A survey on the effects of free and boolean convolutions on Cauchy-Stieltjes Kernel families

Abstract

In the setting of noncommutative probability theory and in analogy with the theory of natural exponential families (NEFs), a theory of Cauchy-Stieltjes Kernel (CSK) families has been recently introduced. It is based on the Cauchy-Stieltjes kernel (1 − θx)−1. In this paper, after presenting some basic concepts on NEFs and CSK families and pointing out some similarities and differences between the two families, we review the present state and developments regarding the effects of free and boolean convolutions powers on CSK families. MSC2020 subject classifications: 60E10, 46L54.