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If the time series process is linear, then the ARIMA model is specified. The residuals from this model are $(1.)$ no autocorrelation $(2.)$ mean equals zero $(3.)$ constant variance. We say that this process is a white noise process. But under white Gaussian noise, how to proof no autocorrelation = independent? And how to proof the white noise from Gaussian distribution?

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  • $\begingroup$ Gaussian distribution does not imply white noise, so you cannot prove that. $\endgroup$ May 25, 2023 at 15:58

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No autocorrelation is not independent. To statistically check if the given time series is autocorrelated you can use Ljung-Box-Test, there is also many other test that have the same target. You can also plot the histogram, the Q-Q-Normal, and the Residuals vs Predicted graph, with those you can visually check if the residuals follow a normal distribution or not.

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