Fisher-Tippett-Gnedenko Theorem

Theorem
In general, for iid $\set {X_i} _{i\mathop = 1}^n$ with cdf $\map F x$ we have $\mathbb P \sqbrk {\max_{i \mathop = 1}^n \set {X_i} > x} = \map {F^n} x$.

Let $x_* := \sup \set {x: \map F x < 1}$.

Then as $n \to \infty$:
 * $\displaystyle \max_{i \mathop = 1, \mathop \ldots, n} \set {X_i} \stackrel {prob} {\to} x_*$

We normalize it by constants $a_n > 0, b_n$ such that:


 * $\dfrac {\displaystyle\max_{i \mathop = 1, \mathop \ldots, n} \set {X_i} - b_n} {a_n} \stackrel {dis} {\to} X_*$

for some non-degenerate $X_*$ with distribution $\map G x$.

We have the following three possible limits


 * Frechet: If $x > 0$, $\map G x = \map \exp {-x^{-a} }$ for some $a > 0$. If $x \le 0$, $\map G x = 0$.


 * Gumbel: If $x \in \R$, $\map G x = \map \exp {-e^{-\frac {x - \mu} {\sigma} } }$ for $\mu\in \R, \sigma>0$.


 * Weibull: If $x \ge 0$, $\map G x = 1 - \map \exp {-\paren {\dfrac x \lambda}^k}$ for $\lambda, k > 0$. If $x < 0$, $\map G x = 0$.

Fisher-Tippet-Gnedenko theorem
 * $(1): \quad \map G x$ will be Frechet $\map F x < 1$ and $\displaystyle \lim_{t \mathop \to +\infty} \dfrac {1 - \map F {t x} } {1 - \map F t} = x^{-\theta}$ for $x > 0$ and $\theta > 0$. Here we can set $b_n := 0$ and $a_n := F^{-1} \paren {1 - \dfrac 1 n}$.


 * $(2): \quad \map G x$ will be Weibull $x_* : = \sup \set {x: \map F x < 1} < \infty$ and $\displaystyle \lim_{t \mathop \to +\infty} \dfrac {1 - \map F {x_* + t x} } {1 - \map F {x_* - t} } = \paren {-x}^{-\theta}$ for $x < 0$ and $\theta > 0$. Here we can set $b_n := x_*$ and $a_n := x_* - \map {F^{-1} } {1 - \dfrac 1 n}$.


 * $(3): \quad G$ will be Gumbel if the density $\map f x := \dfrac \d {\d x} \map F x > 0$ and it is differentiable in $\openint {x_1} {x_*}$ for some $x_1$, and $\displaystyle \lim_{x \mathop \to x_*} \map {\dfrac \d {\d x} } {\dfrac {1 - \map F x} {\map f x} } = 0$. Here we can set $b_n := \map {F^{-1} } {1 - \dfrac 1 n}$ and $a_n := \dfrac 1 {n \map f {b_n} }$.

Proof
Proof under construction