On the problem of reconstructing a summands distribution by the distribution of their sum

The uniqueness and stability conditions of reconstructing it distribution of independent identically distributed random variables X-1......X-m by a distribution of the sum S = X-1 + ... + X-m for fixed m are given. This paper considers two generalizations of the problem of reconstructing the random variables X-j: by the distribution S = gamma(1)X(1) + .... + gamma(m) X-m, where the random variables gamma(j) take values 0 and 1 with some fixed probabilities, and by the distribution of the sum S-N = X-1 + ... + X-N of the random number N of summands X-j. In these problems there are given not only sufficient stability conditions of reconstructing but quantitative stability estimators.