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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
Published electronically: February 8, 2002
MathSciNet review: 1896047
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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 $\leq 2.$

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Additional Information

Gérard Letac
Affiliation: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062, Toulouse, France

Dhafer Malouche
Affiliation: 24 Av. Mongi Slim, 1004 El Menzah V, Tunisie

Stefan Maurer
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22902

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