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

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.