My question is about the maximum domains of attraction $MDA(\xi)$ from extreme value theory. I would like to be able to say statements such as "since $f$ and $g$ both are in $MDA(\xi)$, $f+g$ is also in $MDA(\xi)$", or "since $f\in MDA(\xi_1)$ and $g\in MDA(\xi_2)$ with $\xi_1>\xi_2$, $f+g\in MDA(\xi_1)$.

Given that $f_1\in MDA(\xi_1)$ and $f_2\in MDA(\xi_2)$ are sufficiently smooth, what can we say about:

  1. The domain of attraction that $f_1+f_2$ belongs to
  2. The domain of attraction that $f_1*f_2$ (with $*$ denoting convolution) belongs to
  3. At a minimum, can we say that $f_1+f_2$ and $f_1*f_2$ belong to some maximum domain of attraction, or are there perverse cases where $f_1, f_2$ are, e.g., twice continuously differentiable and belong to some maximum domains of attraction but $f_1+f_2$ does not belong to any maximum domain of attraction?

Ideally, I would like some citeable reference to some useful lemma that I could refer to in my own work. Proving from first principle that e.g. the mixture of a normal and an exponential belongs to $MDA(0)$ is maybe an interesting exercise but I feel like there must be some user-friendly propositions that could help. I am in particular interested in cases including $\xi=0$ (i.e., the thin-tailed distributions such as the normal and exponential distribution).

  • $\begingroup$ For the mixture case the result for $\xi_1 \neq \xi_2$ can be found by using the characterization of the tail index in terms of moments, see my answer easily adapted for a finite mixture. $\endgroup$ – Yves Jun 4 at 6:49

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