## Announcements

From now on, classes will take place in WWH 1302.

## Grading

Problem sets: 50%

Final exam: 50%

## Schedule (tentative)

Notes (updated after class)

- Measure theory: Introduction, Caratheodory extension theorem (existence).
- Measure theory: Caratheodory extension theorem (uniqueness), Lebesgue's characterization for ℝ, Integration (random variables).
- Measure theory: Integration (convergence theorems).
- Measure theory: Transformations, product spaces.
- Measure theory: Distributions and expectations.
- Weak convergence: Characteristic functions, Lévy's theorem
- Weak convergence: Bochner's theorem.
- Independent sums: Convolutions, weak law of large numbers.
- Independent sums: Central limit theorem.
- Independent sums: Borel-Cantelli, 0-1 laws.
- Independent sums: Weak and strong law of large numbers.
- Independent sums: Accompanying laws and infinite divisibility.
- Dependent random variables: Conditioning.
- Dependent random variables: The Radon-Nikodym Theorem.
- Dependent random variables: Conditional expectation and conditional probability.
- Dependent random variables: Markov chains 1.
- Dependent random variables: Markov chains 2.
- Dependent random variables: Markov chains 3.
- Dependent random variables: Markov chains 4.
- Dependent random variables: Markov chains 5.
- Martingales 1.
- Martingales 2.
- Martingales 3.
- Martingales 4.
- Martingales 5.
- Stationary processes 1: ergodic theorems.
- Stationary processes 2: stationary measures.
- Stationary processes 3: the Markov case.

## Problem sets

- Homework 1 (due September 17)
- Homework 2 (due September 24)
- Homework 3 (due October 1)
- Homework 4 (due October 8)
- Homework 5 (due October 17)
- Homework 6 (due October 27)
- Homework 7 (due November 3)
- Homework 8 (due November 10)

## Textbook

- S.R.S. Varadhan, Probability Theory, Available online from NYU network here