Boolean multiplicative convolution and Cauchy-Stieltjes kernel families

Bull. Korean Math. Soc. 2021 Vol. 58, No. 2, 515-526 https://doi.org/10.4134/BKMS.b200380 Published online December 28, 2020 Printed March 31, 2021

Raouf Fakhfakh Sfax University

Abstract : Denote by $\mathcal{M}_+$ the set of probability measures supported on $\mathbb{R}_+$. Suppose $V_{\nu}$ is the variance function of the \CSK family ${\mathcal{K}_{-}}(\nu)$ generated by a non degenerate probability measure $\nu\in\mathcal{M}_+$. We determine the formula for variance function under boolean multiplicative convolution power. This formula is used to identify the relation between variance functions under the map $\nu\mapsto\mathbb{M}_t(\nu)=\left(\nu^{\boxtimes (t+1)}\right)^{\utimes\frac{1}{t+1}}$ from $\mathcal{M}_+$ onto itself.