这是一本概率论的经典教材，内容很详尽，正如作者所说的，他很不喜欢大多数书上都会写得一句话：显而易见。

[此贴子已经被wesker1999于2009-2-21 17:24:54编辑过]

 

 

 

Preface .................................................. V
OutlineofContents ...................................... XVII
NotationandSymbols ...................................XXI
1IntroductoryMeasureTheory ............................ 1
1ProbabilityTheory:AnIntroduction ......................1
2BasicsfromMeasureTheory.............................2
2.1Sets..............................................3
2.2CollectionsofSets..................................5
2.3Generators ........................................7
2.4AMetatheoremandSomeConsequences ..............9
3TheProbabilitySpace ...................................10
3.1LimitsandCompleteness............................11
3.2AnApproximationLemma..........................13
3.3TheBorelSetson R ................................14
3.4TheBorelSetson Rn ...............................16
4Independence;ConditionalProbabilities ...................16
4.1TheLawofTotalProbability;Bayes’Formula .........17
4.2IndependenceofCollectionsofEvents.................18
4.3Pair-wiseIndependence.............................19
5TheKolmogorovZero-oneLaw ...........................20
6Problems..............................................22
2RandomVariables ....................................... 25
1DefinitionandBasicProperties...........................25
1.1FunctionsofRandomVariables......................28
2Distributions...........................................30
2.1DistributionFunctions..............................30
2.2Integration:APreview..............................32
2.3DecompositionofDistributions.......................36
2.4SomeStandardDiscreteDistributions.................39
2.5SomeStandardAbsolutelyContinuousDistributions....40
2.6TheCantorDistribution............................40
2.7TwoPerverseExamples.............................42
3RandomVectors;RandomElements.......................43
3.1RandomVectors...................................43
3.2RandomElements..................................45
4Expectation;DefinitionsandBasics .......................46
4.1Definitions........................................46
4.2BasicProperties....................................48
5Expectation;Convergence ................................54
6IndefiniteExpectations ..................................58
7AChangeofVariablesFormula...........................60
8Moments,Mean,Variance................................62
9ProductSpaces;Fubini’sTheorem ........................64
9.1Finite-dimensionalProductMeasures.................64
9.2Fubini’sTheorem..................................65
9.3PartialIntegration..................................66
9.4TheConvolutionFormula...........................67
10Independence..........................................68
10.1IndependenceofFunctionsofRandomVariables........71
10.2Independenceof σ-Algebras.........................71
10.3Pair-wiseIndependence.............................71
10.4TheKolmogorovZero-oneLawRevisited ..............72
11TheCantorDistribution.................................73
12TailProbabilitiesandMoments ...........................74
13ConditionalDistributions................................79
14DistributionswithRandomParameters....................81
15SumsofaRandomNumberofRandomVariables...........83
15.1Applications.......................................85
16RandomWalks;RenewalTheory..........................88
16.1RandomWalks.....................................88
16.2RenewalTheory....................................89
16.3RenewalTheoryforRandomWalks...................90
16.4TheLikelihoodRatioTest ...........................91
16.5SequentialAnalysis.................................91
16.6ReplacementBasedonAge..........................92
17Extremes;Records......................................93
17.1Extremes..........................................93
17.2Records...........................................93
18Borel-CantelliLemmas..................................96
18.1TheBorel-CantelliLemmas1and2..................96
18.2Some(Very)ElementaryExamples...................98
18.3Records...........................................101
18.4RecurrenceandTransienceofSimpleRandomWalks...102
18.5

∞
n=1 P(An)= ∞ and P(An i.o.)=0.................104
18.6Pair-wiseIndependence.............................104
18.7GeneralizationsWithoutIndependence................105
18.8Extremes..........................................107
18.9FurtherGeneralizations.............................109
19AConvolutionTable....................................113
20Problems..............................................114
3Inequalities .............................................. 119
1TailProbabilitiesEstimatedviaMoments ..................119
2MomentInequalities....................................127
3Covariance;Correlation..................................130
4Interludeon Lp-spaces...................................131
5Convexity..............................................132
6Symmetrization.........................................133
7ProbabilityInequalitiesforMaxima .......................138
8TheMarcinkiewics-ZygmundInequalities..................146
9Rosenthal’sInequality...................................151
10Problems..............................................153
4CharacteristicFunctions ................................. 157
1DefinitionandBasics....................................157
1.1Uniqueness;Inversion ...............................159
1.2Multiplication.....................................164
1.3SomeFurtherResults...............................165
2SomeSpecialExamples..................................166
2.1TheCantorDistribution............................166
2.2TheConvolutionTableRevisited.....................168
2.3TheCauchyDistribution............................170
2.4SymmetricStableDistributions......................171
2.5Parseval’sRelation.................................172
3TwoSurprises..........................................173
4Refinements............................................175
5CharacteristicFunctionsofRandomVectors................180
5.1TheMultivariateNormalDistribution................180
5.2TheMeanandtheSampleVarianceAreIndependent...183
6TheCumulantGeneratingFunction.......................184
7TheProbabilityGeneratingFunction ......................186
7.1RandomVectors...................................188
8TheMomentGeneratingFunction........................189