Three-Toed Sloth

Will There Be a Text in My Class?

Attention conservation notice: An appeal to the reader's knowledge of textbooks on stochastic processes; also a plea not to be thrown into the briar-patch.

In the spring, I'm going to be teaching the department's advanced course on stochastic processes (36-754, for those keeping track at home). The catalogue description of the course reads, in full, as follows:

This course introduces advanced topics in Probability Theory such as Brownian motion, Markov processes, stationary processes, stochastic integration, etc.

It's intended for students from math or statistics who've had a first course in measure-theoretic probability, such as our 36-752, which goes up through the laws of large numbers for independent variables, a little martingale theory, and the central limit theorem. Most if not all of them will have already had a course on stochastic processes at the level of Grimmett and Stirzaker. My plan is to take advantage of the "etc." in the description, and teach a course on my favorite topics some interesting and highly useful material which is perhaps otherwise under-emphasized. To be more concrete, I'd like to start with the Wiener process, stochastic calculus and stochastic differential equations (which is pretty standard), but then do a lot on ergodic theory and mixing, Markov operators and their asymptotics, large deviations, information theory (as it connects to hypothesis testing and the like, not so much to coding), and spatial processes, including the Hammersley-Clifford (-Griffeath-Grimmett-Preston-...) theorem. Ideally, we'd end with some interesting spatio-temporal models, e.g. cellular automata or interacting particle systems. Markovian representations of non-Markovian processes (per Knight, rediscovered by various smart computer scientists and some confused physicists) would also be nice, but maybe too much. (Since any one of those topics could be stretched to cover a semester, the whole class is too much!)

I am looking for a textbook which covers all of this, or at least most of it; I'd be willing to change the material to match a good text. The students currently in 752 are using Ash and Doleans-Dade, which is good, and the last two chapters (which they won't get to) introduce a little ergodic theory and a little stochastic calculus, respectively, but not in enough depth. No one book I know seems to fit, and making them buy more than one expensive book doesn't seem right. If you have any suggestions, please mail them to me at cshalizi [at] oryx [dot] cmu [dot] edu (removing the name of a genus of antelope, which is there only to confuse spammers). I am going to have to spend a lot of time on my lecture notes; I really don't want that to have to grow into, in effect, writing my own book.

Update, next day: Thanks to Bill Tozier, Anand Sarwate and Wolfgang Beirl for writing with suggestions. Wolfgang, in particular, pointed me to Alexandre Stefanov's useful collection of online probability texts and notes (part of a bigger collection of mathematics resources). One of these, Robert Gray's Probability, Random Processes and Ergodic Properties, is something I was already planning to mine, along with his Entropy and Information Theory.

Update, Halloween: We will be using Olav Kallenberg's Foundations of Modern Probability as a reference, with the primary text being my lecture notes.

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Enigmas of Chance; Corrupting the Young