Let T be any set and (Ω, A, P) a probability space. Recall that a real-valued stochastic process indexed by T is a function (t, ω) ↦ Xt(ω) from T × Ω into ℝ such that for each t ∈ T, Xt (·) is measurable from Ω into ℝ. A modification of the process is another stochastic process Yt defined for the same T and Ω such that for each t, we have P(Xt = Yt) = 1. A version of the process Xt, is a process Zt, t ∈ T, for the same T but defined on a possibly different probability space (Ω1, B, Q) such that Xt and Zt, have the same laws, that is, for each finite subset F of Clearly, any modification of a process is also a version of the process, but a version, even if on the same probability space, may not be a modification. For example, for an isonormal process L on a Hilbert space H, the process M(x) ≔ L(−x) is a version, but not a modification, of L.

One may take a version or modification of a process in order to get better properties such as continuity. It turns out that for the isonormal process on subsets of Hilbert space, what can be done with a version can also be done by a modification, as follows.

TheoremLet L be an isonormal process restricted to a subset C of Hilbert space. For each of the following two properties, if there exists a version M of L with the property, there also is a modification N with the property.

Let A be the set of all possible empirical distribution functions F1 for one observation x ∈ [0, 1], namely F1(t) = 0 for t < x and F1{t) = 1 for t ≥ x. We noted previously that A in the supremum norm is nonseparable: it is an uncountable set, in which any two points are at a distance 1 apart. Thus A and all its subsets are closed. If x ≔ X1 has a continuous distribution such as the uniform distribution U[0, 1] on [0, 1], then x → (t → 1t≥x) takes [0, 1] onto A, but it is not continuous for the supremum norm. Also, it is not measurable for the Borel σ-algebra on the range space. So, in Chapter 3, functions f* and upper expectations E* were used to get around measurability problems.

Here is a different kind of example. It is related to the basic “ordinal triangle” counterexample in integration theory, showing why measurability is needed in the Tonelli-Fubini theorem on Cartesian product integrals. Let (Ω, ≤) be an uncountable well-ordered set such that for each x ∈ Ω, the initial segment Ix ≔ {y : y ≤ x} is countable. (In terms of ordinals, Ω is, or is orderisomorphic to, the least uncountable ordinal.) Let S be the σ-algebra of subsets of Ω consisting of sets that are countable or have countable complement. Let P be the probability measure on S which is 0 on countable sets and 1 on sets with countable complement.

Let (S, ∥·∥) be a Banach space (in general nonseparable). A subset of the unit ball {f ∈ S′ : ∥f∥′ ≤ 1} is called a norming subset if and only if for all s ∈ S. The whole unit ball in S is always a norming subset by the Hahn-Banach theorem (RAP, Corollary 6.1.5).

Conversely, given any set, let be the set of all bounded real functions on, with the supremum norm

Then the natural map f ↦ (s ↦ s(f)) takes one-to-one onto a norming subset of S′.

So, limit theorems for empirical measures, uniformly over a class of functions, can be viewed as limit theorems in a Banach space S with norm Conversely, limit theorems in a general Banach space S with norm ∥ · ∥ can be viewed as limit theorems for empirical measures on S, uniformly over a class of functions, such as the unit ball of S′, since for f ∈ S′ and x1, …, xn ∈ S,

Suppose that Xj are i.i.d. real random variables with mean 0 and variance 1. Let One form of “invariance principle” will say that on some probability space, there exist such Xj and also i.i.d. N(0, 1) variables Y1, Y2, …, with such that as n → ∞ in probability. Since Tn/n½ also has a N(0, 1) distribution for each n, the invariance principle implies that Sn/n½ is close to Tn/n½, which implies the central limit theorem. Although it is not as obvious, central limit theorems generally imply invariance principles.

A Gaussian Hilbert space is a (complete) linear space of random variables with (centred) Gaussian distributions. This simple notion combines probability theory and Hilbert space theory into a rich and powerful structure, and Gaussian Hilbert spaces and connected notions such as the Wiener chaos decomposition and Wick products appear in several areas of probability theory and its applications, for example in stochastic processes and fields, stochastic integration, quantum field theory and limit theory for various statistics. There are also applications to non-probabilistic analysis, for example Banach space geometry and partial differential equations.

Although there are many references dealing with such applications where Gaussian spaces are treated and used, see for example Hida and Hitsuda (1976), Hida, Kuo, Potthoff and Streit (1993), Holden, Øksendal, Ubøe and Zhang (1996), Ibragimov and Rozanov (1970), Kahane (1985), Kuo (1996), Major (1981), Malliavin (1993, 1997), Meyer (1993), Neveu (1968), Nualart (1995, 1997+), Obata (1994), Pisier (1989), Simon (1974, 1979a), Watanabe (1984), there seems to be a shortage of works dealing with the basic properties of Gaussian spaces in general, without connecting them to a particular application. (One exception is the paper by Dobrushin and Minlos (1977).) This book is an attempt to fill the gap by providing a collection of the most important definitions and results for general Gaussian spaces, together with some applications to special Gaussian spaces.

In this chapter we use the theory of Gaussian Hilbert spaces to obtain the asymptotic distributions of some important random variables. In the first section, we study U-statistics. In the second section we extend the results to asymmetric statistics. In the third section we extend the results further; as a special case we obtain results for random graphs.

Note that the original variables are defined without any reference to normal variables or Gaussian Hilbert spaces; the Gaussian Hilbert space is introduced as a convenient tool to treat the asymptotic distribution.

A common theme in these results is that ‘typically’ the asymptotic distribution is normal, but in some ‘degenerate’ cases other limits occur; these other limits can be represented as variables in a Wiener chaos H:n: for some Gaussian Hilbert space H, and can for example be expressed as multiple Gaussian stochastic integrals. The explanation for this phenomenon that emerges from the proofs below is that the variable in question may be expanded as a sum, where each term converges in distribution to some chaos, and the first term is asymptotically normal. Typically, the first term dominates the sum and all others are asymptotically negligible, but in degenerate cases the first term vanishes and the sum is dominated by one or several later terms, which may converge to a higher order chaos.