Why Weak Stationarity Does Not Guarantee Ergodicity

Summary

The document distinguishes weak stationarity from ergodicity. Weak stationarity constrains a process’s mean and variance over time, but those stable moments alone do not guarantee that averages along one observed time series converge to the population values. Ergodicity concerns whether time averages can represent the process’s ensemble behavior, and is related here to mixing and a limiting distribution.

A counterexample is a process that draws a random starting value and remains fixed there forever. Its distribution is unchanged over time, so it is stationary, yet a realized path’s time average is simply its initial draw. Across paths, these averages can differ, showing why stationarity alone is insufficient. The explanation is conceptual and brief; it does not state formal conditions for ergodicity or establish a general relationship between mixing and ergodicity.

Key ideas

• Weak stationarity keeps specified moments stable over time but does not ensure convergence of time averages to population moments.
• Ergodicity concerns when time averages can represent ensemble properties.
• A process fixed at its random initial value is stationary but can have different time averages across realizations.
• The example shows why stationarity alone does not establish ergodicity.

Tags

# Does Weak stationarity imply ergodicity ?


# Does Weak stationarity imply ergodicity ?












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 ?

## Answer by userid is i (score 3, accepted)

https://quant.stackexchange.com/a/40119

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.