Why Gaussian process is not Ergodic in general?

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Can anyone use a simple way to explain this? I heard this in class but I do not know why.


By Wiki: a random process is ergodic if its statistical properties can be deduced from a single, sufficiently long sample of the process.

Our note: A random process is ergodic if for all invariant event F, after time shift, either P(F) = 1 or P(F) = 0.


I have difficulty to explain the why through the def. of WIKI or our note.

Thanks