On a Hypothesis Proposed by B. V. Gnedenko

Several years ago Academician B. V. Gnedenko proposed the following: Let $\xi_n=(1/B_n)(\xi_1+\cdots+\xi_n)-A_n$ be a sequence of normed sums of independent stochastic quantities having a nondegenerate limit distribution $G(x)$ for appropriately selected constants $A_n$ and $B_n$. If among the distributions of stochastic quantities $\xi _i $ there are only $s$ different ones, then the limit distribution $G(x)$ is a composition of not more than stable laws.
In the paper the hypothesis proposed by B. V. Gnedenko is proved for $s=2$ and an example is presented showing that the theorem by E. Lebedintseva [2] does not prove this hypothesis in its entirety.