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
- Classical independence, moments and cumulants
- Non-commutative probability
- Free independence
- 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
CLT a cumulants
Cumulants to moments
Moments to cumulants, the Möbius function
Classical Isserlis’ formula
Free Gaussian or semi-circle law
Free Poisson law
Non-commutative and *-probability spaces
Moments and probability laws of variables
Free product of *-probability spaces
Free Isserlis’ formula
Circular and elliptic variables
Free additive convolution
Moments of free variables
0 Compound free Poisson
Free central limit theorem
Free Poisson convergence
Sums of triangular arrays
Free infinite divisibility
C* -probability space
Distribution of a self-adjoint element
Free product of C* -probability spaces
Free additive and multiplicative convolution
Empirical spectral measure
Limiting spectral measure
Moment and trace
Some important matrices
A unified treatment
Wigner matrix: semi-circle law
S-matrix: Marcenko-Pastur law
IID and elliptic matrices: circular and elliptic variables
Reverse Circulant matrix: symmetrized Rayleigh law
Symmetric Circulant: Gaussian law
Almost sure convergence of the ESD
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
Multiple patterns: colors and indices
Two or more patterns at a time
Sum of independent patterned matrices
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
Möbius function on NC(n)
Equivalence of two freeness definitions
Free product construction
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).