A course for Ph. D. Students

Course Blog Spring 2015

Statistics Quals

The quals will cover the material of 531-532. Naturally there will be an emphasis on the most fundamental results, especially but not exclusively those in the text. It's often the case that one master fundamentals is by doing more advanced materials.

Here is some guidance.

1. Look at each chapter in the text. What is the most significant theorem? Can you prove it! Can you give more than one proof? If you can give good answers to this question for each chapter you are in fine shape.

2. Look at each chapter and look over the problems. Some of these are dull as dishwater. Do these in your head--- they are too boring or too tedious to do on paper, yet they may still reminded you of useful definitions. Now look at the other problems and in each chapter find two three that are honestly interesting. Solve these problems.

3. Definitions. One way to create a disaster for yourself is to show up without total mastery of the definitions. Why would one even skate near that thin ice?

4. Old quals. These do differ a bit from year to year, but at least read them over and ask: "Did we put much energy into this material this year?" If so, that is a good study topic.

5. Organized thoughts. OK, you "know the SLLN"--- ah, hum, are you a MASTER of the SLLN? Can you prove it six ways from Sunday? How about the CLT? How about the theory of martingales? How about Brownian motion? How about Markov chain? How about renewal theory? You get the picture. Now. Prioritize and master the material. Be able to explain yourself. If you think you have intuition about something and you "cannot explain it", then you have not taken the steps to organize your knowledge. Take them!

Final Up-Date

I have graded the revisions of the last Toolbox Essays, and your course grades have been submitted. Also, since the earlier essays had a fair number of slips, I have removed all of the essays from the web. I may post a few of these at some later date, but I will ask your permission if I do. In the meanwhile, feel free to post them on your own website if you like.

Revisions

"The best writing is rewriting." - E. B. White

Finding Bugs in Your Work

It's Saturday morning and I am working on a paper of mine that I have not looked at in a couple of days. On page 5, I see I have written "As an immediate consequence of Proposition 4 and Lemmas 1 and 2..."

Naturally what then follows is a bug! Here is what I learn, over and over. Bugs take place when you see words like "obviously", "trivially", "as usual" or any of the other phrases that we use (even on ourselves) to intimidate rather than illuminate.

Search your essays for these words. They are especially dangerous when used with a "clumped explanation". One is not too likely make a mistake saying "from (4) we see" but when one says "by (3), (8), and (15)" the likelihood of a mistake skyrockets.

Tool Box Essays: Self-evaluation Form

Self-evaluations are very useful for take home finals, and I think they will be useful for the final Tool Box Essay. Naturally, the evaluations are more qualitative. Each essay needs to be accompanied with a completed self-evaluation form.

Semi-Group Papers --- One You Can't Miss

It finally dawns on me that some of the links I have posted may be a bit too far down the trail to spark your interests. Here is a change of pace: Using one of the essential ideas of semi-groups ---- the re solvent ---- you can give, slick, fast, informative proof of all of the big results in classical linear algebra in just under 17 pages. You can't afford not to look at this.

It will be a few days until we get to it, but ere long we'll be looking at the product formula of P. Chernoff. It is a theorem that gives a one-line proof of the CLT as well as several other goodies. We'll start with a very easy version which is almost as easy as (1+t/n)^n.

One of our sources is a recent paper by D. Pfeifer. It may look a little remote at first glance, but it does have some intuitive content if you take the details with a drop of honey. His related article has applications of semi-groups to approximations, including the semi-group proof of Le Cam's inequality for Poisson approximation.

Words, Words, Words

The lead echoes Shakespeare, but that is not my point. The point is that although it looks like we need a lot of new words to set up the theory of semi-groups, this overhead is peanuts in comparison to any little slice of genetics. I needed a little gloss up and it took me to the Wiki page on the genetic code. OMG, you need a fountain of definitions to parse this.

