Time: Monday, Wednesday, Friday at 2:00 p.m. - 2:50 p.m.
Location: Taft Hall 300
Instructor: Jie Yang
Office: SEO 513
Phone: (312) 413-3748
E-Mail: jyang06 AT math DOT uic DOT edu
Office Hours: Monday, Friday at 1:00 p.m. - 2:00 p.m.; Wednesday at 11:00 a.m. - 12:00 p.m.
Textbook: Sidney I. Resnick,
A Probability Path, Birkäuser, 1999.
Content: Convergence concepts, laws of large numbers, convergence in distribution, characteristic functions, central limit theorem, conditional expectation, Martingale, fundamental theorems of mathematical finance
Prerequisite: STAT 501 or consent of instructor
Homework:
Turn in every Wednesday before class;
half of the grade counts for completeness;
half of the grade counts for correctness of one selected problem.
Short Presentations:
Each student is required to do two short presentations during the course period.
One presentation should be done in front of the whole class. Another presentation should be
done in office hours. Class presentation may last up to 15 minutes.
Office hour presentation may last up to 20 minutes.
The topics of presentations may come from the optional part of homework assignments.
Grading: Homework 50%, presentations 25% each
Grading Scale: 90% A , 75% B , 60% C , 30% D
| WEEK | SECTIONS | BRIEF DESCRIPTION |
| 01/09 - 01/13 | 6.1, 6.2; 6.3; 6.4, 6.5 | Almost Sure Convergence, Convergence in Probability; Connections between a.s. and i.p. Convergence; Quantile Estimation, Lp Convergence |
| 01/16 - 01/20 | Holiday; 6.5; 6.6 | Lp Convergence; More on Lp Convergence |
| 01/23 - 01/27 | 7.1; 7.2; 7.3 | Truncation and Equivalence; A General Weak Law of Large Numbers; Almost Sure Convergence of Sums |
| 01/30 - 02/03 | 7.3; 7.4; 7.5 | Almost Sure Convergence of Sums; Strong Laws of Large Numbers; Strong Law of Large Numbers for IID Sequences |
| 02/06 - 02/10 | 7.6; 8.1; 8.2 | Kolmogorov Three Series Theorem; Definition of Convergence in Distribution |
| 02/13 - 02/17 | 8.3; 8.4; 8.5 | Baby Skorohod Theorem; Weak Convergence Equivalences, Portmanteau Theorem; More Relations among Modes of Convergence |
| 02/20 - 02/24 | 8.6; 8.7; 8.7 | New Convergences from Old; Convergence to Types Theorem |
| 02/27 - 03/02 | 9.1, 9.2; 9.3; 9.4 | Moment Generating Functions and Central Limit Theorem, Characteristic Functions; Expansions; Moments and Derivatives |
| 03/05 - 03/09 | 9.5; 9.6; 9.7 | Uniqueness and Continuity; Selection Theorem, Tightness, and Prohorov's Theorem; Classical CLT for IID Random Variables |
| 03/12 - 03/16 | 9.8; 10.1; 10.2 | Lindeberg-Feller CLT; Radon-Nikodym Theorem; Definition of Conditional Expectation |
| 03/26 - 03/30 | 10.3; 10.4; 10.5 | Properties of Conditional Expectation; Martingales; Examples of Martingales |
| 04/02 - 04/06 | 10.6; 10.7; 10.8 | Connections between Martingales and Submartingales; Stopping Times; Positive Super Martingales |
| 04/09 - 04/13 | 10.8; 10.9; 10.10, 10.11 | Positive Super Martingales; Examples; Martingale and Submartingale Convergence, Regularity and Closure |
| 04/16 - 04/20 | 10.12; 10.13; 10.14 | Regularity and Stopping; Stopping Theorems; Wald's Identity and Random Walks |
| 04/23 - 04/27 | 10.15; 10.16; 10.16 | Reversed Martingales; Fundamental Theorems of Mathematical Finance |