MA 5260: Probability Theory II (Jan 2012 -- May 2012)

Prerequisites:

ST 5214 (Advanced Probability Theory), or equivalently: Measure Theory, results from a first graduate course in probability, including almost sure convergence and convergence in distributions, Law of Large Numbers, Central Limit Theorem, Borel-Cantelli lemma, conditional probability and expectation.

Course Description:

Following Probability Theory I, this course continues to develop some of the fundamental tools in modern probability theory. The topics covered include the following. Discrete time martingales: Doob's inequality, martingale convergence theorems, optional stopping theorem. Countable state Markov Chains: classification of states, recurrence and transience, existence, uniqueness and convergence to stationary distributions. Ergodic theory: almost sure and mean ergodic theorem, structure of stationary Markov processes. Brownian motion: construction and characterization, path properties, strong Markov property.

References:

  1. Lecture notes on Limit Theorems by S.R.S. Varadhan.

  2. Probability: Theory and Examples by Richard Durrett.

  3. Probability by Leo Breiman.

Time and Location:

Friday 19:00-22:00, S17-0512. Final: Apr 26 (Thursday morning).

Grade makeup:

Final exam: 60%. Homeworks: 40%.

Homeworks:

  1. Homework 1, due Mar 2.
  2. Homework 2, due Apr 13.

Lecture Notes:

  1. Jan 13, 2012. Lecture 1. Regular conditional distributions and probabilities, definition of martingale, martingale transform, martingale decomposition, Azuma-Hoeffding inequality.

  2. Jan 20, 2012. Lecture 2. Doob's maximal inequalities, stopping time, upcrossing inequality, almost sure martingale convergence, second Borel-Cantelli lemma, Polya's urn.

  3. Jan 27, 2012. Lecture 3. L_p martingale convergence theorems, Levy's 0-1 law, Doob's decomposition, quadratic variation process.

  4. Feb 3, 2012. Lecture 4. Non-negative martingales as changes of measure, optional stopping theorem, backwards/reversed martingales.

  5. Feb 10, 2012. Lecture 5. Markov chains, the Markov and strong Markov property, irreducibility, transience and recurrence.

  6. Feb 17, 2012. Lecture 6. Classification of states, criteria for transience and recurrence, simple random walks, birth-death chains, Dirichlet problem and Poisson equation.

  7. Mar 2, 2012. Lecture 7. Existence and uniqueness of stationary measures and convergence.

  8. Mar 9, 2012. Lecture 8. Periodic Markov chains, Perron-Frobenius theorem, reversible Markov chains.

  9. Mar 16, 2012. Lecture 9. Ergodic theorems.

  10. Mar 23, 2012. Lecture 10. Ergodic decomposition, structure of stationary Markov chains, Harris chains.

  11. Mar 20, 2012. Lecture 11. Brownian motion: characterization, invariance, path properties.

  12. Apr 13, 2012. Lecture 12. Brownian motion: Markov and strong Markov property, Blumenthal's 0-1 law, reflection principle, martingale property, recurrence and transience.

  13. Apr 18, 2011. Lecture 13. Weak convergence, Donsker's invariance principle.