Stochastic Processes (Advanced Probability II), 36-754

Spring 2006

See here for the current version of this course

MWF 10:30--11:20, in 232Q Baker Hall

Prof. Cosma Shalizi

Snapshot of a non-stationary spatiotemporal stochastic process (the Greenberg-Hastings model)

Stochastic processes are collections of interdependent random variables. This course is an advanced treatment of such 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 yet intuitive understanding of the material, rather than on analytic rigor.

The first part of the course will cover some foundational topics which belong in the toolkit of all mathematical scientists working with random processes: random functions; stationary processes; Markov processes; the Wiener process and the elements of stochastic calculus. These will be followed by a selection of more advanced and/or abstract topics which, while valuable, tend to be neglected in the graduate curriculum: ergodic theory, which extends the classical limit laws to dependent variables; the closely-related theory of Markov operators, including the stochastic behavior of deterministic dynamical systems (i.e., "chaos"); information theory, as it connects to statistical inference and to limiting distributions; large deviations theory, which gives rates of convergence in the limit laws; spatial and spatio-temporal processes; and, time permitting, predictive, Markovian representations of non-Markovian processes.

Prerequisites: As mentioned, measure-theoretic probability, at the level of 36-752, is essential. I will also presume some familiarity with basic stochastic processes, at the level of 36-703 ("Intermediate Probability"), though I will not assume those memories are very fresh.

Audience: The primary audience for the course are students of statistics, and mathematicians interested in stochastic processes. I hope, however, that it will be useful to any mathematical scientist who uses probabilistic models; in particular we will cover stuff which should help physicists interested in statistical mechanics or nonlinear dynamics, computer scientists interested in machine learning or information theory, engineers interested in communication, control, or signal processing, economists interested in evolutionary game theory or finance, population biologists, etc.

Grading: One to three problems will be assigned weekly. These will either be proofs, or simulation exercises. Students whose performance on homework is not adequate will have the opportunity to take an oral final exam in its place.


In HTML and PDF. Includes information on textbooks and a detailed outline.


For Wednesday, 22 February
David Pollard, "Asymptotics via Empirical Processes", Statistical Science 4 (1989): 341--354 [PDF, 2.2 M]
Note that Pollard's 1984 book, to which this paper makes some references, is available online.
For Wednesday, 12 April (optional)
E. B. Dynkin, "Sufficient Statistics and Extreme Points", The Annals of Probability 6 (1978): 705--730 [PDF, 2M]
Massimiliano Badino, "An Application of Information Theory to the Problem of the Scientific Experiment", Synthese 140 (2004): 355--389 = phil-sci/1830

Homework Assignments

Due in my mailbox at 5pm on the date stated, unless otherwise noted.
  1. Exercise 1.1 and Exercise 3.1 from the notes. Exercise 3.2 is not required. (27 January) Solutions
  2. Exercises 5.3, 6.1 and 6.2. (6 February) Solutions
  3. Exercises 10.1 and 10.2. (20 February) Solutions
  4. Exercises 16.1, 16.2 and 16.4 (13 March)

Lecture Notes in PDF

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