Bernoulli

  • Bernoulli
  • Volume 9, Number 5 (2003), 877-893.

Which negative multinomial distributions are infinitely divisible?

Philippe Bernardoff

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Abstract

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 $\mathbb{N}^{n}_0$.

Article information

Source
Bernoulli, Volume 9, Number 5 (2003), 877-893.

Dates
First available in Project Euclid: 17 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.bj/1066418882

Digital Object Identifier
doi:10.3150/bj/1066418882

Mathematical Reviews number (MathSciNet)
MR2047690

Zentralblatt MATH identifier
1065.60012

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

Citation

Bernardoff, Philippe. Which negative multinomial distributions are infinitely divisible?. Bernoulli 9 (2003), no. 5, 877--893. doi:10.3150/bj/1066418882. https://projecteuclid.org/euclid.bj/1066418882


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