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 Boolean multiplicative convolution and Cauchy-Stieltjes kernel families Bull. Korean Math. Soc.Published online December 28, 2020 Raouf Fakhfakh University of Sfax, Sfax, Tunisia 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 $\mu\longmapsto\mathbb{M}_t(\mu)=\left(\mu^{\boxtimes (t+1)}\right)^{\utimes\frac{1}{t+1}}$ from $\mathcal{M}_+$ onto itself. Keywords : Variance function; Cauchy-Stieltjes kernel; boolean multiplicative convolution MSC numbers : AMS Mathematics Subject Classification 2010 : 60E10; 46L54. Full-Text :