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.