Probability in Banach Spaces

Probability in Banach spaces is a branch of modern mathematics that emphasizes the geometric and functional analytic aspects of Probability Theory. Its probabilistic sources may be found in the study of regularity of random processes (especially Gaussian processes) and Banach space valued random variables and their limiting properties, whose functional developments revealed and tied up strong and fruitful connections with classical Banach spaces and their geometry.

In this first chapter, we present the isoperimetric inequalities which now appear as the crucial concept in the understanding of various concentration inequalities, tail behaviors and integrability theorems in Probability in Banach spaces. These inequalities often arise as the final and most elaborate forms of previous, weaker (but already efficient) inequalities which will be mentioned in their framework throughout the book. In these final forms however, the isoperimetric inequalities and associated concentration of measure phenomena provide the appropriate ideas for an in depth comprehension of some of the most important theorems of the theory.

This chapter collects in rather an informal way some basic facts about processes and infinite dimensional random variables. The material that we present actually only appears as the necessary background for the subsequent analysis developed in the next chapters. Only a few proofs are given and many important results are only just mentioned or even omitted. It is therefore recommended to complement, if necessary, these partial bases with the classical references, some of which are given at the end of the chapter.

With this chapter, we really enter into the subject of Probability in Banach spaces. The study of Gaussian random vectors and processes may indeed be considered as one of the fundamental topics of the theory since it inspires many other parts of the field both in the results themselves and in the techniques of investigation. Historically, the developments also followed this line of progress.

This chapter is devoted to Rademacher averages Σ _i ε _i x _i with vector valued coefficients as a natural analog of the Gaussian averages Σ _i g _i x _i . The properties we examine are entirely similar to those investigated in the Gaussian case. In this way, we will see how isoperimetric methods can be used to yield strong integrability properties of convergent Rademacher series and chaos. This is studied in Sections 4.3 and 4.4. Some comparison results are also available in the form, for example, of a version of Sudakov’s minoration presented in Section 4.5. However, we start in the first two sections with some basic facts about Rademacher averages with real coefficients as well as with the so-called contraction principle, a most valuable tool in Probability in Banach spaces.

After Gaussian variables and Rademacher series, we investigate in this chapter another important class of random variables and vectors, namely stable random variables. Stable random variables are fundamental in Probability Theory and, as will be seen later, also play a rôle in structure theorems of Banach spaces. The literature is rather extensive on this topic and we only concentrate here on the parts of the theory which will be of interest and use to us in the sequel. In particular, we do not attempt to study stable measures in the natural more general setting of infinitely divisible distributions. We refer to [Ar-G2] and [Li] for such a study. We only concentrate on the aspects of stable distributions analogous to those developed in the preceding chapters on Gaussian and Rademacher variables. In particular, our study is based on a most useful representation of stable random variables detailed in the first paragraph. The second section examines integrability properties and tail behavior of norms of infinite dimensional stable random variables. Finally, the last section is devoted to some comparison theorems.

Sums of independent random variables already appeared in the preceding chapters in some concrete situations (Gaussian and Rademacher averages, representation of stable random variables). On the intuitive basis of central limit theorems which approximate normalized sums of independent random variables by smooth limiting distributions (Gaussian, stable), one would expect that results similar to those presented previously should hold in a sense or in another for sums of independent random variables. The results presented in this chapter go in this direction and the reader will recognize in this general setting the topics covered before: integrability properties, equivalence of moments, concentration, tail behavior, etc. We will mainly describe ideas and techniques which go from simple but powerful observations such as symmetrization (randomization) techniques to more elaborate results like those obtained from the isoperimetric inequality for product measures of Theorem 1.4. Section 6.1 is concerned with symmetrization, Section 6.2 with Hoffmann-Jørgensen’s inequalities and the equivalence of moments of sums of independent random variables. In the last and main section, martingale and isoperimetric methods are developed in this context. Many results presented in this chapter will be of basic use in the study of limit theorems later.

