Department of Statistics

Recent Instructors	ST 779	Measure Theory and Advanced Probability
	Course Description Classes of events, random variables, probability measures, integration and expectation, inequalities, Lp-spaces, product spaces, independence, zero-one laws, convergence notions, characteristic function, simplest limit theorems, absolute continuity, conditional expectation and conditional probability, martingales. Course Syllabus Sets, functions, classes, countability. Events, semifields, fields, sigma-fields, generators, Borel sigma-field, good sets principle, Dynkin's theorems. Random variables, measurability, simple functions. Probability measures, Caratheodory extension theorem, Lebesgue measure. Expectation, Lebesgue integral, monotone convergence theorem, Fatou's lemma, dominated convergence theorem, change of variable rule, uniform integrability Inequalities: Cauchy-Schwarz , Holder, Markov, Chebyshev, Minkowski, moment, Jensen. Lp-spaces. Product spaces, product measure, joint measurability, Fubini's theorem, finite products. Infinite product spaces, measurable rectangles and measurable cylinders, transition probability, Ionescu-Tulceea theorem, uncountable product space, Kolmogorov consistency theorem. Independence, independent random variables. Zero-one laws: Kolmogorov and Hewitt-Savage, Borel-Cantelli lemmas, tail events, symmetric events. Stochastic convergences, almost sure convergence, convergence in probability, convergence in distribution, connection between convergence notions, Slutsky's theorem. Characteristic function, relation between moments and differentiability, inversion and continuity theorems, duality. Simplest limit theorems, weak law, strong law and central limit theorem for i.i.d. random variables Absolute continuity, substitution theorem, Radon--Nikodym theorem, likelihood. Conditional expectation, regular conditional distribution. Martingales (sub-, super- and reverse), Doob's maximal inequality, martingale convergence theorem, Lp-convergence and uniform integrability, stopping time, optional sampling theorem, decompositions. Course Prerequisites MA 425 Course Corequisites None Recent Textbooks No required text. Course notes are given prior to each lecture. A suggested reference is Real Analysis and Probability by Ash.