### Section 600, Fall 2014, Prof. D. Cline

Welcome!  This page announces and describes my course for the Fall 2014 semester.

This is a course in probability at the measure theoretic level, with emphasis both on understanding measure theory and its relationship to probability and on understanding the crucial roles that probability plays for statistics.  Topics include probability measures, Lebesgue-Stieltjes integration, sigma-fields, random variables, expectation, moment inequalities, independence, convergence of random variables and sample moments, characteristic functions, convergence of distributions, the central limit theorem and the delta method, and conditional expectation.  The intention is to lay a foundation of theory that will enhance the student's ability to read advanced probability and statistics literature and to write a dissertation.

For comments or questions, e-mail me (dcline@stat.tamu.edu) or contact the TAMU Statistics Department.

PDF file of the Syllabus (updated 08/13/14)

### Course Outline

 Topic Textbook Section 1. Events and Classes of Events outcomes, events, review of set theory 1.2 limits of events 1.3, 1.4 π-classes, fields and σ-fields 1.5, 1.6 Borel σ-fields 1.7, 1.8 Dynkin's π-λ class theorem 2.2 2. Probability Measures and Measures probability measures and general measures 2.1 – 2.3 properties of measures, probability distributions, uniqueness 2.1 distribution functions, Lebesgue-Stieltjes measures extension and existence theorems 2.4, 2.5 3. Random Variables and Measurability random variables and measurable functions, σ-field generated by a random variable 3.1 – 3.3 induced measures, distribution of a random variable 3.2 sufficient conditions for measurability 3.2, 5.1 4. Expectation and Integration definitions, consistency of definitions 5.1, 5.2, 5.4 properties of expectation and integrals 5.2 Lebesgue-Stieltjes integration, absolute continuity, Riemann integrals 5.6 monotone and dominated convergence theorems, extensions 5.2, 5.3 5. Advanced Integration equating differently defined integrals 5.5, 5.6 moments and inequalities, finiteness of moments 5.2 product spaces, multiple integration, Tonelli-Fubini theorem 5.7 – 5.9 characteristic functions, inversion formulas 9.1 – 9.5 6. Independence independence of finitely many events or random variables 4.1, 4.2 infinite collections of independent events or random variables 4.3, 4.4 Borel-Cantelli lemmas, tail events, Kolmogorov's 0-1 law 4.5 7. Convergence of Sequences of Random Variables modes of convergence and their relationships 6.1 – 6.3, 6.5 moment inequalities, Jensen's inequality 6.5 uniform integrability, convergence in mean 6.5, 6.6 8. The Law of Large Numbers and Convergence of Random Series strong law of large numbers 7.1, 7.4 Glivenko-Cantelli theorem 7.5 convergence of series 7.3, 7.6 9. Convergence of Distributions weak convergence, Scheffé's theorem 8.1, 8.2, 8.5 Slutsky's theorem, delta method, Skorohod's lemma, continuous mapping 8.3, 8.6 tightness, Prohorov's theorem, the continuity theorem 9.5, 9.6 portmanteau theorem, multivariate convergence 8.4 10. Weak Convergence for Sums and Maxima central limit theorem, Lindeberg-Feller and Lyapunov theorems 9.7, 9.8 convergence to types, infinitely divisible and stable distributions 8.7 extreme value distributions 8.7 11. Absolute Continuity and Conditional Expectation conditional expectation 10.2, 10.3 regular conditional distributions conditional independence and exchangeablilty absolute continuity, Radon-Nikodym theorem, Lebesgue decomposition 10.1

### Lecture Notes (PDF) (Login required)

 Lecture Notes: The updated lecture notes will be posted here, as they are ready.

### Homework Assignments (PDF) (Login required)

 Assignments: Assignments will be posted regularly here.  Please see the homework policy above.  Partial solutions are meant to provide the main ideas, but may not include all necessary details or full justification. They may suggest, however, shortcuts or notational devices that could improve presentation. For suggestions about what makes a good presentation, please see these homework pointers.  Also, here is a list of questions useful to keep in mind when developing and critiquing your solutions. Ask them habitually of yourself and you will be more likely to avoid common pitfalls.