On a class of limit distributions for normed sums of independent random variables

Let $\zeta_n=\frac{\xi_1+\xi_2+\dots+\xi_n}{B_n}-A_n$ ($n=1,2,\dots$) be a sequence of normed sums $n$ of independent random variables which has a nondegenerate limit distribution $G(x)$ for appropriately selected constants $A_n$, $B_n$.
This paper is devoted to the characterization of the class $\{G(x)\}$ named here $\mathscr P_r$ arizing when among the distributions of the random variables $\xi^i$ there are only $r$ different ones. Three theorems describing the class $\mathscr P_r$ are proved