One-sided Cauchy-Stieltjes Kernel Families

TL;DR

This paper extends the theory of Cauchy-Stieltjes kernel families to unbounded supports, linking cubic pseudo-variance functions to free-infinitely divisible laws without first moments and analyzing the domain of means.

Contribution

It broadens the existing framework to include unbounded supports and characterizes the domain of means for these kernel families.

Findings

Cubic pseudo-variance functions relate to free-infinitely divisible laws without first moments.

The domain of means is determined for compactly supported generating measures.

Extended the theory to support unbounded measures on one side.

Abstract

This paper continues the study of a kernel family which uses the Cauchy-Stieltjes kernel in place of the celebrated exponential kernel of the exponential families theory. We extend the theory to cover generating measures with support that is unbounded on one side. We illustrate the need for such an extension by showing that cubic pseudo-variance functions correspond to free-infinitely divisible laws without the first moment. We also determine the domain of means, advancing the understanding of Cauchy-Stieltjes kernel families also for compactly supported generating measures.