- 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, Mm-Estimators and Resampling (with Snigdhansu Chatterjee) and Random Circulant Matrices (with Koushik Saha).
