## MAT 385 Probability Theory (Fall 2017)

**Instructor** Allan
Sly (asly@princeton.edu)

**Grader** Alexandros Eskenazis (ae3@math.princeton.edu)

**Syllabus** On blackboard or here.

**Class time** 1:30 - 2:50 on Mondays and Wednesdays at 601 Fine Hall.

**Office hours**

Instructor: Monday 10:30-12:00 or by arrangement. Room 405 Fine Hall.

Grader: Wednesday 4:30-5:30 or by arrangement. Room 504 Fine Hall.

**Course Description**

An introduction to probability theory. The course begins with the measure theoretic foundations of probability theory, expectation, distributions and limit theorems. Further topics include concentration of measure, Markov chains, martingales and Brownian motion.

### Grading

The grade for the course will be made up of problem sets (50%) and a final take home exam (50%) after scaling.

Probability is a subject where actively working on problems is essential to your understanding of the material, it is not enough simply to listen in class or read the text. There will be a weekly problem set which will be posted each Wednesday (starting September 20) on blackboard and will be due the following Wednesday. These can be done in groups of up to 3. You should list which students you worked with on the problem set. Each of you should write your own solutions separately - this is important to make sure you understand the solutions yourself. I also strongly encourage you to try the problems yourself first before working with your group.

There will be a take home final exam covering all the material of the course. This must be done individually and may be taken during any 48 hour period between January 17 and January 22.

## Text Books

We will use the text __Theory of Probability and Random Processes__ by Koralov and Sinai. You can access a free to download copy through the library website. We will cover most of chapters 1-10.

Other texts which may be useful

## Schedule

**Sep 13**: Introduction. Measures, sigma-algebras. KS 1.1-1.4, Durrett 1.1 [Notes]

**Sep 18**: Random variables. KS 1.5, Durrett 1.2, 1.3 [Notes]

**Sep 20**: Lesbegue Integration and Expected value KS 3.1, 3.2, 3.5 Durrett 1.4-1.6 [Notes]

**Sep 25 **: Independence KS 2.1 Durrett 2.1 [Notes]

**Sep 27 **: Limits of Random Variables and Law of large numbers KS 7.1, 7.2 Durrett 2.2, 2.3, 2.4 [Notes]

**Oct 2 **: Weak Convergence KS 8.1, 8.2 Durrett 3.1, 3.2 [Notes]

**Oct 4 **: Central Limit Theorem KS 9.1 Durrett 3.4

**Oct 9 **: Central Limit Theorem continued

**Oct 11 **: Markov Chains: Introduction and transition matrices KS 5.2 [Notes]

**Oct 16 **: Markov Chains: Recurrent and Transient Random Walks KS 6.1, 5.3 [Notes]

**Oct 18 **: Markov Chains: Convergence to stationary distributions KS 5.3 [Notes]

**Oct 23 **: Markov Chains: Reversibility, MCMC [Notes]

**Oct 25 **: Markov Chains: Card Shuffling [Notes]

**Nov 6 **: Conditional Expectations and Branching processes KS 13.1, 13.2 [Notes]

**Nov 8 **: Martigales KS 13.4 Durrett 5.2 [Notes]

**Nov 13 **: Martingale Convergence Theorem, Uniform Integrability KS 13.7 Durrett 5.5

**Nov 15 **: Optional Stopping Theorem, KS 13.5 Durrett 5.7 [Notes]

**Nov 20 **: Martingale Concentration: Azuma Hoeffding KS 13.5 Durrett 5.7

**Nov 27 **: Poisson Processes KS 12.3 Durrett 3.6 [Notes]

**Nov 29 **: Gaussian Vectors KS 8.3, 9.3 Durrett 3.9 [Notes]

**Dec 4 **: Multivariate CLT and Prohorov's Theorem KS 8.3, 9.3 Durrett 3.9 Durrett 8.1

**Dec 6 **: Brownian Motion KS 18.1-18.3 Durrett 8.1 [Notes]