Weakly correlated Random variables

Question

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 actual 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.

Answer 1

Take a look at autoregressive processes, such as AR(p) where correlation $Corr[X_i,X_k]=\gamma_{i-k}\sim e^{-|i-k|}$

They end up being stationary. For instance $x_i=c+\phi x_{i-1}+\varepsilon_i$ will have $E[x_i]=\frac{c}{1-\phi}$.

There are many papers on extrema of the time-series.

Answer 2

Firstly you will not easily obtain the exact distribution of $M_n$. You may obtain an asymptotic result for an appropriately normalized maxima under certain conditions (e.g. the Gumbel you allude to, although the class of limiting distributions for normalized maxima is larger than just the Gumbel).

As Aksakal points out, the exponential decay in correlations you refer to is well studied and falls within the class of stationary processes specified under the so-called $D(u_n)$ condition (amongst others too). The canonical reference with proofs would probably be Leadbetter, Lindgren and Rootzen (1983) but the basic principles of EVT for stationary processes is covered in many introductory EVT texts too.

Note that the limit distribution under $D(u_n)$ could be any extreme value distribution (not Gumbel exclusively - the danger of thinking simulations are "truth").