" . . . the authors should be congratulated for producing two highly relevant and well-written books. Statisticians would probably gravitate to LCAM in the first instance and those working in linear algebra would probably gravitate to PRM." ~Jonathan Gillard, Cardiff University
"The book represents a monograph of the authors recent results about the theory of large covariance and autocovariance matrices and contains other important results from other research papers and books in this topic. It is very useful for all researchers who use large covariance and autocovariance matrices in their researches. Especially, it is very useful for post-graduate and PhD students in mathematics, statistics, econometrics and computer science. It is a well-written and organized book with a large number of solved examples and many exercises left to readers for homework. I would like to recommend the book to PhD students and researchers who want to learn or use large covariance and autocovariance matrices in their researches." ~ Miroslav M. Risti (Ni), zbMath
"This book brings together a collection of recent results on estimation of multidimensional time series covariance matrices. In the case where the time series consists of a sequence of independent (Chapter 1) or weakly dependent (Chapter 2) random vectors, the authors call it covariance estimation, whereas in the general case where the time series is only stationary, they call it autocovariance estimation. The framework of the results presented here is the one where the dimension of the observations (as well as the observation window size, otherwise nothing can be said) is high. The prerequisites include knowledge of elementary multivariate analysis, basic time series analysis, and basic results in stochastic convergence.
In Chapter 1, the authors consider the case where we have at our disposal a large time series of iid high-dimensional observations with common covariance matrix C and want to estimate C. They provide some results on how to regularize the empirical sample covariance matrix in order to accurately estimate C in the case where C is either quickly decaying away from its diagonal (or \bendable"), Toeplitz or sparse. The regularization techniques involved are the \banding" (zero-out of all entries above a certain distance from the diagonal), the tapering (instead of turning to zero, multiply by a factor which gets small as the distance to the diagonal gets large), and the thresholding (zero-out all entries smaller, in absolute value, than a certain well-chosen threshold).
In Chapter 2, the same questions and techniques are discussed in the case where the independence between the observations is replaced by weak dependence.
In Chapter 3, the authors suppress completely the hypothesis of independence of the observations, replace it by a stationarity hypothesis, and show how the techniques presented earlier allow one to still get estimations of the (auto)covariance matrix in the case of MA(r) and IVAR(r) models.
Chapters 4 and 5 collect the basic concepts and results from respectively random matrix theory (RMT), about the empirical spectral distribution of various random matrix models, and Voiculescu's free probability theory that are needed in Chapters 6 to 10.
Chapters 6 to 9, among other analogous questions, revisit the covariance matrix estimation results from Chapters 1 to 3 from the point of view of empirical spectral distribution (thanks to the framework defined in Chapters 4 and 5). In Chapter 10, it is demonstrated how the limiting spectral distribution (LSD) results obtained in Chapters 8 and 9 can be used in statistical graphical inference of high-dimensional time series. This includes estimation of unknown order of high-dimensional MA and AR processes.
In Chapter 11, central limit theorems (CLTs) for linear spectral statistics of random matrices are used in signi cance tests for di erent hypotheses on coecient matrices. ~Florent Benaych-Georges - Mathematical Reviews Clippings February 2019
"Most of the materials covered in the book are at an advanced level. Fortunately, their exposition is clear, rigorous and highly self-contained. The book assumes a working knowledge in multivariate analysis, multivariate time series analysis and in stochastic convergence, that should be possessed by graduate students in econometrics, statistics or probability theory. A significant number of exercises are included in each chapter to help the reader master the introduced concepts, methods and results. The book is also an important reference for experienced researchers in the area of high-dimensional multivariate and time series analyses. One particular strength of the book is a thorough presentation of the most relevant concepts of non-commutative probability theory. In recent years and using this theory, the authors have developed several important results on the limiting proprieties of large sample covariance and autocovariance matrices. These results are now very accessible in this book." ~Journal of Time Series Analysis
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