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 7→ Mt(ν) = ( ν (t+1) )∪× 1 t+1 from M+ onto itself.