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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".

kjetil b halvorsen
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gamerx
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1 Answers1

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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 can't think of any counter-examples, but there must be some.

kjetil b halvorsen
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    +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](http://stats.stackexchange.com/q/85902/1005) (2) When the process is a ARMA(p,q) process, when will its acf decay to zero? [When will it be ergodic](http://stats.stackexchange.com/q/95713/1005)? (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](http://stats.stackexchange.com/q/86695/1005)? – Tim Apr 30 '14 at 17:39
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    +1 Here is an example that might explain the rule of thumb: https://stats.stackexchange.com/questions/341814/acf-and-pacf-for-a-unit-root-process/343104#343104 – Christoph Hanck Apr 17 '20 at 13:31