu_x$ does not depend on $t$.

Now, if I have a two variable function, how would the ergodicity must be understood? I mean, for the autocorrelation should I perform a surface integral over an arbitrary 2D region, like $\lim_{T \to \infty} \frac{1}{T^2} \int_{-T}^{T}\int_{-T}^{T}R_{xx}(t_1, t_2)dt_1dt_2 = R_{xx}(t_1, t_2)$ or how would be the "precise" definition?

Actually, the reason of why I ask this is because my teacher said in class that most of the authors beg for the process to be WSS "in order to be able to give a definition of ergodicity" but there could be an ergodic process in some parameter that is not WSS. One of my problems is that when the autocorrelation ergodicity is defined only on WSS process, $R_{xx}$ becomes automatically dependent on just one variable so I don't understand what would happend if I have to work with ergodicity on multiple variable parameters.

Then, I found in all places that if a process is ergodic on mean and autocorrelation it must be WSS. For the mean ergodicity, the consequence is clear. But for the autocorrelation I don't see how it follows that (that if a process is ergodic on autocorrelation, then $R_{xx}$ depends only on $t_2 - t_1$); and most authors instead of giving a proof they just define autocorrelation ergodicity on a WSS process but they do not define autocorrelation ergodicity on any process.

Therefore, by all of what I said now I'm flummoxed. If someone understood the doubt, I would appreciate any help, specially on how to define ergodicity with parameters dependent on multiple variables and how does it follow that "Rxx ergodicity implies WSS".

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  • $\begingroup$ I might have some perspective for you here. $\endgroup$ Commented May 4 at 22:39
  • $\begingroup$ @robertbristow-johnson So, as I understand the concept of ergodicity, i.e, "time average" and "ensemble average" of a parameter makes sense only if the parameter depends on one time variable. $\endgroup$
    – coal
    Commented May 5 at 20:30

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A process that is stationary is one whose unconditional joint probability distribution doesn't change with time. Consequently, all statistical moments are time-invariant. Wide-sense stationary (WSS) describes a process whose first and second moments (mean and autocorrelation) are time invariant. So, it is WSS that leads to the conclusion that the mean is constant.

As for the autocorrelation, I don't necessarily agree that it becomes a single variable function if you assume WSS. What WSS means for an autocorrelation is \begin{equation} r_{xx}(t_{1},t_{2}) = r_{xx}(t_{2}-t_{1}) \end{equation} The time difference is still dependent on two variables (in order to take a difference you need at least two things you are taking the difference of). However, WSS implies we can ignore the true time location and only focus on the difference in time.

As for why WSS is a necessary but not sufficient condition for wide sense ergodic process, if the statistical properties change over time, then the statistical properties of one independent realization of the process will be different from the next independent realization. Therefore, you couldn't guarantee that the time average all independent realizations all converge to the statistical average. If all the time averages are the same, then it is possible that they converge to the statistical average, although it's not necessary that it does.

If there is a process that is wide sense ergodic and not wide sense stationary I haven't heard of it.

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