**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 emailed to your UMN account before 12:00pm on Wednesday, May 8. The
deadline is 11:59 am, Friday May 10.

** 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 | |

Apr 29, May 1 | |

May 6 |