Variance function of boolean additive convolution

Abstract

Suppose Vν is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family K+(ν) generated by a non degenerate probability measure ν with support bounded from above. We determine the formula for pseudo-variance function (or variance function Vν in case of existence) under boolean additive convolution power. This formulas is used to identify the relation between variance functions under Boolean Bercovici-Bata Bijection between probability measures. We also gives the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measures to Catalan numbers and Fuss Catalan numbers.

We determine the formula for pseudo-variance function (or variance function Vν in case of existence) under boolean additive convolution power