On the structure of monoids of admissible translates of multivariate probability measures
The paper under review is devoted to investigations of admissible translates of probability distributions on \({\mathbb{R}}^ n\) without Gaussian component. \((x\in {\mathbb{R}}^ n\) is an admissible translate of \(\mu \in M^ 1({\mathbb{R}}^ n)\) iff \(\mu *\delta_ xy\ll \mu\). Let \(A(\mu)\) be the set of admissible translates, let \(S(\mu)\) be the support of \(\mu\) and let \(\lambda_ n\) be the Lebesgue measure on \({\mathbb{R}}^ n).\) The author obtains in theorem 1 a detailed description of the structures of \(A(\mu)\) and \(S(\mu)\) for \(\mu \ll \lambda_ n\). If \(\mu\) is infinitely divisible (without Gaussian part) in the following theorems 2- 5 necessary and sufficient conditions for \(\mu\) being absolutely continuous resp. being singularly continuous are obtained. E.g. (theorem 3) the following are equivalent: (i) \(\mu \ll \lambda_ n\), (ii) \(\lambda_ n(A(\mu))>0\), (iii) \(A(\mu)^ 0\neq \emptyset\).
admissible translates of probability distributions
scientific article; zbMATH DE number 3858066