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1、SIT?TVERUILNMU··SCDOINENADROUC··CODODNEMarkovChainsandMonte–CarloSimulationUlmUniversityInstituteofStochasticsLectureNotesProf.Dr.VolkerSchmidtSummer2010Ulm,July2010CONTENTS2Contents1Introduction42MarkovChains52.1Speci?cationoftheModelandExamples.................................52.1
2、.1StateSpace,InitialDistributionandTransitionProbabilities.................52.1.2Examples..............................................82.1.3RecursiveRepresentation.....................................102.1.4TheMatrixofthen–StepTransitionProbabilities.......................122.2Ergo
3、dicityandStationarity........................................162.2.1BasicDe?nitionsandQuasi-positiveTransitionMatrices....................162.2.2EstimatesfortheRateofConvergence;Perron–Frobenius–Theorem.............202.2.3IrreducibleandAperiodicMarkovChains..........................
4、..242.2.4StationaryInitialDistributions..................................302.2.5DirectandIterativeComputationMethods...........................352.3Reversibility;EstimatesfortheRateofConvergence..........................382.3.1De?nitionandExamples....................................
5、..382.3.2RecursiveConstructionofthe?Past”...............................422.3.3DeterminingtheRateofConvergenceunderReversibility...................432.3.4MultiplicativeReversibleVersionoftheTransitionMatrix;SpectralRepresentation.....452.3.5AlternativeEstimatefortheRateofConvergence
6、;?2-Contrast................462.3.6Dirichlet–FormsandRayleigh–Theorem.............................502.3.7BoundsfortheEigenvalues?2and?`..............................533Monte–CarloSimulation583.1GenerationofPseudo-RandomNumbers.................................593.1.1SimpleApplication
7、s;Monte–CarloEstimators..........................593.1.2LinearCongruentialGenerators..................................613.1.3StatisticalTests...........................................633.2TransformationofUniformlyDistributedRandomNumbers......................683.2.1InversionMethod.
8、...................
