# Fall 2010

Instructor: Antar Bandyopadhyay (Email: antar (at) isid (dot) ac (dot) in Office: 208 Faculty Building).

Class Time: Tue Thu 14:00 - 16:00 in Class Room 23.

Instructor's Office Hours: Tue Thu 16:00 - 17:30

Course Duration: July 19 - November 19, 2010.

Midterm Examination: September 17, 2010 (Friday) 10:00 - 14:00 hours

Final Examination: TBA

Course Outline:

• Basic notion of cardinality of sets, finite, countable, uncountable sets.
• Algebra of sets, σ-algebra, examples, Borel σ-algebra.
• Definition of measure, set functions, finite and countable additivity. Finite and infinite measures, probability measures, basic laws of probability.
• Measurable functions, sum, product, maximum, minimum of measurable functions. Limits of measurable functions. Random variables, simple functions, Monotone Approximation Theorem.
• Monotone classes, π-systems, λ-systems. Monotone Class Theorem and Dinkin's π-λ Theorem. Uniqueness theorems.
• Existence and extension theorems. Outer measure, Carathéo Dory'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.
• Product space, basic definitions, sections. Product measure, existence and uniqueness. Fubini's Theorem, applications.
• 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).
• Characteristic functions, definition, examples. Moment expansion, characteristics function for Normal distribution. Inversion formula, uniqueness theorem, density formula.
• 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.

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: 20% of the total credit.
• Quizes: 10% of the total credit.
• Midterm Exam: 20% of the total credit.
• Final Exam: 50% of the total credit.

Assignment Policies:

• There will be a total of 7 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, and it will be due in class on Thursday of the following week. Each assignment set will be based on course materials covered in the lectures in the week before and the week it is assigned. For example, the assignment given on July 27 (Tuesday) will be due on August 5 (Thursday), and will be on materials covered during July 19 - 30, 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 the next Tuesday following the due date. For example, the assignment which is due on August 5 (Thursday) will be returned after grading on August 10 (Tuesday).
• Click here for downloading the assignments.

Quiz Policies:

• There will be at least four quizes as surprise tests given in the class. This means there shall be no pre-scheduling. A quiz will cover all the materials done in the lectures prior to it.
• 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 other.
• Each quiz will be a closed books, closed notes test.
• Quiz # 1.
• Quiz # 2.
• Quiz # 3.
• Quiz # 4.

Exam Policies:

• Each examination (except for the quizes) will be an open note examination. That means, students are allowed to bring his/her own hand written notes, study materials, list of theorems etc.

Last modified October 13, 2010.