Three Bears Problem

In Goldie Locks, there are three bowls of porridge. One too hot, one too cold and one just right. Topics for Toolbox essays face a similar problem. If you pick a topic that is too far from your genuine knowledge base, it is too hot for you to handle. In the time you have, you will not achieve enough mastery to create a good essay. If you pick a topic that is routine, textbook material (or just a little more), you probably have material that is too cold. There is just not enough genuine new knowledge in the topic to provide an essay that is interesting to a reader ---- or even an author.

There is no surefire way to get that bowl of porridge that is just right. Still, there are some things to try, and there are some "tells".

One "tell" is that if the only references are textbooks, or if the journal references are just "name dropped." Such an essay is almost certainly weak. Perhaps it can be acceptable if the topic advanced (or specialized) and if the exposition is truly thoughtful. A textbook topic was not a great choice in the first essay, and it would be a dreadful choice in the third essay.

In fact, almost any essay that shows some thoughtful engagement with material that goes beyond the textbook level will be a good essay. I have given many links to papers on this blog, and simply using mathscinet will lead you to many more papers than you can ever read. To "get" this course you have to spend at lease some time engaging a paper from a journal (or ArXive).

The trick is often to look at a paper to see if there is anything in it that honestly interests you. It will seldom be the whole paper. It might just be a nice lemma, inequality, or example. If you look into this small piece and make it your own, you will have the raw material that you need for a good toolbox essay.

This is about as much coaching as I can offer on finding a topic, except to add that you should make your essay as relevant as possible to our course --- i.e. Stat 531 (not Stat 530 or some other course). You can "go off road" if that is what interests you, but you need to have an honest link to our course ---- specifically, Martingale CLT, compactness arguments like the Beuring subsequence argument, Wiener's Tauberian theorem, the Erdos Feller Pollard arguments, the ergodic theorem for Markov chains and processes, the notion of a semi-group --- Dynkin's formula, P. Chernoff's formula, the Kolmogorov equations, BM, Poisson process, etc.

Finally, there is the "authenticity" that I have talked about at great length. Please read what I have written before, but --- in a nutshell, you have to work on what interests YOU and you have to explain why YOU found the information in your essay to be worth sharing. Essays that are authentic are always interesting. Even if they have slips or gaps, they are still the genuine thoughts of a human being at a specific time in life.

Three Semi-group Papers

One of our "target papers" is

Central Limit Theorem for Products of Random Matrices by Marc A. Berger
Source: Transactions of the American Mathematical Society, Vol. 285, No. 2 (Oct., 1984).

This is a very instructive paper that puts the finger on why "product formulas" and the CLT are natural partners. The main fact that it uses from the theory of semi-groups is contained in a classic six page paper.


Note on Product Formulas for Operator Semigroups by PAUL R. CHERNOFF, JOURNAL OF FUNCTIONAL ANALYSIS 2, 238-242 (1968)

Finally, there is the article by Chung that I mentioned where he gives easy proofs of some classic product formula; this one I can offer by link! The Berger paper tells us what semi-groups can do in probability theory, and the Chung paper tells use what probability estimates of a simple kind can tell us about semi-groups.

Markov Decision Problems

One of the sources of importance for Markov processes is that they lay down the ground rules for Markov decision problems. Moreover, MDPs give the critical synthesis between probability and optimization. After we have our feet wet with more general Markov processes, we'll want to spend at least some time with MDPs. It may look a little abstract at first, but a nice source for us is the elegant paper by Karatzas and Sudderth. We'll spend at least one lecture on it, perhaps with another for warm up. This will come near the "end" but you can look ahead.

Burkholder's Inequality

Burkholder's Inequality is a major refinement of the identity for the variance that one gets from orthogonality of the MDS. David Pollard has posted a proof; some review of Pollard's special notation may be needed to parse his exposition, but it is quite nice. We'll go over this in class, and --- except for another look at Hoeffding's inequality --- this will complete our work with discrete time martingales. Soon we'll be on to continuous time processes for good.

Noted in Passing

Lee Kuon Yew, one of the most brilliant leaders of the post-WWII era, is gone. If you study what he did, you will be amazed --- and thankful.

