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