Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(2): 515-526

Online first article December 28, 2020      Printed March 31, 2021

https://doi.org/10.4134/BKMS.b200380

Copyright © The Korean Mathematical Society.

Boolean multiplicative convolution and Cauchy-Stieltjes kernel families

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.

Keywords: Variance function, Cauchy-Stieltjes kernel, boolean multiplicative convolution

MSC numbers: 60E10, 46L54

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