## Bernoulli

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

### Which negative multinomial distributions are infinitely divisible?

Philippe Bernardoff

#### 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

https://projecteuclid.org/euclid.bj/1066418882

Digital Object Identifier
doi:10.3150/bj/1066418882

Mathematical Reviews number (MathSciNet)
MR2047690

Zentralblatt MATH identifier
1065.60012

#### 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