Chapter 6 opens with extensions of martingale theory in two directions: to σ-finite measures and to random variables with values in a Banach space. In §6.2 I prove Burkholder’s Inequality for martingales with values in a Hilbert space. The derivation that I give is essentially the same as Burkholder’s second proof, the one that gives optimal constants. Finally, the results in §6.1 are used in §6.3 to derive Birkhoff’s Individual Ergodic Theorem and a couple of its applications.

We introduce here basic concepts concerning equivalence relations, morphisms, and reductions, in general and in the Borel category.

Section 7.1 provides a brief introduction to the theory of martingales with a continuous parameter. As anyone at all familiar with the topic knows, anything approaching a full account of this theory requires much more space than a book like this can provide. Thus, I deal with only its most rudimentary aspects, which, fortunately, are sufficient for the applications to Brownian motion that I have in mind. Namely, in §7.2 I first discuss the intimate relationship between continuous martingales and Brownian motion (Lévy’s martingale characterization of Brownian motion), then derive the simplest (and perhaps most widely applied) case of the Doob–Meyer Decomposition Theory, and finally show what Burkholder’s Inequality looks like for continuous martingales. In the concluding section, §7.3, the results in §7.1 and §7.2 are applied to derive the Reflection Principle for Brownian motion.

This chapter is devoted to the study of infinitely divisible laws. It begins in §3.1 with a few refinements (especially the Lévy Continuity Theorem) of the Fourier techniques introduced in §2.3. These play a role in §3.2, where the Lévy–Khinchine formula is first derived and then applied to the analysis of stable laws.

Because they are not needed earlier, conditional expectations do not appear until Chapter 5. The advantage gained by this postponement is that, by the time I introduce them, I have an ample supply of examples to which conditioning can be applied; the disadvantage is that, with considerable justice, many probabilists feel that one is not doing probability theory until one is conditioning. Be that as it may, Kolmogorov’s definition is given in §5.1 and is shown to extend naturally to both σ-finite measure spaces and random variables with values in a Banach space. Section 5.2 presents Doob’s basic theory of real-valued, discrete parameter martingales: Doob’s Inequality, his Stopping Time Theorem, and his Martingale Convergence Theorem. In the last part of §5.2, I introduce reversed martingales and apply them to DeFinetti’s theory of exchangeable random variables.

The central topic here is the abstract theory of weak convergence of probability measures on a Polish space. The basic theory is developed in §9.1. In §9.2 I apply the theory to prove the existence of regular conditional probability distributions, and in §9.3 I use it to derive Donsker’s Invariance Principle (i.e., the pathspace statement of the Central Limit Theorem).

Chapter 1 contains a sampling of the standard, point-wise convergence theorems dealing with partial sums of independent random variables. These include the Weak and Strong Laws of Large Numbers as well as Hartman–Wintner’s Law of the Iterated Logarithm. In preparation for the law of the iterated logarithm, Cramér’s theory of large deviations from the law of large numbers is developed in §1.3. Everything here is very standard, although I feel that my passage from the bounded to the general case of the law of the iterated logarithm has been considerably smoothed by the ideas that I learned in conversation with M. Ledoux.

We study invariant and quasi-invariant measures for countable Borel equivalence relations.

Chapter 2 is devoted to the classical Central Limit Theorem. The initial presentation is based on Lindeberg’s non-Fourier techniques. This is followed by a derivation of the Berry–Esseen estimate based on ideas of C. Stein. Fourier techniques are introduced in §2.3, and in the final section the CLT is used to derive W. Beckner’s sharp Lpestimates for the Fourier transform.