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

Random Circulant Matrices

1st Edition

By Arup Bose, Koushik Saha

Chapman and Hall/CRC

192 pages

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Hardback: 9781138351097
pub: 2018-10-25
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Description

Circulant matrices have been around for a long time and have been extensively used in many scientific areas. This book studies the properties of the eigenvalues for various types of circulant matrices, such as the usual circulant, the reverse circulant, and the k-circulant when the dimension of the matrices grow and the entries are random.

In particular, the behavior of the spectral distribution, of the spectral radius and of the appropriate point processes are developed systematically using the method of moments and the various powerful normal approximation results. This behavior varies according as the entries are independent, are from a linear process, and are light- or heavy-tailed.

Arup Bose obtained his B.Stat., M.Stat. and Ph.D. degrees from the Indian Statistical Institute. He has been on its faculty at the Theoretical Statistics and Mathematics Unit, Kolkata, India since 1991. He is a Fellow of the Institute of Mathematical Statistics, and of all three national science academies of India. He is a recipient of the S.S. Bhatnagar Prize and the C.R. Rao Award. He is the author of three books: Patterned Random Matrices, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee) and U-Statistics, M_m-Estimators and Resampling (with Snigdhansu Chatterjee).

Koushik Saha obtained a B.Sc. in Mathematics from Ramakrishna Mission Vidyamandiara, Belur and an M.Sc. in Mathematics from Indian Institute of Technology Bombay. He obtained his Ph.D. degree from the Indian Statistical Institute under the supervision of Arup Bose. His thesis on circulant matrices received high praise from the reviewers. He has been on the faculty of the Department of Mathematics, Indian Institute of Technology Bombay since 2014.

Table of Contents

  1. Circulants
  2. Circulant

    Symmetric circulant

    Reverse circulant

    k-circulant

    Exercises

  3. Symmetric and reverse circulant
  4. Spectral distribution

    Moment method

    Scaling

    Input and link

    Trace formula and circuits

    Words and vertices

    (M) and Riesz’s condition

    (M) condition

    Reverse circulant

    Symmetric circulant

    Related matrices

    Reduced moment

    A metric

    Minimal condition

    Exercises

  5. LSD: normal approximation
  6. Method of normal approximation

    Circulant

    k-circulant

    Exercises

  7. LSD: dependent input
  8. Spectral density

    Circulant

    Reverse circulant

    Symmetric circulant

    k-circulant

    Exercises

  9. Spectral radius: light tail
  10. Circulant and reverse circulant

    Symmetric circulant

    Exercises

  11. Spectral radius: k-circulant
  12. Tail of product

    Additional properties of the k-circulant

    Truncation and normal approximation

    Spectral radius of the k-circulant

    k-circulant for sn = kg +

    Exercises

  13. Maximum of scaled eigenvalues: dependent input
  14. Dependent input with light tail

    Reverse circulant and circulant

    Symmetric circulant

    k-circulant

    k-circulant for n = k +

    k-circulant for n = kg + , g >

    Exercises

  15. Poisson convergence
  16. Point Process

    Reverse circulant

    Symmetric circulant

    k-circulant, n = k +

    Reverse circulant: dependent input

    Symmetric circulant: dependent input

    k-circulant, n = k + : dependent input

    Exercises

  17. Heavy tailed input: LSD
  18. Stable distribution and input sequence

    Background material

    Reverse circulant and symmetric circulant

    k-circulant: n = kg +

    Proof of Theorem

    Contents vii

    k-circulant: n = kg

    Tail of the LSD

    Exercises

  19. Heavy-tailed input: spectral radius
  20. Input sequence and scaling

    Reverse circulant and circulant

    Symmetric circulant

    Heavy-tailed: dependent input

    Exercises

  21. Appendix

Proof of Theorem

Standard notions and results

Three auxiliary results

About the Authors

Arup Bose obtained his B.Stat., M.Stat. and Ph.D. degrees from the Indian Statistical Institute. He has been on its faculty at the Theoretical Statistics and Mathematics Unit, Kolkata, India since 1991. He is a Fellow of the Institute of Mathematical Statistics, and of all three national science academies of India. He is a recipient of the S.S. Bhatnagar Prize and the C.R. Rao Award. He is the author of three books: Patterned Random Matrices, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee) and U-Statistics, M_m-Estimators and Resampling (with Snigdhansu Chatterjee).

Koushik Saha obtained a B.Sc. in Mathematics from Ramakrishna Mission Vidyamandiara, Belur and an M.Sc. in Mathematics from Indian Institute of Technology Bombay. He obtained his Ph.D. degree from the Indian Statistical Institute under the supervision of Arup Bose. His thesis on circulant matrices received high praise from the reviewers. He has been on the faculty of the Department of Mathematics, Indian Institute of Technology Bombay since 2014.

Subject Categories

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