THIS IS THE PAGE FROM SPRING 2017 -- will be updated for SPRING 2018. The topics will change, so they do not repeat 205A topics.

STAT C205B/MATH C218B : Probability Theory (Spring 2017)

Instructor: David Aldous

GSI: Zsolt Bartha

Class time: TuTh 12.30 - 2.00 in room 330 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

This roughly coincides with Chapters 3, 6, 7, 8 in Durrett's book.

Lecture notes

Sinho Chewi has kindly provided these lecture notes.

Weekly schedule Spring 2017

WeekdatestopicsDurrett sections
1 Jan 17/19 Convergence in distribution, Elementary examples. Tightness, Helly selection theorem. 3.2
2 Jan 24/26 Characteristic functions, inversion, dual pairs, Parseval identity, continuity theorem. 3.3
3 Jan 31/Feb 2 Central limit theorems. Poisson limits; method of moments; 3.4, 3.6
4 Feb 7/9 Overview of weak convergence in metric spaces. Weak convergence and CLT in R^d. 3.9
5Feb 14/16Measure-theory background to Markov chains. Examples and elementary properties. 6.1, 6.2
6 Feb 21/23Markov chains. Strong Markov property. Classification of states; recurrence and transience. 6.3, 6.4
7 Feb 28/Mar 2 Existence and convergence results for invariant measures.6.5, 6.6
8 Mar 7/9 Coupling. a.s. ergodic theorem; mixing times and coupling.6.5, 6.6
9 Mar 14/16 Markov chains and martingales; general state spaces and harris chains; iterated function systems. 6.8
10 Mar 21/23 Ergodic theorem; applications to RW. 7.1, 7.2, 7.3
Spring break
11Apr 4/6 Entropy; subadditive ergodic theorem and applications. 7.4, 7.5
12 Apr 11/13 Brownian motion. Law of iterated logarithm. Skorokhod embedding. Donsker's invariance principle and applications. 8.6, 8.8
13 Apr 18/20 . . . . . . . . . . . . No classes this week . . . . . . . . . .
14 Apr 25/27The three arc sine laws. Martingale central limit theorem via Brownian embedding. Local time and its relevance. 8.6
15 May 4 - 8 Take-home final exam, given 12.30 Thursday May 4, due 12.30 Monday May 8.


Here are the weekly homework assignments due in class on Tuesdays. First homework due Tuesday January 31. Solutions posted here.




R. Durrett Probability: Theory and Examples (4th Edition) is the required text, and the single most relevant text for the whole year's course. Most of the homework problems are from there (note: 4th edition). The style is deliberately concise.

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

P. Billingsley Probability and Measure (3rd Edition) Chapters 25-30 make a nice treatment of the "convergence in distribution" part of 205B.

J.R. Norris Markov Chains is a more leisurely account.

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.

R. Bhattacharya and E. C. Waymire Stochastic Processes with Applications gives a broader account of Markov chains, Brownian motion and diffusions, downplaying measure theory.

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

Jim Pitman has his very useful lecture notes linked to the Durrett text (note 3rd edition -- chapter numbers have changed); these notes cover more ground than my course will! Also some lecture notes by Amir Dembo for the Stanford courses equivalent to our 205AB.


There will be a take-home final exam: May 4 - 8.

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

Office Hours

David Aldous ( Thursday 2.30 - 4.00 in 351 Evans.

Zsolt Bartha ( Mondays 2.00 - 3.00pm in 428 Evans.

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