Boolean multiplicative convolution and cauchy-stieltjes kernel families

Abstract

Denote by M+ the set of probability measures supported on ℝ+. Suppose Vν is the variance function of the Cauchy-Stieltjes Kernel (CSK) family K−(ν) generated by a non degenerate probability measure ν ∈ 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 ν ↦→ Mt(ν) =(ν⊠(t+1))∪× t+11 from M+ onto itself.