Limit laws for cumulative sums of independent random variables with distributions of a finite number of types

Let $Z_n=\frac1{B_n}\sum_{j=1}^nX_j-A_n$ ($n=1,2,\dots$) be a sequence of normalized sums of random variables with a non-degenerate limit distribution function $G(x)$. The paper describes classes $\mathfrak G_r$ of possible $G(x)$ when the distributions of $X_j$ ($j=1,2,\dots$) belong to at most $r$ ($r=1,2,\dots$) different types.