{"product_id":"random-matrices-and-non-commutative-probability-9780367700812","title":"Random Matrices and Non-Commutative Probability","description":"\u003cp\u003e • Author(s): Arup Bose\u003cbr\u003e • Publisher: Taylor \u0026amp; Francis\u003cbr\u003e • Publisher Imprint: CRC Press\u003cbr\u003e • BISAC: Probability \u0026amp; Statistics - General\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eThis 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. \u003c\/p\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eCombinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability. \u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eFree independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants.\u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eFree cumulants are introduced through the Möbius function. \u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eFree product probability spaces are constructed using free cumulants. \u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eMarginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed. \u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eConvergence of the empirical spectral distribution is discussed for symmetric matrices. \u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eAsymptotic 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. \u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eAsymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. \u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003c\/p\u003e \u003cul\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cli\u003eExercises, at advanced undergraduate and graduate level, are provided in each chapter. \u003c\/li\u003e \u003cp\u003e\u003c\/p\u003e\n\u003c\/ul\u003e","brand":"Taylor \u0026 Francis","offers":[{"title":"Hardcover","offer_id":45241417007255,"sku":"9780367700812","price":15904.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9780367700812.webp?v=1769228673","url":"https:\/\/atlanticbooks.com\/products\/random-matrices-and-non-commutative-probability-9780367700812","provider":"Atlantic Books","version":"1.0","type":"link"}