UID:
almahu_9949385121102882
Format:
1 online resource (xxii, 264 pages)
Edition:
First edition.
ISBN:
9781000458817
,
1000458814
,
9781003144496
,
1003144497
,
9781000458824
,
1000458822
Content:
This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful. Combinatorial properties of non-crossing partitions, including the Mbius function play a central role in introducing free probability. Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants. Free cumulants are introduced through the Mbius function. Free product probability spaces are constructed using free cumulants. Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed. Convergence of the empirical spectral distribution is discussed for symmetric matrices. Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices. Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. Exercises, at advanced undergraduate and graduate level, are provided in each chapter.
Note:
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
Additional Edition:
Print version: ISBN 0367700816
Additional Edition:
ISBN 9780367700812
Language:
English
Keywords:
Electronic books.
DOI:
10.1201/9781003144496
URL:
https://www.taylorfrancis.com/books/9781003144496
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