# 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?

• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? – Richard Hardy Feb 12 '17 at 12:49