• Bernoulli
  • Volume 12, Number 2 (2006), 371-379.

A characterization of Poisson-Gaussian families by generalized variance

Célestin C. Kokonendji and Afif Masmoudi

Full-text: Open access


We show that if the generalized variance of an infinitely divisible natural exponential family F =F(μ) in a d -dimensional linear space is of the form det K μ ' '(θ )=exp(θ T b +c) , then there exists k in { 0,1,...,d} such that F is a product of k univariate Poisson and ( d -k )-variate Gaussian families. In proving this fact, we use a suitable representation of the generalized variance as a Laplace transform and the result, due to Jörgens, Calabi and Pogorelov, that any strictly convex smooth function f defined on the whole of R d such that det f ' '(θ ) is a positive constant must be a quadratic form.

Article information

Bernoulli, Volume 12, Number 2 (2006), 371-379.

First available in Project Euclid: 25 April 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

affine variance function determinant infinitely divisible measure Laplace transform Monge-Ampère equation r-reducibility


Kokonendji, Célestin C.; Masmoudi, Afif. A characterization of Poisson-Gaussian families by generalized variance. Bernoulli 12 (2006), no. 2, 371--379. doi:10.3150/bj/1145993979.

Export citation


  • [1] Bar-Lev, S., Bschouty, D., Enis, P., Letac, G., Lu, I. and Richard, D. (1994) The diagonal multivariate natural exponential families and their classification. J. Theoret. Probab., 7, 883-929.
  • [2] Caffarelli, L. and Li, Y.Y. (2004) A Liouville theorem for solutions of the Monge-Ampère equation with periodic data. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21, 97-120.
  • [3] Calabi, E. (1958) Improper affine hyperspheres of convex type and a generalization of a theorem by K Jörgens. Michigan Math. J., 5, 105-126.
  • [4] Casalis, M. (1991) Les familles exponentielles à variance quadratique homogène sont des lois de Wishart sur un cône symétrique. C.R. Acad. Sci. Paris Sér. I, 312, 537-540.
  • [5] Casalis, M. (1996) The 2d + 4 simple quadratic natural exponential families on R d. Ann. Statist., 24, 1828-1854.
  • [6] Cheng, S.Y. and Yau, S.T. (1986) Complete affine hypersurfaces; I The completeness of affine metrics. Comm. Pure Appl. Math., 39, 839-866.
  • [7] Consonni, G., Veronese, P. and Gutiérrez-Pena, E. (2004) Reference priors for exponential families with simple quadratic variance function. J. Multivariate Anal., 88, 335-364.
  • [8] Gikhman, I.I. and Skorohod, A.V. (1973) The Theory of Stochastic Processes II. New York: Springer- Verlag.
  • [9] Gutiérrez, C.E. (2001) The Monge-Ampère Equation. Boston: Birkhäuser.
  • [10] Hassairi, A. (1999) Generalized variance and exponential families. Ann. Statist., 27, 374-385.
  • [11] Jörgens, K. (1954) Über die Lösungen der Differentialgleichung rt - s2 = 1. Math. Ann., 127, 130-134.
  • [12] Kokonendji, C.C. (1995) Une caractérisation des vecteurs gaussiens n-dimensionnels. LAMIFA Technical Report No. 95/03, University of Amiens.
  • [13] Kokonendji, C.C. and Pommeret, D. (2001) Estimateurs de la variance généralisée pour des familles exponentielles non gaussiennes. C.R. Acad. Sci. Paris Sér. I, 332, 351-356.
  • [14] Kokonendji, C.C. and Seshadri, V. (1996) On the determinant of the second derivative of a Laplace transform. Ann. Statist., 24, 1813-1827.
  • [15] Kotz, S., Balakrishnan, N. and Johnson, N.L. (2000) Continuous Multivariate Distributions, Vol 1: Models and Applications, 2nd edn. New York: Wiley.
  • [16] Koudou, A.E. and Pommeret, D. (2002) A characterization of Poisson-Gaussian families by convolution-stability. J. Multivariate Anal., 81, 120-127.
  • [17] Letac, G. (1989) Le problème de la classification des familles exponentielles naturelles sur Rd ayant une fonction variance quadratique. In H. Heyer (ed.), Probability Measures on Groups IX, Lecture Notes in Math. 1306, pp. 194-215. Berlin: Springer-Verlag.
  • [18] Morris, C.N. (1982) Natural exponential families with quadratic variance functions. Ann. Statist., 10, 65-80.
  • [19] Muir, T. (1960) A Treatise on the Theory of Determinants. New York: Dover.
  • [20] Pogorelov, A.V. (1972) On the improper convex affine hyperspheres. Geom. Dedicata, 1, 33-46.
  • [21] Seshadri, V. (1997) A property of natural exponential families in Rn with simple quadratic variance function. J. Statist. Plann. Inference, 63, 351-361.