In this Chapter we present the theory of martingales and amarts indexed by directed sets. After Dieudonné showed that martingales indexed by directed sets in general need not converge essentially, Krickeberg—in a series of papers—proved essential convergence under covering conditions called “Vitali conditions.” This theory is presented in an expository article by Krickeberg & Pauc [1963] and in a book by Hayes & Pauc [1970].

Here we offer a new approach and describe the subsequent progress. The condition (V), introduced by Krickeberg to prove the essential convergence of L1-bounded martingales, was shown not to be necessary. Similarly the condition (Vo), introduced to prove convergence of L1-bounded submartingales, is now also known not to be necessary. The condition (VΦ), which Krickeberg showed to be sufficient for the convergence of martingales bounded in the Orlicz space LΨ, is also necessary for this purpose if the Orlicz function ¸ satisfies the (Δ2) condition.

In each instance, the convergence of appropriate classes of amarts exactly characterizes the corresponding Vitali condition. This is of particular interest for (V) and (Vo) since there is no corresponding characterization in the classical theory. In general, to nearly every Vitali type of covering condition there corresponds the convergence of an appropriate class of “amarts.” The understanding of this fact was helped by new formulations of Vitali conditions in terms of stopping times. Informally, a Vitali condition says that the essential upper limit of a 0-1 valued process (1At can be approximated by the process stopped by appropriate stopping times.

We present here a unified approach to most of the multiparameter martingale and ergodic theory. In one parameter, the existence of common formulations and proofs is a well known old problem; see e.g. J. L. Doob [1953], p. 342. It has been known that the passage from weak to strong maximal inequalities can be done by a general argument applicable to harmonic analysis, ergodic theory, and martingale theory. In this book a very general such approach is presented in Chapter 3, involving Orlicz spaces and their hearts. There exists also a simple unified (martingales + ergodic theorems) passage from one to many parameters using no multiparameter maximal theorems, based on a general argument valid for order-continuous Banach lattices. This approach gives a unified short proof of many known theorems, namely multiparameter versions of theorems of Doob (Cairoli's theorem [1970] in stronger form, not assuming independence), theorems of Dunford & Schwartz [1956] and Fava [1972], the multiparameter pointtransformation case having been earlier proved by Zygmund [1951] and Dunford [1951]. We also obtain multiparameter versions of theorems of Akcoglu [1975], Stein [1961], Rota [1962]. For the Banach lattice argument, the order continuity is needed, which means that the L logkL spaces are not acceptable if the measure is infinite: they fail this property and have to be replaced by their hearts, subspaces HΦ which axe closures of simple integrable functions (see Chapter 2). We will first develop in detail the “multiparameter principle” (Theorem (9,1.3)) that allows the reduction of multiparameter convergence problems to one parameter. There is also a one-sided version of this result, useful to prove “demiconvergence” in many parameters.

In this chapter, we consider martingales, amarts, and related processes that take values in Banach spaces. They are useful in Banach spaces that occur naturally in mathematics, such as function spaces. They have also played an important role in the understanding of some geometric properties of Banach spaces. The reader who knows nothing of Banach spaces should, of course, skip this chapter; but someone with only a minimal knowledge of Banach space theory should be able to work through most of the chapter, with the exception of Section 5.5.

There is a close connection between martingales with values in a Banach space E and measures with values in E. This is not unexpected. More interesting are the connections of these two topics with the geometric properties of the Banach space. These are explored in Section 5.4.

If a theorem is true in the real-valued case and extends to a more general setting without change of argument, this may be useful, but is hardly exciting. One modern approach to probability in Banach space consists in exactly matching the convergence property to the geometry of the space. Such theorems are not only the best possible in terms of convergence, but they may also shed new light on the structure of the space. The remarkable theorem of A. & C. Ionescu Tulcea and S. D. Chatterji (that L1-bounded martingales taking values in a Banach space E converge a.s. if and only if E has the Radon-Nikodým property, Theorem (5.3.30); also (5.3.34)) characterizes a geometrical property of Banach spaces in terms of convergence of martingales. It will be seen that there are many such characterizations in terms of amarts.

The main themes of this book axe: stochastic, almost sure, and essential convergence; stopping times; martingales and amarts; processes indexed by directed sets, multiparameter processes, and Banach-valued processes.

We begin in Chapter 1 with the notion of the stopping time, central to the book. That this notion is important in continuous parameter martingale theory and sequential analysis (briefly touched on in Chapter 3) is well known. This book differs from others in that many of the discrete parameter results are proved via processes (amarts) defined in terms of stopping times—in fact only simple stopping times. The Radon-Nikodým Theorem receives an amart proof. That this theorem follows from the martingale theorem is well known, but here martingales come later, so that the Radon-Nikodým theorem is available to define the conditional expectation and the martingale. In Chapter 4, the rewording of the Vitali covering conditions in terms of stopping times clarifies connections with the theory of stochastic processes.

The main result, the Amart Convergence Theorem, is proved by elementary arguments. This—together with a general Sequential Sufficiency Theorem (1.1.3), showing how in metric spaces the convergence of increasing sequences implies that of nets—is used to obtain stochastic convergence of L1-bounded ordered amarts on directed sets. This in turn implies stochastic convergence of quasimartingales—even on directed sets. Quasimartingales include the L1-bounded submartingales studied by Krickeberg on directed sets. We believe the proofs by this method are shorter than those existing in the literature.

