Fisher–Tippett–Gnedenko theorem

Remarks

The three families combine into the generalised extreme value distributions $G_\xi(x) = \exp\left(-(1 + \xi x)^{-1/\xi}\right)$ for $1 + \xi x > 0$, with $G_0 = \Lambda$ as the limit case $\xi \to 0$. The sign of $\xi$ gives the type: $\xi > 0$ is Fréchet, $\xi = 0$ is Gumbel, and $\xi < 0$ is reversed Weibull.

Idea of the proof. If $F$ is the distribution function of $X_1$, then $P(M_n \le a_n x + b_n) = F(a_n x + b_n)^n$. Comparing the limits along $n$ and $nk$ shows that $G$ is max-stable: for each $k \ge 1$ there are $\alpha_k > 0$ and $\beta_k$ with $G(x)^k = G(\alpha_k x + \beta_k)$. Solving this functional equation gives the three families.

Example: let the $X_k$ be exponential with rate $1$, so $F(x) = 1 - e^{-x}$ for $x \ge 0$. Then $P(M_n - \log n \le x) = (1 - e^{-x}/n)^n \to \exp(-e^{-x})$ for every $x$, so the Gumbel law appears with $a_n = 1$ and $b_n = \log n$.

The hypothesis that a non-degenerate limit exists is needed. For some laws, such as the geometric and the Poisson distributions, no choice of $a_n$ and $b_n$ gives a non-degenerate limit.

Going deeper

For a student

The theorem plays for maxima the role that the central limit theorem plays for sums. The set of laws whose maxima converge to a given type is its domain of attraction, and Gnedenko characterised these domains through conditions on the tail $1 - F$. In statistics one fits the generalised extreme value family to block maxima of data.