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A characterization of Poisson-Gaussian families by convolution-stability. (English) Zbl 1011.62051
Summary: If the convolution of natural exponential families on \(\mathbb{R}^d\) is still a natural exponential family, then the families are all Poisson-Gaussian, up to affinity. This statement is a generalization of the one-dimensional versions proved by G. Letac [Lectures on natural exponential functions and their variance functions. (1992; Zbl 0983.62501)] in the case of two families, and by D. Pommeret [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 10, 929-933 (1999; Zbl 0930.62011)] for more than two families.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62E10 Characterization and structure theory of statistical distributions
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