## Probability for PhD students in the Mathematics Department at UU

Lecture notes and assignments: available to those attending the course

NOTE: If you are attending the lectures and are not in the mailing list doktorand@math.uu.se, send me an email so I can include you to my emails.

An overview of probability, interpreted broadly, its foundations, uses, and applications.

PREREQUISITES: some kind of introductory course to probability, and a course in analysis. (Logic and naive set theory are, of course, assumed, by default.)

PRELIMINARY PLAN: This is a course for all PhD students, so, necessarily, it will not cover all details of a measure-theoretic graduate probability course. (For this, please wait until the spring.) It is meant to give an idea of what probability is about. We will cover the basics of rigorous probability theory and several topics, including (introduction to) Markov chains, ergodic theory, information theory, harmonic analysis on discrete spaces, probability on trees, and the basic random variables of probability, i.e., the Poisson process and Brownian motion. Both "discrete" and "continuous" probability will be discussed. The course will be somewhat iconoclastic in nature.

STYLE OF LECTURES: A mélange of theory, examples and applications.

LITERATURE: Internet and lecture notes.

EVALUATION: Course assignments.

- Assignment 1. Due Friday, 28 September
- Solutions
- Assignment 2. Due Friday, 12 October
- Solutions
- Assignment 3. Due Monday, 29 October
- Solutions
- Assignment 4. Due Friday, 16 November

GOALS: to understand what the subject is about, to be able to obtain working experience in some of its areas, to facilitate your understanding of probability-related literature, and to build bridges.

BIBLIOGRAPHY USED/SUGGESTED, IN RANDOM ORDER:

Rick Durret, Probability: Theory and Examples

Valentin Kolchin, Random Mappings

Patrick Billingsley, Convergence of Probability Measures

James Munkres, Topology

Davar Khoshnevisan, Probability

Olav Kallenberg, Foundations of Modern Probability

Eugene Dynkin and A.A. Yushkevich,
Markov Processes: Theorems and Problems

Terence Tao, An Introduction to Measure Theory

My notes on Markov Chains and Random Walks

My notes on the MSc Probability Course

SCHEDULE. If the link does not work, ask someone in the university. I also check my schedule by clicking there. Alternatively, follow the algorithm: (1) Click on http://www.math.uu.se/ (2) Click on Time tables (3) Select Type: Staff (3) Select Name: Write your name (4) Click on Search (5) A + sign will appear below. Click on it. (6) Look next and select starting week and (7) Click on Text format. N.B. Instead of a date range, in Sweden people assign a number to each week. The numbers are given here.

QUOTES:
*
• It is remarkable that a science which began with the consideration of games
of chance should have become the most important object of human knowledge
(Pierre Simon Laplace)
• The theory of probability as a mathematical discipline can and should
be developed from axioms in exactly the same way as geometry and algebra
(Andrei Nikolaievich Kolmogorov)
*