# Are all ergodic random processes (at least wide sense) stationary?

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

## 1 Answer

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.

• 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....
– rhz
Commented Dec 2, 2019 at 18:16
• @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? Commented Dec 3, 2019 at 3:36
• Yes, of course. My bad.
– rhz
Commented Dec 4, 2019 at 1:35