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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. The core of the book focuses on results established under moment conditions on random variables using probabilistic methods, and is thus easily applicable to statistics and other areas of science. The book introduces fundamental results, most of them investigated by the authors, such as the semicircular law of Wigner matrices, the Marcenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extreme eigenvalues, spectrum separation theorems, convergence rates of empirical distributions, central limit theorems of linear spectral statistics, and the partial solution of the famous circular law. While deriving the main results, the book simultaneously emphasizes the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix identities, moment convergence theorems, and the Stieltjes transform. Its treatment is especially fitting to the needs of mathematics and statistics graduate students and beginning researchers, having a basic knowledge of matrix theory and an understanding of probability theory at the graduate level, who desire to learn the concepts and tools in solving problems in this area. It can also serve as a detailed handbook on results of large dimensional random matrices for practical users. This second edition includes two additional chapters, one on the authors' results on the limiting behavior of eigenvectors of sample covariance matrices, another on applications to wireless communications and finance. While attempting to bring this edition up-to-date on recent work, it also provides summaries of other areas which are typically considered part of the general field of random matrix theory. Zhidong Bai is a professor of the School of Mathematics and Statistics at Northeast Normal University and Department of Statistics and Applied Probability at National University of Singapore. He is a Fellow of the Third World Academy of Sciences and a Fellow of the Institute of Mathematical Statistics. Jack W. Silverstein is a professor in the Department of Mathematics at North Carolina State University. He is a Fellow of the Institute of Mathematical Statistics. .

Statistics. --- Statistical Theory and Methods. --- Mathematical statistics. --- Statistique --- Statistique mathématique --- Random matrices. --- Spectrum analysis. --- Analysis, Spectrum --- Spectra --- Spectrochemical analysis --- Spectrochemistry --- Spectroscopy --- Chemistry, Analytic --- Interferometry --- Optics --- Radiation --- Wave-motion, Theory of --- Absorption spectra --- Light --- Spectroscope --- Matrices, Random --- Matrices --- Qualitative --- Mathematics. --- Random matrices --- Spectrum analysis --- Algebra --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Matrices. --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Statistical methods --- Spectrometry --- Statistics . --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics --- Analytical chemistry

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Random matrix theory has a long history, beginning in the first instance in multivariate statistics. It was used by Wigner to supply explanations for the important regularity features of the apparently random dispositions of the energy levels of heavy nuclei. The subject was further deeply developed under the important leadership of Dyson, Gaudin and Mehta, and other mathematical physicists. In the early 1990's, random matrix theory witnessed applications in string theory and deep connections with operator theory, and the integrable systems were established by Tracy and Widom. More recently,

Random matrices.