# AcF and Stationarity

Very often in time series literature, it is remarked that if a series is non-stationary the AcF will decrease to zero very slowly while the opposite occurs for a stationary series.

What's the basis for this "rule of thumb"? I know that for a strictly stationary process the autocorrelation is independent of time, whereas for a wide-sense stationary process the autocorrelation is a function of the time lag but these don't explain the "rule of thumb".

Stationarity is not enough to guarantee that the acf will decay to zero, ergodicity is needed. A non-ergodix example is $$Z(t) = X \sin(t+\omega)$$ when $$X$$ is, say, normal and $$\omega$$ is uniform on $$[0, 2\pi]$$. This is stationary, but clearly not ergodic! and the acf do not decay.