Large Covariance and Autocovariance Matrices

Arup Bose is a professor at the Indian Statistical Institute, Kolkata, India. He is a distinguished researcher in mathematical statistics and has been working in high-dimensional random matrices for the last fifteen years. He has been editor of Sankhyā for several years and has been on the editorial board of several other journals. He is a Fellow of the Institute of Mathematical Statistics, USA and all three national science academies of India, as well as the recipient of the S.S. Bhatnagar Award and the C.R. Rao Award. His first book Patterned Random Matrices was also published by Chapman & Hall. He has a forthcoming graduate text U-statistics, M-estimates and Resampling (with Snigdhansu Chatterjee) to be published by Hindustan Book Agency. Monika Bhattacharjee is a post-doctoral fellow at the Informatics Institute, University of Florida. After graduating from St. Xavier's College, Kolkata, she obtained her master’s in 2012 and PhD in 2016 from the Indian Statistical Institute. Her thesis in high-dimensional covariance and auto-covariance matrices, written under the supervision of Dr. Bose, has received high acclaim.

" . . . 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. Ristic (Niš), 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

• 1. LARGE COVARIANCE MATRIX I Consistency Covariance classes and regularization Covariance classes Covariance regularization Bandable Σp Parameter space Estimation in U Minimaxity Toeplitz Σp Parameter space Estimation in Gβ (M ) or Fβ (M0, M ) Minimaxity Sparse Σp Parameter space Estimation in Uτ (q, C0(p), M ) or Gq (Cn,p) Minimaxity 2. LARGE COVARIANCE MATRIX II Bandable Σp Models and examples Weak dependence Estimation Sparse Σp 3. LARGE AUTOCOVARIANCE MATRIX Models and examples Estimation of Γ0,p Estimation of Γu,p Parameter spaces Estimation Estimation in MA(r) Estimation in IVAR(r)Gaussian assumption Simulations Part II4. SPECTRAL DISTRIBUTION LSD Moment method Method of Stieltjes transform Wigner matrix: semi-circle law Independent matrix: Marˇcenko-Pastur law Results on Z: p/n → y > 0 Results on Z: p/n → 0 5. NON-COMMUTATIVE PROBABILITY NCP and its convergence Essentials of partition theory M¨obius function Partition and non-crossing partition Kreweras complement Free cumulant; free independence Moments of free variables Joint convergence of random matrices Compound free Poisson 6. GENERALIZED COVARIANCE MATRIX I Preliminaries Assumptions Embedding NCP convergence Main idea Main convergence LSD of symmetric polynomials Stieltjes transform Corollaries 7. GENERALIZED COVARIANCE MATRIX II Preliminaries Assumptions Centering and Scaling Main idea NCP convergence LSD of symmetric polynomials Stieltjes transform Corollaries8. SPECTRA OF AUTOCOVARIANCE MATRIX I Assumptions LSD when p/n → y ∈ (0, ∞) MA(q), q < ∞ MA(∞) 10. GRAPHICAL INFERENCE MA order determination AR order determination Graphical tests for parameter matrices 11. TESTING WITH TRACEOne sample traceTwo sample traceTesting12. SUPPLEMENTARY PROOFS Proof of LemmaProof of Theorem (a)Proof of TheoremProof of LemmaProof of Corollary (c)Proof of Corollary (c)Proof of Corollary (c)Proof of Lemma Proof of Lemma Lemmas for Theorem