Inversion of the Lyapunov central limit theorem (Q1360731)

It is well-known that P. L. Chebyshev started the process of defining the probability distribution of a sum of many independent random variables. \textit{A. M. Lyapunov's} famous theorem [``Gesammelte Werke. Bd. 1'' (1954; Zbl 0059.00105)] states that the final distribution function will be normal if the independent random variables having some definite probability characteristics are summed. The inversion problem involves determining the possibility of representing random variables that have the given probability distribution as the sums of random variables that have normal distribution. Naturally, this set-up is not universal, but it can be starting point for solving such problems. Here, this problem is solved for continuous distribution laws. In addition, the connection between the summation of random variables and the approximation of the distribution function is clarified.