## STAT 205A (= MATH 218A): Probability Theory (Fall 2016)

Homework solutions now posted -- see below.

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): Wenpin Tang (also assisted by Raj Agrawal)

Class time: TuTh 11.00 - 12.30 in room 88 Dwinelle.

Optional Lab section: Mondays 10.00 - 11.00 in room 334 Evans (starting 12 September).

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.

## Lecture notes

Sinho Chewi has kindly agreed to post his lecture notes here.

## Weekly schedule

WeekdatestopicsBillingsleyDurrett
1/2Aug 25, 30, Sep 1 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 6/8 Probability spaces, random variables, expectation, inequalities 4,5,20 1.3 - 1.6
4 Sep 13/15 Independence, WLLN, Bernstein's theorem, Borel-Cantelli lemmas, 4'th moment SLLN, Glivenko-Cantelli, gambling on favorable game. 6,20 2.1, 2.2
5 Sep 20/22 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
6 Sep 27/29 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
7 Oct 4 Large deviations 22, 9 4.1, 2.6
7/8 Oct 6/11/13 joint distributions correspond to marginals and a kernel. Product measure, Fubini's theorem and examples. Kolmogorov consistency theorem. 32, 33 5.1, A4
9Oct 18/20 Conditional expectation. Some real data instances of theory. Definition and examples of martingales. Convexity. 18, 34, 36 1.7, 5.1
10 Oct 25/27 Doob decomposition, martingale transforms, stopping times, bounded version of Optional Sampling Theorem. Maximal and upcrossing inequalities. 35 5.2
11Nov 1/3 MG convergence theorems, Levy 0-1 law, L^p convergence, conditional Borel-Cantelli, Kakutani's theorem, general form of optional sampling, MG analog of Wald. 35 5.3, 5.7
12/13 Nov 8/10/15/17 Boundary crossing examples. Patterns in coin-tossing. MG proof of Radon-Nikodym theorem. Azuma's inequality; examples. Reversed MGs and SLLN. Exchangeability and de Finetti's theorem. Galton-Watson processes. L^2 theory. 35 5.3, 5.4, 5.5, 5.7
14/15Nov 22/29, Dec 1 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. Reflection principle and formulas derived from it. Mention bridge, excursion, meander. 37 8.1 - 8.5
Take-home final. 12.30pm Thursday 12/1 -- 12.30pm Monday 12/5

## 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.
If you haven't seen any measure theory it is helpful to read a little before the start of the course, for instance from the Billingsley or Leadbetter et al books below.

## 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. 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 some of the topics of this semester, in particular the measure-theory background. Recommended for students who have not done measure theory.

R. Leadbetter et al A Basic Course in Measure and Probability: Theory for Applications is a new book giving a careful treatment of the measure-theory background.

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.

R. Bhattacharya and E. C. Waymire A Basic Course in Probability Theory is another well-written account, mostly on the 205A topics.

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: Independence, Interchangeability, Martingales . 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.

George Roussas An Introduction to Measure-Theoretic Probability. Recent treatment of classical content.

Santosh Venkatesh The Theory of Probability: Explorations and Applications. Unique new book, intertwining a broad range of undergraduate and graduate-level topics for an applied audience.

I. Florescu Probability and Stochastic Processes. Very clearly written, and with 550 pages gives a broad coverage of topics including intro to SDEs.

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.

## HOMEWORK

Here are the 11 weekly homework assignments, due in class on Tuesdays. You can pick up the graded homeworks at GSI office hours.

## Final

There will be a take-home final exam .