2014/12/03 by Michael Anshelevich, John D. Williams, Anshelevich, Michael +1
Mathematics · #05A19 #46L54 #Combinatorics (math.CO) #FOS: Mathematics #Operator Algebras (math.OA) #Point processes and geometric inequalities #Random Matrices and Applications #Statistical Methods and Bayesian Inference #math.CO #math.OA #msc:05A19 #msc:46L54
paper · pdf · doi:10.48550/arxiv.1412.1280
v3: A major revision, following comments by a referee
openalex publication_date 2014/12/03 · arxiv created 2015/12/17 · arxiv updated 2015/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.