# Probability theory I and II, fall 2016

**Teacher:** Konstantin Izyurov

**Scope:** 5 cr

**Type:** Advanced studies

**Teaching:**

**Topics: **

The goal of these two courses (Probability theory I and Probability theory II) is to give a rigorous and comprehensive introduction to Probability. We will start by going through Kolmogorov's axioms of Probability and the necessary tools from measure theory. Subsequently, the course will be focused around some of the following topics, roughly in that order of priority:

- Laws of large numbers for sums of independent random variables
- Gaussian random variables, central limit theorem and its extensions
- Markov chains and random walks
- Conditional probability and martingales
- Probabilistic arguments in other branches of mathematics
- Brownian motion

Related concepts, such as various notions of convergence of random variables, characteristic functions (a. k. a. Fourier transforms of random variables), and functional analysis tools, will be discussed along the way.

**Prerequisites:**

Analysis I-II, Topology I, Measure theory. The latter is not strictly necessary; however, if have not taken it, you will have to study some material on your own in parallel with Probability lectures.

## News

## Teaching schedule

**Probability theory I:**

Weeks 36-42, Tuesday and Wednesday 10-12 in room C124. Exercises: Thursday 10-12 in room C123.

**Probability theory II:**

Weeks 43-49, Tuesday and Wednesday 10-12 in room C124. Exercises: Thursday 10-12 in room C123.

## Exams

Exam for the second part of the course will take place on Friday, December 16-th, 10:15-14:00, at C124 Exactum. You may also take the Probability theory I exam at that time.

Exam lasts 3 hours 45 minutes.

You may not use anything but pen and paper in the exam. The exam will consist of several problems; solutions to these problems may require any material covered in the lectures and/or exercises. The material of the last to lectures of Probability theory II course is optional.

You may take the exam as many times as you wish, even if you passed it.

## Course material

Lecture notes (02.12. All the material required for the exam is now contained in the recture notes.)

Probability: theory and examples by R. Durrett definitely covers all the necessary material."Probability with martingales" by D. Williams is also highly recommended.

## Registration

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## Exercises

The exercises are not formally compulsory; however, I expect it to be very difficult to learn the material if you don't do any exercises.

Participation in the exercise class and/or returning the exercises may increase your grade.

### Assignments

The graded solutions to the past exercise sets can be retrieved from my office.

**Probability theory I:**

**Probability theory II:**

### Exercise classes

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

1. | Thursday | 10-12 | C123 | Konstantin Izyurov |

## Course feedback

Course feedback can be given at any point during the course. Click here.