# If the square of a time series is stationary, is the original time series stationary?

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?

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.
• @Firebug The mean isn't zero. The mean is $-1$ for odd $t$ and $1$ for even. – Acccumulation May 13 at 17:10