Recently I read a proof in  of the main theorem of extreme value statistics: the Fisher–Tippett–Gnedenko theorem. In this post I give an outline.
Here we are interested in the maximum of many independent and identically distributed random variables with distribution function . Let which may be infinite. Then as ,
tends to 0 if and 1 if .
Therefore converges in probability to as .
In order to avoid this degenerate limiting distribution for all extreme value distributions, it is necessary to normalise the distribution.
To this end, suppose there exist real numbers such that
approaches a non-degenerate limiting distribution.
In other words, there exists a distribution function such that
Taking logarithms of both sides, this is equivalent to
This requires as . Using for close to 1, the above is also equivalent to
Next we use the following definition.
A non-decreasing function has left-continuous inverse defined by
One can use this definition to prove
Lemma 1: If for non-decreasing functions then for each continuity point we have .
Next we claim that (1) is equivalent to
where is the left-continuous inverse of . To see this, let . Then by the definition of , . Then for any ,
By (1) and the lemma as this tends to
proving the claim. We can also write
where and denotes the integer part of .
We are now ready to prove the main theorem of extreme value theory.
Theorem (Fisher, Tippet, Gnedenko):
where the right side is equal to its limiting value if .
This will involve numerous substitutions but the main idea is to arrive at a differential equation that can be solved to obtain the above. Suppose is a continuity point of . Then for any continuity point ,
We can write
The claim is that both and have limits as . If they had more than one limit point, say then for (4) in the limit gives us
Subtracting gives which implies as we know is non-constant (since we seek a non-degenerate solution).
We conclude that
This is a functional equation that we wish to solve. We let to obtain
which using implies
Since is monotone (following from the monotonicity of ), it is differentiable at some . By (6) it is differentiable at all . Indeed from (6) we obtain
Let . Then .
From (6) and (7),
Similarly, and upon subtraction from (8) we obtain from which
Taking the limit as and using gives the following differential equation for .
To solve (9), differentiate both sides with respect to : from we see that
Hence and since , .
Recalling , this leads to .
Recalling this means
Taking the left-continuous inverse of both sides,
Since , .
In other words,
If 1 is not a continuity point, we repeat the above proof with replaced by where is a continuity point. This completes the proof.
This generalised extreme value distribution encaptures three distributions depending on the nature of the tail of the original distribution :
- Type I – : Gumbel (double exponential) distribution (exponential tail – e.g. normal or exponential distribution)
- Type II – : Fréchet distribution (polynomial tail – e.g. power law distribution to model extreme flood levels or high incomes)
- Type III – : reverse-Weibull distribution (finite maximum – e.g. uniform distribution)
Sample density functions are plotted below for specific values of .
 L. De Haan, A. Ferreira, Extreme Value Theory: An Introduction, Springer, 2006.