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Random Circulant Matrices de Arup Bose

Random Circulant Matrices

Autor Arup Bose, Koushik Saha
en Limba Engleză Hardback – 25 oct 2018
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.
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Specificații

ISBN-13: 9781138351097
ISBN-10: 1138351091
Pagini: 212
Dimensiuni: 156 x 234 x 19 mm
Greutate: 0.39 kg
Ediția:1
Editura: CRC Press
Colecția Chapman and Hall/CRC

Cuprins

  1. CirculantsCirculant
    Symmetric circulant
    Reverse circulant
    k-circulant
    Exercises
  2. Symmetric and reverse circulantSpectral 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 approximationMethod of normal approximation
    Circulant
    k-circulant
    Exercises
  4. LSD: dependent inputSpectral density
    Circulant
    Reverse circulant
    Symmetric circulant
    k-circulant
    Exercises
  5. Spectral radius: light tailCirculant and reverse circulant
    Symmetric circulant
    Exercises
  6. Spectral radius: k-circulantTail 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 inputDependent 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 convergencePoint 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: LSDStable 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 radiusInput sequence and scaling
    Reverse circulant and circulant
    Symmetric circulant
    Heavy-tailed: dependent input
    Exercises
  11. Appendix
          Proof of Theorem
          Standard notions and results
          Three auxiliary results

Notă biografică

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.

Descriere

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.