Invariance Property of Cauchy–Stieltjes Kernel Families Under Free and Boolean Multiplicative Convolutions

This article delves into some properties of free and Boolean multiplicative convolutions, in connection with the theory of Cauchy–Stieltjes kernel (CSK) families and their respective variance functions (VFs). Consider K − ( μ ) = { Q m μ ( d s ) : m ∈ ( m − μ , m 0 μ ) } , a CSK family induced by a non-degenerate probability measure μ on the positive real line with a finite first-moment m 0 μ . For γ > 1 , we introduce a new family of measures: K − ( μ ) ⊠ γ = Q m μ ⊠ γ ( d s ) : m ∈ ( m − μ , m 0 μ ) . We show that if K − ( μ ) ⊠ γ represents a re-parametrization of the CSK family K − ( μ ) , then μ is characterized by its corresponding VF V μ ( m ) = c m 2 ln ( m ) , with c > 0 . We also prove that if K − ( μ ) ⊠ γ is a re-parametrization of K − ( D 1 / γ ( μ ⊞ γ ) ) (where ⊞ is the additive free convolution and D a ( μ ) denotes the dilation μ by a number a ≠ 0 ), then μ is characterized by its corresponding VF V μ ( m ) = c 1 ( m ln ( m ) ) 2 , with c 1 > 0 . Similar results are obtained if we substitute the free multiplicative convolution ⊠ with the Boolean multiplicative convolution ⨃.