What type of stationarity preferred during the time series analysis: strong stationary or week stationary? I was googling the answer for the above-mentioned question. I had the definition of both but I don't know which one is preferred during the analysis. So which one is recommended? Reason? Thank you!
In this blog, it mentioned that
"Note that statistics assumes that the variable examined is strong stationary (serial independent)."

Well, I got a short answer from the professor. According to him, weak stationary is the basic and it will be the same for Gaussian distributions. WHY??? And my question was which one would be more preferred. Anybody would like to elaborate a little bit:)
 A: In statistical modelling there is not really a "preferred" type of stationarity --- we prefer whatever model fits the data well and is most justifiable in the analysis.  When conducting data analysis we fit models to the data and see which models are plausible.  Sometimes a strongly stationary model will be a plausible fit to the data and sometimes a model which is only weakly stationary will be a better fit.  Weakly stationary models are stationary up to second order (i.e., up to variance), so the only time a weakly stationary model (that is not also strongly stationary) will be necessary is when there is non-stationarity at moments of high order (skewness and above).$^\dagger$
As a practical matter, standard models for time-series analysis are usually Gaussian models (e.g., Gaussian ARMA).  Since Gaussian models are fully determined by their first two moments, there is a correspondence between strong and weak stationarity in this case --- every weakly stationary Gaussian model is also strongly stationary.$^\dagger$  If you use non-Gaussian models then it is possible to formulate models that are weakly stationary but not strongly stationary.  In some cases it is possible that the weakly stationary models might fit the data better than the strongly stationary models, but it would be unusual.

$^\dagger$ It is important to note a slight technicality here.  Weak stationary is implied by strong stationarity if the moments of the process exist up to second order.  For Gaussian models the moments exist, so strong stationarity also implies weak stationarity.
