Review: Theory of Probability and Random Processes

Theory of Probability and Random Processes L. Koralov, University of Maryland, College Park, MD, USA; Y.G. Sinai, Princeton University, NJ, USA

It took me half an eternity but I eventually finished the book "Theory of Probability and Random Processes".

I initially intended to read it both as a refresher and as a warm up to read a couple of higher level books. And it was indeed an interesting read, though, I must admit, more difficult than I thought.

The obvious shortcoming of this book should be clear for any people having been to university: it is aknowledged as being based on a course... and I reckon it's reminiscent of those handouts you get when the teachers consider the pedagogy's place to be in the classroom only ((rest assured it's not handwritten, LaTeX has at least saved us from that )). Fair enough for Springer's "Universitext" collection. Anyway once you get over this, there are quite a few  impressive features about this book:

  • an overwhelming proportion of the discussed theorems and properties are demonstrated
  • it covers a very broad area of the whole probability field within its 400 pages going from the most basic theoretical definitions (random variables,  Lebesgue integrals etc) up to stochastic integrals, skipping through stationary random processes, Markov processes and Gibbs fields.
  • it's publicized as self contained which is fairly right and impressive considering the previous two points
  • the real-life applications of the most important mathematical objects are explained
At the end of the day,  this book requires, I think, to be already a bit familiar with the first 3 or 4 sections (out of 22) for the reader not be lost too soon, but once you've been through it, it proved to be a pretty good reference on all matters related to the construction of random processes and a fair shoehorn for physics built on those.

Last but not least, this book is also a great hommage to Chebyshev whose inequalities easily are at the heart of half the demonstrations (not kidding) !