What Distinguishes Ergodic Processes from Stationary Ones?

[wil3]

Can someone concisely clarify the distinction between an ergodic process and a stationary one? Specifically, can anyone provide examples of processes that are ergodic but not stationary or vice-versa?

You don't need to provide the definitions; I know what the words mean. But it seems to me that a stationary time series (such as one with the same mean and variance for any sub-interval) would automatically have these same parameters if one took an infinitely long sample of the process, implying ergodicity. What am I missing here?

Thanks!

 

[viraltux]

Can someone concisely clarify the distinction between an ergodic process and a stationary one? Specifically, can anyone provide examples of processes that are ergodic but not stationary or vice-versa?

You don't need to provide the definitions; I know what the words mean. But it seems to me that a stationary time series (such as one with the same mean and variance for any sub-interval) would automatically have these same parameters if one took an infinitely long sample of the process, implying ergodicity. What am I missing here?

Thanks!

Hi Wil,

I had to check for the definition of ergodic process myself and I got

a process is said to be ergodic if its statistical properties (such as its mean and variance) can be deduced from a single, sufficiently long sample (realization) of the process.

So, turns out that you can have stationary processes without mean or variance (e.g. one following a Cauchy distribution), so in this case this process would not be ergodic.

On the other hand, you might have a process increasing linearly its mean over time, that means that with a sufficiently long sample you can deduce its linear mean behavior and thus fitting the definition of ergodic yet, since the mean is changing overtime, it would not be stationary.