On continuation of max-stable laws (Q1286608)

A probability law \(P\) is said to be ``max-stable'' if it is the limit in distribution of a sequence \(a_n^{-1} \max\{X_1,X_2,\dots,X_n\} -b_n\), where the \(X_i\) are iid random variables with distribution \(P\) and the \(a_n\) and \(b_n\) are real constants (\(a_n\) positive). A result of Gnedenko and Senusi-Bereksi says that the convergence in law can be ascertained from the convergence of the distribution functions restricted to an interval. This paper presents a variant of the above result, namely, that if the sequences \(a_n\) and \(b_n\) satisfy a certain regularity condition, convergence in a single point may be extended to all of \(R\). It is also shown that a max-stable distribution \(G\) is determined by the choice of \(x_1<x_2<x_3\) and \(0<y_1<y_2<y_3<1\) such that \(G(x_i)=y_i\).