Let the probability mass function of X be f(x). How will $\bar{X}_n$ be distributed?
$\bar{X}_n$ Is the sample mean of the random variable X and sample size, number of subjects in the sample, of n.
I am looking for the distribution of $\bar{X}_n$ in the most general case when they are correlated not just dependent and the correlation is unknown and not even constant(different subjects have a different unknown correlation). The only thing the subjects have is in common is that they come from the same distribution(which itself is unknown and needs to be estimated), they are identically distributed with an unknown distribution which is to be estimated( not just the parametres of the distribution but the whole distribution we know nothing other than that the subjects come from the same distribution and the value of the subjects).