大维随机矩阵的谱分析（英文版）

1Introduction.
1.1LargeDimensionalDataAnalysis
1.2RandomMatrixTheory
1.2.1SpectralAnalysisofLargeDimensionalRandomMatrices
1.2.2LimitsofExtremeEigenvalues
1.2.3ConvergenceRateofESD
1.2.4CircularLaw
1.2.5CLTofLinearSpectralStatisticslinearspectralstatistics
1.2.6LimitingDistributionsofExtremeEigenvaluesandSpacings
1.3Methodologies
1.3.1MomentMethod
1.3.2StieltjesTransform
1.3.3OrthogonalPolynomialDecomposition
2WignerMatricesandSemicircularLaw
2.1SemicircularLawbytheMomentMethod
2.1.1MomentsoftheSemicircularLaw
2.1.2SomeLemmasofCombinatorics
2.1.3SemicircularLawforiidCase
2.2GeneralizationstotheNon-lidCase
2.2.1ProofofTheorem2.9
2.3SemicircularLawbyStieltjesTransform
2.3.1StieltjesTransformofSemicircularLaw
2.3.2ProofofTheorem2.9
3SampleCovarianceMatrices,Marcenko-PasturLaw
3.1MPLawforiidCase
3.1.1MomentsoftheMPLaw
3.1.2SomeLemmasonGraphTheoryandCombinatorics
3.1.3MPLawforiidCase
3.2GeneralizationtotheNon-iidCase
3.3ProofofTheorem3.9byStieltjesTransform
3.3.1StieltjesTransformofMPLaw
3.3.2ProofofTheorem3.9
4ProductofTwoRandomMatrices
4.1SomeGraphTheoryandCombinatoricResults
4.2ExistenceoftheESDofSnTn
4.2.1TruncationoftheESDofTn
4.2.2Truncation,CentralizationandRescalingoftheX-variables
4.2.3ProofofTheorem4.3
4.3LSDofFmatrix
4.3.1GeneralFormulafortheProduct
4.3.2LSDofMultivariateFMatrices
4.4ProofofTheorem4.5
4.4.1TruncationandCentralization
4.4.2ProofbyStieltjesTransform
5LimitsofExtremeEigenvalues
5.1LimitofExtremeEigenwaluesoftheWignerMatrix
5.1.1SufficiencyofConditionsofTheorem5.1
5.1.2NecessityofConditionsofTheorem5.1
5.2LimitsofExtremeEigenvaluesofSampleCovarianceMatrix
5.2.1ProofofTheorem5.10
5.2.2TheProofofTheorem5.11
5.2.3NecessityoftheConditions
5.3Miscellanies
6SpectrumSeparation
6.1WhatisSpectrumSeparation
6.1.1MathematicalTools
6.2Proofof(1)
6.2.1TruncationandSomeSimpleFacts
6.2.2APreliminaryConvergenceRate
6.2.3ConvergenceofSn—Esn
6.2.4ConvergenceofExpectedValue
6.2.5CompletingtheProof
6.3Proofof(2)
6.4Proofof(3)
6.4.1ConvergenceofaRandomQuadraticForm
6.4.2SpreadofEigenvalues
6.4.3Dependenceony
6.4.4CompletingtheProofof(3)
7SemicircleLawforHadamardProducts
7.1RenormalizedSampleCovarianceMatrix
7.2SparseMatrixandHadamardProduct
7.3ProofofTheorem7.4..
