Given a sample path from a process supposed to be stationary, I saw the sample autocorrelation function of the sample path is used to estimate the autocorrelation function of the process. But this requires that the stationary process is ergodic. So
is ergodicity checked on the sample path before estimation of the autocorrelation function? If yes, how is ergodicity checked?
is the estimation done assuming ergodicity without checking it, and after the estimation, testing/checking if the estimation fits the data? If yes, how is the testing done?
A side question: how is stationarity is tested for estimation of autocorrelation function?