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Can one say anything about the "Gaussiness" of a time series merely by looking at its autocorrelations?

I find this hard to reconcile. Let us say I find that there are significant autocorrelations in the time series. This doesn't imply the series is not Gaussian, because there are processes with dependencies but which nevertheless are Gaussian (for example fractional Brownian Motion).

If there are no autocorrelations, this doesn't tell me much either. Then the series might be a pure random walk and hence be Gaussian.

Edit: To clarify, I am aware that this is not the usual way to check the Normality of the distribution of the increments, which can be done via Jarque–Bera test, prob-plot etc. My question is whether the autocorrelations alone give any information about the Gaussianality.

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  • $\begingroup$ Looking at the autocorrelations is not the way to check for Gaussianity. Searching for normality test on this site you may find some ideas on how to do it. $\endgroup$ – javlacalle Feb 16 '15 at 9:47
  • $\begingroup$ @javlacalle yes, I should probably add that I am aware of the usual normality tests $\endgroup$ – Slug Pue Feb 16 '15 at 11:34
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The relationship between the observations over time as measured by the sample autocorrelations does not give information about the distribution of the underlying random variable.

The probability of the possible values that can take the random variable cannot be inferred from the autocorrelations. Hence, the sample autocorrelations are not helpful to determine whether the process is Gaussian or not.

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