1st Edition

# Large Covariance and Autocovariance Matrices

**Also available as eBook on:**

**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 *Sankhyā* 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.

**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

### Biography

**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 *Sankhyā* 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.

" . . . the authors should be congratulated for producing two highly relevant and well-written books. Statisticians would probably gravitate to LCAM in the first instance and those working in linear algebra would probably gravitate to PRM."

~Jonathan Gillard, Cardiff University"The book represents a monograph of the authors’ recent results about the theory of large covariance and autocovariance matrices and contains other important results from other research papers and books in this topic. It is very useful for all researchers who use large covariance and autocovariance matrices in their researches. Especially, it is very useful for post-graduate and PhD students in mathematics, statistics, econometrics and computer science. It is a well-written and organized book with a large number of solved examples and many exercises left to readers for homework. I would like to recommend the book to PhD students and researchers who want to learn or use large covariance and autocovariance matrices in their researches."

~ Miroslav M. Ristic (Niš), zbMath"This book brings together a collection of recent results on estimation of multidimensional time series covariance matrices. In the case where the time series consists of a sequence of independent (Chapter 1) or weakly dependent (Chapter 2) random vectors, the authors call it covariance estimation, whereas in the general case where the time series is only stationary, they call it autocovariance estimation. The framework of the results presented here is the one where the dimension of the observations (as well as the observation window size, otherwise nothing can be said) is high. The prerequisites include knowledge of elementary multivariate analysis, basic time series analysis, and basic results in stochastic convergence.

In Chapter 1, the authors consider the case where we have at our disposal a large time series of iid high-dimensional observations with common covariance