# Spring 2010

Email: antar (at) isid (dot) ac (dot) in
Office: 208 Faculty Building

Class Time: Tuesday, Thursday 14:00 - 15:30

Class Room # 23 (on the first floor of the Class-Room Block)

Course Duration: January 05 - April 16, 2010.

Mid-Term Examination:
Date: TBA
Venue: TBA
Note: There will be no class in the week of February 22 - 26, 2010.

Final Examination Date:
Date: TBA
Venue: TBA

Course Outline:

• Conditional expectation, definition, examples and special cases. Properties of conditional expectation, linearity, order-preserving, MCT, DCT, Jensen inequality. Conditional expectation as a projection.
• Regular conditional probability, existence for reals. Regular conditional distribution, conditional expectation as integral with respect to the regular conditional distribution.
• Ionescu-Tulcea Theorem.
• Definition of a filtration and adapted sequence.
• Martingales, definitions and examples of discrete parameter sub-martingale, martingale, super-martingale. Basic properties. Doob's Maximal Inequality, Kolmogorov's Maximal Inequality.
• Definition of stopping time, stopped process, stopped σ-algebra, examples and properties.
• Predictable processes, Discrete martingale transform, Doob's Decomposition Theorem.
• Concept of upcrossings. Doob's upcrossing inequality, The (sub) Martingale Convergence Theorem, Convergence theorem for non-negative super-martingale.
• Uniform integrability, convergence in L1.
• Backward Martingales, Levy's Upward and Downward Theorems.
• Optional Stopping Theorems.
• Applications: SLLN for i.i.d. random variables, Hewitt-Savage 0-1 Law, de Finetti's Theorem, SLLN for U-Statistics.
• Randon-Nikodym Theorem through martingale.
• Martingale Central Limit Theorem.
• Azuma's Inequality and some applications.
• Concentration Inequalities.
• Introduction to continuous parameter martingales and basic properties.

Prerequisites:

• Liner Algebra (at the level of Finite Dimensional Vector Spaces, by P. R. Halmos).
• Measure Theoretic Probability (at the level of Probability: Theory and Examples by R. Durrett and/or Probability and Measures by P. Billingsley).
• Real Analysis (at the level of Principles of Mathematical Analysis by W. Rudin).

References:

• Probability and Measure by P. Billingsley.
• Probability Theory and Examples by R. Durrett.
• Foundations of Calculus of Probability by J. Nevue.
• Discrete Parameter Martingales by J. Nevue.

• Assignments: 20% of the total credit.
• Midterm Exam: 20% of the total credit.
• Surprise Tests/Quizzes: 20% of the total credit
• Final Exam: 40% of the total credit.

Assignment Policies:

• There will be a total of 6 sets of homework assignments each carrying a total of 20 points. 5 best assignment scores will be taken for the final grading.
• The assignments will be given in class on every alternate Tuesday, starting from January 12, 2010. Each assignment will be due in class on the Thursday of the following week. For example the first assignment is due on January 21, 2010.
• Each assignment will cover the course material done in class in the previous week as well as in what will be done in class in that week. For example, the first assignment which will be given on Tuesday, January 12, 2010 will cover materials from lectures given in the week of January 4-8, 2010 and week of January 11-15, 2010.
• Late submission of an assignment will not be accepted. If you can not submit an assignment on time, don't worry about it, and try to do well in the others. It will not count in your final grade since you have few extra assignments anyway.
• Graded assignments will be returned in the class on Tuesday of week following their submission. For example, the first assignment which is due on Thursday, January 21, 2010 will be returned after grading on Tuesday, January 26, 2010.