7.3.1TruncationandCentralization
7.4ProofofTheorem7.4byMomentApproach
8ConvergenceRatesofESD
8.1SomeLemmasAboutIntegralsofStieltjesTransforms
8.2ConvergenceRatesofExpectedESDofWignerMatrices
8.2.1LemmasonTruncation,CentralizationandRescaling
8.2.2ProofofTheorem8.6
8.2.3SomeLemmasofPreliminaryCalculation
8.3FurtherExtensions
8.4ConvergenceRatesofExpectedESDofSampleCovarianceMatrices
8.4.1AssumptionsandResults
8.4.2TruncationandCentralization
8.4.3ProofofTheorem8.16
8.5SomeElementaryCalculus
8.5.1IncrementofM-PDensity
8.5.2IntegralofTailProbability
8.5.3BoundsofStieltjesTransformsofM-PLaw
8.5.4Boundsforbn
8.5.5IntegralsofSquaredAbsoluteValuesofStieltjes
Transforms
8.5.6HigherCentralMomentsofStieltjesTransforms
8.5.7Integralofδ
8.6RatesofConvergenceinProbabilityandAlmostSurely
9CLTforLinearSpectralStatistics
9.1MotivationandStrategy
9.2CLTofLSSforWignerMatrix
9.2.1StrategyoftheProof
9.2.2TruncationandRenormalization
9.2.3MeanFunctionofMn
9.2.4ProofoftheNonrandomPartof(9.2.13)forj=l,r
9.3ConvergenceoftheProcessMn-EMn
9.3.1Finite-DimensionalConvergenceofMn-EMn
9.3.2LimitofS1
9.3.3CompletionofProofof(9.2.13)forj=l,r
9.3.4TightnessoftheProcessMN(z)-EMn(z)
9.4ComputationoftheMeanandCovarianceFunctionofG(f)
9.4.1MeanFunction
9.4.2CovarianceFunction
9.5ApplicationtoLinearSpectralStatisticsandRelatedResults
9.5.1TchebychevPolynomials
9.6TechnicalLemmas
9.7CLTofLSSforSampleCovarianceMatrices
9.7.1Truncation
9.8ConvergenceofStieltjesTransforms
9.9ConvergenceofFiniteDimensionalDistributions
9.10TightnessofMn1(z)
9.11ConvergenceofMn2(z)
9.12SomeDerivationsandCalculations
9.12.1Verificationof(9.8.8)
9.12.2Verificationof(9.8.9)
9.12.3DerivationofQuantitiesinExample(1.1)
9.12.4VerificationofQuantitiesinJonsson'sResults
9.12.5Verificationof(9.7.8)and(9.7.9)
10CircularLaw
10.1TheProblemandDifficulty.
10.1.1FailureofTechniquesDealingwithHermitianMatrices
10.1.2RevisitofStieltjesTransformation
10.2ATheoremEstablishingaPartialAnswertotheCircularLaw
10.3LemmasonIntegralRangeReduction
10.4CharacterizationoftheCircularLaw
10.5ARoughRateontheConvergenceofvn,(x,z)
10.5.1TruncationandCentralization
10.5.2AConvergenceRateoftheStieltjesTransformof
10.6Proofsof(10.2.3)and(10.2.4)
10.7ProofofTheorem10.3
10.8CommentsandExtensions
10.8.1RelaxationofConditionsAssumedinTheorem10.3
10.9SomeElementaryMathematics
11AppendixA.SomeResultsinLinearAlgebra
11.1InverseMatricesandResolvent
11.1.1InverseMatrixFormula
11.1.2HolingaMatrix
11.1.3TraceofInverseMatrix
11.1.4DifferenceofTracesofaMatrixAandItsMajorSubmatrices
11.1.5InverseMatrixofComplexMatrices
11.2InequalitiesInvolvingSpectralDistributions
11.2.1SingularValueInequalities
11.3HadamardProductandOdotProduct
11.4ExtensionsofSingularValueInequalities
11.4.1DefinitionsandProperties
11.4.2Graph-AssociatedMultipleMatrices
11.4.3FundamentalTheoremonGraph-AssociatedMM
11.5PerturbationInequalities
11.6RankInequalities
11.7ANormInequality
12AppendixB.MomentConvergenceTheoremandStieltjesTransform
12.1MomentConvergenceTheorem
12.2StieltjesTransform
12.2.1PreliminaryProperties
12.2.2InequalitiesofDistancebetweenDistributionsinTermsofTheirStieltjesTransforms
12.2.3LemmasConcerningLevyDistance
References
Index...