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 Teor. Veroyatnost. i Primenen., 1957, Volume 2, Issue 1, Pages 34–59 (Mi tvp4957)

On the Decomposition of the Convolution of Gaussian and Poissonian Laws

Yu. V. Linnik

Abstract: The paper contains a detailed proof for the following theorem: the convolution of a Gaussian and a Poissonian law can be decomposed only into similar convolutions. More precisely, let $X=X_1+X_2$, where $X_1$ is a Gaussian component and $X_2$ a Poissonian component independent of $X_1$. If there is some other decomposition: $X=Y_1+Y_2,Y_1$ being independent of $Y_2$, then
$$Y_1=Y_{11}+Y_{12}, Y_2=Y_{21}+Y_{22}.$$
where $Y_{11},Y_{21}$ are Gaussian and $Y_{12},Y_{22}$ Poissonian, all mutually independent, and
$$\mathbf D(Y_{11})+\mathbf D(Y_{21})=\mathbf D(X_1), \mathbf D(Y_{12})+\mathbf D(Y_{22})=\mathbf D(X_2).$$
Thus, H. Cramer's theorem on decomposing the normal law, and D. A. Raykov's theorem on decomposing the Poissonian law are special cases of this theorem.

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English version:
Theory of Probability and its Applications, 1957, 2:1, 31–57

Citation: Yu. V. Linnik, “On the Decomposition of the Convolution of Gaussian and Poissonian Laws”, Teor. Veroyatnost. i Primenen., 2:1 (1957), 34–59; Theory Probab. Appl., 2:1 (1957), 31–57

Citation in format AMSBIB
\Bibitem{Lin57} \by Yu.~V.~Linnik \paper On the Decomposition of the Convolution of Gaussian and Poissonian Laws \jour Teor. Veroyatnost. i Primenen. \yr 1957 \vol 2 \issue 1 \pages 34--59 \mathnet{http://mi.mathnet.ru/tvp4957} \transl \jour Theory Probab. Appl. \yr 1957 \vol 2 \issue 1 \pages 31--57 \crossref{https://doi.org/10.1137/1102002} 

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This publication is cited in the following articles:
1. G. M. Feldman, A. E. Fryntov, “Expansion of a convolution of gaussian and poisson distributions of locally compact Abelian groups”, Funct. Anal. Appl., 13:4 (1979), 316–317
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