2010/08/30 by Serban T. Belinschi, Belinschi, Serban T., Mihai Popa +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #46L53 #46L54 #Bernoulli's principle #Cauchy distribution #Commutative property #Connection (principal bundle) #Discrete mathematics #FOS: Mathematics #Free probability #Mathematical analysis #Mathematics #Monotone polygon #Operator (biology) #Operator Algebras (math.OA) #Probability distribution #Probability theory #Pure mathematics #Random Matrices and Applications #Reciprocal #Scalar (mathematics) #Statistical Mechanics and Entropy #Statistics #Stochastic processes and financial applications #math.OA #msc:46L53 #msc:46L54
paper · pdf · doi:10.48550/arxiv.1008.5205
18 pages, LaTeX. Preliminary version
openalex publication_date 2010/08/30 · arxiv created 2010/10/15 · arxiv updated 2010/10/18 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We study of the connection between operator valued central limits for\nmonotone, Boolean and free probability theory, which we shall call the arcsine,\nBernoulli and semicircle distributions, respectively. In scalar-valued\nnon-commutative probability these measures are known to satisfy certain\narithmetic relations with respect to Boolean and free convolutions. We show\nthat generally the corresponding operator-valued distributions satisfy the same\nrelations only when we consider them in the fully matricial sense introduced by\nVoiculescu. In addition, we provide a combinatorial description in terms of\nmoments of the operator valued arcsine distribution and we show that its\nreciprocal Cauchy transform satisfies a version of the Abel equation similar to\nthe one satisfied in the scalar-valued case.\n