Let $\{\xi_n\}$ be a sequence of independent identically distributed random variables. Put \begin{equation} \eta_{nj}=\frac{1}{b_j}(\xi_1+\xi_2+\dots+\xi_{nj})\div a_j \tag{1} \end{equation} and assume that \begin{equation} n_j<n_{j+1}, \quad \lim_{j\to\infty}\frac{n_{j+1}}{n_j}=r\geq 1, \qquad r<\infty. \tag{2} \end{equation} In the paper, the class of limit distributions for the variables (1) under the conditions (2) is studied. This class is shown to possess some properties of the class of stable distributions. A general form of the spectral function of distributions from this class is given (Theorem 1).