Spring 2022 MATH 8652 Theory of Probability



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Time and location: 2:30-3:45pm MW Vincent Hall 1


Zoom office hour: 7:30-8:30pm MTu


About this course: This is the second half of a yearly sequence of graduate probability theory. We aim to cover the following main topics:

    1. Discrete time martingales

    2. Markov chains

    3. Ergodic theory

    4. Brownian motion

Prerequisite: Math 8651


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. Probability with Martingales by D. Williams

    4. Brownian Motion by P. Morters and Y. Peres

Grades:

  1. Homework (50%)

  2. Take Home Final (50%)


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.

Take home final:

  1. The exam will be on Canvas at 8am on May 2 (Monday). The deadline is 8pm on May 4 (Wednesday).

  2. Your solution should be uploaded to Canvas by the deadline.

  3. The material on May 2 will not be covered.

  4. I will not hold the usual office hour, but you can send me an email in case the problems are not clear to you.

Weekly schedule:

  • Lecture 1 (Jan 19): Martingale, examples

  • Lecture 2 (Jan 24): Doob's decomposition, stoping times

  • Lecture 3 (Jan 26): Doob's upcrossing inequality, martingale convergence theorem

  • Lecture 4 (Jan 31): Maximal inequalities, L^p-martingale convergence theorem

  • Lecture 5 (Feb 2): Galton-Watson Process

  • Lecture 6 (Feb 7): Uniform integrability, L^1-convergence of martingales

  • Lecture 7 (Feb 9): Optional stopping theorem, Gambler's ruin problem

  • Lecture 8 (Feb 14): Backward martingales, SLLN via reverse martingale, Hewitt-Savage 0-1 Law

  • Lecture 9 (Feb 16): Proof of Hewitt-Savage 0-1 Law, exchangeability, de Finetti's theorem

  • Lecture 10 (Feb 21): Markov Chains

  • Lecture 11 (Feb 23): Strong Markov property

  • Lecture 12 (Feb 28): Recurrent and transient states

  • Lecture 13 (Mar 2): Random walk on Z^d

  • Lecture 14 (Mar 14): Existence and uniqueness of stationary measure

  • Lecture 15 (Mar 16): Existence of stationary distribution, reversible Markov chain

  • Lecture 16 (Mar 21): Convergence of Markov chains I

  • Lecture 17 (Mar 23): Convergence of Markov chains II, mixing times

  • Lecture 18 (Mar 28): Measure preserving systems, stationary sequence, ergodicity, examples

  • Lecture 19 (Mar 30): Birkhoff's ergodic theorem and its applications

  • Lecture 20 (Apr 4): Kingman's subadditive ergodic theorem

  • Lecture 21 (Apr 6): First passage percolation, longest increasing sequence, range of random walks

  • Lecture 22 (Apr 11): Brownian motion, construction of BM

  • Lecture 23 (Apr 13): Construction of BM, Properties of BM

  • Lecture 24 (Apr 18): Nowhere differentiability, Markov and strong property of BM, Blumenthal's 0-1 law

  • Lecture 25 (Apr 20): Properties of zero set of BM, reflection principle

  • Lecture 26 (Apr 25): The distribution of the maximum value of BM, arcsine law for the last zero in the unite interval,

  • Lecture 27 (Apr 27): Martingales for BM, optional stopping theorem, Donsker's theorem

  • Lecture 28 (May 2): Skorokhod's embedding, proof of Donsker's theorem