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Short Communications

Convergence of types under monotonous mappings

E. Pancheva

Institute of Mathematics, Sofia, Bulgaria

Abstract: Let $\mathcal F$ be the set of all D.F. on $\overline{\mathbf R}{}^d=[-\infty,\infty)^d$. Denote by $GMA$ the group of all max-automorphisms of $\overline{\mathbf R}{}^d$, i.e. such one-to-one mappings $L$ that preserve the max-operation in $\overline{\mathbf R}{}^d$, $L(x\vee y)=L(x)\vee L(y)$. We define type $(F):=\{G\in\mathscr{F}:\exists T\in GMA,G=F\circ T\}$. Hеге the convergence to type theorem is proved for distributions in $\mathcal F$ and norming sequences $\{L_n\}$ in $GMA$.

Keywords: Convergence of types, extreme values, max-automorphisms.

Received: 20.02.1991

Language: English

 English version: Theory of Probability and its Applications, 1993, 38:3, 551–556

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