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My intuition of ergodicity is the Law of Large Numbers for time series i.e. Given sufficient, data points, their mean and standard deviation would converge to population mean and standard deviation.

Does weak stationarity imply this inherently? Weak stationarity says the mean and standard deviation do not vary with time.

If it weak stationarity does imply, then why is the concept of ergodicity necessary at all ?

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Ergodicity is connected to mixing, meaning there is one limiting distribution and it is used for time averages too. If you take a process in the real numbers that starts at a random value and then just stays at its initial point, it is stationary but not ergodic because there is not a unique distribution for time averages.

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  • $\begingroup$ @whisperer: You see, in this counterexample even given a very long observation of one trajectory, it gives you no useful information about the mean of the population of all trajectories. Each trajectory becomes "trapped" or "stuck" in a state that is not statistically representative of what the other trajectories are doing. It is in a sense the complete opposite of ergodicity. In an ergodic p. every trajectory tells you all the stats you need to know and there are no "trapped" states, in the long run you will visit the entire state space. $\endgroup$ – Alex C Jun 4 '18 at 13:50

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