INDIAN STATISTICAL INSTITUTE

(203 B. T. Road, Kolkata 700 108)

Martingale Theory

M.Stat. IInd Year & M.Math. IInd Year

Compulsory Course for: Theoretical Statistics Specialization and Probability Specialization of M.Stat. IInd Year
Optional Course for: Applied Statistics Specialization of M.Stat. IInd Year & M.Math. IInd Year

Academic Year 2021 - 2022: Semester I

Class Time: Monday 14:00 - 16:00 hours & Thursday 16:00 - 18 hours.

Lecture Hall: Online Google Classroom and online class via Zoom Meetings

Course Duration: September 20, 201 - January 27, 2022
(Total of 18 weeks = 14 weeks of classes + 1 Midterm weeks + 1 week of Puja break + 2 Final weeks).

Midterm Examination:
Date: November 15 - 22, 2021
Time: TBA
Venue: TBA

Final Examination:
Date: January 17 - 27, 2022
Time: TBA
Venue: TBA

Instructor: Antar Bandyopadhyay
Email: antar (at) isid (dot) ac (dot) in
Office:

Room # 208 on the First Floor of the Faculty Block (when in Delhi); and
Room # 3.4 on the Third Floor of the A. N. Kolmogorov Bhavan (when in Kolkata).

Teaching Assistant (TA): Deborshi Das
Email: deborshidas6 (at) gmail (dot) com

Course Outline:

Absolute continuity and singularity of measures. Signed 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, criterion for existence. Regular conditional distribution, conditional expectation as integral with respect to the regular conditional distribution.
Definition of a filtration and adapted sequence.
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.
L_p-bounded martingales, for p > 1 and L_p-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.
Introduction to continuous parameter martingales: definition, examples and basic properties.
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.

Grading Policy:

Assignments: 10% of the total credit. (the assignments will be graded by the TA)
Quizzes: 10% of the total credit.
Midterm Exam: 30% 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 Thursday, starting from September 23, 2021. Each assignment will be due on the following Thursday by 11:59 PM. For example the first assignment is due on Thursday, September 30, 2022 by 11:59 PM.
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. In particular, they will not be graded even if you submit in the Google Classroom. 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 Google Classroom.

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 all materials done in the lectures given in the weeks prior to it.
Each quiz will be of 10 points and will be of 10 minutes duration at the most.
Final grade for the quizzes will be (Best Score Before Midterm + Best Score Between Midterm and Final)/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.

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.

Regrading Policy:

Regrading of homework or exams will only be undertaken in cases where, you believe there has been a genuine error or misunderstanding. Please note that our primary aim in grading is consistency, so that all students are treated the same; for this reason, we will not adjust the score of one student on an issue of partial credit, unless the score allocated clearly deviates from the grading policy we adopted for that problem.
If you wish to request a regrading of a homework or exam, you must return it to the instructor with a written note on a separate piece of paper explaining the problem.
The entire assignment or the exam may be regraded, so be sure to check the solutions to ensure that your overall score will go up after regrading.
All such requests must be received within one week from the date on which the homework or exam was made available for return.