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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.

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.

Edit: the comments below refer to the contents of the wikipedia page when this answer was original written but it's since been updated to fix this issue

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.

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 says 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.

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.

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 says 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 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.

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 says 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.