Patterned Random Matrices: 1st Edition (Hardback) book cover

Patterned Random Matrices

1st Edition

By Arup Bose

Chapman and Hall/CRC

269 pages

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Hardback: 9781138591462
pub: 2018-05-17
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pub: 2018-05-23
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Description

Large dimensional random matrices (LDRM) with specific patterns arise in econometrics, computer science, mathematics, physics, and statistics. This book provides an easy initiation to LDRM. Through a unified approach, we investigate the existence and properties of the limiting spectral distribution (LSD) of different patterned random matrices as the dimension grows. The main ingredients are the method of moments and normal approximation with rudimentary combinatorics for support. Some elementary results from matrix theory are also used. By stretching the moment arguments, we also have a brush with the intriguing but difficult concepts of joint convergence of sequences of random matrices and its ramifications.

This book covers the Wigner matrix, the sample covariance matrix, the Toeplitz matrix, the Hankel matrix, the sample autocovariance matrix and the k-Circulant matrices. Quick and simple proofs of their LSDs are provided and it is shown how the semi-circle law and the Marchenko-Pastur law arise as the LSDs of the first two matrices. Extending the basic approach, we also establish interesting limits for some triangular matrices, band matrices, balanced matrices, and the sample autocovariance matrix. We also study the joint convergence of several patterned matrices, and show that independent Wigner matrices converge jointly and are asymptotically free of other patterned matrices.

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 the Editor of Sankyhā 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 forthcoming books are the monograph, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee), to be published by Chapman & Hall/CRC Press, and a graduate text, U-statistics, M-estimates and Resampling (with Snigdhansu Chatterjee), to be published by Hindustan Book Agency.

Reviews

". . . this book can be recommended for students and researchers interested in a broad overview of random matrix theory. Each chapter ends with plenty of problems useful for exercises and training." ~ Statistical Papers

Table of Contents

  1. A unified framework
  2. Empirical and limiting spectral distribution

    Moment method

    A metric for probability measures

    Patterned matrices: A unified approach

    Exercises

  3. Common symmetric patterned matrices
  4. Wigner matrix

    Toeplitz and Hankel matrices

    Reverse Circulant matrix

    Symmetric Circulant and related matrices

    Additional properties of the LSDs

    Exercises

  5. Patterned XX matrices
  6. A unified setup

    Aspect ratio y = 0

    Aspect ratio y = 0

    Exercises

  7. Circulant matrices
  8. Normal approximation

    Circulant matrix

    k-Circulant matrices

    Exercises

  9. Wigner-type matrices
  10. Wigner-type matrix

    Exercises

  11. Balanced Toeplitz and Hankel matrices
  12. Main results

    Exercises

  13. Patterned band matrices
  14. LSD for band matrices

    Proof

    Exercises

  15. Triangular matrices
  16. General pattern

    Triangular Wigner matrix  

  17. Joint convergence of i.i.d. patterned matrices
  18. Non-commutative probability space

    Joint convergence

    Nature of the limit

    Exercises

  19. Joint convergence of independent patterned matrices
  20. Definitions and notation

    Joint convergence

    Freeness

    Sum of independent patterned matrices

    Proofs

    Exercises

  21. Autocovariance matrix

Preliminaries

Main results

Proofs

Exercises

About the Author

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 the Editor of Sankyha 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 forthcoming books are the monograph, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee), to be published by Chapman & Hall/CRC Press, and a graduate text, U-statistics, M-estimates and Resampling (with Snigdhansu Chatterjee), to be published by Hindustan Book Agency.

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis