Syllabus Topics in Mathematical Probability – MATH 6534

Review on Measure Theory

1. Sigma Algebra
2. Measure and Measure space
3. Measurable sets and functions
4. Integration
5. Properties of the Integral
6. Product Measures, Fubini’s Theorem

Probability Theory v.s. Measure Theory

1. Events
2. Probability Measure and Probability space
3. Distributions
4. Random Variables
5. Almost surely convergence
6. Convergence in Probability
7. Expected Value
8. Properties of the Expected Value
9. Useful Theorem for Computing Expected Value
   1. Inequalities
   2. Integration to the Limit

Law of Large Numbers

1. Independence
   • Constructing Independent Random Variables
2. Weak Laws of Large Numbers
3. Borel-Cantelli Lemmas
4. Strong Law of Large Numbers
   1. Applications in statistics:
      • Random Sample
      • Estimators: consistency and examples
   2. Proof

Central Limit Theorem

1. Normal distribution
2. De Moivre-Laplace Theorem
3. Weak Convergence
4. Characteristic Functions
   • Inversion Formula
   • Weak Convergence
   • Moments and Derivatives
5. The Central Limit Theorem
   • i.i.d Sequences
   • Rates of Convergence
   • Application in Statistics: estimating errors and confidence intervals

Markov Chains

1. Definitions and Basic Properties
2. The Markov Property
3. Seeing MC as a RW on graphs
4. Recurrence and Classification
5. Equilibrium: Stationary Distributions
6. Approach to Equilibrium
7. Simulating Markov Chains
8. Markov Chain Monte Carlo
9. Graph Inference*

Martingales

1. Conditional Expectation
2. Almost sure convergence of Martingales
3. Classical Examples
   1. Polya’s Urn Scheme
   2. Branching Process
4. Convergence in $L^p$
   1. Uniform integrability
5. Optional Stopping Theorems
6. Concentration Inequalities
   • Azuma’s inequality
7. Application on network analysis
   • Barabási-Álbert model: degree analysis