Frechet and subexponentiality

Question

In the context of extreme samples from distributions, I note that some subexponential distributions (such as the lognormal) are in the "domain of attraction" of the Gumbel/exponential class of distributions.

But are all distributions in the "domain of attraction" of the Frechet/power-law class necessarily subexponential?

Answer 1

I have found a citation (Embrechts et al. 1997, Chapter 3.3.1) indicating that all regularly varying distributions are in the domain of attraction of the Frechet distribution, and all distribution in the domain of the Frechet distribution are regularly varying. Since regular variation is a subset of subexponential, I conclude that distributions in the domain of the Frechet are, in fact, subexponential.