Existence of multivariate max-universal laws (Q2488466)

A multivariate distribution function \(F:R^d\to[0,1]\) belongs to a domain of partial max-attraction of d.f. \(G\) iff for some \(k_n\to\infty\), \(a_n\in R\), \(b_n\in R^d\), \((F(a_n x+b_n))^{k_n}\to G(x)\) at all continuity points \(x\) of \(G\). The author demonstrates that there exists a max-universal d.f. \(F\) which belongs to partial domains of max-attraction of all max-infinitely divisible laws \(G\). The proof follows the outline of the corresponding result for sums of i.i.d. random vectors.