Some Results on the Free Poisson Distribution

Let K+(μi)={Qsiμi,si∈(m0μi,m+μi)}, i=1,2, be two CSK families generated by the nondegenerate probability measures μ1 and μ2 with support bounded from above. Define the set of measures L=K+(μ1)•K+(μ2)={Qs1μ1•Qs2μ2,s1∈(m0μ1,m+μ1)ands2∈(m0μ2,m+μ2)}, where Qs1μ1•Qs2μ2 denotes the Fermi convolution of Qs...

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Bibliographic Details
Published in:Axioms 2024-08, Vol.13 (8), p.496
Main Authors: Alanzi, Ayed. R. A., Alqasem, Ohud A., Elwahab, Maysaa Elmahi Abd, Fakhfakh, Raouf
Format: Article
Language:English
Subjects:
Cauchy–Stieltjes transform
Convolution
Fermi convolution
free Poisson distribution
Mathematical research
Poisson distribution
Probability
Statistical analysis
variance function
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Summary:Let K+(μi)={Qsiμi,si∈(m0μi,m+μi)}, i=1,2, be two CSK families generated by the nondegenerate probability measures μ1 and μ2 with support bounded from above. Define the set of measures L=K+(μ1)•K+(μ2)={Qs1μ1•Qs2μ2,s1∈(m0μ1,m+μ1)ands2∈(m0μ2,m+μ2)}, where Qs1μ1•Qs2μ2 denotes the Fermi convolution of Qs1μ1 and Qs2μ2. We prove that if L is still a CSK family (that is, L=K+(σ) for some nondegenerate probability measure ()σ), then the probability measures σ, μ1 and μ2 are of the free Poisson type and follow the free Poisson law up to affinity. The same result, regarding the free Poisson measure, is obtained if we consider the t-deformed free convolution t replacing the Fermi convolution • in the family of measures L.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms13080496