His son Lee Hsien Long is a contemporary leader in Singapore and (in an earlier life) he was a talented mathematician with a top Cambridge degree. My friend and co-author David Aldous was a contemporary of H. L. Lee in the small math group at Cambridge. About twenty years ago David said to me, "Everybody knew who was the smartest in the group --- Lee --- by a mile."

Let's hope that the positive experiences of the past are continued and multiplied.

Strange Suggestion ... Maybe Too Strange, but Maybe Not

It can be fun to have some generic variations in your head so that when you see a new result you can ask if it works in the generic variation. We'll cover Chaterjee's identity on Monday (3/23) and you might ask what variations are possible. Since covariance is a special inner product, any variation on inner products can offer the possibility of a variation of Chaterjee's identity You then need two pieces of luck: (1) that there is an analog in the new context and (2) that the analog is more than a curiosity. Typically (1) is easier than (2), and for a home run you need both. Still, a journey of a thousand miles ...

So, you might ask, what is the relativistic version of Chaterjee's identity? Here is useful to have a quick gloss on Einstein Addition. Thankfully no knowledge of physics is needed; independent of any application story the formulas are cool. At a minimum, if you let c go to infinity, you recover your old formulas.

Revisions

Some revisions to the second essay have been posted.

Third Essay --- and Other Things

The third Toolbox Essay will be due on April 15.

The rule are all as before. Now it seems that everyone has the hang of things. I'll keep posting papers and throwing out suggestions in class.

To keep everyone's "head in the game" I may also need to give some short quizzes. These will be just 10 minute affairs, where you are asked something straightforward about some recent lecture.

Here's a little idea for an essay. Use the method of my old paper to prove the Chaterjee's covariance identity which we proved in class (3/18). This suggestion will not necessarily give the easiest or shortest proof. It can't be as short as Chaterjee's. Still, the I'll bet the proof you find will be suggestive of other results, some of which might be new and interesting.

Results of Second Toolbox Essays

I've read the second essays, and there are two prize papers --- and a three way tie for runner up:

  • First Prize. What's a Lengendre-Fenchel Transformation and Why Should I Care by Hasegawa and Karamakar. This is a thoughtful essay that led me to change my opinion about "Young's Inequality," which I had never given much respect. I definitely learned something from this essay, and I will discuss what I learned in class.
  • Second Prize. A Proof of Convergence for the Stochastic Gradient Descent Method by Fazlyab, Gamma, and Paternain. This is a clearly written paper on a useful topic, and it gives a nice application of martingale theory. Definitely worth reading.
  • Runner Up Prizes. There were three papers that came close to the prize, but due to some slip or some rough spot they came in "out of the money". The runners up (in no special order) are the following:
    • A Note on the Maximum of a Random Walk with Negative Drift by Zhang and Dong. [Current post is a revision; which puts it in the "prize money"]
    • A Note on Stable and Mixing Convergence of Random Measures by team Monkey D.
    • The Frobenius Number of a Numerical Semigroup with Two Generators by the Penn Steele Workers' Union.

Everyone should look at as many of the essays as possible. This time the formats were all acceptable. Now we can focus all our energy on content --- and correctness!

One BIG piece of advice: "Spill the Beans about the Secret Sauce." If you have done something that is new or different, then you have identified what I call "secret sauce." In particular, if you claim that you have done something a little better than someone else, you need to say what you did that was different. Without such an explanation, the reader will be very confused.

Sometimes in student papers, there is the claim that a simpler proof is given, when in fact the "simpler proof" is simply incorrect. Now it is true that published proofs can have lots of defects, but if you claim to do better, you need to explain carefully to yourself and to others what causes the improvement. I'm not saying that this has happened this time, but it has happened often enough to be worth a word of advice: be skeptical of yourself!

Harris Processes

After we've had our fill of Poisson processes and Poisson approximation, we'll look harder at Markov processes on more general state spaces --- though the real line is general enough for us. In this context, the coupling argument starts to become technical, so the slickest way to proceed is actually by the method of contraction mappings. The art is to define the appropriate metric for the contraction. We will look at an elegant paper which suggests that the famous Lyapunov function is the key to the design of a good space.

