Fisher–Tippett–Gnedenko theorem

Let ${\displaystyle X_{1},X_{2}\ldots ,X_{n}\ldots }$ be a sequence of independent and identically-distributed random variables, and ${\displaystyle M_{n}=\max\{X_{1},\ldots ,X_{n}\}}$. If a sequence of pairs of real numbers ${\displaystyle (a_{n},b_{n})}$ exists such that each ${\displaystyle a_{n}>0}$ and ${\displaystyle \lim _{n\to \infty }P\left({\frac {M_{n}-b_{n}}{a_{n}}}\leq x\right)=F(x)}$, where ${\displaystyle F}$ is a non-degenerate distribution function, then the limit distribution ${\displaystyle F}$ belongs to either the Gumbel, the Fréchet or the Weibull family.[4] These can be grouped into the generalized extreme value distribution.

If G is the distribution function of X, then M_n can be rescaled to converge in distribution to

• a Fréchet if and only if G (x) < 1 for all real x and ${\displaystyle {\frac {1-G(tx)}{1-G(t)}}\,{\xrightarrow[{t\to +\infty }]{\,}}x^{-\theta },\quad x>0}$. In this case, possible sequences are

b_n = 0 and ${\displaystyle a_{n}=G^{-1}\left(1-{\frac {1}{n}}\right).}$

• a Weibull if and only if ${\displaystyle \omega =\sup\{G<1\}<+\infty }$ and ${\displaystyle {\frac {1-G(\omega +tx)}{1-G(\omega -t)}}\,{\xrightarrow[{t\to 0^{+}}]{\,}}(-x)^{\theta },\quad x<0}$. In this case possible sequences are

b_n = ω and ${\displaystyle a_{n}=\omega -G^{-1}\left(1-{\frac {1}{n}}\right).}$

Convergence conditions for the Gumbel distribution are more involved.

• Extreme value theory
• Gumbel distribution
• Generalized extreme value distribution
• Pickands–Balkema–de Haan theorem
• Generalized Pareto distribution
• Exponentiated generalized Pareto distribution

• de Haan, Laurens; Ferreira, Ana (2006). Extreme Value Theory: An Introduction. New York: Springer. pp. 6–12. ISBN 0-387-34471-3.