# Fisher–Tippett–Gnedenko theorem

In statistics, the **Fisher–Tippett–Gnedenko theorem** (also the **Fisher–Tippett theorem** or the **extreme value theorem**) is a general result in extreme value theory regarding asymptotic distribution of extreme order statistics. The maximum of a sample of iid random variables after proper renormalization can only converge in distribution to one of 3 possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution. Credit for the extreme value theorem (or convergence to types theorem) is given to Gnedenko (1948)^{[1]}, previous versions were stated by Ronald Fisher and Leonard Henry Caleb Tippett (1928)^{[2]} and Fréchet (1927).^{[3]}

The role of the extremal types theorem for maxima is similar to that of central limit theorem for averages, except that the central limit theorem applies to the average of a sample from any distribution with finite variance, while the Fisher–Tippet–Gnedenko theorem only states that *if* the distribution of a normalized maximum converges, *then* the limit has to be one of a particular class of distributions. It does not state that the distribution of the normalized maximum does converge.

## Statement[edit]

Let be a sequence of independent and identically-distributed random variables, and . If a sequence of pairs of real numbers exists such that each and
,
where is a non-degenerate distribution function, then the limit distribution belongs to either the Gumbel, the Fréchet or the Weibull family.^{[4]} These can be grouped into the generalized extreme value distribution.

## Conditions of convergence[edit]

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 . In this case, possible sequences are

*b*= 0 and_{n}

- a Weibull if and only if and . In this case possible sequences are

*b*=_{n}*ω*and

Convergence conditions for the Gumbel distribution are more involved.

## See also[edit]

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

## Notes[edit]

**^**Gnedenko, B.V. (1943), "Sur la distribution limite du terme maximum d'une serie aleatoire",*Annals of Mathematics*,**44**(3): 423–453, doi:10.2307/1968974, JSTOR 1968974**^**Fisher, R.A.; Tippett, L.H.C. (1928), "Limiting forms of the frequency distribution of the largest and smallest member of a sample",*Proc. Camb. Phil. Soc.*,**24**(2): 180–190, Bibcode:1928PCPS...24..180F, doi:10.1017/s0305004100015681**^**Fréchet, M. (1927), "Sur la loi de probabilité de l'écart maximum",*Annales de la societe Polonaise de Mathematique*,**6**(1): 93–116**^**Mood, A.M. (1950). "5. Order Statistics".*Introduction to the theory of statistics*. New York, NY, US: McGraw-Hill. pp. 251–270.

## References[edit]

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

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