In this chapter we will prepare some of the tools to be used later.

One important possibility is the use of an infinite measure space, rather than a probability space. For probability in the narrow sense, only finite measure spaces are normally used. Attempts to do probability in infinite measure spaces have had little success. In most of the book, we consider primarily the case of finite measure space. But there are reasons for considering infinite measure spaces. The techniques to be developed for pointwise convergence theorems can tell us something also in infinite measure spaces. The ergodic theorems in Chapter 8 often have their natural setting in infinite measure spaces. J. A. Hartigan [1983] argues that an infinite measure space provides a rigorous foundation for Bayesian statistics. Our material on derivation requires the possibility of infinite measures to cover the most common cases, such as Euclidean space IRn with Lebesgue measure.

Another important tool to be used is the Orlicz space. The class of Orlicz spaces generalizes the class of function spaces Lp. If the primary concern is finite measure space, then Orlicz spaces are basic. A necessary and sufficient condition for uniform integrability of a set of functions is boundedness in an Orlicz norm. Zygmund's Orlicz space L log L, and, more generally, Orlicz spaces L logkL, appear naturally in considerations of integrability of supremum and related multiparameter convergence theorems. For consideration of infinite measure spaces, Orlicz spaces will not suffice, unless property (Δ2) holds.

In this chapter, we will prove some of the pointwise convergence theorems from ergodic theory. The main result of this chapter is the superadditive ratio ergodic theorem. It implies the Chacon-Ornstein theorem, the Kingman subadditive ergodic theorem, and, for positive operators, the Dunford-Schwartz theorem and Chacon's ergodic theorem involving “admissible sequences.” We consider positive linear contractions T of L1.

Our plan is as follows. We first prove weak maximal inequalities, from which we obtain the Hopf decomposition of the space Ω into the conservative part C and the dissipative part D (8.3.1). Assuming T conservative (that is, Ω = C, we prove the Chacon-Ornstein theorem (8.5.4), i.e., the convergence to a finite limit of the ratio of sums of iterates of T applied to functions f and g The limit is identified in terms of f, g, and the σ-algebra C of absorbing sets. The superadditive operator ergodic theorem is proved in the conservative case (8.4.6). It is then observed that the total contribution of Ω to C is a superadditive process with respect to the conservative operator Tc induced by T on C. Since the behavior of the ergodic ratio on the dissipative part D is obvious, the Chacon-Ornstein theorem (8.6.10) and, more generally, the superadditive ratio theorem (8.6.7) will follow on Ω This affords considerable economy of argument, since the direct study of the contribution of D to C is not obvious even for additive processes.

The superadditive theory (or, equivalently, the subadditive theory) is mostly known for its applications, but in fact the notion of a superadditive process is shown to shed light on the earlier additive theory of L1 operators.

We will begin with the material that will be used throughout the book. The idea of the stopping time, especially the simple stopping time, is central. The setting in which this naturally occurs involves Moore-Smith convergence, or convergence of nets or generalized sequences. This will be useful even if we are interested only in sequences of real-valued random variables; but will be even more useful when we consider derivation (Chapter 7) and processes indexed by directed sets (Chapter 4).

Given a stochastic process (Xn), a stopping time is a random variable τ taking values in IN ∪ {∞} such that, for each k, the event {τ = k} is determined by the first k random variables X1, X2, …, Xk A process (Xn) is an amart iff for every increasing sequence τn of bounded stopping times, E[Xτn] converges. (For variants of this definition, see Section 1.2.) The main result of this chapter is the amart convergence theorem for the index set IN, proved in Section 1.2. The argument, using stopping times, is elementary, and may be followed by a reader with only a basic knowledge of the measure theory. To make the point, we will sketch the proof of almost sure convergence of an amart (Xn) with integrable supremum. The basic observation is that there is an increasing sequence τn of simple stopping times such that Xτn converges in probability to X* = lim sup Xn. The reason for this is that lim sup or any other accumulation point manifests itself infinitely often on the way to infinity; it is like a light shining on the horizon.

INTRODUCTION

Let M be a Riemannian manifold. Our aim is to study differential forms on the following infinite dimensional manifolds:

1. (1) the path space PM consisting of paths w : [0, 1] → M,

2. (2) the loop space LM consisting of paths w such that w(0) = w(1),

3. (3) the based loop space LxM consisting of loops w such that w(0) = w(1) = x where x is a chosen base point in M.

One consequence of the fact that these manifolds are infinite dimensional is that there are infinite sequences αn of forms with each αn homogeneous of degree n. These infinite sequences are very important in the geometrical applications of loop spaces; for example they are essential in the theory of equivariant cohomology in infinite dimensions as is made quite clear in [25].

In [11] Chen describes the theory of “iterated integrals”; this is a method of constructing differential forms on these infinite dimensional manifolds. We will refer to forms constructed by this means as Chen forms. The purpose of this paper is to study some of the analytical properties of Chen forms; in particular to make estimates for suitable LP-norms and to consider various decay conditions which one might put on the terms in an infinite sequence of the kind mentioned in the previous paragraph.