**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: ** MW 11:40-12:40pm or 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. Also available
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.

- Homework 1. Due Feb 15 (Monday) in class. Solution.
- Homework 2. Due Feb 24 (Wednesday) in class. Solution.
- Homework 3. Due Mar 9 (Wednesday) in class. Solution.
- Homework 4. Due Mar 30 (Wednesday) in class. Solution.
- Homework 5. Due April 25 (Monday) in class. Solution.
- Homework 6. Due May 4 (Wednesday) in class. Solution.

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

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

**Announcements:** 1. The deadline of Homework 6 has been extended to May 4.

2. Your take home final exam will be emailed to your UMN account on Sunday 9 am. The deadline is 12 noon, Wednesday May 11.

** Weekly schedule: **

Jan 20 | martingales, stopping times. |

Jan 25, 27 | optional stopping theorem, Doob's decomposition, Doob's upcrossing inequality, martingale convergence theorem |

Feb 1, 3 | Polya's urn, recurrence of random walk on Z, more on optional stopping theorem, maximal inequalities, L^p-martingale convergence theorem, uniform integrability, martinagle convergence in L^1, Lévy's forward law |

Feb 8, 10 | Reverse martingale, proof of SLLN using reverse martingale, gambler's ruin problem using optional stopping theorem, Ballot theorem |

Feb 15, 17 | Discrete Dirichlet problem, Galton-Watson process. [You can find more about random walks on graphs and electrical networks here] , Markov chains : construction and examples, Kolmogorov's extension theorem |

Feb 22, 24 | proof of existence of Markov chains, Markov and strong Markov property, recurrent and transient states |

Feb 29, Mar 2 | characterization of recurrent and transient states for finite state Markov chain, recurrence/transience of random walks on Z^d, birth death chains |

Mar 7, 9 | existence and uniqueness of stationary measure and stationary distribution, asymptotic density of states |

Mar 14, 16 | spring break |

Mar 21, 23 | periodicity, convergence to equilibrium. An introduction to mixing time of Markov chains: coupling and strong stationary time RW on cycle, RW on hypercube and Top-to-random shuffle [You can find much more about this in this wonderful book by Levin, Peres and Wilmer]. |

Mar 28, 30 | Measure preserving systems, stationary sequence, ergodicity, examples: i.i.d. sequence, function of stationary sequence, Markov chain, rotation on circle, doubling map on circle |

April 4, 6 | examples continued, Birkhoff's ergodic theory, asymptotic frequency of a fixed pattern in Markov chain, range of random walk |

April 11, 13 | Kingman's subadditive ergodic theory, applications: product of i.i.d. random matrices and first passage percolation. Introduction to Brownian Motion. An attempt to construct BM on the product space R^[0, \infty). |

April 18, 20 | Construction of BM. Properties of BM, non-differentiability, Holder exponent of Brownian paths |

April 25, 27 | Markov and strong property of BM, Blumenthal's 0-1 law, reflection principle, distribution of hitting times and maxima of BM, arsine law for last zero of BM. |

May 2, 4 | Properties of zero set of BM, Brownian motion as a martingale, Donsker's theorem (without proof) |