On the Problem of Reconstructing a Summands Distribution by the Distribution of Their Sum

The uniqueness and stability conditions of reconstructing a distribution of independent identically distributed random variables $X_1,\dots,X_m$ by a distribution of the sum $S=X_1+\dots+X_m$ for fixed $m$ are given. This paper considers two generalizations of the problem of reconstructing the random variables $X_j$: by the distribution $S=\gamma_1X_1+\dots+\gamma_mX_m$, where the random variables $\gamma_j$ take values 0 and 1 with some fixed probabilities, and bythe distribution of the sum $S_N=X_1+\dots+X_N$ 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.