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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.
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참고문헌 (18)
참고문헌 신청
Michael Anshelevich / 2014 / Local limit theorems for multiplicative free convolutions / Journal of Functional Analysis 267(9):3469~3499
Octavio Arizmendi / 2013 / Semigroups related to additive and multiplicative, free and Boolean convolutions / Studia Mathematica 215(2):157~185
Serban T. Belinschi / 2008 / On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution / Indiana University Mathematics Journal 57(4):1679~1714
H. Bercovici / 2006 / Operator theory 20 / Theta :7~13
Hari Bercovici / 1999 / Stable Laws and Domains of Attraction in Free Probability Theory / The Annals of Mathematics 149(3):1023~1060
Octavio Arizmendi / 2013 / Semigroups related to additive and multiplicative, free and Boolean convolutions / Studia Mathematica 215(2):157~185
Serban T. Belinschi / 2008 / On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution / Indiana University Mathematics Journal 57(4):1679~1714
H. Bercovici / 2006 / Operator theory 20 / Theta :7~13
Hari Bercovici / 1999 / Stable Laws and Domains of Attraction in Free Probability Theory / The Annals of Mathematics 149(3):1023~1060
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