大維隨機矩陣的譜分析

內(nèi)容概要

《大維隨機矩陣的譜分析(英文版)》:The aim of the book is to introduce basic concepts, main results, and widely applied mathematical tools in the spectral analysis of large dimensional random matrices. In it we will introduce many of the fundamental results, such as the semicircular law of Wigner matrices, the Marchenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extremal eigenvalues, spectrum separation theorems, convergence rates of empirical spectral distributions, central limit theorems of linear spectral statistics and the partial solution of the famous circular law. While deriving the main results, the book will simultaneously emphasize the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix transformations, moment convergence theorems, and the Stieltjes transform. Thus, its treatment is especially fitting to the needs of mathematics and statistic graduate students, and beginning researchers, who can learn the basic methodologies and ideas to solve problems in this area.

書籍目錄

1 Introduction1.1 Large Dimensional Data Analysis1.2 Random Matrix Theory1.2.1 Spectral Analysis of Large Dimensional Random Matrices1.2.2 Limits of Extreme Eigenvalues1.2.3 Convergence Rate of ESD1.2.4 Circular Law1.2.5 CLT of Linear Spectral Statisticslinear spectral statistics1.2.6 Limiting Distributions of Extreme Eigenvalues and Spacings1.3 Methodologies1.3.1 Moment Method1.3.2 Stieltjes Transform1.3.3 Orthogonal Polynomial Decomposition2 Wigner Matrices and Semicircular Law2.1 Semicircular Law by the Moment Method2.1.1 Moments of the Semicircular Law2.1.2 Some Lemmas of Combinatorics2.1.3 Semicircular Law for iid Case2.2 Generalizations to the Non-iid Case2.2.1 Proof of Theorem 2.92.3 Semicircular Law by Stieltjes Transform2.3.1 Stieltjes Transform of Semicircular Law2.3.2 Proof of Theorem 2.93 Sample Covariance Matrices, Marˇcenko-Pastur Law3.1 MP Law for iid Case3.1.1 Moments of the MP Law3.1.2 Some Lemmas on Graph Theory and Combinatorics3.1.3 MP Law for iid Case3.2 Generalization to the Non-iid Case3.3 Proof of Theorem 3.9 by Stieltjes Transform3.3.1 Stieltjes Transform of MP Law3.3.2 Proof of Theorem 3.94 Product of Two Random Matrices4.1 Some Graph Theory and Combinatoric Results4.2 Existence of the ESD of SnTn4.2.1 Truncation of the ESD of Tn4.2.2 Truncation, Centralization and Rescaling of the X-variables4.2.3 Proof of Theorem 4.34.3 LSD of F matrix4.3.1 General Formula for the Product4.3.2 LSD of Multivariate F Matrices4.4 Proof of Theorem 4.54.4.1 Truncation and Centralization4.4.2 Proof by Stieltjes Transform5 Limits of Extreme Eigenvalues5.1 Limit of Extreme Eigenvalues of the Wigner Matrix5.1.1 Sufficiency of Conditions of Theorem 5.15.1.2 Necessity of Conditions of Theorem 5.15.2 Limits of Extreme Eigenvalues of Sample Covariance Matrix5.2.1 Proof of Theorem 5.105.2.2 The Proof of Theorem 5.115.2.3 Necessity of the Conditions5.3 Miscellan

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