On limit distribution of sums of random variables

Centered Rademacher sequences and centered sequences of lattice random variables with a non-trivial weak limit of the sums $\frac{1}{\sqrt{n}}\sum\limits_{i=1}^n\xi_i$ are considered in the article. A general form of limit distribution is found for these sequences. It is shown that the form of limit distribution depends only on the average mixed moments of the first order characterizing random variables of the sequence. In the case of lattice random variables we mean a sequence of Rademacher random variables in which we can distribute the elements of the given sequence.