Measure Theoretic Probability

Lecture Hall: Room # 23 Academic Block (First Floor).

Course Duration (including examinations): January 08 - May 03, 2024
(Total of 17 Weeks = 7 Weeks of Classes + 1 Midterm Examination Week + 7 Weeks of Classes + 2 Final Examination Weeks).

Midterm Examination:
Date: TBA (During February 26 - March 01, 2024)
Duration: 10:00 AM - 12:00 noon (Two Hours)
Venue: TBA

Final Examination:
Date: TBA (during April 22 - May 03, 2024)
Duration: 10:00 AM - 01:00 PM (Three Hours)
Venue: TBA

Course Outline:

• Basic Measure Theory:
  • Review of the notion of cardinality of sets, finite, countable, uncountable sets.
  • Algebra of sets, σ-algebra, examples, Borel σ-algebra.
  • Monotone classes, π-systems, λ-systems. Monotone Class Theorem and Dinkin's π-λ Theorem. Uniqueness theorems.
  • Measurable functions, sum, product, maximum, minimum of measurable functions. Limits of measurable functions. Random variables, simple functions, Monotone Approximation Theorem.
  • Definition of measure, set functions, finite and countable additivity. Finite and infinite measures, probability measures, basic laws of measures. σ-finite measures.
  • Existence and extension theorems. Outer measure, Carathéodory's Extension Theorem. Construction of the Lebesgue measure on unit interval, real line and d-dimensional Euclidean space. Properties of the Lebesgue measure.
  • Distribution function, inverse distribution function, the Fundamental Theorem of Probability.
  • Lebesgue Theory of Integration. MCT, Fatou's Lemma, DCT, Schéffe's Theorem. Calculus of measure zero sets. Concept of "almost surely (a.s)". Change of variable formula. Expectation of a random variable, law of unconscious statistician.
  • Product space, basic definitions, sections. Product measure, existence and uniqueness. Fubini's Theorem, applications.
  
  

• Measure Theoretic Probability:
  • Independence. Borel-Cantelli Lemmas. Tail σ-algebra, Kolmogorov's 0-1 Law.
  • Various modes of convergence and their interrelations. WLLN, SLLN. Fundamental Theorem of Statistics (Glivenko-Cantelli Lemma).
  • Kolmogorov's maximal inequality and applications. Kolmogorov's Three Seres Theorem, proof of the sufficiency part.
  • Characteristic functions, definition, examples. Moment expansion, characteristics function for Normal distribution. Inversion formula, uniqueness theorem, density formula.
  • Revision of weak convergence, definition, representation theorem, Portmanteau Theorem. Hally-Bray Theorem, tightness.
  • Characteristic functions and weak convergence, Lévy's Continuity Theorem (statement only).
  • Central Limit Theorems: De Moiver-Laplace CLT, I.I.D. CLT, Lindeberg CLT, Lyapounov's CLT, proofs using Lideberg CLT. Applications. Proof of Lindeberg CLT. Proof of the necessity part of the Kolmogorov's Three Series Theorem.

References:

• Probability and Measure by P. Billingsley.
• Probability: Theory and Examples by R. Durrett.
• Measure and Probability by S. Athreya and V. S. Sunder.
• Introduction to Probability and Measure by K. R. Parthasarathy.
• Measure Theory by P. R. Halmos.
• Probability Theory, Vol - I & II by W. Feller.
• Real Analysis and Probability by R. B. Ash.

We will mostly follow Billingsley, only occasionally the others may be needed. Though they are all very good and important sources to learn measure theory/probability theory.

Prerequisites:

• Real Analysis (at the level of Principles of Mathematical Analysis, W. Rudin).
• Liner Algebra (at the level of Finite Dimensional Vector Spaces, P. R. Halmos).
• Basic Probability (at the level of Introduction to Probability Theory, P. G. Hoel, S. C. Port and C. J. Stone).

Grading Policy:

• Assignments: 10% of the total credit.
• 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 Assignment component of the final grading.
• The assignments will be uploaded in the Google Classroom on every Thursday, starting from January 11, 2024. Each assignment will be due on the following Thursday by 11:59 PM and you have to upload your scanned hand written solutions in the Google Classroom. For example the first assignment is due on Thursday, January 18, 2024.
• Each Assignment will be based on the course materials which will be covered in the class in the week of its 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 assignments anyway.
• Assignments will be graded by the TA and the Graded assignments will be returned in the Google classroom.

Quiz Policies:

• There will be at least four 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 10 points and will be of 20 minutes duration.
• 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.
• 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.

Regrading Policy:

• Regrading of homeworks 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.

Last modified February 11, 2024.