# 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.

For the non-stationary part of the question, I think that is really only an empirical rule-of-thumb. I cant think of any counter-examples, but there must be some.

• +1, thanks. (1) For acf to dacay to zero, did you mean that wide-sense stationarity and ergodicity are necessary and sufficient condition, or just necessary or sufficient? Also see my question and replies (2) When the process is a ARMA(p,q) process, when will its acf decay to zero? When will it be ergodic? (3) Same questions for a white noise (defined as uncorrelatedness, mean zero and variance one): when its acf decays to zero, and when it is ergodic? – Tim Apr 30 '14 at 17:39