Some Results on the Free Poisson Distribution

Let K+(mu i)={Qsi mu i,si is an element of(m0 mu i,m+mu i)}, i=1,2, be two CSK families generated by the nondegenerate probability measures mu 1 and mu 2 with support bounded from above. Define the set of measures L=K+(mu 1)center dot K+(mu 2)={Qs1 mu 1 center dot Qs2 mu 2,s1 is an element of(m0 mu 1,m+mu 1)ands2 is an element of(m0 mu 2,m+mu 2)}, where Qs1 mu 1 center dot Qs2 mu 2 denotes the Fermi convolution of Qs1 mu 1 and Qs2 mu 2. We prove that if L is still a CSK family (that is, L=K+(sigma) for some nondegenerate probability measure ()sigma), then the probability measures sigma, mu 1 and mu 2 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 center dot in the family of measures L.