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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.

**Classical independence, moments and cumulants**- Non-commutative probability
- Free independence
- Convergence
- Transforms
*C**-probability space- Random matrices
- Convergence of some important matrices
- Joint convergence I: single pattern
- Joint convergence II: multiple patterns
- Asymptotic freeness of random matrices
- Brown measure
- Tying three loose ends

Classical independence

CLT a cumulants

Cumulants to moments

Moments to cumulants, the Möbius function

Classical Isserlis’ formula

Exercises

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

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

Algebraic convergence

Free central limit theorem

Free Poisson convergence

Sums of triangular arrays

Exercises

Stieltjes transform

R transform

Interrelation

*S*-transform

Free infinite divisibility

Exercises

*C* *-probability space

Spectrum

Distribution of a self-adjoint element

Free product of *C* *-probability spaces

Free additive and multiplicative convolution

Exercises

Empirical spectral measure

Limiting spectral measure

Moment and trace

Some important matrices

A unified treatment

Exercises

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

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

Multiple patterns: colors and indices

Joint convergence

Two or more patterns at a time

Sum of independent patterned matrices

Discussion

Exercises

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

Brown measure

Exercises

Möbius function on *NC*(*n*)

Equivalence of two freeness definitions

Free product construction

Exercises

Bibliography

Index

### 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, M _{m}-Estimators and Resampling* (with Snigdhansu Chatterjee) and

*Random Circulant Matrices*(with Koushik Saha).