Testing for stationarity We know that the definition of stationarity (either weak or strong) of a random time series involves having the same joint distribution or statistic (like mean or variance) for "any" set of time points, extending to infinity. 
Since we can only observe finite series in reality, how can any stationarity test work? Let me rephrase: it seems to me that many seemingly non-stationary time series we acquire may be generated by a doubly-stochastic time series where the parameters that describe the series are coming from a stationary random process that changes over an arbitrarily larger time-scale. Then how do the stationarity tests work from this perspective?
 A: Precisely by your argument

the definition of stationarity (either weak or strong) of a random time series involves having the same joint distribution or statistic (like mean or variance) for "any" set of time points

we should have the same joint distributions for small sets of points, too. Since one counterexample is enough to reject a statement that applies to all possible examples, if we reject for small sets we no longer care about any other sets (not rejecting for large sets will not change the conclusion, so why even inspect that).
In your example of doubly-stochastic process, you seem to have a hierarchy. There is a "superprocess" generating the population parameters and a "subprocess" generating the series for the particular population parameters that come from the "superprocess". When testing for stationarity you should then specify whether you are interested in the "superprocess" or the "subprocess" and act accordingly. If this structure is not imposed, we treat the outcome of this hierarchical structure as coming out of one fixed process; if it is nonstationary we do not care whether it is due to the nonstationarity of the "superprocess" or the "subprocess", we only care about their combination (which is nonstationary).
