Measure Theoretic Probability
Spring 2013
Instructor: Antar Bandyopadhyay
(Email: antar (at) isid (dot) ac (dot) in
Office: 208 Faculty Building).
Class Time: Monday and Thursday 15:30 - 17:30 hours in Class Room 23.
Instructor's Office Hours: TBA
Course Duration: January 7, 2013 - April 19, 2013.
Midterm Examination: February 25, 2013 (Monday) Time: 10:00 - 12:00 hours
Final Examination: During April 22, 2013 - May 03, 2013
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, law of unconscious statistician.
- 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).
- Kolmogorov's maximal inequality and applications. Kolmogorov's Three Seres Theorem, proof of 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 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: 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 biweekly 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 Monday,
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.
- 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
Monday following the due date.
- 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.
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 April 08, 2013.