[For the entire collection see Zbl 0428.00021.]
For every \(n\geq 1\) let \(\{X_{1,n},...,X_{n,n}\}\) be a set of independent i.i.d. multivariate r.v., with distribution depending on n; if \(X_{k,n}=(X_{k,n}(1),...,X_{k,n}(p))\in {\mathbb{R}}^ p\), and \(Y_ n=\{\max_{1\leq k\leq n}X_{k,n}(1),...,\max_{1\leq k\leq n}X_{k,n}(p)\}\), we say that a probability distribution P is of infinite order if there exists a sequence of \(Y_ n\) whose distributions converge weakly to P.
We develop here the theory of infinite order probability distributions which includes, as a particular case the extreme value distributions. We obtain similar results as in the theory of infinite divisible distributions, a representation theorem analogous to the de Finetti theorem, and a canonical representation in terms of dependence functions, which gives, for infinite order distributions: \[ D(u_ 1,...,u_ p)= \]
\[ u_ 1...u_ p\exp \{\sum^{p}_{k=2}(-1)^ k\sum_{1\leq i_ 1<...<i_ k\leq p}\int^{-Log u_{i_ 1}}_{0}...\int^{-Log u_{i_ k}}_{0}d\mu_{k;i_ 1,...,i_ k}\} \] We study as examples the normal, Gumbel, Marshall-Olkin, Morgenstern distributions.