Probability Theory

MATH/STAT7032, Spring semester, 2024

Instructor: Yizao Wang
Email: yizao.wang@uc.edu
Office: 4302 French Hall
Office Hours: by appointment.
Class meeting: MWF 12:20pm - 1:15pm, 60WCharlton 270.

Textbook

• Lecture notes (based on the textbook below) updated at Canvas.
• (Recommended) Probability: Theory and Examples, 4th Edition (Cambridge Series in Statistical and Probabilistic Mathematics) by Rick Durrett.
  You may find a PDF version of the 5th edition from the website of the author. (Either edition is good.)
  

Course description

We will cover the following materials: measure-theoretic foundations of probability; laws of large numbers; weak convergence; characteristic functions; central limit theorems. The course is required for the Probability Preliminary Exam.

Pre-req: Advanced Calculus (MATH 6001/6002) or equivalent. Real Analysis would be very helpful, but not required.

Grades

• Grading: homework 60%, midterm 15%, final 25%.
• Letter grades: A (90), A- (85), B+ (80), B (75), B- (70), etc.

Tentative Schedule and Homework

HWs will be posted, downloaded, submitted and graded via Canvas. See info/deadlines there.

• Week 1 (Jan 8, Mon): 1.1 Measure spaces, 1.2 Measurable functions. HW1.
• Week 2 (Jan 15): No class on MLK, Jan 15. 1.3 Integration.
• Week 3 (Jan 22): 1.4 Properties of integration. HW2.
• Week 4 (Jan 29): 1.5 Product spaces, Fubini's theorem. 2.1 Random variables as measurable functions.
• Week 5 (Feb 5): 2.2 Distributions, 2.3 Expected values. 2.4 Important examples of distributions. HW3.
• Week 6 (Feb 12): 2.5 Independent random variables. 3.1 Convergence in probability and in $L^p$. HW4.
• Week 7 (Feb 19): 3.2 $L^2$-convergence for partial sums of random variables. 3.3 Triangular arrays. 3.4 Weak law of large numbers. HW5
• Week 8 (Feb 26): midterm.
• Week 9 (Mar 4): 3.5 Borel-Cantelli Lemmas.
• Week 10 (Mar 11): Spring break
• Week 11 (Mar 18): 3.6 Strong law of large numbers. HW6
• Week 12 (Mar 25): 4.1 Weak convergence. HW7
• Week 13 (Apr 1): 4.2 Central limit theorem. 4.3 A multivariate central limit theorem. HW8.
• Week 14 (Apr 8): 4.4 Random walks and Brownian motion. 4.5 Poisson convergence.
• Week 15 (Apr 15): 3.7 Convergence of random series, and review.
• Week 16 (Apr 22): Final, TBA.

Sections 4.4, 4.5: expository lectures, not covered by final exam and the prelim.

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