On the Continuity of Minimum Stable Distributions

It is a well known result in extreme value theory that there are only three possible
non-degenerate limiting distributions for the sequence Mn = min{X1,...,Xn} , n >=1,
where X1, X2, ... are independent and identically distributed random variables defined on
the same probability space. The result, due to Fisher and Tippett (1928) and Gnedenko
(1943), appears in many textbooks of probability theory and its proof relies on the
solution of certain functional equations. It is of great instructional value, however, the
direct derivation of many interesting properties of these limiting distributions, from
the notion of minimum stability and in this article a proof of their necessary continuity
is presented.