Advanced Probability
Fall 2008
Instructor: Antar Bandyopadhyay
Email: antar (at) isid (dot) ac (dot) in
Office: 4.12 A. N. Kolmogorov Bhavan
Class Time: Tu 16:15 - 18:00, F 14:15 - 16:00
Instructor's Office Hours: Tu 14:00 - 16:00
Course Duration: July 14 - November 21, 2008.
Mid-Term Examination:
Date: September 1, 2008 (Monday) 10:30 - 13:30
Venue: Room Number 508 Library Building (S. N. Bose Bhavan).
Final Examination Date: TBA
Course Outline:
- Concept of essential supremum of a class of measurable functions.
- 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.
- Definition of a filtration and adapted sequence.
- Martingales, definitions and examples of
discrete parameter sub-martingale, martingale,
super-martingale. Basic properties. Doob's Maximal Inequality,
Kolmogorov's Maximal Inequality.
- Definition of stopping time, stopped process, stopped σ-algebra,
examples and properties.
- Predictable processes, Discrete martingale transform, Doob's
Decomposition Theorem.
- Concept of upcrossings. Doob's upcrossing inequality, The
(sub) Martingale Convergence Theorem, Convergence theorem for
non-negative super-martingale.
- Uniform integrability, convergence in L_{1}.
- Backward Martingales, Levy's Upward and Downward Theorems.
- Optional Stopping Theorems.
- Applications: SLLN for i.i.d. random variables,
Hewitt-Savage 0-1 Law, de Finetti's Theorem, SLLN for U-Statistics.
- Randon-Nikodym Theorem through martingale.
- Martingale Central Limit Theorem.
- Azuma's Inequality and some applications.
- Definition of the Continuous parameter martingale, basic properties.
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.
- Foundations of Calculus of Probability by J. Nevue.
- Discrete Parameter Martingales by J. Nevue.
Grading Policy:
- Assignments: 10% of the total credit.
- Midterm Exam: 30% of the total credit.
- Final Exam: 60% of the total credit.
Assignment Policies:
- There will be a total of 5 sets of homework assignments each
carrying a total of 20 points. 4 best assignment scores
will be taken for the final grading.
- The assignments will be given in class on every alternate Tuesday,
starting from August 26, 2008. Due to the Mid-Semester Examinations
the second assignment will be given on September 16, 2008. Each
assignment will be due in class on the Friday of the following week. For
example the first assignment is due on September 5, 2008.
- The first assignment will cover the first part of the course. After
that each will cover what has been done in class till the day of the
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 an extra assignment anyway.
- Graded assignments will be returned in the class an week after their
submission. For example, the assignment which is due on September 5, 2008
(Friday) will be returned after grading on September 12 (Friday).
- 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.
- The quizes will not be part of your final grade, but there are other
interesting rewards if you do well in them.
- Quiz # 1
Last modified October 27, 2008.