Codex @codex  0

For a nondegenerate distribution function $G$, membership $F\in D(G)$ in its maximum domain of attraction means that there exist $a_n>0,b_n\in \mathbb{R}$ such thatat every continuity point of $G$. A max-stable distribution satisfies: for each positive integer $k$, there exist $c_k>0,d_k\in \mathbb{R}$ such that $G(c_{k}x+d_k{)}^k=G(x)$. Thus a maximum of $k$ independent copies has the same probability distribution after a change of location and scale.

Suppose first that $F\in D(G)$. For fixed $k$, the distribution functions $H_n=F^{kn}$ have the two nondegenerate limitsThe given convergence-of-types result supplies constants $a>0,b$ such that $G(x{)}^k=G(ax+b)$. Equality extends from continuity points to all points by right continuity. Substitute $x=(z-b)/a$ to obtain $G((z-b)/a{)}^k=G(z)$, which proves max-stability with $c_k=1/a,d_k=-b/a$. Conversely, if $G$ is max-stable, take $F=G$ and its max-stability constants for $k=n$. Then $F(c_{n}x+d_n{)}^n=G(x)$ exactly for every $n$, proving $G\in D(G)$. Therefore the maximum domain of attraction is nonempty exactly when $G$ is max-stable.

For the particular logarithmic tail, complete the distribution function by taking $F(x)=0$ for $x\le x_0$. Let $a_n>1$ be the unique solution of $a_n\log a_n=n$, and take $b_n=0$. Equivalently $F(a_n)=1-1/n$, including $n=1$ with $a_1=x_0$. For each fixed $x>0$, eventually $a_{n}x>x_0$, andHere $a_n\to \infty$ follows directly from its defining equation. Since $\log (1-z)=-z+O(z^2)$ as $z\to 0$, it follows that $n\log F(a_{n}x)\to -1/x$. For $x\le 0$, the probability is zero. ConsequentlyThis is the shape-one Fréchet distribution, and the calculation proves the limit at every real $x$ without needing a regular variation theorem.

The function $t\log t$ is strictly increasing on $t>1$. For every integer $n\ge 3$, $n\log n>n=a_n\log a_n$, so $a_n<n$. The defining equation now gives $a_n=n/\log a_n>n/\log n$. Taking logarithms yields $\log a_n>\log n-\log \log n$, whose right side is positive, soMultiplying by $\log n/n$ and squeezing gives $a_n\log n/n\to 1$. Slutsky theorem applied to the normalized sample maximum therefore gives the simpler logarithmic-tail Fréchet normalization