IMPORTANT. The best reference, and some of the homeworks, are from R. Durrett Probability: Theory and Examples 4th Edition. An online version is available here. But I strongly recommend you get an actual book version of one of this or an earlier edition.
Note: First class is Thursday August 23.
Instructor: David Aldous
Teaching Assistant: TBA
Class time: TuTh 12.30 - 2.00 in room 70 Evans
Optional Lab section: TBA
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.
In brief, the course will cover
| Week | start | topics | Billingsley | Durrett |
|---|---|---|---|---|
| 1 - 2 | Aug 23/28/30 | Fields, sigma-fields, measurable functions, measures, Lebesgue measure, distribution functions, coin-tossing, abstract integration | 2,10,13; 3,12,15 | 1.1, 1.2 |
| 3 | Sep 4/6 | Probability spaces, random variables, expectation, inequalities | 4,5,20 | 1.3 - 1.6 |
| 4 | Sep 11/13 | Independence, WLLN, Bernstein's theorem | 6,20 | 2.1, 2.2 |
| 5 | Sep 18/20 | Borel-Cantelli lemmas, 4'th moment SLLN, Glivenko-Cantelli, gambling on favorable game, a.s. limit theorems for maxima, 2nd moment SLLN, modes of convergence, dominated convergence, Fatou, | 6,21 | 2.3, 2.4 |
| 6 | Sept 25/27 | SLLNs, maximal inequality, convergence of random series, Renewal SLLN. | 22 | 2.4, 2.5 |
| 7 | Oct 2/4 | Stopping times, Wald's equation; Kolmogorov 0-1 law; large deviations | 22, 9 | 4.1, 2.6 |
| 8 | Oct 9/11 | Radon-Nikodym; joint distributions correspond to marginals and a kernel | 32, 33 | 5.1, A4 |
| 9 | Oct 16/18 | Product measure, Fubini's theorem and examples. Kolmogorov consistency theorem. Conditional expectation | 18, 34, 36 | 1.7, 5.1 |
| 10 | Oct 23/25 | Conditional expectation (continued). Definition and examples of martingales | 35 | 5.2 |
| 11 | Oct 30/Nov 1 | Convexity, optional sampling, MG analog of Wald, RW examples. | 35 | 5.3, 5.7 |
| 12 | Nov 6/8 | Maximal and upcrossing inequalities, MG convergence theorems, Levy 0-1 law, L_p convergence, Kakutani's theorem. | 35 | 5.3, 5.4, 5.5 |
| 13 | Nov 13/15 | Azuma's inequality; examples. Brownian motion. Existence and path continuity. | 35 | 5.3, 5.7, 8.1 |
| 14-15 | Nov 20/27/29 | Brownian motion. Invariance properties. Path non-differentiability. Associated martingales and their use in finding distributions, e.g. of hitting time for BM with drift. Reflection principle. | 37 | 8.1 - 8.5 |
| Take-home final. tentatively 2.00pm Thursday 11/29 -- 2.00pm Monday 12/3 |
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.
There are many other books at roughly the same ``first year graduate" level. Here are my personal comments on some.
D. Khoshnevisan Probability is a well-written concise account of the key topics in 205AB.
K.L. Chung A Course in Probability Theory covers many of the topics of 205A: more leisurely than Durrett and more focused than Billingsley.
D. Williams Probability with Martingales has a uniquely enthusiastic style; concise treatment emphasizes usefulness of martingales.
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.
O. Kallenberg Foundations of Modern Probability. Quoting an amazon.com reviewer: ``.... a compendium of all the relevant results of probability ..... similar in breadth and depth to Loeve's classical text of the mid 70's. It is not suited as a textbook, as it lacks the many examples that are needed to absorb the theory at a first pass. It works best as a reference book or a "second pass" textbook."
Jim Pitman has his very useful lecture notes linked to the Durrett text; these notes cover more ground than my course will! Also some lecture notes by Amir Dembo for the Stanford courses equivalent to our 205AB.
Here is an old page with extra problems, homework hints, etc
Grading 60% homework, 40% final.
GSI TBA
if you email us put "STAT 205A" in subject.