Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The real powers of the convolution of a negative binomial distribution and a Bernoulli distribution


Authors: Gérard Letac, Dhafer Malouche and Stefan Maurer
Journal: Proc. Amer. Math. Soc. 130 (2002), 2107-2114
MSC (1991): Primary 60E10; Secondary 33A65
DOI: https://doi.org/10.1090/S0002-9939-02-05352-2
Published electronically: February 8, 2002
MathSciNet review: 1896047
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For $c>0,$ this note computes essentially the set of $(x,y)$ in $[0,+\infty)^2$ such that the entire series in $z$ defined by $(1+z/c)^x(1-z)^{-y}$ has all its coefficients non-negative. If $X$ and $Y$ are independent random variables which have respectively Bernoulli and negative binomial distributions, denote by $\mu$ the distribution of $X+Y$. The above problem is equivalent to finding the set of $p>0$ such that $\mu^{*p}$ exists; this set is a finite union of intervals and may be the first example of this type in the literature. This gives the final touch to the classification of the natural exponential families with variance functions of Babel type, i.e. of the form $aR(m)+(bm+c)\sqrt{R(m)}$, where $R$ is a polynomial with degree $\leq 2.$


References [Enhancements On Off] (What's this?)

  • [1] ERDELYI, A., MAGNUS, W., OBERHETTINGER, F., TRICOMI, F. G., et al (1953), Higher Transcendental Functions. Vol. 2. New York, McGraw-Hill Book Co. MR 15:419i
  • [2] HAYMAN, W. K. (1956), ``A generalisation of Stirling's formula." J. Reine Angew. Math., 196, 67-95. MR 18:293f
  • [3] JORGENSEN, B. (1987), ``Exponential dispersion models." J. Roy. Statist. Soc. Ser. B, 49, 127-162. MR 88m:62094
  • [4] KOKONENDJI, C. C. (1994), ``Exponential families with variance functions in $\sqrt{\Delta}P(\sqrt{\Delta}):$ Seshadri's class." Test 3 123-172. MR 97c:62033
  • [5] KOKONENDJI, C. C. (1995), ``Sur les familles exponentielles naturelles de grand-Babel." Annales de la Faculté des Sciences de Toulouse, IV,4, 763- 799. MR 99f:62024
  • [6] LETAC, G. and MORA, M. (1990), ``Natural real exponential families with cubic variances." Ann. Statist. 18, 1-37. MR 91b:62032
  • [7] LETAC, G. (1992), Lectures on Natural Exponential Families and their Variance Functions. Monografias de Matemática, Vol 50. Rio de Janeiro: Instituto de Matemática Pura e Aplicada. MR 94f:60020
  • [8] LÉVY, P. (1937), ``Sur les exponentielles de polynômes." Annales de l'Ecole Normale Supérieure, 73, 231-292.
  • [9] LUKACS, E. (1970), Characteristic Functions. 2nd edition. Charles Griffin and Co., London. MR 49:11595
  • [10] MORRIS, C. N. (1982), ``Natural exponential families with quadratic variance functions." Ann. Statist. 10, 65-80. MR 83a:62037
  • [11] SZEGÖ, G. (1975), Orthogonal Polynomials (4 th edition) AMS, Providence, R.I. MR 51:8724
  • [12] WIDDER D. V. (1941), The Laplace Transform. Princeton University Press, New Jersey. MR 3:232d

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 60E10, 33A65

Retrieve articles in all journals with MSC (1991): 60E10, 33A65


Additional Information

Gérard Letac
Affiliation: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062, Toulouse, France
Email: letac@cict.fr

Dhafer Malouche
Affiliation: 24 Av. Mongi Slim, 1004 El Menzah V, Tunisie
Email: dhafer_malouche@yahoo.fr

Stefan Maurer
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22902
Email: srm4x@virginia.edu

DOI: https://doi.org/10.1090/S0002-9939-02-05352-2
Keywords: Exponential family, Meixner polynomials, Jorgensen set
Received by editor(s): May 1, 1998
Received by editor(s) in revised form: November 4, 1998
Published electronically: February 8, 2002
Communicated by: Stanley Sawyer
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society