I am a bit confused with stationary time series. Data transformations (detrending, difference etc) are used to seek stationary time series so that we can treat correlation as a constant over time. Using the ACF to compare before and after data transformed. But is the stationary really important? This question comes from two sides: first if no transformation can find satisfied ACF, we still need to analyze the series. Second is that various models can handle different ACFs like long tails, cutoff lags etc so that the ACF can help pick the right model and model orders.
Therefore, stationary seems not very needed but it is a "good-to-have". Does this make sense?