Estimation and testing of unconditional distribution in time series models

In time series models, such as ARMA-GARCH, is it possible to estimate what the unconditional distribution is? Given that the time series is auto-correlated / persistent, how many observations would be required to get a good coverage of the underlying space (of course, the extreme tails cannot be accurately estimated, but what if we restricted ourselves to say, 95% or 99% of the mass around median)? Are there some estimates for this in terms of the parameters of the models?

Also, if it is possible to estimate the unconditional distribution, how does one then assess a fit of that distribution? KS/AD/CFM tests rely on independent observations, which for the time series models they are not, so these tests (at least with the standard critical values) don't seem to be suitable here. From 1000 simulations of AR-GARCH process with 2500 time steps, I found that the two-sample KS test returns p-value of, for example, less than 5% on about 500 runs instead of expected 50.

Presumably, due to the auto-correlation / persistence in the time series, each observation contributes a fraction of the information about the distribution compared to an independent sample. Maybe this can be used to re-scale various goodness-of-fit statistic to reflect the "effective" number of observations in the time series which would make these standard tests usable again. So, I suppose it comes down to this "information content fraction" of observations in the time series. Any references on this topic would be very welcome.