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

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.