Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples

TL;DR

This paper explores the relationships between generalized Stieltjes transforms of certain beta distributions and elementary functions of the classical Stieltjes transform, revealing new examples and connections to representation theory and free probability.

Contribution

It introduces new examples of generalized Stieltjes transforms expressed as elementary functions of the classical transform, extending previous results and linking them to symmetric groups and free probability.

Findings

Generalized Stieltjes transforms of specific beta distributions relate to powers and fractions of the Wigner distribution's transform.

The power of the Stieltjes transform of the symmetric Bernoulli distribution is a generalized Stieltjes transform only if the power exceeds one.

The free Poisson distribution corresponds to the product of two independent Beta distributions in [0,1].

Abstract

For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the symmetric Bernoulli distribution is a generalized Stietljes transform of a probability distribution if and only if the power is greater than one. As to the free Poisson distribution, it corresponds to the product of two independent Beta distributions in $[0,1]$ while another example of Beta distributions in $[-1,1]$ is found and is related with the Shrinkage process. We close the exposition by considering the generalized Stieltjes transform of a linear functional related with Humbert polynomials and generalizing the symmetric Beta distribution.