Lois de probabilite infiniment divisibles sur les hypergroupes commutatifs, discrets, denombrables

Abstract

This paper is a contribution to the study of infinitely divisible probability measures on hypergroups. In the case of a discrete, infinite countable hypergroup X without non trivial compact subhypergroups, we prove that the only infinitely divisible probability measures are of Poisson type as soon as (roughly speaking) we have a Levy continuity Theorem. More precisely : μ ε M₁ (X) is infinitely divisible if and only if there exists a unique positive bounded Radon measure ν on X with ν({e})=0 and such that :

$$\hat \mu (\chi ) = exp\left[ {\sum\limits_{x \in X} {(\overline {\chi (X)} - 1)\nu (\{ x\} )} } \right](\chi \in \hat \chi )$$

, where $\hat \mu$ denotes the Fourier transform of μ (which is defined on the set $\hat X$ of hermitian characters of X). This result is true in one of the two following cases : C 1 : The support of the Plancherel measure on $\hat X$ contains the point II (= the character identically 1). C 2 : II is not an isolated point in $\hat X$ and there exists a neighbourhood V of II such that $\mathop {lim}\limits_{\chi \to \infty } \chi (x) = 0$χ(x)=0 for every χ ε V−{II}.

These conditions are clearly only sufficient. They are illustrated by exemples in paragraph 6. In such cases we study also the problem of triangular arrays of probability measures on X.