This Lyapunov business also shows up in Foster's criterion for the recurrence of a chain. We'll have more to say about this too.

Poisson Approximation

When we get back from Spring break, we'll work for a while with the Poisson process in d=1 and d>1. After some elementary reminders, we'll see how John Hammersley used the Poisson process in d=2 to make important progress on a combinatorial problem. We'll then look at a magical characterization of the Poisson process by Renyi; it is one of the few places where one proves independence without an independence assumption that is baked into the cake. We'll then look as some more sophisticated Poisson approximation theorems due to Svante Janson (a probabilistic paper) and to Zacharovas and Hwang (an analytical paper).

After dealing with nth Poisson process we'll look at other Levy processes including Brownian motion. This will complete our list of "must does" after which we are at liberty to what interest us most. I am inclined to spend more time on renewal theory, random walk, and Markov decision problems.

Beta Integrals

We had a little sidebar fun with differential forms and the representation of the Beta integral in terms of the Gamma function. Playing a bit, I looked at the Wikipedia article on the Beta function, and it is lovely. In particular, it suggests a proof of the identity by consideration of the integral of a convolution. This is a sweet trick, and there are several others one can learn from the article. It's too far down stream for us, but there is also a very elegant continuation of the Beta function to complex values of alpha and beta. This involves the picture hanger puzzle --- which I may mention.

Perron-Frobenius

The Perron-Frobenius theorem for matrices is a close cousin of the theorem on the existence of a stationary distribution of a finite MC. I mention this because it came up in a talk in CS on Friday. This is also a topic for which the Wikipedia article is rather good. The Perron-Frobenius theorem went being time when it was used in the first paper on Google Rank. You can find tons of related material through mathscinet.

Some Perspective

"One of the big lessons from social science in the last 40 years is that environment matters. If you go to a buffet and the buffet is organized in one way, you will eat one thing. If it’s organized in a different way, you’ll eat different things. We think that we make decisions on our own but the environment influences us to a great degree. Because of that we need to think about how to change our environment." --- Google for Source.

Problem added to the Problem List

I added a problem to the problem list. It suggests a "generic" way to try to generalize many problems that make an assertion about conditional expectations. Sometimes this recipe can lead to a novel fact. It's often worth a try.

Hints about the Midterm (Feb. 25)

I will write the exam by looking over the ten days of lectures notes. For each lecture I will try to invent a question or two based on what I think were the major "take always". I'll then review the list, and select 5 to 8 questions --- fewer if there are lots of parts.

It goes without saying that you should master every definition and every theorem statement. The hard part is knowing when mastery is reached. This requires going back and forth with yourself, all the while being skeptical of your own statements. We also discussed some generic "tricks" for mastering a theorem. Here are some steps that seem worth recalling:

  1. Alternate between trees and forest i.e. work out all the little facts of a proof, then make sure you know how they fit together.
  2. Read with examples in mind. Draw pictures. Read skeptically. Try to foil the proof.
  3. Create lemmas to replace and refine the assertions that may clutter up the proof.
  4. Get a graphical view of the proof of the theorem where your new lemmas are vertices and the edges are implications.

I’ll think of more hints by Monday, but these will give you something to ponder now.

No Stationary Measure

I wrote out the details of the example that we discussed toward the end of the class today. It now all checks neatly.

Lalley on Markov Chains

Steve Lalley has written an elegant chapter on the elementary theory of Markov Chains which everyone should read. I have been assuming that people in the class is familiar with this material, but --- familiar or not --- it is lovely material to review. Even if you are an expert on elementary MCs, you will find many interesting observations. The material on the contraction mapping is particularly nice --- and not universally noted.

More than Can Be Imagined

It is sometimes the case with an old and natural idea that there features of the idea that have been developed far beyond what one might have imagined. The 2014 American Mathematical Monthly article by Moree on "Numerical Semi-groups..." certainly illustrates this point.

