Let us have a set of random variables $\{X_i\}_{i=1,2,...,n}$, where $X_i$s are identically distributed and are not a Gaussian random variables.
The central limit theorem states that if $X_i$ and $X_j$ are independent for all $i\not =j$, then $$\bar X = \sum_i X_i/n$$ is (approximately) a Gaussian random variable.
Question:
However, what would the necessary and sufficient conditions for the distribution of $\bar X$ be the same as $X_i$s ?
For example, if we know that $X_i$s are mutually correlated in a specific way - hence they are not independent, the central theorem would not be valid, hence the distribution of $\bar X$ would (possible) be different that of Gaussian one.
Edit: My main goal by asking this question is that if I have a bunch of identically distributed random variables, then I would like to see the same distribution in their mean, i.e $\overline X$. In a way, carry the distribution to the distribution of their mean.