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E-raamat: Random Matrices and Non-Commutative Probability [Taylor & Francis e-raamat]

(Indian Statistical Institute, Kolkata)
  • Formaat: 286 pages, 1 Tables, black and white
  • Ilmumisaeg: 27-Oct-2021
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-13: 9781003144496
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  • Taylor & Francis e-raamat
  • Hind: 240,04 €*
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Random Matrices and Non-Commutative ProbabilitySuurem pilt
  • Formaat: 286 pages, 1 Tables, black and white
  • Ilmumisaeg: 27-Oct-2021
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-13: 9781003144496
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9781003144496
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.
Preface xi
About the Author xiii
Notation xv
Introduction xvii
1 Classical independence, moments and cumulants
1(14)
1.1 Classical independence
1(3)
1.2 CLT via cumulants
4(2)
1.3 Cumulants to moments
6(3)
1.4 Moments to cumulants, the Mobius function
9(2)
1.5 Classical Isserlis' formula
11(1)
1.6 Exercises
12(3)
2 Non-commutative probability
15(16)
2.1 Non-crossing partition
15(2)
2.2 Free cumulants
17(1)
2.3 Free Gaussian or semi-circular law
18(3)
2.4 Free Poisson law
21(1)
2.5 Non-commutative and *-probability spaces
22(3)
2.6 Moments and probability laws of variables
25(4)
2.7 Exercises
29(2)
3 Free independence
31(18)
3.1 Free independence
31(3)
3.2 Free product of probability spaces
34(1)
3.3 Free binomial
35(1)
3.4 Semi-circular family
35(1)
3.5 Free Isserlis' formula
36(1)
3.6 Circular and elliptic variables
37(2)
3.7 Free additive convolution
39(1)
3.8 Kreweras complement
40(4)
3.9 Moments of free variables
44(2)
3.10 Compound free Poisson
46(1)
3.11 Exercises
47(2)
4 Convergence
49(12)
4.1 Algebraic convergence
49(4)
4.2 Free central limit theorem
53(2)
4.3 Free Poisson convergence
55(2)
4.4 Sums of triangular arrays
57(1)
4.5 Exercises
58(3)
5 Transforms
61(26)
5.1 Stieltjes transform
61(6)
5.2 1Z transform
67(2)
5.3 Interrelation
69(5)
5.4 5-transform
74(5)
5.5 Free infinite divisibility
79(6)
5.6 Exercises
85(2)
6 C*-probability space
87(20)
6.1 C*-probability space
87(1)
6.2 Spectrum
88(11)
6.3 Distribution of a self-adjoint element
99(3)
6.4 Free product of C*-probability spaces
102(1)
6.5 Free additive and multiplicative convolution
103(2)
6.6 Exercises
105(2)
7 Random matrices
107(16)
7.1 Empirical spectral measure
107(1)
7.2 Limiting spectral measure
108(1)
7.3 Moment and trace
109(1)
7.4 Some important matrices
110(7)
7.5 A unified treatment
117(4)
7.6 Exercises
121(2)
8 Convergence of some important matrices
123(26)
8.1 Wigner matrix: semi-circular law
123(4)
8.2 5-matrix: Marcenko-Pastur law
127(8)
8.3 IID and elliptic matrices: circular and elliptic variables
135(3)
8.4 Toeplitz matrix
138(2)
8.5 Hankel matrix
140(1)
8.6 Reverse Circulant matrix: symmetrized Rayleigh law
141(3)
8.7 Symmetric Circulant: Gaussian law
144(2)
8.8 Almost sure convergence of the ESD
146(2)
8.9 Exercises
148(1)
9 Joint convergence I: single pattern
149(22)
9.1 Unified treatment: extension
149(5)
9.2 Wigner matrices: asymptotic freeness
154(3)
9.3 Elliptic matrices: asymptotic freeness
157(3)
9.4 S-matrices in elliptic models: asymptotic freeness
160(6)
9.5 Symmetric Circulants: asymptotic independence
166(1)
9.6 Reverse Circulants: asymptotic half-independence
167(2)
9.7 Exercises
169(2)
10 Joint convergence II: multiple patterns
171(16)
10.1 Multiple patterns: colors and indices
171(3)
10.2 Joint convergence
174(3)
10.3 Two or more patterns at a time
177(7)
10.4 Sum of independent patterned matrices
184(1)
10.5 Discussion
185(1)
10.6 Exercises
186(1)
11 Asymptotic freeness of random matrices
187(38)
11.1 Elliptic, IID, Wigner and 5-matrices
187(1)
11.2 Gaussian elliptic, IID, Wigner and deterministic matrices
188(7)
11.3 General elliptic, IID, Wigner and deterministic matrices
195(2)
11.4 5-matrices and embedding
197(3)
11.5 Cross-covariance matrices
200(12)
11.5.1 Pair-correlated cross-covariance; p/n → 4 y ≠ 0
201(7)
11.5.2 Pair correlated cross-covariance; p/n → 0
208(4)
11.6 Wigner and patterned random matrices
212(10)
11.7 Discussion
222(1)
11.8 Exercises
223(2)
12 Brown measure
225(8)
12.1 Brown measure
225(6)
12.2 Exercises
231(2)
13 Tying three loose ends
233(18)
13.1 Mobius function on NC(n)
233(6)
13.2 Equivalence of two freeness definitions
239(4)
13.3 Free product construction
243(6)
13.4 Exercises
249(2)
Bibliography 251(8)
Index 259
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).
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