# Probability Theory II

## Spring Semester 2012

Last update: 04/18/2012
• ## Course Announcement

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

• ## Course Syllabus

 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

• ## Relevant Course Materials

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