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 $i \ll k$. If I assume independence, then it would be Gumbel distribution and through simulation it works. But need to justify the results.

  • $\begingroup$ Approximate them as independent so as to derive what conclusion? The first thing that comes to mind is the stronger versions of the classical central limit theorem. $\endgroup$ – Ian Dec 13 '14 at 22:13
  • $\begingroup$ To a conclusion that these samples are independent and identically distribute. $\endgroup$ – upol94 Dec 13 '14 at 22:16
  • $\begingroup$ You can't do that, they are weakly correlated, not iid. You do the approximation in order to claim that they have or at least approximately have some useful property of iid variables (the central limit theorem being a canonical example). $\endgroup$ – Ian Dec 13 '14 at 22:29
  • $\begingroup$ Per your edit: what sort of distribution do the $X_i$ have? In particular how many finite moments do they have? Also, I think rather than actually approximating by iid variables, you should approximate the maximum itself as a submartingale plus some hopefully small term. $\endgroup$ – Ian Dec 13 '14 at 22:33
  • $\begingroup$ If I use Gumbel distribution, it does not matter what type distribution $X_i$ has. In my case, it is sum of non identically distributed gamma variables. $\endgroup$ – upol94 Dec 13 '14 at 22:35

If it works when you simulate it, then that is pretty good evidence that it works! However, if you feel the need to appeal to theoretical arguments, you can imagine choosing a subsequence of your $X_i$ such that the difference in indices is $k$. Lets call a sequence like this with $i$ terms $S_k^i$.

Using this notation, and your assumption of exponentially decaying correlation between variables in the sequence, we can see that $S_k^i$ is an asymptotically independent sequence of RVs as $k\to \infty$. Therefore, the normalized distribution of the maximum in $S_k$ will approach a gumbel as $k,i\to \infty$. Since we are talking about limiting distributions, the fact that we ommitted some of our original RVs is not important.

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