Invariance Property of Cauchy–Stieltjes Kernel Families Under Free and Boolean Multiplicative Convolutions
Year: 2025
Type: article
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)={Qmμ(ds):m∈(m−μ,m0μ)}, a CSK family induced by a non-degenerate probability measure μ on the positive real line with a finite first-moment m0μ. For γ>1, we introduce a new family of measures: K−(μ)⊠γ=Qmμ⊠γ(ds):m∈(m−μ,m0μ). 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 Vμ(m)=cm2ln(m), with c>0. We also prove that if K−(μ)⊠γ is a re-parametrization of K−(D1/γ(μ⊞γ)) (where ⊞ is the additive free convolution and Da(μ) denotes the dilation μ by a number a≠0), then μ is characterized by its corresponding VF Vμ(m)=c1(mln(m))2, with c1>0. Similar results are obtained if we substitute the free multiplicative convolution ⊠ with the Boolean multiplicative convolution ⨃.
Source: Mathematics
DOI: 10.3390/math13071044
Institutions: Jouf University, Princess Nourah bint Abdulrahman University
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W4408797979