Random Circulant Matrices

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.

1. Circulants

Circulant

Symmetric circulant

Reverse circulant

k-circulant

Exercises

2. Symmetric and reverse circulant

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

3. LSD: normal approximation

Method of normal approximation

Circulant

k-circulant

Exercises

4. LSD: dependent input

Spectral density

Circulant

Reverse circulant

Symmetric circulant

k-circulant

Exercises

5. Spectral radius: light tail

Circulant and reverse circulant

Symmetric circulant

Exercises

6. Spectral radius: k-circulant

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

7. Maximum of scaled eigenvalues: dependent input

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

8. Poisson convergence

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

9. Heavy tailed input: LSD

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

10. Heavy-tailed input: spectral radius

Input sequence and scaling

Reverse circulant and circulant

Symmetric circulant

Heavy-tailed: dependent input

Exercises

11. Appendix

Proof of Theorem

Standard notions and results

Three auxiliary results

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.