Note: This is page from last year -- will be edited over the summer.
STAT 205A: Probability Theory (Fall 2012)Cross-listed also as Math 218A.
IMPORTANT. The best reference, and some of the homeworks, are from R. Durrett Probability: Theory and Examples 4th Edition.
Instructor: David Aldous Teaching Assistant (GSI): Jorge Banuelos Class time: TuTh 12.30 - 2.00 in room 70 Evans Optional Lab section /GSI Office Hours M 10.00-11.00am in 332 Evans and F 9.00 - 10.00am in 313 Evans. 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

• 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, 2, 5 and (first half of ) 8 in Durrett's book. Weekly schedule

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 [/td]
3	Sep 4/6	Probability spaces, random variables, expectation, inequalities	4,5,20	1.3 - 1.6 [/td]
4	Sep 11/13	Independence, WLLN, Bernstein's theorem, Borel-Cantelli lemmas, 4'th moment SLLN, Glivenko-Cantelli, gambling on favorable game.	6,20	2.1, 2.2 [/td]
5	Sep 18/20	a.s. limit theorems for maxima, 2nd moment SLLN, modes of convergence, dominated convergence, maximal inequality, convergence of random series, 1st moment SLLN	6,21	2.3, 2.4 [/td]
6	Sept 25/27	variant SLLNs, Fatou, Renewal SLLN. Stopping times, Wald's equation, Kolmogorov 0-1 law; Radon-Nikodym; Cantor measure, decomposition of measures on R.	22	2.4, 2.5 [/td]
7	Oct 2	Large deviations	22, 9	4.1, 2.6 [/td]
7/8	Oct 4/9/11	joint distributions correspond to marginals and a kernel. Product measure, Fubini's theorem and examples. Kolmogorov consistency theorem.	32, 33	5.1, A4[/td]
9	Oct 16/18	Conditional expectation. Definition and examples of martingales. Convexity.	18, 34, 36	1.7, 5.1[/td]
10	Oct 23/25	Doob decomposition, martingale transforms, stopping times, bounded version of Optional Sampling Theorem. Maximal and upcrossing inequalities.	35	5.2 [/td]
11	Oct 30/Nov 1	MG convergence theorems, Levy 0-1 law, L_p convergence, conditional Borel-Cantelli, Kakutani's theorem,	35	5.3, 5.7[/td]
12	Nov 6/8	[prediction markets], regular conditional distributions, general form of optional sampling, MG analog of Wald. Boundary crossing examples. Galton-Watson processes. Patterns in coin-tossing.	35	5.3, 5.4, 5.5[/td]
13	Nov 13/15	Azuma's inequality; examples. Reversed MGs and SLLN [briefly]. Brownian motion. Existence and path continuity. Invariance properties.	35	5.3, 5.7, 8.1[/td]
14-15	Nov 20/27/29	Brownian motion. Path non-differentiability. Associated martingales and their use in finding distributions, e.g. of hitting time for BM with drift. Reflection principle and formulas derived from it. Distribution of last zero before/first zero after t=1. Mention bridge, excursion, meander.	37	8.1 - 8.5[/td]
		Take-home final. 2.00pm Thursday 11/29 -- 2.00pm Monday 12/3

PrerequisitesIdeally

• 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.
  

If you haven't seen any measure theory it is helpful to read a little before the start of the course.
Note: There is a parallel first year graduate course in probability theory, STAT 204, taught by Allan Sly, which does not use measure theory. BooksR. Durrett Probability: Theory and Examples 4th edition. is the required text, and the single most relevant text for the whole year's course. The style is deliberately concise. Quite a few of the homework problems are from there,
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."
John B. Walsh Knowing the Odds: An Introduction to Probability. New in 2012. Looks very nice -- concise treatment with quite challenging exercises developing part of theory.
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.
HOMEWORKHere are the 11 weekly homework assignments, due in class on Tuesdays. You can pick up the graded homeworks at GSI office hours.
FinalThere will be a take-home final exam: dates TBA.
Grading 60% homework, 40% final.
Office HoursDavid Aldous (aldous@stat) Thursdays 2.15 - 3.45 in 351 Evans
GSI M 10.00-11.00am in 332 Evans and F 9.00 - 10.00am in 313 Evans.
if you email us put "STAT 205A" in subject.