Combinatorics of Free Probability Theory
－Roland Speicher
英文版


Contents
Part 1. Basic concepts 5
Chapter 1. Basic concepts of non-commutative probability theory 7
1.1. Non-commutative probability spaces and distributions 7
1.2. Haar unitaries and semicircular elements 10
Chapter 2. Free random variables 17
2.1. Definition and basic properties of freeness 17
2.2. The group algebra of the free product of groups 21
2.3. The full Fock space 23
2.4. Construction of free products 27
Part 2. Cumulants 35
Chapter 3. Free cumulants 37
3.1. Motivation: Free central limit theorem 37
3.2. Non-crossing partitions 46
3.3. Posets and M¨obius inversion 50
3.4. Free cumulants 52
Chapter 4. Fundamental properties of free cumulants 59
4.1. Cumulants with products as entries 59
4.2. Freeness and vanishing of mixed cumulants 66
Chapter 5. Sums and products of free variables 75
5.1. Additive free convolution 75
5.2. Description of the product of free variables 84
5.3. Compression by a free projection 87
5.4. Compression by a free family of matrix units 91
Chapter 6. R-diagonal elements 95
6.1. Definition and basic properties of R-diagonal elements 95
6.2. The anti-commutator of free variables 103
6.3. Powers of R-diagonal elements 105
Chapter 7. Free Fisher information 109
7.1. Definition and basic properties 109
7.2. Minimization problems 112
Part 3. Appendix 123
Chapter 8. Infinitely divisible distributions 125
8.1. General free limit theorem 125
8.2. Freely infinitely divisible distributions 127



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