Department:

MATH
Course Number:

6221
Hours - Lecture:

3
Hours - Lab:

0
Hours - Recitation:

0
Hours - Total Credit:

3
Typical Scheduling:

Every Spring Semester Description:

Classical introduction to probability theory including expectation, notions of convergence, laws of large numbers, independence, large deviations, conditional expectation, martingales and Markov chains.

Prerequisites:

Math 4221 or consent of instructor.

Course Text:

At the level of Grimmett and Stirzaker, *Probability and Random Processes*

Topic Outline:

- Distribution Functions and Random Variables Definition and examples of discrete and continuous distribution functions, discrete and continuous random variables, independence
- Expectation and Mode of Convergence Expectation and conditional expectation; Markov, Chebychev, Holder, Minkowski and other inequalities Various notions of convergence
- Laws of Large Numbers and Convergence of Series Borel-Cantelli lemmas, Kolmogorov three series theorem, Kolmogorov's strong law
- Large Deviations Elements of large deviations, the theorems of Cramer, Hoeffding, Chernoff
- The Central Limit Theorem Characteristic functions The Central Limit Theorem and its rate of convergence (Berry-Esseen inequality)
- Conditional Expectations and Discrete Time Martingales Definition and examples of martingales (super and sub) The martingale convergence theorem L2 bounded and uniformly integrable martingales
- Markov Chains Definitions and examples of Markov chains Invariant measure Rate of convergence, transience and recurrence