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:

    1. Elements from measure theory (bare minimum which is needed to give measure-theoretic formulation of probability theory).

    2. Random variables, integration and independence.

    3. Different notion of convergences.

    4. Weak and strong laws of large numbers.

    5. Central limit theorems.

    6. Conditional distribution and conditional expectation.

    7. 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:

    1. Probability Theory by S.R.S. Varadhan

    2. Probability and Measure (3rd ed) by P. Billingsley

    3. Real Analysis for Graduate Students (4.2 version) by R. F. Bass. You can download this book from author's website

Grades:

  1. Homework (60%)

  2. Final Exam (40%)


Homework:

    1. Biweekly homework will be assigned on Canvas.

    2. 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.

    3. 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:

    1. Zoom exam, 1:30-3:30pm on Dec. 18 (Saturday)

    2. 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.

    3. 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.