I understand the definition of a random process $X(t)$ being ergodic in mean (first-order ergodic) is that the expectation of the sample mean $<u_X>_T=\frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} X_w(t)dt $ should converge to the ensemble mean as $T \rightarrow \infty$. Can we define ergodic in mean property for processes which do not have constant mean.

I know that the property of being ergodic in ACF, second-order ergodic, requires WSS assumption.

My question is motivated by the fact that the requirement of WSS is not given in the wiki article describing ergodic process but used in the example illustrating the same.

  • $\begingroup$ If the mean function $\mu_X(t)$ is not a constant, what number should the time-average of the realization $X_w(t)$ of the process converge to? $\endgroup$ Jul 19, 2015 at 15:55
  • $\begingroup$ @DilipSarwate : Yes, you are right. My oversight was the fact that a process with constant mean does not imply WSS. $\endgroup$
    – Abraham
    Jul 27, 2015 at 18:35

1 Answer 1


Time average is defined over a time interval, T which will always result in a constant value. Keeping this constraint in mind, ensemble mean is always forced to be constant for satisfying the condition of ergodicity.


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