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On a problem of a Khinchin-type decomposition theorem for extreme values

E. Pancheva

Institute of Mathematics, Sofia, Bulgaria

Abstract: Traditionally, extreme value theory has been treated in the multiplicative semigroup $\mathcal{P}$ of distribution functions (d.f's) on $\mathbf{R}^d$ endowed with the Lévy metric $L$ (which metrizes the weak convergence in $\mathcal{P}$). Unfortunately, in $(\mathcal{P},L,\cdot)$ there is no Khinchin-type decomposition theorem, as is shown in [7]. We choose another approach to extreme values, namely, we consider the multiplicative semigroup $\mathcal{F}$ of distributions on $\overline{\mathbf R}=[-\infty,\infty)^d$, introduce in it a metric $\mathcal{L}$, corresponding to the weak convergence in $\mathcal{F}$, and show that in the structure $(\mathcal{F},L,\cdot)$ there are analogues of the well known first and second Khinchin's theorems.

Keywords: extreme values, Khinchin-type decomposition, class max-$I_0$.

Received: 11.05.1993

Language: English

 English version: Theory of Probability and its Applications, 1994, 39:2, 329–336

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