Anyone interested in attending should email james.duffy [at] economics.ox.ac.uk
Format: We meet once a week, again rescheduled for Sundays at 6:30-8:00pm in the Conference Room (L staircase) at Nuffield. The first meeting will be held on 13 July. Each week I will assign reading and problems from Pollard (2002) [first two chapters: here], and you will be asked to present your solutions to some of these to the rest of the group.
Note on the text: Pollard (2002) is a very well-written textbook, and completely rigorous despite its somewhat dinky title. The main issue some people have with it is Pollard's idiosyncratic "de Finetti" notation, which is explained in Chapter 1. My advice is to learn and stick to his notation, rather than attempting to translate everything into more familiar notation. You will very quickly get used to it ...
We will aim to go through the material in Chapters 1-6 of Pollard (2002). This means we will stop just short of proving the CLT: those of you who want to see the proof (and what course in probability theory would be complete without it?) will have a chance to do so in Michaelmas, when Bent and I will run a reading group on it.
Summaries: will be made available at least for the earlier weeks, if I have sufficient time. If you want to edit the supplied sources, you will need to also download this macro file. You may also need to download LyX. Please report any errors to me.
Reading:
Skim through Chapter 1, Sections 1-4
Carefully read Chapter 2, Sections 1-2
Problems:
Verify property (i) on page 19
Example <5>: explain why H is not in E_delta, but H_N is
Example <7>: prove the assertion in the first paragraph
Problems from Chapter 2: do 3, 4, 5, 6 [and 12 if you are sufficiently motivated]
Reading:
Carefully read Chapter 2, Sections 3 and 5: the real meat here is the dominated convergence theorem (Theorem <22>) and what is sometimes termed the dominated derivatives theorem (Example <23>).
Skim Section 4 if you have time: do not worry about understanding every assertion (it is very terse); the main thing to understand is that the integral for general measurable functions is constructed by using the approximating sequence of simple functions from Lemma <11>.
Problems:
Problems from Chapter 2: do 6 (from last week!), 10, 7 and 8.
Construct a counterexample to <24>, i.e. a case in which the integral of the derivative does not equal the derivative of the integral. (Of course, this can only occur if the domination condition fails.)
Prove Lemma 3.2 in the summary
Reading: Chapter 2, Sections 6 and 7. (Section 7 is notably terse, but covers important ground. I will try to lay out the key results somewhat more clearly in my summary.)
Problems: see here.
Reading:
Chapter 2, Sections 8 and 9.
At some point in the (possibly distant) future, we are going to need to apply Theorem <38> from Section 10. So if you have time, you might want to read that section down to the statement of the theorem (but ignore the proof). But don't worry about this too much -- it is rather abstract.
Problems: see here.
I don't think that there is so much to do this week, so you might want to use your time to undertake some unattempted problems from previous weeks, to consolidate your knowledge of the material in Chapter 2.
Reading:
Read Chapter 3, Sections 1 and 2 carefully.
Skip over Example <1>, at least on your first reading: unfortunately, I think Pollard's explanation is simply too terse here. I might say something about the Cantor set at our meeting. The main thing to take away from this example is that there exist measures whose associated distribution functions are continuous, but which do not have densities with respect to Lebesgue measure.
The most important parts to understand are the proof of Lemma <5> and the proof of the Radon-Nikodym theorem (a special case of the Lebesgue decompostion). You should write up a fuller account of these for your notes.
Problems: see here.
Reading:
Chapter 4, Sections 1 and 2.
Note that last week's material is in no way a prerequisite for this week's: so please move onto Chapter 4 immediately, even if you haven't mastered last week's work.
Problems: see here.
Reading:
Chapter 4, Sections 6 and 7: these take you through the proof of the Kolmogorov SLLN
You may find Theorem <39> a little mysterious: we will discuss it at our next meeting, and hopefully build some more intuition for the blocking argument
Problems: see here
There no reading set for this week. To aid your revision of the material that we have covered thus far, please attempt some of the problems here (UPDATED: 28/8). (I have set quite a large number of these, in the hope that you will all be able to successfully complete at least some of these.) You might also want to attempt some of the questions that I assigned in previous weeks.
Reading:
Chapter 4, Sections 3 and 4.
On your first reading, skip the proof of Corollary <22>, Examples <26>--<27>, Exercise <31> and Example <32>. Don’t bother looking at these until after you have attempted some of the problems. [There is also no need to consult Section 2.11 for more details about lambda-cones.]
You may find this material a little abstract. Its utility will become more apparent next week, when we study conditional distributions, which are themselves just probability kernels. You will also learn about the conditions under which the order of an iterated integral may be changed (the importance of which can hardly be overstated ...)
Problems: see here
Reading:
Chapter 5, Section 1--3.
First read Sections 1--2, skipping Examples <6> and <7>. Then attempt questions (i)--(iii), possibly skipping the parts that ask you to use lambda-cones. Then read Section 3, and attempt the remaining questions. If you have the appetite (and time) for more, read Example <6>.
Problems: see here
Reading:
Chapter 5, Section 6.
Skip the proof of Theorem <19> on your first reading, then attempt problems (i)--(iv). Then read the proof of Theorem <19>, and attempt the remaining problems.
Note that the really important material is that which concerns conditioning on a sub-sigma-field (pp. 126--128): we will need this, and only this, when we come to martingales (the final two weeks of this course).
Problems: see here
Reading:
Chapter 6, Sections 1--2
Read Section 1, skipping Example <7>. Then attempt problems (i) and (ii). Then read Section 2, skipping Exercises <15> (and the paragraph immediately following it) and <18>. Then attempt the remaining problems.
To be honest, I have never developed a good intuition for why the pre-tau sigma-field is defined the way it is -- in spite of Pollard’s best efforts to give some sort of explanation in the paragraph preceding Defintion <11>. So please do not get too hung up on this. What ultimately matters is that the definition ‘works’, in the sense that it entails the stopping time lemma (Lemma <16>).
Problems: see here
Reading:
Chapter 6, Sections 3--6
To build intuition for what follows, first make an attempt at problem (i). [It does not require anything that we haven’t already covered.] Then read Sections 3--4, skipping the (long!) Example <25>, and attempt problems (ii)--(v). Then read Sections 5--6, omitting Example <37>, and attempt the remaining problems. If you have time, read Example <37>, which is quite intriguing.
Problems: see here