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 :
, 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.
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Gallardo, L., Gebuhrer, O. (1984). Lois de probabilite infiniment divisibles sur les hypergroupes commutatifs, discrets, denombrables. In: Heyer, H. (eds) Probability Measures on Groups VII. Lecture Notes in Mathematics, vol 1064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073638
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DOI: https://doi.org/10.1007/BFb0073638
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