**Course Instructor:** Arnab Sen

Office: 238 Vincent Hall

Email: arnab@math.umn.edu

**Class time:** MWF 10:10 am -11:00 am

** Location: ** Vincent Hall 1

** Office hours: ** M 2-3pm, W 11-12pm 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.
- Discrete time martingales (if time permits).

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

3. Probability with Martingales by David Williams.

** Homework:** There will be biweekly homework assignments (around
7 in total). The lowest score will be dropped in calculating the final
score.

- Homework 1. Due Sep 23 (Wed) in class. Solution
- Homework 2. Due Oct 5 (Mon) in class. Solution
- Homework 3. Due Oct 21 (Wed) in class. Solution
- Homework 4. Due Nov 6 (Fri) in class Solution
- Homework 5. Due Nov 20 (Fri) in class Solution
- Homework 6. Due Dec 4 (Fri) in class. Solution
- Homework 7. Due Dec 14 (Mon) in class. Solution

** Final Exam:** 1:30-3:30pm, Wednesday, December 23.

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

**Announcements:**

** Final Exam ** (1:30 p.m.-3:30 p.m., 1:30-3:30pm, Wednesday, December 23
in class). You are allowed
to bring one sheet of A4 paper (both sides) with notes written by
yourself.

** Weekly schedule **

Sep 9, 11: Sigma-field, measure, Lebesgue measure, Dynkin's π-λ theorem, Carathéodory's Extension Theorem. |

Sep 14, 16, 18: Construction of Lebesgue measure, random variables. Distributions. |

Sep 21, 23, 25: CDF, Lebesgue integration, MCT, change of variable formula |

Sep 28, 30, Oct 2: DCT, inequalities, product measures, Fubini. |

Oct 5, 7, 9: independence, sum of independent random variables, weak law of large numbers |

Oct 12, 14, 16: Weierstrass approximation, coupon collector, Borel-Cantelli Lemmas and its applications, almost sure convergence |

Oct 19, 21, 23: Strong law of large numbers, Renewal theorem, Glivenko-Cantelli Theorem, Kolmogorov's 0-1 law |

Oct 26, 28, 30:Kolmogorov's maximal inequality, convergence of random series, Kolmogorov's three series theorem |

Nov 2, 4, 6: CLT, proof of CLT using Lindeberg's replacement lemma, Lindeberg-Feller CLT, weak convergence and its properties |

Nov 9, 11, 13: Helly's seclection theorem, tightness, characteristic function and its properties, inversion formula |

Nov 16, 18, 20: Continuity theorem for characteristic funcions, proof of CLT using characteristic funcion, weak convergence in R^d, multivariate CLT |

Nov 23, 25: Hoeffding's inequality, conditional expectation |

Nov 30, Dec 2, 4: Properties of conditional expectation, Radon-Nikodym theorem |

Dec 7, 9, 11: proof of Radon-Nikodym theorem, regular conditional distribution |

Dec 14, 16: some examples on probablistic methods, first and second moment method. |