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Zusatztext This textbook is suitable for a one-semester course on measure theory and probability for beginning graduate students in mathematics! probability and statistics. It can also be used as a textbook for advanced undergraduate students in mathematics ? The topics are well selected to meet the needs of students who are interested in graduate studies in areas related to analysis! probability! stochastic processes and statistics ? This makes the book student-friendly. A motivated student can use it by him- or herself to learn the topics well. -Yimin Xiao! Mathematical Reviews! 2010 Informationen zum Autor Siva Athreya Klappentext Covering the fundamentals of measure theory and probability theory! this title begins with the construction of Lebesgue measure via Caratheodory's outer measure approach and goes on to discuss integration and standard convergence theorems. Zusammenfassung Covers the fundamentals of measure theory and probability theory. This title begins with the construction of Lebesgue measure via Caratheodory's outer measure approach and goes on to discuss integration and standard convergence theorems. It also discusses discrete time Markov chains, stationary distributions, and limit theorems. Inhaltsverzeichnis Probabilities and MeasuresIntroduction?-algebras as eventsAlgebras! monotone classes! etc.Preliminaries on measuresOuter measures and Caratheodory extensionLebesgue measureRegularityBernoulli trials IntegrationMeasurable functionsIntegrationa.e. considerationsRandom VariablesDistribution and expectationIndependent events and tail ?-algebraSome distributionsConditional expectationProbability Measures on Product SpacesProduct measuresJoint distribution and independenceProbability measures on infinite product spacesKolmogorov consistency theorem Characteristics and ConvergencesCharacteristic functionsModes of convergenceCentral limit theoremLaw of large numbersMarkov ChainsDiscrete time MCExamplesClassification of statesStrong Markov propertyStationary distributionLimit theoremsSome AnalysisComplex measuresLp spacesRadon-Nikodym theoremChange of variablesDifferentiationThe Riesz representation theoremAppendixMetric spacesTopological spacesCompactnessThe Stone-Weierstrass theoremTablesReferencesIndex ...
