Advanced Probability
(Part - I)
Fall 2011
Instructor: Antar Bandyopadhyay
Email: antar (at) isid (dot) ac (dot) in
Office: 3.16 A. N. Kolmogorov Building
Class Time: Tuesday and Thursday 14:15 - 15:55 hours (new time table)
Instructor's Office Hours: Friday 14:15 - 15:15 hour
Course Duration: July 04 - November 04, 2011.
Mid-Term Examination:
Date: TBA
Venue: TBA
Final Examination Date:
Date: TBA
Venue: TBA
Course Outline(only for the part I'm teaching):
- Absolute continuity and singularity of 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, existence for reals. Regular conditional
distribution, conditional expectation as integral with respect to the
regular conditional distribution.
- Countable infinite product of measurable spaces.
Product of probabilities.
Kolmogorov Consistency Theorem, Ionescu-Tulcea Theorem. Uncountable
product.
- Concept of essential supremum of a class of measurable functions.
Prerequisites:
- Liner Algebra (at the level of Finite Dimensional Vector Spaces,
by P. R. Halmos).
- Measure Theoretic Probability (at the level of Probability: Theory and
Examples by R. Durrett).
- Real Analysis (at the level of Principles of Mathematical Analysis
by W. Rudin).
References:
- Probability and Measure by P. Billingsley.
- Probability Theory and Examples by R. Durrett.
Grading Policy(for the whole course):
- Assignments: 10% of the total credit.
- Midterm Exam: 30% of the total credit.
- Quizes: 10% of the total credit.
- Final Exam: 50% of the total credit.
Assignment Policies:
- In my part there will be exactly two sets of homework assignments each
carrying a total of 40 points.
- The assignments will be given in class on every alternate Monday,
starting from July 11, 2011. Each
assignment will be due in class on the Wednesday of the following week. For
example the first assignment is due on July 20, 2011. So you get 10
days to work on an assignment.
- Each assignment will cover the topics done in class in the previous week
and what will be taught in 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.
- Graded assignments will be returned in the class on the Monday of the week which
is after their submission. For example, the assignment which is due on July 20, 2011
(Wednesday) will be returned after grading on July 25, 2011 (Monday).
- Click here
for downloading the assignments.
Quiz Policies:
- There will be two or three
quizes as surprised 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.
- 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.
- All quizes will be closed note.
Last modified July 26, 2011.