STAT 205B/MATH C218B: Probability Theory (Spring 2012)
Instructor Allan
Sly
Teaching Assistant Subhroshekhar Ghosh
Office hours
Wednesday 12:30-1:30
Friday 1:30-3:30 Room 789 Evans
Class time 12:30-2:00 on TuTh at 334 Evans
This is the second half of a year course in mathematical probability at the
measure-theoretic level. It is designed for students whose ultimate research
will involve rigorous proofs in mathematical probability. It is aimed at Ph.D.
students in the Statistics and Mathematics Depts, but is also taken by Ph.D.
students in Computer Science, Electrical Engineering, IEOR, Business and
Economics who expect their thesis work to involve probability.
The course will cover
- Markov Chains and Random Walks.
- Ergodic theory and applications.
- Brownian motions.
- Other topics.
Much of the material is covered in
Chapters 6, 7, 8 in Durrett's book, Probability: Theory and Examples (4th
Edition) which is the required text. Quite a few of the homework problems
are from there and it is available online here. Using the 3rd
edition shouldn't be a problem, but be aware that the exercises are numbered
differently than in the 4th edition. Jim Pitman has his very
useful lecture notes linked to the Durrett text; these notes cover more
ground than my course will.
Students who are interested in more advanced
reading are encouraged to consult the comprehensive book by Kallenberg,
Foundations of Modern Probability.
For other relevant books, see
Aldous list
at the old course homepage.
Prerequisites
- STAT 205A - familiarity with measure-theoretic approach to mathematical
probability.
- Undergraduate-level familiarity with Markov chains.
- Upper division analysis, e.g. uniform convergence of functions, basics of
complex numbers. Basic properties of metric and function spaces.
Schedule
Jan 17: Foundations of Markov Chains, Markov Property (6.1, 6.2, 6.3)
Jan 19: Irreducibility, Recurrence and Transience (6.4)
Jan 24: Recurrence and Transience continued (6.4, 4.2)
Jan 26: Class canceled
Jan 31: Stationary Measures (6.5)
Feb 2: Coupling
Feb 7: Convergence to stationarity, mixing times (6.6)
Feb 9: Markov chains, mixing etc
Feb 14: Large deviations and concentration of measure (2.6)
Feb 16: Azuma–Hoeffding inequality and concentration of measure
Feb 21: Ergodic Theorem (7.1, 7.2)
Feb 23: Subadditive Ergodic Theorem (7.4)
Feb 28: Brownian Motion (8.1)
Mar 1: Markov property, Blumenthal 0-1 Law (8.2)
Mar 6: Strong Markov property (8.3)
Mar 8: Path properties, martingales (8.4, 8.5)
Mar 13: Martingales, Donsker's Theorem (8.5,8.6)
Mar 15: Donsker's Theorem (8.6)
Mar 20: Law of the iterated logarithm (8.8)
Mar 22: Stochastic Calculus (PM 7.1.1)
Apr 3: Ito's Formula (PM 7.1.2)
Apr 5: Conformal Invariance (PM 7.2)
Apr 10: Stochastic differential equations
Apr 12: Existence and uniqueness of solution, martingale problem
Apr 17: Convergence of discrete Markov chains to diffusions
Homework
Week 2 due February 2. Questions 6.2.7, 6.2.8, 6.2.9, 6.3.5, 6.3.10, 6.4.9
Week 3 due February 9. Link
Week 4 due February 16. Link
Week 5 due February 23. Link
Week 6 due March 1. Link
Week 7 due March 8. Link
Week 8 due March 15. Questions 8.3.6, 8.3.7, 8.4.2 from Durrett. From Brownian Motion by Morters and Peres questions 2.10, 2.13. Note: These questions numbers are from the online edition, they correspond to Q 2.12, 2.15 in the print edition
Week 9 due March 22. Link
Week 11 due April 12. Link
Week 12 due April 19. Link
Final
There will be a take-home final exam.
Grading 55% homework, 45% take-home final.
Office Hours
Allan Sly (sly@stat)
Tuesday 3:00-5:00 at 333 Evans
TA Subhroshekhar Ghosh (subhro@math) Wednesday 12:30-1:30
Friday 1:30-3:30 Room 789 Evans
If you email us, please put STAT 205B in subject.