Abstract
In this paper we present two different results in the theory of Cauchy-Stieltjes Kernel (CSK) families. We firstly provide the construction of free Sheffer systems with the theory of CSK families. We associate a free additive convolution semigroup of probability measures to any free Sheffer systems and we prove that this is the only one that leads to an orthogonal free Sheffer systems. We also show that the orthogonality of free Sheffer systems occurs if and only if the associated free additive convolution semigroup of probability measures generates CSK families with quadratic variance function. Secondly, we are interested in the study of boolean additive convolution. Based on the criteria of convergence for a sequence of variance functions we give an approximation of elements of the CSK family generated by the boolean Gaussian distribution and an approximation of elements of the CSK family generated by the boolean Poisson distribution.
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