Bernoulli

  • Bernoulli
  • Volume 12, Number 1 (2006), 169-189.

Which multivariate gamma distributions are infinitely divisible?

Philippe Bernardoff

Full-text: Open access

Abstract

We define a multivariate gamma distribution on R n by its Laplace transform ( P( -θ ) ) - λ , λ >0, where

P ( θ )= T { 1,,n }p T i Tθ i .

Under p { 1,,n } 0 , we establish necessary and sufficient conditions on the coefficients of P , such that the above function is the Laplace transform of some probability distribution, for all λ >0, thus characterizing all infinitely divisible multivariate gamma distributions on R n .

Article information

Source
Bernoulli, Volume 12, Number 1 (2006), 169-189.

Dates
First available in Project Euclid: 28 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1141136656

Mathematical Reviews number (MathSciNet)
MR2202328

Zentralblatt MATH identifier
1101.60008

Keywords
Bell polynomials exponential families Frullani integral generalized hypergeometric series infinitely divisible distribution Laplace transform multivariate gamma distribution Stirling numbers of the second kind

Citation

Bernardoff, Philippe. Which multivariate gamma distributions are infinitely divisible?. Bernoulli 12 (2006), no. 1, 169--189. https://projecteuclid.org/euclid.bj/1141136656


Export citation

References

  • [1] Bar-Lev, S.K., Bshouty, D., Enis, P., Letac, G., Lu, I. and Richards, D. (1994) The diagonal multivariate natural exponential families and their classification. J. Theoret. Probab., 7, 883-929.
  • [2] Barndorff-Nielsen, O.E. (1980) Conditionality resolutions. Biometrika, 67, 293-310.
  • [3] Bernardoff, P. (2003) Which negative multinomial distributions are infinitely divisible? Bernoulli, 9, 877-893.
  • [4] Berndt, B.C. (1985) Ramanujan´s Notebooks, Part I. New York: Springer-Verlag.
  • [5] Comtet, L. (1970a) Analyse Combinatoire, Vol. 1. Paris: PUF.
  • [6] Comtet, L. (1970b) Analyse Combinatoire, Vol. 2. Paris: PUF.
  • [7] Griffiths, R.C. (1984) Characterization of infinitely divisible multivariate gamma distribution. J. Multivariate Anal., 15, 13-20.
  • [8] Johnson, N., Kotz, S. and Balakrishnan, N. (1997) Continuous Multivariate Distributions. New York: Wiley.
  • [9] Krob, D. and Legros, S. (1999) Algèbre Générale et Linéaire. Introduction au Calcul Symbolique et aux Mathématiques Expérimentales, Vol 2. Paris: Vuibert.
  • [10] Letac, G. (1991) Lectures on Natural Exponential Families and Their Variance Functions. Rio de Janeiro: Instituto de Matemática Pura e Aplicada.
  • [11] Moran, P.A.P. and Vere-Jones, D. (1969) The infinite divisibility of multivariate gamma distributions. Sankhya Ser. A, 40, 393-398.
  • [12] Sato, K.-I. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press.
  • [13] Seshadri, V. (1987) Contribution to discussion of the paper by B. Jørgensen: Exponential dispersion models. J. Roy. Statist. Soc. Ser. B, 49, 156.
  • [14] Slater, L.J. (1966) Generalized Hypergeometric Functions. Cambridge: Cambridge University Press.
  • [15] Stanley, R.P. (1999) Enumerative Combinatorics, Vol. 1, 3rd edn. Cambridge: Cambridge University Press.
  • [16] Vere-Jones, D. (1967) The infinite divisibility of a bivariate gamma distribution. Sankhya Ser. A, 29, 421-422.