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Unformatted text preview: Syllabus for Advanced Probability II, Stochastic Processes 36-754 Cosma Shalizi Spring 2006 This course is an advanced treatment of interdependent random variables and random functions, with twin emphases on extending the limit theorems of probability from independent to dependent variables, and on generalizing dynamical systems from deterministic to random time evolution. Familiarity with measure-theoretic probability (at the level of 36-752) is essential, but the emphasis will be on developing a sound understanding of the material, rather than on mathematical rigor. The first part of the course will cover topics making up the basic toolkit of stochastic processes: random functions; stationary processes; Markov pro- cesses; the Wiener process; and the elements of stochastic calculus. These will be followed by slightly more unusual topics: ergodic theory, which extends the classical limit laws to dependent variables; the closely-related theory of Markov operators, including the stochastic behavior of deterministic dynami- cal systems (i.e., “chaos”); information theory, as it connects to statistical infer- ence and to limiting distributions; large deviations theory, which gives rates of convergence in the limit laws; and multi-parameter (spatial or spatio-temporal) proccesses. PREREQUISITES: Measure-theoretic probability, at the level of 36-752, is es- sential. So is familiarity with stochastic processes in elementary or intermedi- ate probability. GRADING: One to three problems will be assigned weekly. Some will be proofs, others simulation exercises based on the theorems. Students whose performance on homework is not adequate will have the opportunity to take an oral final exam in its place. TEXTS The primary text will be the lecture notes. Kallenberg’s Foundations of Modern Probability (2nd ed.) will also be used, especially as a source of prob- lems. (It’s on order at the CMU bookstore, but often cheaper online.) Students who took 36-752 last semester, when it used Ash and Doleans-Dade, will also find the last two chapters helpful....
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- Spring '06
- Probability, Probability theory, Wiener process, large deviations, ergodic decompositions