Large Covariance and Autocovariance Matrices

Description

Cover Title Preface Introduction Part I Chapterr1 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 Gf(M) or Ff(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 Chapterr2 LARGE COVARIANCE MATRIX II 2.1 Bandable p 2.1.1 Models and examples 2.1.2 Weak dependence 2.1.3 Estimation 2.2 Sparse p Chapterr3 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 Chapterr4 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: Marcenko{Pastur law 4.3.1 Results on Z: p=n ! y > 0 4.3.2 Results on Z: p=n ! 0 Chapterr5 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 Chapterr6 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 Corollaries Chapterr7 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 Chapterr8 SPECTRA OF AUTOCOVARIANCE MATRIX I 8.1 Assumptions 8.2 LSD when p=n ! y 2 (0;1) 8.2.1 MA(q), q < 1 8.2.2 MA(1) 8.2.3 Application to specifc cases 8.3 LSD when p=n ! 0 8.3.1 Application to specifc cases 8.4 Non-symmetric polynomials Chapterr9 SPECTRA OF AUTOCOVARIANCE MATRIX II 9.1 Assumptions 9.2 LSD when p=n ! y 2 (0;1) 9.2.1 MA(q), q < 1 9.2.2 MA(1) 9.3 LSD when p=n ! 0 9.3.1 MA(q); q < 1 9.3.2 MA(1) Chapterr10 GRAPHICAL INFERENCE 10.1 MA order determination 10.2 AR order determination 10.3 Graphical tests for parameter matrices Chapterr11 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 A.9 Proof of Lemma 8.2.3 A.10 Lemmas for Theorem 8.2.2 Bibliography Index