## Bernoulli

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

### Which negative multinomial distributions are infinitely divisible?

#### 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