# INDIAN STATISTICAL INSTITUTE

## Academic Year 2014 - 2013: Semester I

Class Time: Wednesday and Friday 02:00 PM - 04:00 PM (4 hours per week).

Lecture Hall: Room # 508 on the Fifth Floor of the S. N. Bose Bhavan till September 24, 2014.
Room # 701 on the Seventh Floor of the S. N. Bose Bhavan starting from September 26, 2014 till November 14, 2014.

Course Duration: July 21 - November 28, 2014
(Total of 19 weeks = 14 weeks of classes + 2 Midterm weeks + 1 week of Puja break + 2 Final weeks).

Midterm Examination:
Date: September 16, 2014 (Tuesday)
Duration: 10:30 AM - 01:30 PM (3 hours)
Venue: Room # 716 - 717 on the Seventh Floor of the S. N. Bose Bhavan.

Final Examination:
Date: November 26, 2014 (Wednesday)
Duration: 10:30 AM - 02:30 PM (4 hours)
Venue: Room # 716 - 717 on the Seventh Floor of the S. N. Bose Bhavan.

Email: antar (at) isid (dot) ac (dot) in
Office: Room # 3.15 on the Third Floor of the A. N. Kolmogorov Bhavan.

Instructor's Office Hours: Friday 04:00 PM - 06:00 PM.

Teaching Assistant (TA): Gursharn Kaur
Email: gursharn (dot) kaur12r (at) isid (dot) ac (dot) in
Office: Room # 5.2 on the Fifth Floor of the A. N. Kolmogorov Bhavan.

TA's Office Hours: Monday 5:30 PM - 7:30 PM.

Course Outline:

• Countably infinite product of measurable spaces. Product of probabilities. Kolmogorov Consistency Theorem, Ionescu-Tulcea Theorem. Uncountable product.
• Absolute continuity and singularity of measures. Hahn-Jordon decomposition, Radon-Nikodym Theorem, Lebesgue decomposition. Properties of the Radon-Nikodym derivative.
• 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.
• 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 up crossings. Doob's up crossing inequality, The (sub) Martingale Convergence Theorem, Convergence theorem for non-negative super-martingale.
• Lp-bounded martingales, for p > 1 and Lp-convergence.
• Uniform integrability, convergence in L1.
• Backward Martingales, Levy's Upward and Downward Theorems.
• Applications: SLLN for i.i.d. random variables, Hewitt-Savage 0-1 Law, de Finetti's Theorem, SLLN for U-Statistics.
• Optional Stopping Theorems.
• Randon-Nikodym Theorem through martingale.
• Martingale Central Limit Theorem and applications.
• Azuma's Inequality and some applications.

Prerequisites:

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

References:

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

• Assignments: 10% of the total credit (the assignments will be graded by the TA).
• Quizzes: 15% of the total credit.
• Midterm Exam: 25% of the total credit.
• Final Exam: 50% of the total credit.

Assignment Policies:

• There will be a total of 14 sets of homework assignments each carrying a total of 10 points. The average of the 10 best assignment scores will be taken for the final grading.
• The assignments will be given in class on every Wednesday, starting from July 23, 2014. Each assignment will be due in class on the Wednesday of the following week. For example the first assignment is due on Wednesday, July 30, 2014.
• Each assignment will be based on the course materials which will be covered in the class in the week of the assignment.
• 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 four extra assignment anyway.
• Graded assignments will be returned in the class an week after their submission. For example, the assignment which is due on July 30, 2014 (Wednesday) will be returned after grading on August 7, 2014 (Wednesday).

Quiz Policies:

• There will be four or more quizzes as surprise tests given in the class. This means there shall be no pre-scheduling. A quiz will cover materials done in the lectures given in the weeks prior to it.
• Each quiz will be of 15 points and will be of 15 minutes duration.
• Final grade for the quizzes will be (Maximum + Minimum)/2.
• There will be NO supplementary quiz given for any student who may miss a quiz for whatsoever reason. If you miss one then do not worry, try doing well in the others.
• All quizzes will be part of the final grading.
• All quizzes will be closed note and closed book examinations.

Exam Policies:

• The Midterm and the Final Examinations will be open notes examinations. That means, students will be allowed to bring his/her own hand written notes, study materials, list of theorems etc. But no printed or photocopied materials will be allowed.
• Any unfair means used by any students in the examinations will be dealt with the strictest possible measures, as per the Institute rules. In particular, if any student is found to be using any kind of unfair practice during any of the examinations (including the quizzes) then he/she will be awarded ZERO in that examination.
• Note that the homework assignments are part of the final examination and hence their grading and other policies will be same as that of the final examination.