Syllabus for the Comprehensive exam in Probability

  1. Probability Measures: Including connections to distribution functions
  2. Random Variables: Including random vectors and discrete-parameter stochastic processes
  3. Expectation: Basic properties; convergence theorems; inequalities
  4. Independent Random Variables: Basic properties; connections to infinite-dimensional product measures; Fubini's theorem
  5. Modes of Convergence of Random Variables: Almost sure convergence; the Borel-Cantelli lemma; convergence in probability; convergence in L^p
  6. Laws of Large Numbers: Weak and strong laws; Kolmogorov's inequality; equivalent sequences; random series
  7. Convergence in Distribution: Basic properties; connections to sequential compactness, tightness, and uniform integrability
  8. Characteristic Functions: Basic properties; connections to probability measures
  9. The Classical Central Limit Theorem
  10. Conditional Probability and Expectation: Basic properties; connections to Radon-Nikodym derivatives, projections, etc.
  11. Martingales: Basic inequalities and convergence theorems; optional sampling; backward martingales; applications
  12. 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