# If the density functions $f_1, f_2$ each are in domains of attraction $MDA(\xi_1)$ and $MDA(\xi_2)$, what can we say about $0.5f_1+0.5f_2$?

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).

• 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. – Yves Jun 4 at 6:49