In this chapter and in the next one, we present respectively the strong law of large numbers and the law of the iterated logarithm for sums of independent Banach space valued random variables. In this study, the isoperimetric approach of Section 6.3 demonstrates its efficiency. We only investigate extensions to vector valued random variables of some of the classical limit theorems such as the laws of large numbers of Kolmogorov and Prokhorov.

This chapter is devoted to the classical laws of the iterated logarithm of Kolmogorov and Hartman-Wintner-Strassen in the vector valued setting. These extensions both enlighten the scalar statements and describe various new interesting phenomena in the infinite dimensional setting. As in the previous chapter on the strong law of large numbers, the isoperimetric approach proves to be an efficient tool in this study. The main results described here show again how the strong almost sure statement of the law of the iterated logarithm reduces to the corresponding (necessary) statement in probability, under moment conditions similar to those of the scalar case.

The notion of type of a Banach space already appeared in the last chapters on the law of large numbers and the law of the iterated logarithm. We observed there that, in quite general situations, almost sure properties can be reduced to properties in probability or in L _P , 0 ≤ p < ∞. Starting with this chapter, we will now study the possibility of a control in probability, or in the weak topology, of probability distributions of sums of independent random variables.

The study of strong limit theorems for sums of independent random variables such as the strong law of large numbers or the law of the iterated logarithm in the preceding chapters showed that in Banach spaces these can only be reasonably understood when the corresponding weak property, that is tightness or convergence in probability, is satisfied. It was shown indeed that under some natural moment conditions, the strong statements actually reduce to the corresponding weak ones. On the line, or in finite dimensional spaces, the moment conditions usually automatically ensure the weak limiting property. As we pointed out, this is no longer the case in general Banach spaces.

In Chapter 9 we described how certain conditions on Banach spaces can ensure the existence and the tightness of some probability measures.

In the preceding chapter, we presented some sufficient metric entropy and majorizing measure conditions for the sample boundedness and continuity of random processes satisfying incremental conditions. In particular, these results were applied to Gaussian random processes in Section 11.3. The main concern of this chapter is necessity. We will see indeed, as one of the main results, that the sufficient majorizing measure condition for a Gaussian process to be almost surely bounded or continuous is actually also necessary. This characterization thus provides a complete understanding of the regularity properties of Gaussian paths. The arguments of proof rely heavily on the basic ultrametric structure which lies behind a majorizing measure condition.

In Chapter 11, we evaluated random processes indexed by an arbitrary index set T. In this chapter, we take advantage of some homogeneity properties of T and we investigate in this setting, using the general conclusions of Chapters 11 and 12, the more concrete random Fourier series. The tools developed so far indeed lead to a definitive treatment of those processes with applications to Harmonic Analysis. Our main reference for this chapter is the work by M. B. Marcus and G. Pisier [M-P1], [M-P2] to which we refer for an historical background and accurate references and priorities.

The purpose of this chapter is to present applications of the random process techniques developed so far to infinite dimensional limit theorems, and in particular to the central limit theorem (CLT). More precisely, we will be interested for example in the CLT in the space C(T) of continuous functions on a compact metric space T. Since C(T) is not well behaved with respect to the type or cotype 2 properties, we will rather have to seek for nice classes of random variables in C(T) for which a central limit property can be established. This point of view leads to enlarge this framework and to investigate limit theorems for empirical measures or processes. Random geometric descriptions of the CLT may then be produced via this approach, as well as complete descriptions for nice classes of functions (indicator functions of some sets) on which the empirical processes are indexed. While these random geometric descriptions do not solve the central limit problem in infinite dimension (and are probably of little use in applications), however, they clearly describe the main difficulties inherent to the problem from the empirical point of view.

This last chapter emphasizes some applications of isoperimetric methods and of process techniques of Probability in Banach spaces to the local theory of Banach spaces. The applications which we present are only a sample of some of the recent developments in the local theory of Banach spaces (and we refer to the lists of references, and seminars and proceedings, for further main examples in the historical developments). They demonstrate the power of probabilistic ideas in this context. This chapter is organized along its subtitles of rather independent context. Several questions and conjectures are presented in addition, some with details as in Sections 15.2 and 15.6, the others in the last paragraph on miscellaneous problems.