In the Erdos Feller Pollard "Second Proof" one needs to know that if we have a set of integers S with GCD equal to 1, then from some point on, every integer can be written as a positive integer linear form with respect to a finite integer "basis" of S. Such an S is called Numerical Semi-group, and they come up all over mathematics.

For example, they are relevant to the theory of Chicken McNuggets. What is the smallest number of nuggets that cannot be exactly purchased? Here you need to recall that one can purchase McNuggets in orders of size 4, 6, 9, and 20. Form a conjecture and prove it. Then take a look at Moree's paper (which you find via MathSciNet). There is also a very nice, elementary introduction to this topic by M. Beck. It notes that more that 400 papers have been written on this topic.

EFP Etc

On Monday Feb 16, please bring your copy of Erdos-Feller-Pollard (1949) to class. In the meanwhile, give it as good a reading as you can. This is a pretty dense piece of work. The second proof is especially challenging, but if you keep the ideas that we have reviewed in mind, it should be penetrable.

Our first look turned ups some useful analogies and distinctions. For one, we see that Abel's limit theorem is "in the class" with the DCT, MCT, Fatou in that it tells us about a kind of interchange of limits. The big distinction is that this is about conditionally convergent limits, and such limits are definitely not in the land of the DCT, etc.

Beurling Subsequence

As I mentioned in class, I got the notion of a Beurling subsequence from the paper of Garcia (1963). There is still juice in this old paper, if you are attracted to the asymptotics of sequences.

Results of the First Toolbox Essays

I've read the first essays, and there are two prize papers and a runner up:

  • First Prize. Limiting Properties on the Number of Leaves in the Barabasi-Albert Model by Zhang and Dong. This paper uses two martingales to substantially simplify earlier known results. The paper can benefit from further polishing, but it has serious content and room to grow.
  • Second Prize. Generalization of Cauchy-Schwarz in Probability Theory by Hasegawa and Karamakar. This well written essay shows how Selberg's generalization of Bessel's inequality leads to crisp and well motivated proofs of new and old results like the Chung Erdos inequality.
  • Runner Up. Feller's Proof of the Discrete Arcsine Law---The Essential Steps by Team "Penn Steele Workers' Union". This is a lovely exposition and I learned something from reading it, even though I have known the basic argument for many years.

There were some bright spots in the other essays, but these papers did stand out. Here are some comments that apply to almost all of the papers.

  • Too many of the papers had out-right mathematical errors. Here I am not talking about typos or small slips. There were conceptual errors that sank --- or almost sank --- the essay. This sort of thing simply should not happen.
  • Here are some suggestions on how to do better:
    • Every team member has to be responsible for the whole paper. Delegation does not work. You can't have one partner "do the write up." This generates new errors and perpetuates old errors. Don't do it.
    • Every team member must read every word --- many times! The team should be skeptical of what has been written. Rough spots need to be argued out.
  • Be very careful of belaboring a triviality and glossing over the real issues. This kind of beginners mistake will not happen with material that is well-understood, but if you try to write an essay without first mastering the material, this is what you get. It stands out like a sore thumb.
    • The cure for this is making sure that you have organized your time well. You need a chunk of time reading and talking with each other---honing the issues, isolating the essence of the problem.
    • Before putting pen to paper, the team should lecture itself on the material. Hard questions must be asked and answered.
  • All but a few papers had faulty references --- in more ways than one.
    • At the trivial (but important) level, many people tried to use the BibTeX code given by Google Scholar. This is not up to professional standards, and it makes your paper look amateurish. You need to get the BibTex from MathSciNet. You as lo need some sense of what is correct and what is not. You can learn this by looking at a few published articles. It's not rocket science.
    • The more important issue is that too few people looked at more than one source. If you are trying to understand some theorem, you should read about it in several sources. You should understand that some of these are poorly written; some could even be wrong. A toolbox essay requires the initiative to look at multiple sources; even the humble Wikipedia can help, especially when you sense that some word may have multiple interpretations.
    • Too few people really USED their references. References are not very useful if they are just chanted at the beginning of the paper or the end of the paper. A well written paper typically need to make concrete references several times in the heart of the paper. Sometimes these are needed to back up assertions, and at other times the serve to point out differences between the present exposition and the references.
  • Finally, in too many cases, the essays leave one thinking, "What's new here?" This needs to be said at a general level in the abstract, at a more specific level in the introduction, and in an absolutely clear way at the place of the new material, and then --- at last --- in a general way in the conclusion. The reader should have no way to escape understanding exactly what is new.

