Название: Random Matrices and Non-Commutative Probability Автор: Arup Bose Издательство: Chapman and Hall/CRC Press Год: 2022 Страниц: 287 Язык: английский Формат: pdf (true) Размер: 11.4 MB
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 M?bius 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 M?bius 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.
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Random Matrices and Non-Commutative Probability 
Название: Random Matrices and Non-Commutative Probability
