Bochner's theorem for finite-dimensional conelike semigroups
A subset \(S\) of some real vector space \(V\) is called conelike, if for every \(s \in S\) there exists a number \(a(s) \in \mathbb{R}_ +\) such that \(\alpha s \in S\) for all \(\alpha \geq a(s)\). If \(S\) is also a semigroup and \(V\) is of finite dimension, we can prove the following extension of Bochner's classical theorem: Each bounded continuous positive definite function on \(S\) is a mixture of continuous characters. These continuous characters are identified in terms of a canonically associated subspace of \(V\) and the dual cone of the hermitian part of \(S\). We then give a probabilistic application.
scientific article; zbMATH DE number 537260