# prob1

## Announcement

From Dr. Sean O’Rourke:

I will be organizing an introductory probability theory reading course next semester. We plan to meet Thursdays at 3pm (which is the same time as the probability seminar) when there is no external speaker for the probability seminar. The goal is to go through parts of Rick Durrett’s book “Probability: Theory and Examples.” The book is available free online: https://services.math.duke.edu/~rtd/PTE/PTEfive.pdf

Everyone at any level is welcome to attend. The material will be measure theory-based so the only prerequisite I recommend is a basic understanding of measure theory (for example, a basic understanding of the measure theory concepts from Real Analysis I, MATH 6310). As the department does not often offer a graduate-level introductory probability theory course, I hope this will be a good opportunity for anyone interested in the topic.

## Outline

Fall 2018, Ed Perkins taught Math 418/544 (from Durrett). To borrow their outline:

Together with Math 419/545 in term 2, these courses give a comprehensive introduction to mathematically rigorous and measure-theoretic probability theory for honours undergraduates and graduate students. Math 418/544 will “follow” (at times we will depart from the text presentation) the first 3 Chapters of the above text, and some of Chapter 4:

1. Foundations. Probability spaces, random variables, expectation, some results from measure theory.

2. Laws of Large Numbers. Independence, modes of convergence, Borel-Cantelli Lemma, Kolmogorov Extension Theorem (statement only), weak and strong laws of large numbers, Kolmogorov 0-1 Law, introduction to random walk.

3. Central Limit Theorem. Weak convergence, characteristic functions, Binomial convergence to the Poisson law, central limit theorem, multi-dimensional central limit theorem.

4. Conditional Expectation and Introduction to Martingales.

The course is intended to be useful for those who use probability as a tool in other fields, or planning to do research in probability. Probability theory has applications in analysis, electrical and computer engineering, statistics, economics, finance, applied mathematics, math biology, combinatorics and partial differential equations and has ties to many other fields. Students interested in these fields are encouraged to enrol.

Prerequisite: Background in measure theory (e.g. math 420) is not strictly required, and the requisite notions will be introduced in class. Some results from measure theory will be stated without proofs. It is often rewarding to take a measure theory course at the same time although it is not a corequisite either.

Fall 2014, Dr. O’Rourke taught MATH 6534 Topics in Mathematical Probability:

This course is intended as an introduction to modern probability theory. We will cover the foundations of measure theory, independence and conditioning, limit theorems for independent sums, large deviation estimates, Markov chains, and martingales.