If not, please provide a simple example of a non-stationary process that is ergodic (in mean and covariance).


1 Answer 1


I answered a similar question over on dsp.SE. For the convenience of those too lazy to mosey on over there for a look, or too proud to stoop to reading what mere engineers might be writing, and also for the sake of completeness of stats.SE as a repository of questions and answers on statistics and related subjects, I am reproducing below a part of my answer on dsp.SE.

Can there be a random process that is not stationary but is ergodic? Well, NO, not if by ergodic we mean ergodic in every possible way one can think of: for example, if we measure the fraction of time during which a long segment of the sample path $x(t)$ has value at most $\alpha$, this is a good estimate of $P(X(t) \leq \alpha) = F_X(\alpha)$, the value of the (common) CDF $F_X$ of the $X(t)$'s at $\alpha$ if the process is assumed to be ergodic with respect to the distribution functions. But, we can have random processes that are not stationary but are nonetheless mean-ergodic and autocovariance-ergodic. For example, consider the process $\{X(t)\colon X(t)= \cos (t + \Theta), -\infty < t < \infty\}$ where $\Theta$ takes on four equally likely values $0, \pi/2, \pi$, and $3\pi/2$. Note that each $X(t)$ is a discrete random variable that, in general, takes on four equally likely values $\cos(t), \cos(t+\pi/2)=-\sin(t), \cos(t+\pi) = -\cos(t)$ and $\cos(t+3\pi/2)=\sin(t)$, It is easy to see that in general $X(t)$ and $X(s)$ have different distributions, and so the process is not even first-order stationary. On the other hand, $$E[X(t)] = \frac 14\cos(t)+ \frac 14(-\sin(t)) + \frac 14(-\cos(t))+\frac 14 \sin(t) = 0$$ for every $t$ while \begin{align} E[X(t)X(s)]&= \left.\left.\frac 14\right[\cos(t)\cos(s) + (-\cos(t))(-\cos(s)) + \sin(t)\sin(s) + (-\sin(t))(-\sin(s))\right]\\ &= \left.\left.\frac 12\right[\cos(t)\cos(s) + \sin(t)\sin(s)\right]\\ &= \frac 12 \cos(t-s). \end{align} In short, the process has zero mean and its autocorrelation (and autocovariance) function depends only on the time difference $t-s$, and so the process is wide sense stationary. But it is not first-order stationary and so cannot be stationary to higher orders either. Now, when the experiment is performed and the value of $\Theta$ is known, we get the sample function which clearly must be one of $\pm \cos(t)$ and $\pm \sin(t)$ which have DC value $0$ which equals $E[X(t)] 0$, and whose autocorrelation function is $\frac 12 \cos(\tau)$, same as $R_X(\tau)$, and so this process is mean-ergodic and autocorrelation-ergodic (autocovariance-ergodic too) even though it is not stationary at all. In closing, I remark that the process is not ergodic with respect to the distribution function, that is, it cannot be said to be ergodic in all respects.

  • $\begingroup$ You have demonstrated that strict sense stationarity is not required for mean and covariance ergodicity by providing an example that is only wide sense stationary. Is wide sense stationarity a requirement? Are there non-WSS processes that are mean/covariance ergodic? I wonder if there are cyclostationary processes that are mean/covariance ergodfic.... $\endgroup$
    – rhz
    Dec 2, 2019 at 18:16
  • $\begingroup$ @rhz If a process does not have a constant mean, but is nonetheless mean-ergodic, meaning that for almost all sample paths $x(t)$, the integral $$\frac{1}{2T}\int_{-T}^T x(t)\, \mathrm dt$$ converges to a limit as $T \to \infty$ and this limit is the mean of the process, which of the different means of the random variables comprising the process is being converged to? $\endgroup$ Dec 3, 2019 at 3:36
  • $\begingroup$ Yes, of course. My bad. $\endgroup$
    – rhz
    Dec 4, 2019 at 1:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.