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October 2003 Which negative multinomial distributions are infinitely divisible?
Philippe Bernardoff
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Bernoulli 9(5): 877-893 (October 2003). DOI: 10.3150/bj/1066418882


We define a negative multinomial distribution on $\mathbb{N}^{n}_0$, where $\mathbb{N}_0$ is the set of non-negative integers, by its probability generating function which will be of the form $(A(a_{1},\rm dots,a_{n})/$ $A(a_{1}z_{2},\rm dots,a_{n}z_{n}))^{\lambdaambda}$, where \[A(\mathbf{z})=\sum\lambdaimits_{T\subset\lambdaeft\{1,2,\rm dots,n\right\}}a_{T}\prod\lambdaimits_{i\in T}z_{i},\] \(a_{\rm emptyset}\neq0\), and \(\lambdaambda\) is a positive number. Finding couples \(\lambdaeft( A,\lambdaambda\right)\) for which we obtain a probability generating function is a difficult problem. We establish necessary and sufficient conditions on the coefficients of \(A\) for which we obtain a probability generating function for any positive number \(\lambdaambda\). In consequence, we obtain all infinitely divisible negative multinomial distributions on $\mathbb{N}^{n}_0$.


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Philippe Bernardoff. "Which negative multinomial distributions are infinitely divisible?." Bernoulli 9 (5) 877 - 893, October 2003.


Published: October 2003
First available in Project Euclid: 17 October 2003

zbMATH: 1065.60012
MathSciNet: MR2047690
Digital Object Identifier: 10.3150/bj/1066418882

Keywords: discrete exponential families , infinitely divisible distribution , negative multinomial distribution , probability generating function

Rights: Copyright © 2003 Bernoulli Society for Mathematical Statistics and Probability

Vol.9 • No. 5 • October 2003
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