Some properties of multivariate extreme value distributions and multivariate tail equivalence (Q1099543)

Denote by H a k-dimensional extreme value distribution with marginal distribution \(H_ i(x)=\Lambda (x)=\exp (-e^{-x})\), \(x\in {\mathbb{R}}^ 1\). Then it is proved that \[ H({\mathfrak x})=\Lambda (x_ 1)...\Lambda (x_ k)\quad for\quad any\quad {\mathfrak x}=(x_ 1,...,x_ k)\in {\mathbb{R}}^ k, \] if and only if the equation holds for \({\mathfrak x}=(0,...,0)\). Next some multivariate extensions of the results by \textit{S. I. Resnick} [J. Appl. Probab. 8, 135-156 (1971; Zbl 0217.499)] on tail equivalence and asymptotic distributions of extremes are established.