Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality

TL;DR

This paper explores polynomial variance functions within Cauchy-Stieltjes Kernel families, introduces operations to generate new functions, and links these to generalized orthogonality, using classical and free probability techniques.

Contribution

It characterizes all cubic variance functions, constructs polynomial variance functions of any degree, and connects these to generalized orthogonality.

Findings

Exhaustive classification of cubic variance functions.

Construction methods for polynomial variance functions of arbitrary degree.

Establishment of links between Cauchy-Stieltjes families and generalized orthogonality.

Abstract

This paper studies variance functions of Cauchy-Stieltjes Kernel families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known ones. We construct a class of examples which exhausts all cubic variance functions, and provide examples of polynomial variance functions of arbitrary degree. We also relate Cauchy-Stieltjes Kernel families with polynomial variance functions to generalized orthogonality. Our main results are stated solely in terms of classical probability; some proofs rely on analytic machinery of free probability.