Say that I have n samples from a parameter generated from a MCMC-algorithm, the samples are therefore correlated with eachother. I want to test if the sample distribution that i have obtained of this parameter are equal to the true distribution of that parameter.

The sample distribution are not necessarily normal and we have as above mentioned correlated samples.

At the moment I think that testing for equal mean and variances will suffice for checking equality of distributions. I think sample means are normally distributed for large samples by Central limit theorem be tested with usual t-test if the samples were not correlated. Can I test it with the t-test even if we have correlated samples?

How to do with the sample variance test? Can we use Cental limit theorem on this r.v. as well?

I have thought about consider thinned out sub samples of my chain which are then not as strongly correlated with each other, however one looses much information so I would prefer some other suggestion.