Notes on Cauchy–Stieltjes Kernel Families
The free Meixner family (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">FMF</mi></semantics></math></inline-formula>) is the family of measures...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-03-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/3/189 |
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| Summary: | The free Meixner family (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">FMF</mi></semantics></math></inline-formula>) is the family of measures that produces quadratic Cauchy–Stieltjes Kernel (CSK) families (i.e., meaning that the associated variance function (VF) is a polynomial with degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula> in the mean). Furthermore, a cubic class is introduced in the context of CSK families and is connected to the quadratic class via a reciprocity relation. The associated probability measures are the so-called free analog of the Letac–Mora class (with VF of degree 3). In free probability theory, these two classes of probabilities are crucial. However, a novel transformation of measures is introduced in the setting of free probability, known as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation of probability measures. Denote by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">P</mi></semantics></math></inline-formula> the set of (non-degenerate) real probabilities. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>∈</mo><mi mathvariant="script">P</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, consider the transformation of measure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>, denoted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>a</mi></msub><mrow><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">F</mi><mrow><msub><mi>T</mi><mi>a</mi></msub><mrow><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>=</mo><msub><mi mathvariant="script">F</mi><mi>ν</mi></msub><mrow><mo>(</mo><mi>w</mi><mo>−</mo><mi>a</mi><mo>)</mo></mrow><mo>+</mo><mi>a</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">F</mi><mi>ν</mi></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the inverse of the Cauchy–Stieltjes transformation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>. In this study, we provide important insights into the notion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation of probabilities. We demonstrate that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">FMF</mi></semantics></math></inline-formula> (respectively, the free counterpart of the Letac–Mora class of measures) is invariant under the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation. Furthermore, we develop additional characteristics of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation, which yield intriguing findings for significant free probability distributions such as the free Poisson and free Gamma distributions. |
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| ISSN: | 2075-1680 |