**Course Instructor:** Arnab Sen

Office: 238 Vincent Hall

Email: arnab@math.umn.edu

**Class time:** MW 4:00 - 5:15 pm

** Location: ** Vincent Hall 211

** Office hours: ** M 1:00-2:00 pm, W 11:30 - 12:30 pm, and by appointment.

** Course Description:** This is the second 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:

- Discrete-time martingales.
- Markov chains
- Ergodic theory
- Brownian motion.

** Prerequisite: ** Math 8651.

** Textbook (for both 8651 and 8652): **
Probability: Theory and Examples by Rick Durrett, Cambridge Series in
Statistical and Probabilistic Mathematics, 4th Edition. Available
online. A
fifth edition of the textbook is also avaialble
online.

Other recommended books:

1. Probability and Measure (3rd Edition) by Patrick Billingsley.

2. A Modern Approach to Probability theory by
Bert Fristedt and Lawrence Gray.

3. Probability with Martingales by David Williams.

4. Brownian motion by Peter Morters and Yuval Peres.

** Homework:** There will be 6 homework assignments. The lowest
score will be dropped in calculating the final score.

** Final Exam:** There will be a take-home final exam.

**Grading:** Homework 50%, Final 50%.

** Canvas: ** Homework assignments, lecture
notes and grades will be posted on Canvas.

**Announcements:** The take-home final exam
will be posted on Canvas just before 12:00 noon on Wednesday, May 8. The
deadline is 2pm, Friday May 10.

The materials from the last lecture (on May 6) will not be included in
the final exam.

Barring special circumstances, a paper copy of the final exam should
be submitted before the deadline. I will be in my office between 1-2pm
on Friday if you want hand it in person. Else, you can slide it under
my office door (if I am not around) or in my mailbox.

** Weekly schedule: **

Jan 23 | martingales, examples. |

Jan 28, 30 | Doob's decomposition, stopping times. No class on Jan 30. |

Feb 4, 6 | Doob's upcrossing inequality, martingale convergence theorem, maximal inequalities, L^p-martingale convergence theorem |

Feb 11, 13 | Galton-Watson Process, uniform integrability, uniform integrable martingales, Optional Stopping theorem, Gambler's ruin problem |

Feb 18, 20 | proof of OST, backward martingale, convergence, SLLN using backward martingale, Hewitt-Savage 0-1 law, exchangeable sequence |

Feb 25, 27 | de Finetti's theorem, polya's urn. Markov chains: defintion, construction and example, Markov and strong Markov property. |

Mar 4, 6 | Markov and strong Markov property (contd.), recurrent and transient states, characterization of recurrent and transient states for finite state Markov chain |

Mar 11, 13 | recurrence/transience of random walks on Z^d, existence and uniqueness of stationary measure and stationary distribution, positive and null recurrence |

Mar 18, 20 | spring break |

Mar 25, 27 | reverisble Markov chains, asymptotic density of states, periodicity, convergence to equilibrium, an introduction to mixing time of Markov chains: RW on cycle, RW on hypercube [You can find much more about the mixing time of Markov chains in this book by Levin, Peres and Wilmer]. |

Apr 1, 3 | Measure preserving systems, stationary sequence, ergodicity, examples: i.i.d. sequence, function of stationary sequence, stationary Markov chain, rotation on circle, Birkhoff's ergodic theorem |

Apr 8, 10 | asymptotic frequency of a fixed pattern in Markov chain, range of random walk, Kingman's subadditive ergodic theorem, first passage percolation, longest common subsequence |

Apr 15, 17 | Introduction to Brownian Motion. Construction of BM. Basic properties of BM. Holder exponent of Brownian paths. |

Apr 22, 24 | nowhere differentiability of BM, Markov and strong property of BM, Blumenthal's 0-1 law, Properties of zero set of BM, reflection principle |

Apr 29, May 1 | the distribution of the maximum value of BM, arcsine law for the last zero in the unite interval, martingales for BM, optional stopping theorem, Donsker's theorem |

May 6 | Skorokhod's embedding, proof of Donsker's theorem |