Advanced Classical Probability Theory

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