STAT 205A Home Page

STAT 205A: Probability Theory (Fall 2004)

Cross-listed also as Math 218A.

Shankar's office hours have changed -- see bottom of page

Instructor David Aldous

Teaching Assistant Shankar Bhamidi

Class time MWF 2.00 - 3.00 in room 106 Moffitt

This is the first 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.

NEW!!! Shankar has a page with extra problems, homework hints, etc

Note There is a parallel first year graduate course in probability theory, STAT 204 taught by Evans, which does not have a measure theory prerequisite.

In brief, the course will cover

Sketch of pure measure theory (not responsible for proofs)
Measure-theoretic formulation of probability theory
Classical theory of sums of independent random variables: laws of large numbers
Technical topics relating to proofs of above: notions of convergence, a.s. convergence techniques
Conditional distributions, conditional expectation
Discrete time martingales
Introduction to Brownian motion

This roughly coincides with Chapters 1, 4 and (first half of ) 7 in Durrett's book. See week-by-week schedule for more details. and for the weekly homework assignments.

Prerequisites

Ideally

Upper division probability - familiarity with calculations using random variables.
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 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. The new 3rd edition corrects typos from the 2nd edition; either will be OK to use. The style is deliberately concise.

P. Billingsley Probability and Measure (3rd Edition) Chapters 1-30 contain a more careful and detailed treatment of the topics of this semester, in particular the measure-theory background. Recommended for students who have not done measure theory.

K.L. Chung A Course in Probability Theory covers many of the topics of 205A: more leisurely than Durrett and more focused than Billingsley.

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.

There are some lecture notes for Jim Pitman's Fall 2002 STAT 205A which covers more ground than my course will! Also some lecture notes by Amir Dembo (Stanford) covering Chapters 1-2 of Durrett, and the relevant measure theory.

Final

There will be a take-home final exam: tentatively December 10 - December 14.

Grading 60% homework, 40% final.

Office Hours

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

Shankar Bhamidi (shanky@stat) Tuesday 9.00 -- 11.00, Thursday 9.00 - 10.00, in 307 Evans.

if you email us put "STAT 205A" in subject.