Math 563 - Probability - Topics

The content will be roughly the unstarred sections in the first three chapters of Durrett's book, and a few of the starred sections. Then I plan to cover conditional expectation (section 4.1 in Durrett). These topics will be the bulk of the course. If time permits I will give a very quick overview of martingales and Markov processes. And if time still permits I will end with a look at Brownian motion. Here is a list of the sections in Durrent I plan to cover:

1 Measure Theory
1.1 Probability Spaces
1.2 Distributions
1.3 Random Variables
1.4 Integration
1.5 Properties of the Integral
1.6 Expected Value
1.6.1 Inequalities
1.6.2 Integration to the Limit
1.6.3 Computing Expected Values
1.7 Product Measures, Fubini's Theorem

2 Laws of Large Numbers
2.1 Independence
2.1.1 Sufficient Conditions for Independence
2.1.2 Independence, Distribution, and Expectation
2.1.3 Sums of Independent Random Variables
2.1.4 Constructing Independent Random Variables
2.2 Weak Laws of Large Numbers
2.2.1 L^2 Weak Laws
2.2.2 Triangular Arrays
2.2.3 Truncation
2.3 Borel-Cantelli Lemmas
2.4 Strong Law of Large Numbers
2.7 Large Deviations*

3 Central Limit Theorems
3.1 The De Moivre-Laplace Theorem (may skip this)
3.2 Weak Convergence
3.3 Characteristic Functions
3.3.1 Definition, Inversion Formula
3.3.2 Weak Convergence
3.3.3 Moments and Derivatives
3.4 Central Limit Theorems
3.4.1 i.i.d. Sequences
3.4.2 Triangular Arrays
3.4.4 Rates of Convergence (Berry-Esseen)*
3.5 Local Limit Theorems* - maybe

We will probably skip some of the remaining sections in chapter 3
3.6 Poisson Convergence
3.6.1 The Basic Limit Theorem
3.6.2 Two Examples with Dependence
3.7 Poisson Processes
3.7.1 Compound Poisson Processes
3.7.2 Thinning
3.7.3 Conditioning
3.8 Stable Laws*
3.9 Infinitely Divisible Distributions*
3.10 Limit Theorems in R^d

4 Martingales
4.1 Conditional Expectation

Quick look at martingales and Markov processes

Introduction to Brownian motion

The last two topics are covered in Durrett in great depth. If we have any time left for these topics, our coverage will be pretty superficial. These topics are covered in much greater depth in 565a and 565b.