Fall 2021 MATH 8651 Theory of Probability
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Time and location: 2:30-3:45pm MW Vincent Hall 2
Zoom office hour: 7:30pm-8:30pm MTu or by appointment. You can find the zoom link on Canvas.
About this course: This is the first half of a yearly sequence of graduate probability theory at the measure-theoretic level. There will be an emphasis on rigorous proofs. In this course, we aim to cover the following main topics:
Elements from measure theory (bare minimum which is needed to give measure-theoretic formulation of probability theory).
Random variables, integration and independence.
Different notion of convergences.
Weak and strong laws of large numbers.
Central limit theorems.
Conditional distribution and conditional expectation.
Other topics (depending on time).
Prerequisite: Upper division analysis: Math 5616 (or equivalent) - the students should be familiar with concepts such as uniform convergence, continuity, sequences and series of numbers and functions, Riemann integral and the topology (open, closed, compact sets, etc.) of the real line. No background in measure theory will be assumed. Some familiarity with basic undergrad probability will be helpful.
Textbook: Probability: Theory and Examples, 5th ed. by Richard Durrett. You can download this book from author's website
Other books:
Probability Theory by S.R.S. Varadhan
Probability and Measure (3rd ed) by P. Billingsley
Real Analysis for Graduate Students (4.2 version) by R. F. Bass. You can download this book from author's website
Grades:
Homework (60%)
Final Exam (40%)
Homework:
Biweekly homework will be assigned on Canvas.
Please submit your homework solutions on Canvas before the deadline. This is your responsibility to check that your submission is correct and complete. Late homework or any unsuccessful submissions will not be accepted.
The lowest score will be dropped in the final score calculation. You are allowed and encouraged to discuss homework solutions with your friends. However, you have to write your own solutions. To get full credit, be neat and answer with reasons.
Final exam:
Zoom exam, 1:30-3:30pm on Dec. 18 (Saturday)
The exam will be held on Zoom. During the exam, you will be asked to turn on your Camera. Please be sure that the audio and camera of your device are available during the exam.
We take your solutions via Canvas.
Weekly schedule:
Lecture 1 (Sep 8): Sigma-field, measure, Caratheodory extension theorem
Lecture 2 (Sep 13): Dynkin's π-λ theorem, Proof of Caratheodory's extension theorem
Lecture 3 (Sep 15): Lebesgue measure, Random variables
Lecture 4 (Sep 20): Properties of random variables, Distributions, Cumulative distribution functions
Lecture 5 (Sep 22): Probability measures on R, Expectation, Monotone convergence theorem
Lecture 6 (Sep 27): Limiting theorems of expectations, Transformation of integrals, Inequalities of expectations
Lecture 7 (Sep 29): More inequalities of expectations, Product measures
Lecture 8 (Oct 4): Fubini's theorem, Independence
Lecture 9 (Oct 6): Independence, Sum of independent random variables
Lecture 10 (Oct 11): Weak law of large numbers, Weierstrass approximation, coupon collector
Lecture 11 (Oct 13): Borel-Cantelli Lemmas, Applications
Lecture 12 (Oct 18): Strong law of large numbers, Renewal theorem
Lecture 13 (Oct 20): Glivenko-Cantelli theorem, Kolmogorov's 0-1 theorem, Kolmogorov's maximal inequality
Lecture 14 (Oct 25): First and second moment method
Lecture 15 (Oct 27): Weak convergence
Lecture 16 (Nov 1): Properties of weak convergence, Helle's selection theorem, tightness
Lecture 17 (Nov 3): Central limit theorem, Lindeberg's replacement lemma
Lecture 18 (Nov 8): Lindeberg-Feller CLT, Examples
Lecture 19 (Nov 10): Characteristic function, Inversion formula
Lecture 20 (Nov 15): Continuity theorem, Examples
Lecture 21 (Nov 17): CLT via characteristic function, weak convergence via moment method
Lecture 22 (Nov 22): Poisson convergence
Lecture 23 (Nov 24): Weak convergence in R^d, Multivariate CLT
Lecture 24 (Nov 29): Concentration of measure I
Lecture 25 (Dec 1): Concentration of measure II
Lecture 26 (Dec 6): Stein's method
Lecture 27 (Dec 8): Conditional expectation
Lecture 28 (Dec 13*): Properties of conditional expectations
Lecture 29 (Dec 15*): Radon-Nykodym theorem, Regular conditional distribution
* Not included in the final exam.