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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML"...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2025-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/7/1044 |
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| Summary: | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">K</mi><mo>−</mo></msub><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msubsup><mi>Q</mi><mi>m</mi><mi>μ</mi></msubsup><mrow><mo>(</mo><mi>d</mi><mi>s</mi><mo>)</mo></mrow><mo>:</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mi>m</mi><mo>∈</mo><mrow><mo>(</mo><msubsup><mi>m</mi><mo>−</mo><mi>μ</mi></msubsup><mo>,</mo><msubsup><mi>m</mi><mn>0</mn><mi>μ</mi></msubsup><mo>)</mo></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, a CSK family induced by a non-degenerate probability measure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> on the positive real line with a finite first-moment <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>m</mi><mn>0</mn><mi>μ</mi></msubsup></semantics></math></inline-formula>. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, we introduce a new family of measures: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="(" close=")"><msub><mi mathvariant="script">K</mi><mo>−</mo></msub><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mfenced><mrow><mo>⊠</mo><mi>γ</mi></mrow></msup><mo>=</mo><mfenced separators="" open="{" close="}"><msup><mfenced separators="" open="(" close=")"><msubsup><mi>Q</mi><mi>m</mi><mi>μ</mi></msubsup></mfenced><mrow><mo>⊠</mo><mi>γ</mi></mrow></msup><mrow><mo>(</mo><mi>d</mi><mi>s</mi><mo>)</mo></mrow><mo>:</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mi>m</mi><mo>∈</mo><mrow><mo>(</mo><msubsup><mi>m</mi><mo>−</mo><mi>μ</mi></msubsup><mo>,</mo><msubsup><mi>m</mi><mn>0</mn><mi>μ</mi></msubsup><mo>)</mo></mrow></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> We show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mfenced separators="" open="(" close=")"><msub><mi mathvariant="script">K</mi><mo>−</mo></msub><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mfenced><mrow><mo>⊠</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula> represents a re-parametrization of the CSK family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">K</mi><mo>−</mo></msub><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is characterized by its corresponding VF <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">V</mi><mi>μ</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mo>=</mo><mi>c</mi><msup><mi>m</mi><mn>2</mn></msup><mo form="prefix">ln</mo><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We also prove that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mfenced separators="" open="(" close=")"><msub><mi mathvariant="script">K</mi><mo>−</mo></msub><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mfenced><mrow><mo>⊠</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula> is a re-parametrization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">K</mi><mo>−</mo></msub><mrow><mo>(</mo><msub><mi>D</mi><mrow><mn>1</mn><mo>/</mo><mi>γ</mi></mrow></msub><mrow><mo>(</mo><msup><mi>μ</mi><mrow><mo>⊞</mo><mi>γ</mi></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> (where ⊞ is the additive free convolution and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>a</mi></msub><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> denotes the dilation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> by a number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>), then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is characterized by its corresponding VF <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">V</mi><mi>μ</mi></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><msup><mrow><mo>(</mo><mi>m</mi><mo form="prefix">ln</mo><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Similar results are obtained if we substitute the free multiplicative convolution ⊠ with the Boolean multiplicative convolution ⨃. |
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| ISSN: | 2227-7390 |