1st Edition

Random Circulant Matrices





ISBN 9780367732912
Published December 18, 2020 by Chapman and Hall/CRC
212 Pages

USD $54.95

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Book 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



...
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Author(s)

Biography

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.