Index

∗-algebra, 160

2-Schatten norm, 114

R-transform, 180

S-transform, 181

k-point correlation function, 196

absolutely continuous, 12

absolutely integrable, 13, 16

Airy function, 206, 260

Airy kernel, 206

almost sure convergence, 32, 135

almost surely, 6

amalgamated free product, 176

antichain, 215

approximation to the identity, 144

asymptotic free independence, 174

asymptotic notation, 5

asymptotically almost surely, 6

Atiyah convexity theorem, 44

augmented matrix, 133

Azuma’s inequality, 63

Bai-Yin theorem, 129, 131

Baker-Campbell-Hausdorff formula, 252

Bernoulli distribution, 11

Bernoulli ensemble, 105

Berry-Ess´ een theorem, 87, 95, 100, 101,

213

bijective proof, 122

Bochner’s theorem, 257

Borel-Cantelli lemma, 14, 34

Brown measure, 232, 233

Burgers’ equation, 153

Carleman continuity theorem, 90

Catalan number, 122

Cauchy transform, 142

Cauchy-Binet formula, 253

Cauchy-Schwarz inequality, 164

central limit theorem, 79, 86

characteristic function, 15

Chebyshev’s inequality, 17

Chernoff bound, 79

Chernoff inequality, 61

Christoffel-Darboux formula, 197

circular law, 193, 209, 224

circular unitary ensemble, 268

classical probability theory, 160

compressible, 213, 215, 218

concentration of measure, 56

condition number, 209

conditional expectation, 27

conditional probability, 27

conditional probability measure, 24

conditioning, 24, 29

convergence in distribution, 32

convergence in expectation, 135

convergence in probability, 32, 135

convergence in the sense of ∗-moments,

172

convergence in the sense of moments,

172

correlation, 58

Courant-Fischer minimax theorem, 42

covariance matrix, 86

cumulant, 180

cumulative distribution function, 12

279