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In other words, if $X_t$ is $I(0)$, is $f(X_t)$ also $I(0)$?

I would say yes:

  • The mean stays constant.
  • The autocovariance still depends only on the lag between the terms.
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    $\begingroup$ You have not defined or restricted $f$ in any way. For an arbitrary $f$, why should the mean stay constant or the ACF only depend on the lag between the terms? $\endgroup$ – Richard Hardy May 2 at 11:01
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  1. You will at least need strict stationarity, because even in the $X_t$ all have the same mean (and variance), the distributions can be different and so the $f(X_t)$ does not necessarily have the same mean if $f$ is nonlinear.

  2. So then assume strict stationarity, then the answer becomes yes for marginal transformations $f$. That is really a consequence of the well-known result that $\DeclareMathOperator{\E}{\mathbb{E}} \E f(X) = \inf f(x) \; dF(x)$, see wikipedia.

  3. For a question about the converse see If the square of a time series is stationary, is the original time series stationary?

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