Measure Theoretic Probability
Fall 2007
Instructor: Antar Bandyopadhyay
(Email: antar (at) isid (dot) ac (dot) in
Office: 208 Faculty Building).
Class Time: TuTh 14:00 - 16:00 in Class Room 24.
Course Duration: August 14 -- December 7, 2007
(Note: No classes in the week of September 3rd).
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,
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.
- 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).
- Weak convergence, definition, Portmanteau Theorem.
Hally-Bray Theorem, tightness.
- 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), proof of I.I.D. CLT using Characteristic functions,
proof of all others using Lindeberg CLT.
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, C. J. Stone).
References:
- Probability and Measure by P. Billingsley.
- Probability: Theory and Examples by R. Durrett.
- 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.
Lecture Schedule:
Click here
to get the lecture schedule (please note that this
schedule may change as the semester progress).
Grading Policy:
- Assignments: 80% of the total credit.
- Class Presentations: 20% of the total credit (will be
graded by external examiners).
Assignment Policies:
- There will be a total of 13 sets of homework assignments each
carrying a total of 20 points. 10 best assignment scores
will be taken for the final grading.
- The assignments will be given in class on every Tuesday,
and it will be due in class on Tuesday of the following week. Each assignment
set will be
based on course materials covered in the lectures given in the previous week.
For example, the assignment given on August 21 (Tuesday)
will be due on August 28 (Tuesday), and will be on materials
covered in lectures on August 14 and 16.
- 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 Thursday of the
week they are due. For example, the assignment which is due on August 28
(Tuesday) will be returned after grading on August 30 (Thursday).
- Click here
for downloading the assignments.
Class Presentation Policies:
- At the end of the course the students are required to give one or two
presentations of 50 minutes duration on topics assigned to them.
- The grading of this will be done by a group of faculty members other
than the instructor.
- Click here
for the schedule and
a list of topics for the class presentations.
Last modified December 1, 2007.