If $N$ random variables are identically distributed but weakly correlated, in what condition we can approximate them as independent identically distributed (iid) ? I saw an old paper where based on the exponentially decaying correlation coefficients, author approximate samples as iid, but could not find the paper. Does anybody knows any formula or corollary or paper that clearly explain this type of situations ?
My Problem: I am trying to find the distribution of $M_n$ where $M_n=max_n(X_1,X_2, \dots X_n) $ Here correlation of $X_i, X_k$ are exponentially decaying where $i<<k$. If I assume independence, then it would be Gumbel distribution and through simulation it works. But need to justify the results.