A course for Ph. D. Students

## HW Questions

On the metric space question: just take the max and min over the ratios p_i/q_i where p and q are the two vectors. Sorry about the j business!

## Day Summaries

Well, it looks like were terminally behind on the day summaries. Its SPRING!

## Day 5 and 6

We will continue with renewal theory and morph it into the theory of Markov chains, starting with a nice geometric proof of uniqueness of the stationary measure of a finite Markov chain. We'll do the basic convergence theory of finite state spaces rather quickly so we can get to the more mathematically challenging theory of uncountable state spaces, like the real line. I'll mention problems in class on Monday and Wednesday, and I may make an assignment from the text, but we'll have to see how far we get first.

## Day 3 and 4

We will complete the new proof of the Choque-Deny lemma, and use it to give a proof of a generic special case of the discrete renwal theorem. We'll then look at an alternative proof of the Choquet-Deny lemma that exploits martingale theory and a generalization of the Kolmogorov 0-1 law. We'll sneak up on this result via two other proofs: martingale proof of the law of large numbers and a martingale proof of Kolmogorov's 0-1 law. This will set the stage for our second 0-1 law and the second proof of Choquet-Deny.

HW2 is due Monday, January 30. You do not need to latex the solutions, but if you are here and you do not know latex, you had better learn quickly.

## Day 2

We gave Kakutani's proof of the existence of a stationary measure for a Markov Chain on a finite state space. We then studied the simplest renewal equation, the one from Hardy's Pure Mathematics, and the Mathematical Tripos of 1906. We got the asymptotics by means of the very general "compactness+determining condition method", which we will use many more times.

We also gave a good start on the proof of the Choquet-Deny Lemma, but we have one more fact to justify next time This will be easy via Fubini's theorem, and we will also have time to look at the structure and logic of this new proof.

Since the week is a strangely short one, we will not have HW this week. Nevertheless, you should look at the discussion of the renewal theorem in our text.

## TA: Terry Liang

Our TA is Terry Liang (tengyuan@wharton.upenn.edu). He is doing job interviews during the first half of the semester and will have irregular office hours until February 15. I will post further information as it is available. For quick questions for Terry you can use his email, but keep in mind that he may be in an airplane.

## Day One

We'll review some facts about uniform integrability, proof "Plancherel's phi Lemma" and get a representation theorem for uniformly integrable martingales. I'll post the first HW assigment on Thursday, so you'll need to check back.

I'll go over the course plan and the logistical details in class, but there are two bottom lines: (1) we will have about the same structure we had for 930 and (2) there will be a 4 page essay, but there will not do a "tool box" essay like was done in 2015. This should make life easier for everybody.

## First HW

HW1 is due Wenesday, January 18.

## General Information

This website is the place where one checks in to find all of the additional information about our course, including periodic postings of supplemental material.

You can to look at the course syllabus for general information about the course as well as information about grading, homework, the midterm, and the final exam.

The course will have a less regimented fell than Stat 931, but this actually puts more reponsibility on the student to be accountable for mastering the material of the course. Homework assigments will be less regular, and more creative.

Please do review the course policies.

Feel free to contact me if you have questions about the suitability of the course for you. In general the course will only be appropriate if you have had a solid background in real analysis, preferably at the graduate level.

It will only be in EXREMELY special cases that one can take Stat 931 without having taken Stat 930.