Random Circulant Matrices

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.

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