Large Covariance and Autocovariance Matrices

1. LARGE COVARIANCE MATRIX I

Consistency

Covariance classes and regularization

Covariance classes

Covariance regularization

Bandable ¿p

Parameter space

Estimation in U

Minimaxity

Toeplitz ¿p

Parameter space

Estimation in Gß (M ) or Fß (M0, M )

Minimaxity

Sparse ¿p

Parameter space

Estimation in U¿ (q, C0(p), M ) or Gq (Cn,p)

Minimaxity

2. LARGE COVARIANCE MATRIX II

Bandable ¿p

Models and examples

Weak dependence

Estimation

Sparse ¿p

3. LARGE AUTOCOVARIANCE MATRIX

Models and examples

Estimation of ¿0,p

Estimation of ¿u,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: Mar¿cenko-Pastur law

> 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 Theorem

Proof of Lemma

Proof of Corollary (c)

Proof of Corollary (c)

Proof of Corollary (c)

Proof of Lemma

Proof of Lemma

Lemmas for Theorem