Factorization of probability measures on symmetric hypergroups

Abstract

AbstractGeneralizing known results for special examples, we derive a Khintchine type decomposition of probability measures on symmetric hypergroups. This result is based on a triangular central limit theorem and a discussion of conditions ensuring that the set of all factors of a probability measure is weakly compact. By our main result, a probability measure satisfying certain restrictions can be written as a product of indecomposable factors and a factor in I0(K), the set of all measures having decomposable factors only. Some contributions to the classification of I0(K) are given for general symmetric hypergroups and applied to several families of examples like finite symmetric hypergroups and hypergroup joins. Furthermore, all results are discussed in detail for a class of discrete symmetric hypergroups which are generated by infinitely many joins, for a class of countable compact hypergroups, for Sturm-Liouville hypergroups on [0, ∞[ and, finally, for polynomial hypergroups.