Large Covariance and Autocovariance Matrices

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Content: Cover Half title Editors Title Copyright Dedication Contents Preface Acknowledgments Introduction Part I Chapter 1 LARGE COVARIANCE MATRIX I 1.1 Consistency 1.2 Covariance classes and regularization 1.2.1 Covariance classes 1.2.2 Covariance regularization 1.3 Bandable p 1.3.1 Parameter space 1.3.2 Estimation in U 1.3.3 Minimaxity 1.4 Toeplitz p 1.4.1 Parameter space 1.4.2 Estimation in G (M) or F (M0 M) 1.4.3 Minimaxity 1.5 Sparse p 1.5.1 Parameter space 1.5.2 Estimation in U˝ (q C0(p) M) or Gq(Cn p) 1.5.3 Minimaxity Chapter 2 LARGE COVARIANCE MATRIX II 2.1 Bandable p2.1.1 Models and examples 2.1.2 Weak dependence 2.1.3 Estimation 2.2 Sparse p Chapter 3 LARGE AUTOCOVARIANCE MATRIX 3.1 Models and examples 3.2 Estimation of 0 p 3.3 Estimation of u p 3.3.1 Parameter spaces 3.3.2 Estimation 3.4 Estimation in MA(r) 3.5 Estimation in IVAR(r) 3.6 Gaussian assumption 3.7 Simulations Part II Chapter 4 SPECTRAL DISTRIBUTION 4.1 LSD 4.1.1 Moment method 4.1.2 Method of Stieltjes transform 4.2 Wigner matrix: Semi-circle law 4.3 Independent matrix: Mar cenko{Pastur law 4.3.1 Results on Z: p=n ! y &gt 0 4.3.2 Results on Z: p=n ! 0Chapter 5 NON-COMMUTATIVE PROBABILITY 5.1 NCP and its convergence 5.2 Essentials of partition theory 5.2.1 MŁobius function 5.2.2 Partition and non-crossing partition 5.2.3 Kreweras complement 5.3 Free cumulant free independence 5.4 Moments of free variables 5.5 Joint convergence of random matrices 5.5.1 Compound free Poisson Chapter 6 GENERALIZED COVARIANCE MATRIX I 6.1 Preliminaries 6.1.1 Assumptions 6.1.2 Embedding 6.2 NCP convergence 6.2.1 Main idea 6.2.2 Main convergence 6.3 LSD of symmetric polynomials 6.4 Stieltjes transform 6.5 CorollariesChapter 7 GENERALIZED COVARIANCE MATRIX II 7.1 Preliminaries 7.1.1 Assumptions 7.1.2 Centering and Scaling 7.1.3 Main idea 7.2 NCP convergence 7.3 LSD of symmetric polynomials 7.4 Stieltjes transform 7.5 Corollaries Part III Chapter 8 SPECTRA OF AUTOCOVARIANCE MATRIX I 8.1 Assumptions 8.2 LSD when p=n ! y 2 (0 1) 8.2.1 MA(q), q &lt 1 8.2.2 MA(1) 8.2.3 Application to speci c cases 8.3 LSD when p=n ! 0 8.3.1 Application to speci c cases 8.4 Non-symmetric polynomials Chapter 9 SPECTRA OF AUTOCOVARIANCE MATRIX II 9.1 Assumptions 9.2 LSD when p=n ! y 2 (0 1) 9.2.1 MA(q), q &lt 19.2.2 MA(1) 9.3 LSD when p=n ! 0 9.3.1 MA(q) q &lt 1 9.3.2 MA(1) Chapter 10 GRAPHICAL INFERENCE 10.1 MA order determination 10.2 AR order determination 10.3 Graphical tests for parameter matrices Chapter 11 TESTING WITH TRACE 11.1 One sample trace 11.2 Two sample trace 11.3 Testing Appendix: SUPPLEMENTARY PROOFS A.1 Proof of Lemma 6.3.1 A.2 Proof of Theorem 6.4.1(a) A.3 Proof of Theorem 7.2 A.4 Proof of Lemma 8.2.1 A.5 Proof of Corollary 8.2.1(c) A.6 Proof of Corollary 8.2.4(c) A.7 Proof of Corollary 8.3.1(c) A.8 Proof of Lemma 8.2.2