General Perspective: Almost all of the issues addressed above can be cured by the investment of a little more time. To motivate the investment, one has to "raise the bar" of personal expectations. No one should permit a mathematical error that can possibly be helped. Everyone makes mistakes, but a professional has to catch essentially all of those mistakes before they get out the door. On the other hand, learning how to write clearly is more of a long range challenge. It's not absolutely critical at this stage, but the people who so do learn how to write clearly almost always have much smoother academic careers than those who do not. The important thing to know is that anyone can learn how to write clearly; it's not much harder than proper hygiene.

Toolbox Essays

I have created the index page for the first toolbox essays. Please do read all of the essays and bring your comments and suggestions to class. Everyone benefits by learning how any essay can be improved --- or corrected. Certainly, if you know the team, please fee free to communicate directly with the team. The whole goal is to foster a creative leaning environment. Any team can send me a revision of any essay at any time.

As you read an essay, I encourage you to look hard at what the authors intended to do. Even when an essay drifts off course (or has errors or confusions) you can still learn something from the essay if you are carefully attentive to what the essay hoped to achieve. Good intentions do matter, even if details do sometimes get in the way.

If a team wants to amend an essay with an addition or correction, just email me the essay WITH THE SAME FILE NAME and I will post the up-date.

Sunday Evening 2/8

I have added a three problems to the problem list. The new problems are more open ended and they also ask that you dig into some original papers. I have picked three that are brief and on track with our course, as well as being rich with honest research potential --- even though one of the papers goes back to 1949.

If you can get yourself to dig into one or more of these papers, you will be on a track that you can use to model your efforts for years. After one has been introduced to the basics of some topic, the real way to make progress is to read the original research works. Every page of such a work is an "exercise set".

Ground Hog Day

I have added two problems to the problem list. I also edited Problem 5, first to fix an omission, then to explain more fully why the problem is interesting. Please do let me know anytime you find a bug (or suspected bug) in one of the problems.

Problem List and Toolbox Essays (TEs)

I have added a problem to the problem list. Also, please get cracking on your team's Toolbox essay. It is due at class time February 9. Please provided both hard copy at class time and an electronic copy by email.

Please title the PDF of your essay in the style: TE-Dunford-Pettis, if your essay happened to be about the Dunford-Pettis theorem or call it TE-Vitalli-Saks if it is about he Vitali-Saks theorem. Incidentally, both of these theorems are useful to probabilists, but they are not well known. You could make a nice essay just by explaining the statement of one of these theorems and by illustrating it with an nice example that can be understood and appreciated by a 531 reader.

In writing your essay please be very attentive to the standards of academic writing. Not much "original work" is needed here, but you must give careful and appropriate credit to your sources.

You should underscore, precisely what novelty you bring to the essay, e.g. "We do a 3 by 3 example that has the all of features of Durrett's 4 by 4 example; but, unlike Durrett's example, ours only requires calculations that one can do in ones head."

Your originality can rest mainly in your thoughtful exposition and in any examples or applications that you add.

In writing your essay, you should have a reader in mind. Ideally, you will write an essay that you personally would enjoy reading.

Way Off Topic

It is way off topic, but I was quite entertained this weekend to learn the technique of Vieta Jumping. It lives in the land of divisibility theory, so it is not well connected to probability theory. Still, it is philosophically related to a combinatorial technique that Lovasz calls the "method of the minimal criminal" where to prove a conjecture one assumes that there is a counter example of some minimal size and then one shows that there is a counter example that is even smaller. This gives the contradiction and proves the conjecture.

