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A Characterization of Poisson–Gaussian Families by Convolution-Stability

Authors
Journal
Journal of Multivariate Analysis
0047-259X
Publisher
Elsevier
Publication Date
Volume
81
Issue
1
Identifiers
DOI: 10.1006/jmva.2001.1995
Keywords
  • Convolution
  • Marginal Law
  • Natural Exponential Families
  • Poisson–Gaussian Families
  • Variance Function

Abstract

Abstract If the convolution of natural exponential families on 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 (1992, “Lectures on Natural Exponential Functions and Their Variance Functions,” Instituto de Matemática pura e aplicada: Monografias de matemática, 50, Rı́o de Janeiro) in the case of two families, and by D. Pommeret (1999, C. R. Acad. Sci. Ser. I 328, 929–933) for more than two families.

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