Exercise set 0 - Measure theory clinic (to be discussed 3.09)
The goal of this exercise set is to help those students who have taken the "Mitta ja integraali" course to recall its content, and provide some guidance to those who are new to measure theory.
- if you have taken "Mitta ja integraali", just go ahead and see if you can solve the exercises.
- if you have not, read Chapter 1.2 of lecture notes for definitions, and then try to solve the exercises. Also consult the material for "Mitta ja integraali" course if necessary.
The first exercise class does not count towards exercise bonus; however, if you have troubles with these exercises, it's almost certain that you will have to put in a lot of additional work to be able to follow the course.
Reading assignment: week 1
Material for the first week: Lecture notes, pages 1-10.
Exercise set 1 - due 10.09
Reading assignment, week 2
In Week 2 of the course, we will go through sections 1.4-1.6 of Lecture notes. Note that section 1.5 will be mostly a recap of integration theory; the proofs will be short or not given at all. Please recall this material from "Mitta ja integraali" course if needed.
Week 1, recap
In the first week of the course, we covered the following questions:
What is a sigma-algebra? A Measure? A probability space? A random variable? A distribution of a random variable?
How does one describe a distribution of a scalar random variable by a function rather than by a measure? Which functions can be probability distribution functions?
What are some ways to construct sigma-algebras? To construct measures?
How to prove that two measures are equal without checking the equality on all measurable sets? What is a criterion for two Borel probability measures on the real line to be equal?
Philosophical aspects of Probability: https://plato.stanford.edu/entries/probability-interpret/
What's more important, probability spaces or random variables: Section 1 in https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-proba...
Exercise set II - due 17.09
Exercise set III - due 24.09
Reading assignment, week 3
We will go through the remaining parts of Chapter 1: Fubini and Kolmogorov extension.
Exercise set IV - due 1.10
Check Section 2.1 of lecture notes if you want to start doing these exercises before the lectures.
Weeks 2-3, recap
- what is a semi-ring? what is a pre-measure? Given a semi-ring and a pre-measure, how does one construct a measure from it? How can the sigma-algebra on which the measure is defined be described? Is the resulting measure unique? How does one check that the length is a pre-measure on the semi-ring of half-open intervals?
- how is the expectation - and the integral - defined? Is it always defined for a measurable function? What are some properties of the integral?
- what is the example that shows that one cannot always exchange a limit and an integral? what are some situations where it IS possible to exchange them? For non-negative functions, we always have an inequality between the integral of the limit and the limit of the integral; which way does it go?
- how do we define - and construct - a direct product of two measures? How to compute a product measure of a set using integral? What are some situations in which it is possible to exchange the order of integration?
- how to compute a measure of a Borel-measurable set in a metric space using only open and closed sets?
Reading assignment, week 4
We will discuss Kolmogorov extension theorem (Section 1.7) and move on to independent random variables and their properties (Sections 2.1-2.3).
Exercise set V: due 8.10
Reading assignment, weeks 5-6
We will study the laws of large numbers (sections 2.3 and 2.5), Kolmogorov's 0-1 law (section 2.6) and different notions of convergence for random variables (sections 2.7-2.8). Time permitting, we will start discussing characteristic functions and the proof of the Central limit theorem.
Exercis set VI: due 17.10
Weeks 4-5, recap
What is a consistent family of measures? How does one extend such a family to a single measure on the infinite product space?
What is the definition of independence for finite collection of events/random variables/sigma-algebras? And for infinite collections? What's the relation between independence and pairwise independence? What is the relation between (infinite) direct products of probability spaces and (infinite) sequences of independent random variables?
What are some necessary and sufficient conditions for scalar random variables to be independent?
What is a Gaussian random variable? What are some "surprising" examples of independent random variables related to Gaussians?
What does the weak law of large numbers say? And the strong one? How does one prove the WLLN in the case of finite variance? What is the idea of the proof for the general case? What is Borel-Cantelli lemma? How does one apply it to the proof of SLLN in the case of finite fourth moment? What is a tail event and what can be said about its probability?
The Criticism of Homer.
Master's Programme in Mathematics and Statistics is responsible for the course.
The course belongs to the Mathematics and Applied mathematics module.
The course is available to students from other degree programmes.
Analysis I-II, Topology I, Vector analysis I, Measure and integral
Vector analysis II, Probability II.
The course gives a firm theoretical ground for Probability and introduces some of its classical results
Recommended time/stage of studies for completion: 1. year
Term/teaching period when the course will be offered: varying
Measure theoretic foundations of probability, independence, laws of large numbers, characteristic functions and the central limit theorm, Gaussian measures, recurrence/transience of random walks
Lecture notes; D. Williams: "Probability with martingales", R. Durrett: "Probability: theory and examples"
Lectures and exercise classes, possibly other methods like reading assignments+discussion
Exam and excercises, Course will be graded with grades 1-5
Exam, other methods will be described later