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

Fahad Alsharari, Raouf Fakhfakh () and Fatimah Alshahrani
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Fahad Alsharari: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia
Raouf Fakhfakh: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia
Fatimah Alshahrani: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Mathematics, 2025, vol. 13, issue 7, 1-9

Abstract: 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 ⨃.

Keywords: Cauchy transform; multiplicative Boolean and free convolutions; variance function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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