A Characterization of Poisson–Gaussian Families by Convolution-Stability

If the convolution of natural exponential families on Rd 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. I328, 929–933) for more than two families.