Advanced Classical Probability Theory

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

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