**Course Instructor:** Arnab Sen

Office: 238 Vincent Hall

Email: arnab@math.umn.edu

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

** Location: ** Vincent Hall 211

** Office hours: ** M 11-12pm, T 1-2pm or by appointment.

** Course Description:** 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:

- Elements from measure theory (bare minimum which is needed to give measure-theoretic formulation of probability theory).
- Random variables, integration and independence.
- Different notion of convergences.
- Weak and strong laws of large numbers.
- Central limit theorems.
- Conditional distribution and conditional expectation.
- Other topics (depending on time).

** Prerequisite: ** Upper division analysis: Math 5616 (or
equivalent) - the students should be familier 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 (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. Probability with Martingales by David Williams.

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

** Homework:** There will be biweekly homework assignments (around
7 in total). The lowest score will be dropped in calculating the final
score. Homework are due on the corresponding deadlines in class. Late
homework will not be accepted. You are allowed and even encoraged 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:** 4:00-6:00pm, Monday, December 17.

**Grading:** Homework 60%, Final 40%.

**Announcements:**

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

** Weekly schedule **

Sep 5: Sigma-field, measure, Lebesgue measure |

Sep 10, 12: Dynkin's π-λ theorem, Carathéodory's Extension Theorem, random variables. |

Sep 17, 19: Distributions, CDF, Lebesgue integration, MCT |

Sep 24, 26: DCT, change of variable formula, inequalities, product measures i |

Oct 1, 3: product measure, Fubini, independence, sum of independent random variables |

Oct 8, 10: Weak law of large numbers, Weierstrass approximation, coupon collector, Borel-Cantelli Lemmas and its applications |

Oct 15, 17: Strong law of large numbers, Renewal theorem, Glivenko-Cantelli Theorem, Kolmogorov's 0-1 law |

Oct 22, 24 : Kolmogorov's maximal inequality, convergence of random series, weak convergence and its properties, Helly's seclection theorem, tightness |

Oct 29, 31: Prokhorov's theorem, central limit theorem, proof of CLT using Lindeberg's replacement lemma, |

Nov 5, 7: Lindeberg-Feller CLT, examples, characteristic function, inversion formula |

Nov 12, 14: continuity theorem, CLT via characteristic function, weak convergence via method of moments |

Nov 19, 21: Poisson convergence, weak convergence in R^d |

Nov 26, 28: Multivariate CLT, Hoeffding's inequality, conditional expectation |

Dec 3, 5: properties of conditional expectation, Radon-Nikodym theorem |

Dec 10*, 12*: proof of Radon-Nikodym theorem, regular conditional distribution, first and second moment method. |