Some results in Cauchy-Stieltjes kernel families

On Cauchy-Stieltjes kernel families

We explore properties of Cauchy-Stieltjes families that have no counterpart in exponential families. We relate the variance function of the iterated Cauchy-Stieltjes family to the pseudo-variance function of the initial Cauchy-Stieltjes family. We also investigate when the domain of means can be extended beyond the ”natural domain”.

One-sided Cauchy-stieltjes Kernel Families

This paper continues the study of a kernel family which uses the Cauchy-Stieltjes kernel 1/(1 − θx) in place of the celebrated exponential kernel exp(θx) 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-i...

On Generalized Cauchy-stieltjes Transforms of Some Beta Distributions

We express the generalized Cauchy-Stieltjes transforms (GCST) of some particular Beta distributions depending on a positive parameter λ as λ-powered Cauchy-Stieltjes transforms (CST) of some probability measures. The CST of the latter measures are shown to be the geometric mean of the CST of the Wigner law together with another one. Moreover, they are absolutely continuous and we derive their d...

Generalized Cauchy-stieltjes Transforms of Some Beta Distributions

We express generalized Cauchy-Stieltjes transforms (GCST) of some particular beta distributions depending on a positive parameter λ as λ-powered Cauchy-Stieltjes transforms (CST) of some probability measures. The CST of the latter measures are shown to be the geometric mean of the CST of the Wigner law together with another one. Moreover, these measures are shown to be absolutely continuous and...

Cauchy-stieltjes and Riemann-stielt Jes Integrals