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.

Table of Contents:
Circulants Circulant Symmetric circulant Reverse circulant k-circulant Exercises 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 LSD: normal approximation Method of normal approximation Circulant k-circulant Exercises LSD: dependent input Spectral density Circulant Reverse circulant Symmetric circulant k-circulant Exercises Spectral radius: light tail Circulant and reverse circulant Symmetric circulant Exercises 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 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 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 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 Heavy-tailed input: spectral radius Input sequence and scaling Reverse circulant and circulant Symmetric circulant Heavy-tailed: dependent input Exercises Appendix Proof of Theorem Standard notions and results Three auxiliary results