- Probability Measures: Including connections to distribution functions
- Random Variables: Including random vectors and discrete-parameter stochastic processes
- Expectation: Basic properties; convergence theorems; inequalities
- Independent Random Variables: Basic properties; connections to infinite-dimensional product measures; Fubini's theorem
- Modes of Convergence of Random Variables: Almost sure convergence; the Borel-Cantelli lemma; convergence in probability; convergence in L^p
- Laws of Large Numbers: Weak and strong laws; Kolmogorov's inequality; equivalent sequences; random series
- Convergence in Distribution: Basic properties; connections to sequential compactness, tightness, and uniform integrability
- Characteristic Functions: Basic properties; connections to probability measures
- The Classical Central Limit Theorem
- Conditional Probability and Expectation: Basic properties; connections to Radon-Nikodym derivatives, projections, etc.
- Martingales: Basic inequalities and convergence theorems; optional sampling; backward martingales; applications
- Markov Processes: Basic properties and examples; stopping times and the strong Markov property; use of transition probabilities; applications
Suggested textbooks: A Course in Probability Theory by Chung.
Probability: Theory and Examples by Rick Durrett
Probability and Measure by Patrick Billingsley Probability by Albert Shiryaev
Suggested courses: 6241 and 6242