# Math 641 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.

#### Prerequisites

Measure theory (the equivalent of Math 631 or Stats 711 ).

#### Syllabus

1. Preliminaries:
• Borel-Cantelli theorems
• Rigorous definition of independence for events, random variables, and sigma-fields
• Kolmogorov and Hewitt-Savage 0-1 laws
• Characteristic functions (aka Fourier transforms)
• Weak convergence of random variables, central limit theorem
2. Martingales:
• Almost sure convergence, upcrossing inequality
• L^p convergence, Doob's inequality
• Uniform integrability, L^1 convergence
• Levy 0-1 law
• Optional stopping theorem
• Backward martingales
• Important examples and applications: branching processes, Polya urns, etc.
• Radon-Nikodym derivative, application to decomposition of measures
3. Markov Chains:
• Transition probabilities, Rigorous construction
• Markov property, strong markov property
• Rigorous construction
• Recurrence and transience
• Convergence to equilibrium, coupling
• Important examples
4. Ergodic Theory:
• Stationary sequences, invariant measures
• Birkhoff ergodic theorem, Kac recurrence theorem
• Mutual singularity of ergodic measures, ergodic decomposition via martingale convergence
• Subadditive ergodic theorem (only statement and a few applications)
• Law of large numbers for iid sequence and martingales
5. Brownian Motion:
• Construction of Weiner measure
• Sample path properties, zero set
• Blumenthal 0-1 law
• Strong Markov property, reflection principle
• Donsker's theorem
• Empirical Distribution and Brownian Bridge

#### References

• R. Durrett.  Probability Theory and Examples
• J. Rosenberg.  A First Look at Rigorous Probability Theory
• D. Khoshnevisan. Probability
• Athreya and Lahti. Measure Theoretic Probability Theory
• Fristedt and Gray.  A Modern Approach to Probability Theory
• O. Kallenberg.  Foundations of Modern Probability Theory