Approximation of convolutions by accompanying laws under the existence of moment of low orders

It is shown that if a one-dimensional distribution $F$ has finite moment of the order $1+\beta$ for some $\beta$, $\frac12\le\beta\le1$, then the rate of approximation of the $n$-fold convolution $F^n$ by accompanying laws is $O(n^{-\frac12})$. Moreover, if, in addition, $\mathbf E\xi^2=\infty$, $\frac12<\beta<1$, then this rate of approximation is $o(n^{-\frac12})$. The question about the true rate of approximation of $F^n$ by infinitely divisible and accompanying laws is discussed. Bibl. 27 titles.