Department of Statistics

Recent Instructors Bhattacharyya, Bibhuti	ST 778	Measure Theory and Advanced Probability - I
	Course Description Modern measure and integration theory in abstract spaces. Probability measures, random variables, expectations. Distributions and characteristic functions. Modes of convergence. Independence, zero-one laws, laws of large numbers, three-series theorem. Central limit problem. Conditional expectations, martingales and martingale convergence theorems. Course Syllabus Sigma fields and measures: Review of sets, Classes and functions, Sequences and limits, Fields and sigma-fields, Mappings and inverse mappings, measures, Measurable functions, Simple functions, Random variables. Different kinds of measures: Lebesgue measure, Counting measure, Probability measure, Signed measure, Extensions of measures from fields to sigma-fields, Outer measure. Integration theory and convergence in abstract space: Integration of simple functions, Non-negative functions and Borel measurable functions, Expectations, Relationship between Riemann and Lebesgue integration, Monotone convergence theorem (MCT), Dominated convergence theorem (DCT), Fatou's lemma, Uniform integrability. Absolute continuity and derivatives: Absolute continuity, Singularity and statement of Radon-Nikodym theorem, Likelihood ratios, Substitution theorem. Product measure and conditional expectation: Fubini's theorem, Construction of probability measures in general function space, Kakutani theorem, Conditional expectation and basic properties, Conditional probability. Function spaces and inequalities: Lp-spaces, Hellinger distance, Kullback-Liebler discrepancy, Holder's and Minkowski inequalities, Moment inequalities, Jensen's inequality. Convergence concepts: Sequence of random variables, Almost everywhere convergence, almost uniformly convergence, Lp in measure, Relationships among the various notions of convergence, Strong law of large numbers, Central limit theorem. Course Prerequisites MA 426; ST 521 or MA(ST) 546 or equivalent 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.