Operator-valued Jacobi parameters and examples of operator-valued distributions Academic Article

abstract

• 2017 Elsevier Masson SAS In the setting of distributions taking values in a C -algebra B, we define generalized Jacobi parameters and study distributions they generate. These include numerous known examples and one new family, of B-valued free binomial distributions, for which we are able to compute free convolution powers. Moreover, we develop a convenient combinatorial method for calculating the joint distributions of B-free random variables with Jacobi parameters, utilizing two-color non-crossing partitions. This leads to several new explicit examples of free convolution computations in the operator-valued setting. Additionally, we obtain a counting algorithm for the number of two-color non-crossing pairings of relative finite depth, using only free probabilistic techniques. Finally, we show that the class of distributions with Jacobi parameters is not closed under free convolution.

authors

• Anshelevich, Michael

published proceedings

• BULLETIN DES SCIENCES MATHEMATIQUES

author list (cited authors)

• Anshelevich, M., & Williams, J. D.

citation count

• 1

complete list of authors

• Anshelevich, Michael||Williams, John D

publication date

• January 2018

publisher

• Elsevier  Publisher

published in

• Bulletin des Sciences Mathematiques  Journal

keywords

• Free Meixner Distributions
• Jacobi Parameters
• Operator-valued Distributions

Digital Object Identifier (DOI)

• 10.1016/j.bulsci.2017.09.003

start page

• 1

end page

• 37

volume

• 145

URL

• http://dx.doi.org/10.1016/j.bulsci.2017.09.003