Balkema-de Haan-Pickands theorem, generalized Pareto and lognormal On the wikipedia page on the Balkema-de Haan-Pickands theorem, en.wikipedia.org/wiki/Pickands-Balkema-de_Haan_theorem, it is said the "for a large class of underlying distribution functions", their far-right values can be approximated by a generalized Pareto.
What is and what is not that "large class"?
 A: The standard requirement is that $F$ is in the domain of attraction of the GEV (generalized extreme value distribution).  If F is in said domain of attraction, then the distribution of excesses over a high threshold (conditional on clearing that threshold) is well approximated by a distribution in the GP family.
"In the domain of attraction of the GEV" means that block maxima $M_n = \max_iX_i$, for some sequence of constants $a_n$, $b_n$, where $n$ refers to block size, the distribution of the standardized sample maximum doesn't degenerate.
$$
\lim\limits_{n\to\infty} \text{P}\left(\frac{M_n - b_n}{a_n} \leq x\right) = G(x)
$$
If F is in the domain of attraction of a GEV, then
$$
\lim\limits_{u\to\infty}\text{P}\left(X \geq u + y \mid x \geq u\right) = \lim\limits_{u\to\infty}\frac{1 - F(u + y)}{1 - F(u)} = H(y)
$$
Where $H$ is generalized Pareto.
A: There are some helpful comments on this in a draft textbook by Smith and Weissman, available at https://rls.sites.oasis.unc.edu/s834-2020/s834.html.
Domains of attraction
Section 2 discusses domains of attraction and says:

In practice nearly all continuous distributions are in the domain of attraction of some extreme value limit.

Page 12 presents some counterexamples:

Continuous distributions not in any domain of attraction These are hard to construct but they do exist! Here are two examples:
(a) Any distribution function with slowly varying tail, for instance
$F(x) = 1 − 1/ \log x$ for $x > e$.
(b) Consider
$F_\delta(x) = 1-x^{-1}\{1 + \delta \sin(\log x)\}$ valid for $x \geq$ some $x_0$, with $|\delta|$
small enough to make it a valid distribution function.

The second example is credited to Resnick (1971) "Tail equivalence and its applications".
Log-normal
The question title mentions the log-normal distribution. Page 12 of the book has another example showing the log-normal is in the GPD domain of attraction. It might be worth pointing out that this involves a more general version of the Pickands theorem than is given on the wikipedia page currently, which says $\Pr(X-u \leq y | X>u)$ converges to a GPD (with constant scale $\sigma$) as $u \to \infty$. A fuller vesion is that $\Pr(\frac{X-u }{h(u)} \leq y | X>u)$ converges to a GPD for some scaling function $h(u)$. (The first pages of Tawn and Papastathopolous https://arxiv.org/abs/1111.6899 have a nice summary of this.) The example in the book requires $h(u)$ to be non-constant. So I suspect the log-normal only lies in the GPD domain of attraction for the general theorem, and not for the simplified wikipedia form.
