Random Circulant Matrices

ISBN-13: 9780367732912 / Angielski / Miękka / 2020 / 192 str.

Arup Bose; Koushik Saha

Opis

This book is on properties of the eigenvalues of several different Random Circulant- type matrices as the dimension goes to infinity. In particular, we consider the bulk behavior of the eigenvalues (limiting spectral distribution) and also the edge behavior of the eigenvalues.

Spis treści

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

O autorze

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.