機率論和隨機過程

內容簡介

《機率論和隨機過程(第2版)》是以作者在Princeton大學和Maryland大學的講義為藍本擴充而成，書中的內容正好可作為《機率論和隨機過程》課程一學年的獨立教材。這對於高年級的本科生、研究生和想要了解本科目基礎知識的科研人員都是相當有用的。全書文筆流暢，其中的概念和相關的結果都是生動豐富，並具有啟發性。每章末都包含難易不等的練習題。此書已經被作者用作Princeton大學和Maryland高年級本科生和研究生學習該科目的一學期的教程。目次：（第一部分）機率論：隨機變數及其分布；獨立試驗序列；勒貝格積分和數學期望；條件機率和期望；具有有限數狀態的馬爾科夫鏈；大數定理；測度的弱收斂；特徵函式；極限定理；幾個有趣的問題；（第二部分）隨機過程：基本概念；條件期望和鞅；有限狀態空間的馬爾科夫鏈；廣泛意義上的平穩隨機過程；嚴格平穩隨機過程；廣義隨機過程；布朗運動；馬爾科夫過程和馬爾科夫族；隨機積分和Ito公式；隨機微分方程；Gibbs隨機域。

圖書目錄

PartⅠProbabilityTheory
1RandomVariablesandTheirDistributions
1.1SpacesofElementaryOutcomes，a-Algebras，andMeasures
1.2ExpectationandVarianceofRandomVariablesonaDiscreteProbabilitySpace
1.3ProbabilityofaUnionofEvents
1.4EquivalentFormulationsofa-Additivity，Borela-AlgebrasandMeasurability
1.5DistributionFunctionsandDensities
1.6Problems
2SequencesofIndependentTrials
2.1LawofLargeNumbersandApplications
2.2deMoivre-LaplaceLimitTheoremandApplications
2.3PoissonLimitTheorem.
2.4Problems
3LebesgueIntegralandMathematicalExpectation
3.1DefinitionoftheLebesgueIntegral
3.2InducedMeasuresandDistributionFunctions
3.3TypesofMeasuresandDistributionFunctions
3.4RemarksontheConstructionoftheLebesgueMeasure
3.5ConvergenceofFunctions，TheirIntegrals，andtheFubiniTheorem
3.6SignedMeasuresandtheR，adon-NikodymTheorem
3.7LpSpaces
3.8MonteCarloMethod
3.9Problems
4ConditionalProbabilitiesandIndependence
4.1ConditionalProbabilities
4.2IndependenceofEvents，Algebras，andRandomVariables
4.3
4.4Problems
5MarkovChainswithaFiniteNumberofStates
5.1StochasticMatrices
5.2MarkovChains
5.3ErgodicandNon-ErgodicMarkovChains
5.4LawofLargeNumbersandtheEntropyofaMarkovChain
5.5ProductsofPositiveMatrices
5.6GeneralMarkovChainsandtheDoeblinCondition
5.7Problems
6RandomWalksontheLatticeZd
6.1RecurrentandTransientR，andomWalks
6.2RandomWalkonZandtheRefiectionPrinciple
6.3ArcsineLaw
6.4Gambler'sRuinProblem
6.5Problems
7LawsofLarzeNumbers
7.1Definitions，theBorel-CantelliLemmas，andtheKolmogorovInequality
7.2KolmogorovTheoremsontheStrongLawofLargeNumbers
7.3Problems
8WeakConveraenceofMeasures
8.1DefnitionofWeakConvergence
8.2WeakConvergenceandDistributionFunctions
8.3WeakCompactness，Tightness，andtheProkhorovTheorem
8.4Problems
9CharacteristicFunctions
9.1DefinitionandBasicProperties
9.2CharacteristicFunctionsandWeakConvergence
9.3GaussianRandomVectors
9.4Problems
10LimitTheorems
10.1CentralLimitTheorem，theLindebergCondition
10.2LocalLimitTheorem
10.3CentralLimitTheoremandRenormalizationGrOUDTheorv
10.4ProbabilitiesofLargeDeviations
……
PartⅡRandomProcesses
Index

機率論和隨機過程

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