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Pré-Publication, Document De Travail Année : 2021

Domain of existence of the Laplace transform of infinitely divisible negative multinomial distributions

Domaine d'existence de la transformée de Laplace des lois multinomiales négatives indéfiniment divisibles

Résumé

This article provides the domain of existence Ω of the Laplace transform of infinitely divisible negative multinomial distributions. We define a negative multinomial distribution on Nⁿ, where N is the set of nonnegative integers, by its probability generating function which will be of the form (A(a₁z₁,…,a_{n}z_{n})/A(a₁,…,a_{n}))^{-λ} where A(z)=∑_{T⊂{1,2,…,n}}a_{T}∏_{i∈T}z_{i}, where a_{∅}≠0, and where λ is a positive number. Finding couples (A,λ) for which we obtain a probability generating function is a difficult problem. Necessary and sufficient conditions on the coefficients a_{T} of A for which we obtain a probability generating function for any positive number λ are know by (Bernardoff, 2003). Thus we obtain necessary and sufficient conditions on a=(a₁,…,a_{n}) so that a=(e^{t₁},…,e^{t_{n}}) with t=(t₁,…,t_{n}) belonging to Ω. This makes it possible to construct all the infinitely divisible multinomial distributions on Nⁿ. We give examples of construction in dimensions 2 and 3.
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Dates et versions

hal-03243831 , version 1 (31-05-2021)
hal-03243831 , version 2 (08-06-2022)

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Philippe Bernardoff. Domain of existence of the Laplace transform of infinitely divisible negative multinomial distributions. 2021. ⟨hal-03243831v1⟩
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