Ask a probabilist
 

In an effort to find something interesting to do with what I assume will be my last 21 posts as an "adept" (whatever that is), I decided to start this thread. Some may know already, but for those that do not, I am a probabilist (a mathematician who specializes in probability theory). If you have any burning questions you would like to ask a probabilist, then here is your chance. Post them here and I will try to answer them.

I can obviously answer questions regarding undergraduate mathematics, and basic graduate-level, measure-theoretic probability, including topics such as Markov chains, Brownian motion, and stochastic differential equations.

When it comes to more advanced topics, I can probably only answer questions within my field. I usually work on limit theorems for stochastic processes. This typically involves finding analogues of the law of large numbers and the central limit theorem, which apply to continuous-time processes. A simple example of this type of theorem is Donsker's theorem.

I am not a statistician, I do not work in numerical analysis or computing, and I do not work on discrete probability. I am no expert on philosophical issues related to probability, but I have read and thought about this topic a little bit and can offer my opinions. I can also offer some advice to graduate students about selecting an adviser, looking for a job, and so on.

But feel free to ask anything and I will do my best.

Re: Ask a probabilist
 

All right. Here's one for you. This may be too simple or out of the realm of your expertise (I don't know because I'm not a mathematician), but maybe it'll kick-start this thread:

To your knowledge, has anyone ever proven genetic algorithm to converge to a global optimal solution in the limit?

I know this has been done with simulated annealing, and the proof has to do with hidden Markov models. I also know that some work has been done to show that HMMs exist within GA, but I have never heard of anyone actually proving convergence to a global optimum.

Re: Ask a probabilist
 

Are you a professor? What other types of jobs are available within this field?

Re: Ask a probabilist
 

why did you get into probability?
what do you think the five best probability departments in the united states are?
i've taken a standard two semester graduate course in probability out of durrett. what's the next textbook you'd recommend that i read?
are you familiar with my thesis supervisor (martin barlow) or my advisor (ed perkins)?

e:
in your opinion, what are the hot areas of probability?
how easy is it for someone with a ph.d from a good school in probability to get a quant-type job in finance, and how would one go about doing that?

Re: Ask a probabilist
 

This is outside my expertise, although I have done work with Freidlin-Wentzell large deviations, which is at the heart of proving convergence in simulated annealing. I cannot offer any concrete answers, but maybe you can find something in one of these:

@article {MR2052865,
AUTHOR = {Stannat, Wilhelm},
TITLE = {On the convergence of genetic algorithms---a variational
approach},
JOURNAL = {Probab. Theory Related Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {129},
YEAR = {2004},
NUMBER = {1},
PAGES = {113--132},
ISSN = {0178-8051},
CODEN = {PTRFEU},
MRCLASS = {35J20 (35A15 60J25 92D10 92D15)},
MRNUMBER = {MR2052865 (2005d:35040)},
MRREVIEWER = {Dominique L{\'e}pingle},
}

@article {MR1832784,
AUTHOR = {Schmitt, Lothar M.},
TITLE = {Theory of genetic algorithms},
JOURNAL = {Theoret. Comput. Sci.},
FJOURNAL = {Theoretical Computer Science},
VOLUME = {259},
YEAR = {2001},
NUMBER = {1-2},
PAGES = {1--61},
ISSN = {0304-3975},
CODEN = {TCSDI},
MRCLASS = {90C59 (68T05 90C15)},
MRNUMBER = {MR1832784 (2002j:90117)},
MRREVIEWER = {R. Shonkwiler},
}

@article {MR2020342,
AUTHOR = {Schmitt, Lothar M.},
TITLE = {Theory of genetic algorithms. {II}. {M}odels for genetic
operators over the string-tensor representation of populations
and convergence to global optima for arbitrary fitness
function under scaling},
JOURNAL = {Theoret. Comput. Sci.},
FJOURNAL = {Theoretical Computer Science},
VOLUME = {310},
YEAR = {2004},
NUMBER = {1-3},
PAGES = {181--231},
ISSN = {0304-3975},
CODEN = {TCSDI},
MRCLASS = {68T05},
MRNUMBER = {MR2020342 (2004j:68156)},
}

@article {MR2039188,
AUTHOR = {Zhao, Xiao-yan and Nie, Zan-kan},
TITLE = {The {M}arkov chain analysis of premature convergence of
genetic algorithms},
JOURNAL = {Chinese Quart. J. Math.},
FJOURNAL = {Chinese Quarterly Journal of Mathematics. Shuxue Jikan},
VOLUME = {18},
YEAR = {2003},
NUMBER = {4},
PAGES = {364--368},
ISSN = {1002-0462},
MRCLASS = {60J20 (60J10 92D10)},
MRNUMBER = {MR2039188 (2004k:60201)},
MRREVIEWER = {Ren{\'e} L. Schilling},
}

Re: Ask a probabilist
 

Yes.

