# Is the stationarity property invariant by transformation?

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.
• 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? – Richard Hardy May 2 at 11:01

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.