Time series I need someone to help me with the following Use the attachment to w

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Time series
I need someone to help me with the following Use the attachment to write a clear definition of ergodic Would explain in the definition what is the time average and ensemble average
Some random processes have a property of ergodicity - it means that the time averaging is equvalent to ensemble averaging. This requires that prosess to be stationary. The vise-versa in general is not true stationary process might not be ergodic. To estimate patameters of random process you should have its time observation results. The longer the observation the better precision. But usually you cannot have the total time realization of process, so you take a part of it - the ensemble. Ensemble is any random sample collection of time realization of random procesS.

Explanation / Answer

Time average is the average taken over time. And the ensembel average is the average taken over the space of all the system's states. You can understand these by the following example. Suppose I want to measure the temperature of my room at different points (note, that the temperature is varying inside the room). I put a detector in one corner, one in the center and few others at different points. When I choose one detector and measure the temperature on that particular detector for a period of time. And I take the average over time of that particular detector. That is what is called time average. On the other hand, if I measure temperature on all the detectors at aparticular time. And then I average over all the detectors. That is what we call ensemble everage.

Now, when the time average becomes equal to the ensembe average, this situation is called ergodicity.

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