1st Edition

Random Matrices and Non-Commutative Probability



  • Available for pre-order. Item will ship after October 27, 2021
ISBN 9780367700812
October 27, 2021 Forthcoming by Chapman and Hall/CRC
278 Pages

USD $175.00

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Book Description

This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful.

  • Combinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability.

  • Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants.

  • Free cumulants are introduced through the Möbius function.

  • Free product probability spaces are constructed using free cumulants.

  • Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed.

  • Convergence of the empirical spectral distribution is discussed for symmetric matrices.

  • Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices.

  • Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices.

  • Exercises, at advanced undergraduate and graduate level, are provided in each chapter.

Table of Contents

 

  1. Classical independence, moments and cumulants
  2. Classical independence

    CLT a cumulants

    Cumulants to moments

    Moments to cumulants, the Möbius function

    Classical Isserlis’ formula

    Exercises

  3. Non-commutative probability
  4. Non-crossing partition

    Free cumulants

    Free Gaussian or semi-circle law

    Free Poisson law

    Non-commutative and *-probability spaces

    Moments and probability laws of variables

    Exercises

  5. Free independence
  6. Free independence

    Free product of *-probability spaces

    Free binomial

    Semi-circular family

    Free Isserlis’ formula

    Circular and elliptic variables

    Free additive convolution

    Kreweras complement

    Moments of free variables

    0 Compound free Poisson

    Exercises

  7. Convergence
  8. Algebraic convergence

    Free central limit theorem

    Free Poisson convergence

    Sums of triangular arrays

    Exercises

  9. Transforms
  10. Stieltjes transform

    R transform

    Interrelation

    S-transform

    Free infinite divisibility

    Exercises

  11. C* -probability space
  12. C* -probability space

    Spectrum

    Distribution of a self-adjoint element

    Free product of C* -probability spaces

    Free additive and multiplicative convolution

    Exercises

  13. Random matrices
  14. Empirical spectral measure

    Limiting spectral measure

    Moment and trace

    Some important matrices

    A unified treatment

    Exercises

  15. Convergence of some important matrices
  16. Wigner matrix: semi-circle law

    S-matrix: Marcenko-Pastur law

    IID and elliptic matrices: circular and elliptic variables

    Toeplitz matrix

    Hankel matrix

    Reverse Circulant matrix: symmetrized Rayleigh law

    Symmetric Circulant: Gaussian law

    Almost sure convergence of the ESD

    Exercises

  17. Joint convergence I: single pattern
  18. Unified treatment: extension

    Wigner matrices: asymptotic freeness

    Elliptic matrices: asymptotic freeness

    S-matrices in elliptic models: asymptotic freeness

    Symmetric Circulants: asymptotic independence

    Reverse Circulants: asymptotic half independence

    Exercises

  19. Joint convergence II: multiple patterns
  20. Multiple patterns: colors and indices

    Joint convergence

    Two or more patterns at a time

    Sum of independent patterned matrices

    Discussion

    Exercises

  21. Asymptotic freeness of random matrices
  22. Elliptic, IID, Wigner, and S-matrices

    Gaussian elliptic, IID, Wigner and deterministic matrices

    General elliptic, IID, Wigner, and deterministic matrices

    S-matrices and embedding

    Cross covariance matrices

    Pair-correlated cross-covariance; p/n ! y

    Pair correlated cross covariance; p/n !

    Wigner and patterned random matrices

    Discussion

    Exercises

  23. Brown measure
  24. Brown measure

    Exercises

  25. Tying three loose ends

Möbius function on NC(n)

Equivalence of two freeness definitions

Free product construction

Exercises

Bibliography

Index

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Author(s)

Biography

Arup Bose is on the faculty of the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, India. He has research contributions in statistics, probability, economics and econometrics. He is a Fellow of the Institute of Mathematical Statistics (USA), 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 and holds a J.C.Bose National Fellowship. He has been on the editorial board of several journals. He has authored four books: Patterned Random Matrices, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee), U-Statistics, Mm-Estimators and Resampling (with Snigdhansu Chatterjee) and Random Circulant Matrices (with Koushik Saha).