I just read the topic about Ergodicity but I have ambiguity about its meaning (by intuition). What does mean: (for mean) Statistical average = Time average. Could you please explain it in detail. Thanks.


2 Answers 2


If you sample a random process for a specific t, you will get one realization of a random variable. For another t, you get another realization of that random variable. This random variable has its statistics which is almost impossible to learn in real world because not all sample paths are observable. See the brown rectangle in the figure below.

That becomes possible in ergodic processes. An ergodic process is the one where the time average of a sample path is the same as the statistical mean. See the purple line in the figure below, which was the observed sample path such as a particular realization of noise in a communications receiver.

enter image description here

  • $\begingroup$ Ignore the y-axis ticks. I quickly drew the figure in Matlab and forgot to remove that. $\endgroup$
    – QMC
    Nov 3, 2016 at 10:08
  • $\begingroup$ Thanks. What you mean by "the time average of a sample path"? It means it should be any sample path? $\endgroup$
    – Amin
    Nov 3, 2016 at 12:19
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    $\begingroup$ @Stephen Yes, the time average of any sample path -- if the process is ergodic, all sample paths have the same statistics, so it doesn't matter which particular one you look at. $\endgroup$
    – MBaz
    Nov 3, 2016 at 13:04
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    $\begingroup$ @Stephen dsp.stackexchange.com/questions/34976/… $\endgroup$
    – MBaz
    Nov 3, 2016 at 16:25
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    $\begingroup$ I recently found this link and it contains the best explanation of ergodicity ( for the applied person ) that I've ever seen. larspsyll.wordpress.com/2016/11/23/what-is-ergodicity-2 $\endgroup$
    – mark leeds
    Jun 26, 2019 at 20:56

An eternal well-balanced dice has 1/6 probability for each facet $f$, each time. This uniform probability law yields a mean expectation of 3.5: $\sum_{f=1}^{6} \frac{1}{6}\times f$.

Each time you cast the dice, you get this expectation. Of course, for each throw you'll only get an integer 1, 2, 3, 4, 5, or 6, never a decimal like 3.5. So there is an apparent mismatch between what you can expect (in probability) and what you get (actually). A mismatch that relates the (theoretical) probability space and the (real) time space.

The hypothesis of ergodicity may reconcile the two aspects: it tells you that, averaging over a sufficient number of trials in time, you can get the same results as if you were capable of throwing an infinity of dice at the same time.

But remember that ergodicity is an hypothesis on the processes one want to study, and that non-ergodic phenomena exist (under chaotic mixing conditions for instance).

See also: What is a good example of an ergodic process?

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    $\begingroup$ Is there any study about how to accept or reject ergodicity hypothesis? $\endgroup$
    – AlexTP
    May 24, 2020 at 13:13
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    $\begingroup$ AFAIK, ergodicity can be sustanciated by generative models for data or signals, and there exist test, for instance in econometrics. $\endgroup$ May 24, 2020 at 14:59

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