Large Covariance and Autocovariance Matrices

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 >
 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 <
 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 <
 19.2.2 MA(1)
 9.3 LSD when p=n ! 0
 9.3.1 MA(q)
 q <
 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