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How do you check ergodicity of a wide-sense stationary stochastic processes from its sample path(s)?

Can we check ergodicity from a single sample path? Or do we need multiple sample paths?

One motivation of checking ergodicity is in time series to ensure that you could safely use the average of a sample path over time as estimate for the population mean?

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A signal is ergodic if the time average is equal to its ensemble average. If all you have is one realization of the ensemble, then how can you compute the ensemble average? You can't. Therefore you don't have enough information to know if the signal is ergodic or not.

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    $\begingroup$ To add to this brief answer: Ergodicity is an assumption that cannot be tested on a sample of one time series. It is more of a commitment, it must come from some theory about the process. $\endgroup$
    – kjetil b halvorsen ♦
    Commented Sep 5, 2018 at 13:35

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