Abstract:
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, Poisson distribution, geometric distribution.
Citation:
A. V. Prokhorov, N. G. Ushakov, “On the Problem of Reconstructing a Summands Distribution by the Distribution of Their Sum”, Teor. Veroyatnost. i Primenen., 46:3 (2001), 449–462; Theory Probab. Appl., 46:3 (2002), 420–430
\Bibitem{ProUsh01}
\by A.~V.~Prokhorov, N.~G.~Ushakov
\paper On the Problem of Reconstructing a Summands Distribution by the Distribution of Their Sum
\jour Teor. Veroyatnost. i Primenen.
\yr 2001
\vol 46
\issue 3
\pages 449--462
\mathnet{http://mi.mathnet.ru/tvp3895}
\crossref{https://doi.org/10.4213/tvp3895}
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\zmath{https://zbmath.org/?q=an:1032.60010}
\transl
\jour Theory Probab. Appl.
\yr 2002
\vol 46
\issue 3
\pages 420--430
\crossref{https://doi.org/10.1137/S0040585X97979202}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000179228700003}
Linking options:
https://www.mathnet.ru/eng/tvp3895
https://doi.org/10.4213/tvp3895
https://www.mathnet.ru/eng/tvp/v46/i3/p449
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