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