Geodesic
On the Problem of Reconstructing a Summands Distribution by the Distribution of Their Sum
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 3, pp. 449-462

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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.
Keywords: summands distribution, stability, sum of a random number of summands, linear combinations, characteristic function, geometric distribution.
Mots-clés : Poisson distribution 
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     author = {A. V. Prokhorov and N. G. Ushakov},
     title = {On the {Problem} of {Reconstructing} a {Summands} {Distribution} by the {Distribution} of {Their} {Sum}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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A. V. Prokhorov; N. G. Ushakov. On the Problem of Reconstructing a Summands Distribution by the Distribution of Their Sum. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 3, pp. 449-462. http://geodesic.mathdoc.fr/item/TVP_2001_46_3_a2/