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On a generalization of the Rényi–Srivastava characterization of the Poisson law
- Part of
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- Journal:
- Journal of Applied Probability / Volume 58 / Issue 1 / March 2021
- Published online by Cambridge University Press:
- 25 February 2021, pp. 68-82
- Print publication:
- March 2021
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Free Bessel Laws
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- Journal:
- Canadian Journal of Mathematics / Volume 63 / Issue 1 / 01 February 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 3-37
- Print publication:
- 01 February 2011
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- Article
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We introduce and study a remarkable family of real probability measures
${{\pi }_{st}}$ that we call free Bessel laws. These are related to the free Poisson law 
$\pi $ via the formulae 
${{\text{ }\!\!\pi\!\!\text{ }}_{s1}}={{\text{ }\!\!\pi\!\!\text{ }}^{\boxtimes s}}$ and 
$\text{ }\pi {{\text{ }}_{1t}}=\text{ }\pi {{\text{ }}^{\boxtimes }}^{t}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.
