This article is cited in 10 papers

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

A. V. Prokhorov^a, N. G. Ushakov<sup>b

a</sup> M. V. Lomonosov Moscow State University
^b Institute of Microelectronics Technology and High Purity Materials, Russian Academy of Sciences

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.

Received: 02.09.1999

DOI: 10.4213/tvp3895

 English version: Theory of Probability and its Applications, 2002, 46:3, 420–430

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