NEWS (4/30) Graded homeworks can be picked up from the GSI's office, 5.00-6.00pm either Monday 30 April or Tuesday 1 May.

## STAT C205B/MATH C218B : Probability Theory (Spring 2018)

Instructor: David Aldous

GSI: Sourav Sarkar

Class time: TuTh 12.30 - 2.00 in room 344 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 6, 7, 8 in Durrett's book.

## Lecture notes

Sinho Chewi has kindly provided lecture notes from my 2017 205B course and lecture notes from my 2016 205A course.

## Weekly schedule Spring 2018

WeekdatestopicsDurrett sections
1 Jan 16 Recap of Measure theory as used in Probability theory. And link to Tao's three stages. Chap. 1, Appendix A, sec 2.1.4
1 Jan 18 Joint distributions correspond to marginal and a kernel. 5.1.3
2 Jan 23 Conditional distributions and conditional expectation. The two views of conditional independence. Kolmogorov extension theorem. 2.1.4,
2 Jan 25 Markov chains: Strong Markov, hitting times and generating function identities. Examples. 6.1, 6.2
3 Jan 30 Classification of states; recurrence and transience. 6.3, 6.4
3 Feb 1 Invariant measures and stationary distributions. Some old lecture notes 6.5
4 Feb 6 Existence of, and convergence to, stationary distributions. 6.5
4 Feb 8 Examples of coupling bounds. The Markov chain ergodic theorem. 6.6
5 Feb 13 The fundamental matrix. Hitting time formulas via the occupation time identity. Asymptotic variance rates. Sections 2.1-2.3 of Aldous-Fill
5 Feb 15 Martingale methods for Markov chains. 6.4
6 Feb 20 Metropolis algorithm and rejection sampling; general state spaces and Harris chains. 6.8
6 Feb 22 Iterated random functions and coupling from the past; continuous-time reversible chains. See Diaconis-Freedman and section 3.6 of Aldous-Fill.
7 Feb 27/ Mar 1 Overview of weak convergence in metric spaces. Prohorov's theorem and indirect proof via characterization of limit. C[0,1] and D[0,1] and tightness. Examples of convergence of random "objects". Billingsley Convergence of Probability Measures
8 Mar 6/8 Ergodic theorem; applications to RW. 7.1, 7.2, 7.3
9Mar 13/15 Entropy and Shannon-Breiman-McMillan theorem; subadditive ergodic theorem and applications. 7.4, 7.5
10 Mar 20/22 Brownian motion. Existence and path continuity. Invariance properties. Path non-differentiability. Associated martingales and their use in finding distributions, e.g. of hitting time for BM with drift. 8.1, 8.5
Spring break
11 Apr 3/5 Reflection principle and formulas derived from it. Mention bridge, excursion, meander. BM as a Gaussian process. Law of iterated logarithm. Skorokhod embedding. 8.4
12 Apr 10 Donsker's invariance principle and applications. 8.6
12 Apr 12NO CLASS
13 Apr 17/19 The three arc sine laws. Martingale central limit theorem via Brownian embedding. Local time and its relevance. 8.4; Morters-Peres Chap. 6.
14 Apr 24 Levy's theorem. Absolute maximum of Brownian bridge and the Kolmogorov-Smirnov limit. 8.7; Morters-Peres Chap. 6.
14 Apr 26 de Finetti's theorem and representation of exchangeable arrays. Tim Austin's Exchangeable random arrays notes.
15+ May 3 - 7 Take-home final exam, given 12.30 Thursday May 3, due 12.30 Monday May 7.

## Homeworks

Here are the weekly homework assignments due in class on Tuesdays. Solutions also will be posted there You can pick up graded work at the GSI office hours.

## 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 (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.

## Final

There will be a take-home final exam: May 3 - 7.

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

## Office Hours

David Aldous (aldous@stat.berkeley.edu): Thursday 2.30 - 4.00 in 351 Evans.

Sourav Sarkar (souravs@berkeley.edu): Monday 10.00 - 11.00am and Wednesday 4.00 - 5.00pm on room 444 Evans.

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