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.

We survey the theory of treeable countable Borel equivalence relations.

Chapter 8 provides an introduction to Gaussian measures on a Banach space from the point of view that originated in the work of N. Wiener and was further developed by L. Gross and I. Segal. The underlying idea is that, even though it cannot fit there, the measure would like to live on the Hilbert space (the Cameron–Martin space) for which it would be the standard Gauss measure, and it is in that Hilbert space that its properties are encoded. A good deal of functional analysis is required to carry out this program, and the estimate that makes the program possible is X. Fernique’s remarkable exponential estimate. Included are derivations of M. Schilder’s large deviations theorem for Brownian motion and V. Strassen’s function space version of the law of the iterated logarithm, both of which confirm the importance of the Cameron–Martin space.

We discuss results demonstrating the complex structure of the Borel reducibility order.

We study the poset of bireducibility types of countable Borel equivalence relations and its relation to the theory of cardinal algebras.

In Chapter 4 I construct the Lévy processes (a.k.a. independent increment processes) corresponding to infinitely divisible laws. Section 4.1 provides the requisite information about the pathspace D(ℝN) of right-continuous paths with left limits, and §4.2 gives the construction of Lévy processes with discontinuous paths, the ones corresponding to infinitely divisible laws having no Gaussian part. Finally, in §4.3 I construct Brownian motion, the Lévy process with continuous paths, following the prescription given by Lévy. This section also contains a derivation of Kolmogorov’s continuity criterion for general Banach space-valued stochastic processes.