Random matrices and non-commutative probability. (2021)

• Record Type: Book
• Further Information: Note: Arup Bose.
• Authors: Bose, Arup
• Contents: Classical independence, moments and cumulants Classical independence CLT a cumulants Cumulants to moments Moments to cumulants, the Möbius function Classical Isserlis’ formula Exercises Non-commutative probability Non-crossing partition Free cumulants Free Gaussian or semi-circle law Free Poisson law Non-commutative and *-probability spaces Moments and probability laws of variables Exercises Free independence Free independence Free product of *-probability spaces Free binomial Semi-circular family Free Isserlis’ formula Circular and elliptic variables Free additive convolution Kreweras complement Moments of free variables 0 Compound free Poisson Exercises Convergence Algebraic convergence Free central limit theorem Free Poisson convergence Sums of triangular arrays Exercises Transforms Stieltjes transform R transform Interrelation S -transform Free infinite divisibility Exercises C* -probability space C* -probability space Spectrum Distribution of a self-adjoint element Free product of C* -probability spaces Free additive and multiplicative convolution Exercises Random matrices Empirical spectral measure Limiting spectral measure Moment and trace Some important matrices A unified treatment Exercises Convergence of some important matrices Wigner matrix: semi-circle law S -matrix: Marcenko-Pastur law IID and elliptic matrices: circular and elliptic variables Toeplitz matrix Hankel matrix Reverse Circulant matrix: symmetrized Rayleigh law Symmetric Circulant: Gaussian law Almost sure convergence of the ESD Exercises Joint convergence I: single pattern Unified treatment: extension Wigner matrices: asymptotic freeness Elliptic matrices: asymptotic freeness S -matrices in elliptic models: asymptotic freeness Symmetric Circulants: asymptotic independence Reverse Circulants: asymptotic half independence Exercises Joint convergence II: multiple patterns Multiple patterns: colors and indices Joint convergence Two or more patterns at a time Sum of independent patterned matrices Discussion Exercises Asymptotic freeness of random matrices Elliptic, IID, Wigner, and S -matrices Gaussian elliptic, IID, Wigner and deterministic matrices General elliptic, IID, Wigner, and deterministic matrices S -matrices and embedding Cross covariance matrices Pair-correlated cross-covariance; p/n ! y Pair correlated cross covariance; p/n ! Wigner and patterned random matrices Discussion Exercises Brown measure Brown measure Exercises Tying three loose ends Möbius function on NC (n ) Equivalence of two freeness definitions Free product construction Exercises Bibliography Index
• Edition: 1st
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2021
• Extent: 1 online resource
• Subjects: 512.9434; Random matrices; Free probability theory
• Languages: English
• ISBNs: 9781000458824; 9781000458817; 9781003144496
• Related ISBNs: 9780367700812
• Notes: Note: Description based on CIP data; resource not viewed.
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