This is an attempt. Things may change as we go.

## Probability: Theory and Examples

by Rick Durrett

## Review on Measure Theory

- Sigma Algebra
- Measure and Measure space
- Measurable sets and functions
- Integration
- Properties of the Integral
- Product Measures, Fubini’s Theorem

## Probability Theory v.s. Measure Theory

- Events
- Probability Measure and Probability space
- Distributions
- Random Variables
- Almost surely convergence
- Convergence in Probability
- Expected Value
- Properties of the Expected Value
- Useful Theorem for Computing Expected Value
- Inequalities
- Integration to the Limit

## Law of Large Numbers

- Independence
- Constructing Independent Random Variables

- Weak Laws of Large Numbers
- Borel-Cantelli Lemmas
- Strong Law of Large Numbers
- Applications in statistics:
- Random Sample
- Estimators: consistency and examples

- Proof

- Applications in statistics:

## Central Limit Theorem

- Normal distribution
- De Moivre-Laplace Theorem
- Weak Convergence
- Characteristic Functions
- Inversion Formula
- Weak Convergence
- Moments and Derivatives

- The Central Limit Theorem
- i.i.d Sequences
- Rates of Convergence
- Application in Statistics: estimating errors and confidence intervals

## Markov Chains

- Definitions and Basic Properties
- The Markov Property
- Seeing MC as a RW on graphs
- Recurrence and Classification
- Equilibrium: Stationary Distributions
- Approach to Equilibrium
- Simulating Markov Chains
- Markov Chain Monte Carlo
- Graph Inference*

## Martingales

- Conditional Expectation
- Almost sure convergence of Martingales
- Classical Examples
- Polya’s Urn Scheme
- Branching Process

- Convergence in $L^p$
- Uniform integrability

- Optional Stopping Theorems
- Concentration Inequalities
- Azuma’s inequality

- Application on network analysis
- Barabási-Álbert model: degree analysis