 1st Edition

# Random Matrices and Non-Commutative Probability

By

## Arup Bose

ISBN 9780367700812
Published October 27, 2021 by Chapman and Hall/CRC
286 Pages

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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.

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

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

Free product of C* -probability spaces

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

...

## 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).