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 and its convergence details are given to Fréchet (1927), Ronald Fisher and Leonard Henry Caleb Tippett (1928), Mises (1936) and Gnedenko (1943). 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. Let formula_1 be a sequence of independent and identically-distributed random variables with cumulative distribution function formula_2. Suppose that there exist  two sequences of real numbers formula_3 and formula_4 such that the following limits converge to a non-degenerate distribution function: or equivalently: In such circumstances, the limit distribution formula_7 belongs to either the Gumbel, the Fréchet or the Weibull family. In other words, if the limit above converges we will have formula_8 assume the form: or else for some parameter formula_11 This is the cumulative distribution function of the generalized extreme value distribution (GEV) with extreme value index formula_12. The GEV distribution groups the Gumbel, Fréchet and Weibull distributions into a single one. Note that the second formula (the Gumbel  distribution) is the limit of the first as formula_12 goes to zero. Conditions of convergence. The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution formula_8 above. The study of conditions for convergence of formula_7 to particular cases of the generalized extreme value distribution began with Mises, R. (1936) and was further developed by Gnedenko, B. V. (1943). Let formula_2 be the distribution function of formula_17, and formula_18 an i.i.d. sample thereof. Also let formula_19 be the populational maximum, i.e. formula_20. The limiting distribution of the normalized sample maximum, given by formula_7 above, will then  be: Examples. Fréchet distribution. If we take the Cauchy distribution the cumulative distribution function is: formula_43 is asymptotic to formula_44 or and we have Thus we have and letting formula_48 (and skipping some explanation) for any formula_50 The expected maximum value therefore goes up linearly with . Gumbel distribution. Let us take the normal distribution with cumulative distribution function We have and Thus we have If we define formula_55 as the value that satisfies then around formula_57 As increases, this becomes a good approximation for a wider and wider range of formula_59 so letting formula_60 we find that Equivalently, We can  see that formula_63 and then so the maximum is expected to climb ever more slowly toward infinity. Weibull distribution. We may take the simplest example, a uniform distribution between 0 and 1, with cumulative distribution function Approaching 1 we have Then Letting formula_68 we have The expected maximum approaches 1 inversely proportionally to .