STAT 205B Home Page

STAT 205B: Probability Theory (Spring 2006)

Instructor David Aldous

Teaching Assistant Yun Long.

Class time TuTh 11.00 - 12.30 in room 332 Evans.

This is the second half of a year course in mathematical probability at the measure-theoretic level. It is designed for students whose ultimate research will involve rigorous proofs in mathematical probability. It is aimed at Ph.D. students in the Statistics and Mathematics Depts, but is also taken by Ph.D. students in Computer Science, Electrical Engineering, Business and Economics who expect their thesis work to involve probability.

In brief, the course will cover

Countable state Markov chains
Ergodic theory, emphasising applications of the subadditive ergodic theorem
Brownian motion
Extra (than 205A) aspects of martingales and convergence in distribution

This roughly coincides with Chapters 5, 6, 7 in Durrett's book. See week-by-week schedule for more details. and for the weekly homework assignments.

Prerequisites

Ideally

STAT 205A - familiarity with measure-theoretic approach to mathematical probability.
Undergraduate-level familiarity with Markov chains.
Upper division analysis, e.g. uniform convergence of functions, basics of complex numbers. Basic properties of metric spaces helpful.

Books

R. Durrett Probability: Theory and Examples (3rd Edition) is the required text, and the single most relevant text for the whole year's course. Quite a few of the homework problems are from there (note: 3rd edition). The style is deliberately concise. Here is a list of typos in the 2nd edition (PDF).

P. Billingsley Probability and Measure (3rd Edition) Chapters 25-30 make a nice treatment of the "convergence in distribution" part of the course.

There are many other books at roughly the same ``first year graduate" level. Here are my personal comments on some.

Y.S. Chow and H. Teicher Probability Theory. Uninspired exposition, but has useful variations on technical topics such as inequalities for sums and for martingales.

R.M. Dudley Real Analysis and Probability. Best account of the functional analysis and metric space background relevant for research in theoretical probability.

B. Fristedt and L. Gray A Modern Approach to Probability Theory. 700 pages allow coverage of broad range of topics in probability and stochastic processes.

L. Breiman Probability. Classical; concise and broad coverage.

Final

There will be a take-home final exam: assigned Thursday May 4, due Tuesday May 9.

Grading 60% homework, 40% take-home final.

Office Hours

David Aldous (aldous@stat) Wednesdays 9.30-11.30 in 351 Evans

Yun Long (yunlong@math) Mondays 10.00 - 12.00; Fridays 1.00-2.00; in room 1062 Evans.

If you email us, please put STAT 205B in subject.