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

- 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

- 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

- Markov Chains:
- Transition probabilities, Rigorous construction
- Markov property, strong markov property
- Rigorous construction
- Recurrence and transience
- Convergence to equilibrium, coupling
- Important examples

- 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

- 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*