# Is covariance stationarity a sample property?

I apologize in advance if this proves to be a nonsensical question but it is something I have been struggling with.

Is there a way to prove/discern that covariance stationarity is not an attribute of the sample at hand and is an attribute of the underlying process?

edit:

The motivation of the question comes from unit root testing; when we perform unit root testing on a series, we may accept the null ie that the series in question, has a unit root on one sample, but reject the null on a different (sub-)sample.

Covariance stationarity is described as constancy over time of the first two moments of $X_t$. This is clearly a property of the underlying process: the moments are a process property, not a sample property.