Famous One Page Paper

You can't beat Nash's 1949 PNAS paper connecting games with the Kakutani fixed point theorem. Ironically, I confess that I don't get the big message; von Neumann had much earlier made the connection in the case of 2 by 2 games, so what really is the news here? I'm sure it is there; I just can't see it through the fog of time.

Teams!

Please be sure to have formed your team by class time, Monday January 26. Most teams are expected to be teams of 2. If for logistical or personal reasons you want a team of 1 or a team of three that is OK. Obviously a larger team will have more that is expected in terms of what is delivered, and a smaller team will have less expected.

Put your team members, contact information, and team name (!) on one page to pass along on Monday.

Problem List Up-date (1/22)

There are now seven problems on the problem list. I also edited Problem 6; there was a 2 that was missing from the LHS. I also added a small amount of coaching about this problem. This would make a good toolbox essay, especially if it were combined with a related problem that could be solved with a similar technique.

Problem List and a Nice Two Page Paper

I edited problem 2 on the problem list and I added a third problem. Also, if you want to look at a two page paper that I think is marvelous, take a look at D. J. Newman's, "A Simple Proof of Wiener's 1/f Theorem" (PAMS, 1975). The fastest way to get a copy is to go through MathSciNet --- which is a useful exercise in and of itself. Wiener's theorem will come up when we discuss the renewal theorem.

Calibration

It will only be in extremely special cases that one can take Stat 531 without having taken Stat 530. To qualify as a special case, you will need to have taken substantial courses in graduate probably, graduate real analysis, and complex analysis. If you're on your way to doing a thesis in the theory of semi-groups of operators, and if you know martingale theory, then you're OK. On the other hand, if you are relying on a course like Stat 510 as a replacement for Stat 530; well, the analogy of a bicycle and a jet engine comes to mind. These are enterprises of entirely different kinds.

If you need a course in Stochastic Processes and if you need graduate credit, you can consider taking my undergraduate course Stat 433 like a regular student --- but you can sign up for credit under a 900-reading number. You will need to clear this with your advisor and me before we can set this up. Stat 433 meets at 1:30 MW.

Getting into the Game

I've been looking for new ways to make 531 more creative, more interactive, and more fun --- while still being sure to achieve the benchmarks that are needed to prepare you for your next steps as graduate students.

The new, more creative part of the course comes in three pieces that use TEAMS. We will create teams of two students (or in exceptional cases three students or one student). You can have team names: Zeno's Zombies, West Philly Wizards, Villanova Vampires, ... whatever. The teams will be involved in three kinds of activities.

I will mention many problems in class. You can work on these with your friends. Some of the better problems will be collected on the semester long problem list. You should check it regularly for up-dates. I will not collect or grade problems, but they will remain a vital part of the course. You should also be attentive to reading the text, including the problems in the text. The lectures and the text will not cover the same material in the same ways, but there will always be substantial overlap.

We'll also make light use a variation of the Scribe System. Many course around campus now use some version of the scribe system and it seems to be a richly educational experience. We won't do scribing on a daily basis, and the scribe's work will deal with brief themes or stories that typically take much less than a full lecture.

We'll also try something new: The Toolbox Essay. We'll get going on these pretty quickly, and each team may do three or more. The first one will be assigned on the first day of class and it may be due after three weeks, or so.

Teams may be called upon to present the solution of problems in Class. We'll evolve the details of this process over the first few classes.

Finally, we will have a traditional in-class mid-term and a traditional in-class final. The mid-term exam will be on Wednesday February 25. It will be in our class room.

In accordance with the UPenn Schedule of Exams, our FINAL EXAM will be on Wednesday May 6 from Noon (12pm) until 2pm. The room will be announce when it is available.

Both the final and the midterm will deal with the problems that have been discussed in class or assigned has homework. Original problem solving will not be required on these tests, but serious understanding of previously solved problems is the sine qua non. To do well on the test, one simply has to do ones homework.

We may also have quizzes, either of the no-name type, or of the named type. Other innovations may be implemented as the course evolves.

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.

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 extremely special cases that one can take Stat 531 without having taken Stat 530.