Invariance Property of Cauchy–Stieltjes Kernel Families Under Free and Boolean Multiplicative Convolutions - Fahad Alsharari; Raouf Fakhfakh; Fatimah Alshahrani (Mathematics)
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 <semantics>K−(μ)={<msubsup>Qmμ</msubsup>(ds):<mspace width="4pt"></mspace><mspace width="4pt"></mspace>m∈(<msubsup>m−μ</msubsup>,<msubsup>m0μ</msubsup>)}</semantics>, a CSK family induced by a non-degenerate probability measure <semantics>μ</semantics> on the positive real line with a finite first-moment <semantics><msubsup>m0μ</msubsup></semantics>. For <semantics>γ>1</semantics>, we introduce a new family of measures: <semantics><mfenced separators="" open="(" close=")">K−(μ)</mfenced>⊠γ=<mfenced separators="" open="{" close="}"><mfenced separators="" open="(" close=")"><msubsup>Qmμ</msubsup></mfenced>⊠γ(ds):<mspace width="4pt"></mspace><mspace width="4pt"></mspace>m∈(<msubsup>m−μ</msubsup>,<msubsup>m0μ</msubsup>)</mfenced>.</semantics> We show that if <semantics><mfenced separators="" open="(" close=")">K−(μ)</mfenced>⊠γ</semantics> represents a re-parametrization of the CSK family <semantics>K−(μ)</semantics>, then <semantics>μ</semantics> is characterized by its corresponding VF <semantics>Vμ(m)=cm2ln(m)</semantics>, with <semantics>c>0</semantics>. We also prove that if <semantics><mfenced separators="" open="(" close=")">K−(μ)</mfenced>⊠γ</semantics> is a re-parametrization of <semantics>K−(D1/γ(μ⊞γ))</semantics> (where ⊞ is the additive free convolution and <semantics>Da(μ)</semantics> denotes the dilation <semantics>μ</semantics> by a number <semantics>a≠0</semantics>), then <semantics>μ</semantics> is characterized by its corresponding VF <semantics>Vμ(m)=c1(mln(m))2</semantics>, with <semantics>c1>0</semantics>. Similar results are obtained if we substitute the free multiplicative convolution ⊠ with the Boolean multiplicative convolution ⨃.