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I'm analyzing data about which I have a strong suspicion that it is self-similar (Hurst parameter ranging from 0.60 to 0.78 depending on estimation method and sample sequence). I also observe high realization values much more often (compared to experiments where no self-similarity is suspected) which suggests that the generating distribution can have a long (heavy) tail.

It seems that Extreme Value Theory cannot be used in this case to reason about the tail behavior. In case of self-similarity long range dependence does obviously violate the i.i.d. requirement, but does the inability to use EVT in my case apply more broadly to all long tailed distributions?

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    $\begingroup$ Frechet distribution with $\alpha \leq 1$ has a long tail and can be used to model maxima of long tailed distributions. $\endgroup$
    – moorray
    Commented Sep 11, 2014 at 14:04

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I believe it is true that EVT does not apply to self-similar processes (i.e. processes with dependence, violating iid). However, not all heavy tailed distributions exhibit dependence. (I say "I believe..." because I don't have expertise in EVT specifically.)

I can suggest several resources that could be helpful, both to answer your question and to offer alternative methods:

Here's a resource about varieties of correlation and dependence, which might help you select a method that is appropriate to the dependence in your data:

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