University of SaskatchewanHARVEST
  • Login
  • Submit Your Research
  • About
    • About HARVEST
    • Guidelines
    • Browse
      • All of HARVEST
      • Communities & Collections
      • By Issue Date
      • Authors
      • Titles
      • Subjects
      • This Collection
      • By Issue Date
      • Authors
      • Titles
      • Subjects
    • My Account
      • Login
      JavaScript is disabled for your browser. Some features of this site may not work without it.
      View Item 
      • HARVEST
      • Electronic Theses and Dissertations
      • Graduate Theses and Dissertations
      • View Item
      • HARVEST
      • Electronic Theses and Dissertations
      • Graduate Theses and Dissertations
      • View Item

      Some Results on the Distributions of Operator Valued Semicircular Random Variables

      Thumbnail
      View/Open
      SOLTANIFAR-THESIS.pdf (559.5Kb)
      Date
      2011-09-13
      Author
      Soltanifar, Mohsen
      Type
      Thesis
      Degree Level
      Masters
      Metadata
      Show full item record
      Abstract
      The operator-valued free central limit theorem and operator-valued semi-circular random variables were first introduced by D. Voiculescu in 1995 as operator-valued free analogues of the classical central limit theorem and normal random variables, respectively. In 2007, R. Speicher and others showed that the operator-valued Cauchy transform of the semicircular distribution satisfies a functional equation involving the variance of the semicircular distribution. In this thesis, we consider a non - commutative probability space (A;E_B;B) where in which A is a unital C -algebra, B is a C -subalgebra of A containing its unit and E_B A 􀀀 B is a conditional expectation. For a given B−valued self-adjoint semicircular random variable s > A with variance ; it is still an open question under what conditions the distribution of s has an atomic part. We provide a partial answer in terms of properties of when B is the algebra of n × n complex matrices. In addition, we show that for a given compactly supported probability measure its associated Cauchy transform can be represented in terms of the operator-valued Cauchy transforms of a sequence of finite dimensional matrix-valued semicircular random variables in two ways. Finally, we give another representation of its Cauchy transform in terms of operator-valued Cauchy transform of an in finite dimensional matrix-valued semicircular random variable.
      Degree
      Master of Science (M.Sc.)
      Department
      Mathematics and Statistics
      Program
      Mathematics
      Supervisor
      Belinschi, Serban; Samei, Ebrahim
      Committee
      Steel, Tom; Choi, Yemon; Soteros, Chris
      Copyright Date
      August 2011
      URI
      http://hdl.handle.net/10388/ETD-2011-08-60
      Subject
      Semicircular distributions, Atoms, Cauchy transform, Continued Fraction
      Collections
      • Graduate Theses and Dissertations
      University of Saskatchewan

      University Library

      © University of Saskatchewan
      Contact Us | Disclaimer | Privacy