Advanced Probability
Spring 2010
Instructor: Antar Bandyopadhyay
Email: antar (at) isid (dot) ac (dot) in
Office: 208 Faculty Building
Class Time: Tuesday, Thursday 14:00 - 15:30
Class Room # 23 (on the first floor of the Class-Room Block)
Course Duration: January 05 - April 16, 2010.
Mid-Term Examination:
Date: TBA
Venue: TBA
Note: There will be no class in the week of February 22 - 26, 2010.
Final Examination Date:
Date: TBA
Venue: TBA
Course Outline:
- 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.
- Ionescu-Tulcea Theorem.
- 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.
- Concentration Inequalities.
- Introduction to continuous parameter martingales and 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 and/or Probability and Measures by P. Billingsley).
- 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: 20% of the total credit.
- Midterm Exam: 20% of the total credit.
- Surprise Tests/Quizzes: 20% of the total credit
- Final Exam: 40% of the total credit.
Assignment Policies:
- There will be a total of 6 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,
starting from January 12, 2010. Each assignment will be due in class on
the Thursday of the following week. For
example the first assignment is due on January 21, 2010.
- Each assignment will cover the course material done in class in the
previous week as well as in what will be done in class in that week. For example,
the first assignment which will be given on Tuesday, January 12, 2010 will
cover materials from lectures given in the week of January 4-8, 2010 and week of
January 11-15, 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 Tuesday of week following their
submission. For example, the first assignment which is due on Thursday, January 21, 2010
will be returned after grading on Tuesday, January 26, 2010.
- Click here
for downloading the assignments.
- Click here
for downloading the extra assignments.
Quiz Policies:
- There will be two to four
quizzes 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.
Last modified March 10, 2010.