A lot of people with a background in probability go into the financial industry. Also, with a little work, a probabilist can usually pass for a statistician, which obviously opens up a lot of jobs everywhere.

Re: Ask a probabilist
 

What's the solution to the Two Envelopes Problem?

Re: Ask a probabilist
 

I flip a coin and record whether or not I get heads or tails. I do this many times and calculate the average number of flips before I get the pattern HTH. Call this number A. I do it again and calculate the average number of flips before I get HTT. Call this number B.

Is A==B, A > B or A < B?

Re: Ask a probabilist
 

A=10, B=8 (i calculated A in two different ways, and the equations for B only differ from the equations for A very slightly, so i'm pretty sure this is correct)

of course it is obvious that A>B (if you 'fail' in the second case, you're already part of the way to a success, whereas if you 'fail' in the first case, you have to start from scratch)

i'm a graduate student in probability so your question is actually being answered by a probabilist.

Re: Ask a probabilist
 

I'm pretty sure you're correct. I like that question; it's somewhat counterintuitive.

Re: Ask a probabilist
 

I took a few years off between undergrad and grad school to basically be a bum and develop a lot of ill-founded opinions. (Okay, it was not really that bad.) Anyway, one of those opinions was that the future of scientific progress would center around probabilistic analysis, and not deterministic analysis. It was at that time I decided to study probability theory and learn about stochastic differential equations. Luckily for me, when I actually learned what those were, I found them interesting.

I feel like the best answer to this is, "it depends." But I do not want to avoid the question. So here is a somewhat arbitrary list: Berkeley, Columbia, Cornell, Princeton, and Stanford. I think, though, that a better answer would be this: decide what topics interest you, find the best probabilists in that area, and try to go where they are.

Karatzas and Shreve. From it, you will learn about Brownian motion, continuous martingales, the Ito integral, and some basics about stochastic diff eqs. These are important topics no matter what area of probability you end up in. It is a tough book, though. You may want to supplement it with something more intuitive, like Oksendal.

Yes and yes. Ed, particularly, I know quite well. But in the US, we typically do not have two separate people filling these roles. Do you do it differently in Canada, or is yours a special case?

SLE is obviously hot, given the recent Fields Medal. Mathematical biology is another very hot topic with a good future, in my opinion. There also seems to be a lot of activity regarding random walks in random environment, but this is pretty far from my area, so it is hard for me to judge its hotness. (But I can point you to this survey article.)

Oh, I think it would be pretty easy. I had a friend in grad school with a PhD in something to do with PDEs. After he finished, he spent a few months learning probability and Ito calculus, and then landed a fancy, high-paying job on Wall Street. I think he is in Japan now. Of course, now he works non-stop, and also he is not legally allowed to talk to me about his work. I very rarely hear from him these days.

I do not really know how one goes about it. I think he just found the job listings somewhere, applied in the usual way, had some phone interviews, and the process just took off from there.

Re: Ask a probabilist
 

I find the proposed solutions in the Wikipedia article to be quite satisfactory.

Re: Ask a probabilist
 

This:

The above can be made rigorous by a coupling. Build a chain of iid coin flips and let T_A be the time until first seeing HTH. Now build a second chain out of the first. Make it identical to the first, except swap what appears after any occurrence of HT. (For example, if the first begins with HTTTHTTHTH..., then the second begins with HTHTHTHHTT...) Let T_B be the time until first seeing HTT in the second chain.

Then

A - B = E[T_A] - E[T_B] = E[T_A - T_B].

Now you just have to show that T_A - T_B is always non-negative, and has a positive probability of being positive. I will leave that part as an exercise.

Cool question, by the way.

