Are return time series ergodic?

It seems intuitive to me that return time series would be ergodic. Is there a test statistic that I can use to check this? Would this be affected by sampling rate?

One way I can think of checking ergodicity is to plot the ACF and check if the autocorrelation decays with sufficiently large lag (which it will).

• My fear is that someday there will be a big stock market crash from which we will never recover. That would be a failure of ergodicity. That it has never happened yet is no guarantee it might not happen in the future. But maybe I worry too much. May 6 at 16:11
• @noob2 quite a good point, especially as it is showing that any kind of model / assumption is specific to the space and time. May 6 at 16:59
• Why would such a thing be intuitive? May 6 at 17:18
• @rubikscube09 because it makes sense that autocorrelation decays with time which would make the series ergodic.
– s5s
May 6 at 18:18
• Ergodicity is a (to me) rather tricky property. It asks that average over all states is equal to average over time. It is generally imposed as an assumption, not tested empirically. We cannot see all states and observe the system for all time. May 6 at 20:13

It has the interesting property that any point reached $$(\alpha,\beta)$$ will be returned to an infinite number of times, but also that there will exist many holes that will never be reached.