Let 
be independent and identically distributed random
variables with common distribution function 
,
and let
 |
(1)
|
The Fisher-Tippett-Gnedenko theorem states that if there are constants 
and 
such that
 |
(2)
|
at every continuity point of a nondegenerate distribution function 
, then, up to a change of location and scale, 
is one of the three distribution functions
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  | ![{exp[-(-x)^alpha] if x<=0; 1 if x>0,](/images/equations/Fisher-Tippett-GnedenkoTheorem/Inline15.svg) |
(5)
|
where 
for 
and 
for 
and 
.
These are known as the Gumbel, Fréchet,
and Weibull types, respectively. Conversely,
each of the three types can occur as such a limit (Fisher and Tippett 1928, Gnedenko
1943).
The three types can be combined into the generalized extreme value form
![G_xi(x)=exp[-(1+xix)^(-1/xi)]](/images/equations/Fisher-Tippett-GnedenkoTheorem/NumberedEquation3.svg) |
(6)
|
on the set 
, with the case 
defined by the limit
 |
(7)
|
Positive, zero, and negative values of 
correspond to the Fréchet,
Gumbel, and Weibull
types, respectively.
The classification is closely related to max-stability. Dividing a large sample into 
blocks and taking the maximum within each block leads, after changes of location
and scale, to the functional relation
 |
(8)
|
for suitable 
and 
. The three types above are the possible nondegenerate solutions
arising from this relation. The theorem is therefore the extreme-value analog of
the central limit theorem (Leadbetter et
al. 1983, de Haan and Ferreira 2006).