Re: Ask a probabilist
 

first, thanks for all the replies!

i wish that my story was as good. i studied a lot of analysis and PDE as an undergrad and had a good grasp of measure theory so i enrolled in Ed's graduate probability I course while i was an undergrad at the 'other' university in the vancouver area. i thought that it would be easy since my measure theory background was strong. in short i found it very difficult and frustrating, especially since i wasn't able to attend many classes, so of course it was what i decided to study in graduate school.

your answer is all i really wanted :) what's your opinion of courant and UCLA for probability?

i have both, although i have read the first 9 chapters or so of oksendal and almost none of karatzas and shreve (i think i glanced at it once to get an answer for a homework assignment). Martin currently has me reading port and stone's book on brownian motion and potential theory in addition to oksendal.

do you have an opinion on this book?

small world! actually to be honest i'm not even 100% sure how it works. i asked Ed to be my advisor when i started my masters a year ago, but i wasn't even thinking of my thesis at that point. i chose Martin (on Ed's recommendation) as my thesis supervisor because his interests looked the closest to my interests. i chat with both of them on 'advisor-type' topics like what courses i should take.

i think my situation is somewhat unique because a lot of the Ph.D students come to work with a specific person and hence their advisor/thesis supervisor end up being the same person. i haven't decided which Ph.D programs i'm going to apply to after i finish my masters next summer.

cool. i took a course from Grimmett this summer on 'discrete spatial processes in probability' and he talked briefly about SLE. i also attended a series of lectures by Varadhan at CMU on random walks in random environment a couple of summers ago, but as an undergrad without much probability background it went way over my head :(

i'm sure i'll come up with more questions in the next few days if it's not too much trouble.

Re: Ask a probabilist
 

"This statement is a lie" is a Russell paradox. What about "This statment is 90% to be a lie"?

Re: Ask a probabilist
 

What do you think of Karatzas and Shreve's "Methods of
Mathematical Finance" ? ( already own a copy of their
"Brownian Motion and Stochastic Calculus")

Also, if normality of asset returns isn't assumed, isn't it
worthwhile to consider Lévy processes? If so, what's a
decent book or paper on Lévy processes?

Re: Ask a probabilist
 

I recall you mentioning at some point that you were reading Jaynes's book. What do you think of it?

Re: Ask a probabilist
 

They almost made my list, but you forced me to pick five! Obviously, McKean and Varadhan are very big names at Courant. Scott Sheffield is also there, and he seems to be a very strong probabilist. It looks like Liggett is the big name at UCLA. Other than that, I recognize Biskup and Berger on the UCLA roster, and they certainly have very good reputations. You should look at what these folks work on and whether or not it interests you.

I used it once as a reference for a theorem I could not find anywhere else. Other than that, I am not too familiar with it. My first impression was that it would make a better reference book than a textbook.

Keep 'em coming. I'm here to serve.

Re: Ask a probabilist
 

In my opinion, the above sentence in bold is not even meaningful, let alone a paradox.

Re: Ask a probabilist
 

I only thumbed through that book, and it was a while ago. I remember thinking it would be tough to read without some prior background in finance.

Frankly, I would like to see a financial model that is derived from some underlying principles, rather than one which is justified only by data fitting. But I think that may be too much to expect for now. So, yes, if you are looking for a model that fits the data, and you do not want to assume normality, then of course you should try Lévy processes on for size.

I am pretty sure that Protter covers Levy processes. But that is another tough book to wade through. Unfortunately, I do not have a light supplement to suggest for that.

Re: Ask a probabilist
 

I gave up on that. The first part was interesting and thought provoking. But somewhere in the middle, it became painful to read. Since it was supposed to be recreational reading, I dropped it in favor of "Pebble in the Sky."

Re: Ask a probabilist
 

Is there a p-adic version of probability? Other versions?

Re: Ask a probabilist
 

I think Andrei Khrennikov is your man. You might also check out this arXiv posting of his. (The published version is available somewhere.) Also, Jaynes discusses probabilities taking values in spaces other than the real numbers. The discussion is in Appendix A, and begins on page A-6, under the heading "Comparative Probability." It might be interesting to compare what he had to say back then to Khrennikov's results.

Re: Ask a probabilist
 

do you play poker?

how would you solve the following problem?

Let X_i be iid and such that X_i converges almost surely to X. Then X is a.s. constant.

Re: Ask a probabilist
 

What is the most general form of the Lovasz Local Lemma? Reference?