Large Covariance and Autocovariance Matrices

Large Covariance and Autocovariance Matrices brings together a collection of recent results on sample covariance and autocovariance matrices in high-dimensional models and novel ideas on how to use them for statistical inference in one or more high-dimensional time series models. The prerequisites include knowledge of elementary multivariate analysis, basic time series analysis and basic results in stochastic convergence. Part I is on different methods of estimation of large covariance matrices and auto-covariance matrices and properties of these estimators. Part II covers the relevant material on random matrix theory and non-commutative probability. Part III provides results on limit spectra and asymptotic normality of traces of symmetric matrix polynomial functions of sample auto-covariance matrices in high-dimensional linear time series models. These are used to develop graphical and significance tests for different hypotheses involving one or more independent high-dimensional linear time series. The book should be of interest to people in econometrics and statistics (large covariance matrices and high-dimensional time series), mathematics (random matrices and free probability) and computer science (wireless communication). Parts of it can be used in post-graduate courses on high-dimensional statistical inference, high-dimensional random matrices and high-dimensional time series models. It should be particularly attractive to researchers developing statistical methods in high-dimensional time series models. 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 Sankhya 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.

Table of Contents:
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 II 4. 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 Corollaries 8. SPECTRA OF AUTOCOVARIANCE MATRIX I Assumptions LSD when p/n → y ∈ (0, ∞) MA(q), q < ∞ MA(∞) Application to specific cases LSD when p/n → 0 Application to specific cases Non-symmetric polynomials 9. SPECTRA OF AUTOCOVARIANCE MATRIX II Assumptions LSD when p/n → y ∈ (0, ∞) MA(q), q < ∞ MA(∞) LSD when p/n → 0 MA(q), q < ∞ MA(∞) 10. GRAPHICAL INFERENCE MA order determination AR order determination Graphical tests for parameter matrices 11. TESTING WITH TRACE One sample trace Two sample trace Testing 12. SUPPLEMENTARY PROOFS Proof of Lemma Proof of Theorem (a) Proof of Th