Shalizi Cosma Rohilla, Kontorovich Aryeh. Almost None of the Theory of Stochastic Processes

Unpublished, 2010. — 347 p.This is intended to be a second course in stochastic processes (at least!); I am going to assume you have all had a first course on stochastic processes, using elementary probability theory. You might then ask what the added benefit is of taking this course, of re-studying stochastic processes within the framework of measure-theoretic probability, a framework I am going to assume that you already know. There are a number of reasons to do this.First, the measure-theoretic framework allows us to greatly generalize the range of processes we can consider. Topics like empirical process theory and stochastic calculus are basically incomprehensible without the measure-theoretic framework. Much of the impetus for developing measure-theoretic probability in the first place came from the impossibility of properly handling continuous random motion, especially the Wiener process, with only the tools of elementary probability.Second, even topics like Markov processes and ergodic theory, which can be discussed without it, greatly benefit from measure-theoretic probability, because it lets us establish important results which are beyond the reach of elementary methods.Third, many of the greatest names in twentieth century mathematics have worked in this area, and the theories they have developed area, and the theories they have developed are profound, useful and beautiful. Knowing them will make you a better person.