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I found a solution that stated that if the square of a time series is stationary, so is the original time series, and vice-versa. However I don't seem able to prove it, anyone has an idea if this is true, and if it is how to derive it?

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That conjecture is false. A simple counter-example is the deterministic time-series $X_t = (-1)^t$ over times $t \in \mathbb{Z}$. This time series is not even mean stationary, but its square is strictly stationary.

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  • $\begingroup$ How about positive numbers only? $\endgroup$ – smci May 13 at 10:36
  • $\begingroup$ interesting. Is it possible to deduce nonstationarity from a single realization? That time series looks nonstationary only on paper. $\endgroup$ – Cowboy Trader May 13 at 10:53
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    $\begingroup$ @Firebug The mean isn't zero. The mean is $-1$ for odd $t$ and $1$ for even. $\endgroup$ – Acccumulation May 13 at 17:10
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    $\begingroup$ @Acccumulation It's zero through time. $\endgroup$ – Firebug May 13 at 18:49
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    $\begingroup$ @Firebug It appears that at least one of doesn't understand what the word "stationary" means. $\endgroup$ – Acccumulation May 13 at 18:50

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