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".


1 Answer 1


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.

  • 2
    $\begingroup$ +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? $\endgroup$
    – Tim
    Apr 30, 2014 at 17:39
  • 1
    $\begingroup$ +1 Here is an example that might explain the rule of thumb: stats.stackexchange.com/questions/341814/… $\endgroup$ Apr 17, 2020 at 13:31

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