R. M. Dudley

The sizes of compact subsets of Hilbert space and continuity of Gaussian processes

The first two sections of this paper are introductory and correspond to the two halves of the title. As is well known, there is no complete analog of Lebesue or Haar measure in an infinite-dimensional Hilbert space H, but there is a need for some measure of the sizes of subsets of H. In this paper we shall study subsets C of H which are closed, bounded, convex and symmetric (— x e C if x e C). Such a set C will be called a Banach ball, since it is the unit ball of a complete Banach norm on its linear span. In most cases in this paper C will be compact.

Sample Functions of the Gaussian Process

This is a survey on sample function properties of Gaussian processes with the main emphasis on boundedness and continuity, including Holder conditions locally and globally. Many other sample function properties are briefly treated.

Distances of Probability Measures and Random Variables

Let (S, d) be a separable metric space. Let \( P; > \left( S \right)\) be the set of Borel probability measures on S. \(C\left( S \right)\) denotes the Banach space of bounded continuous real-valued functions on S, with norm

Invariance Principles for Sums of Banach Space Valued Random Elements and Empirical Processes

\left\| f \right\|_\infty= \sup \left\{ {\left| {f\left( x \right)} \right|:x{\text{ }}\varepsilon {\text{ }}S} \right\}.

On seminorms and probabilities, and abstract Wiener spaces

When \(\mathfrak{F}\) is a universal Donsker class, then for independent, indetically distributed (i.i.d) observation \(\mathbf{X}_1,\ldots,\mathbf{X}_n\) with an unknown law P, for any \(\mathfrak{f}_i\)in \(\mathfrak{F},\) \(i=1,\ldots,m,\quad n^{-1/2}\left\{ \mathfrak{f}_1\left(\mathbf{X}_1\right)+\ldots+\mathfrak{f}_i\left(\mathbf{X}_n\right)\right\}_{1\leq i\leq m}\) is asymptotically normal with mean Vector \(n^{1/2}\left\{\int\mathfrak{f}_i\left(\mathbf{X}_n\right)d\mathbf{P}\left(x\right)\right\}_{1_\leq i\leq m}\) and covariance matrix \(\int\mathfrak{f}_i\mathfrak{f}_j d\mathbf{P}-\int\mathfrak{f}_id\mathbf{P}\int\mathfrak{f}_jd\mathbf{P},\) uniformly for \({\mathfrak{f}_i}\in \mathfrak{F}.\) Then, for certain Statistics formed frome the \(\mathfrak{f}_i\left(\mathbf{X}_k\right),\) even where \(\mathfrak{f}_i\) may be chosen depending on the \(\mathbf{X}_k\) there will be asymptotic distribution as \(n \rightarrow \infty.\) For example, for \(\mathbf{X}^2\) statistics, where \(f_i\) are indicators of disjoint intervals, depending suitably on \(\mathbf{X}_1,\ldots,\mathbf{X}_n\), whose union is the real line, \(\mathbf{X}^2\) quadratic forms have limiting distributions [Roy (1956) and Watson (1958)] which may, however, not be \(\mathbf{X}^2\) distributions and may depend on P [Chernoff and Lehmann (1954)]. Universal Donsker classes of sets are, up to mild measurability conditions, just classes satisfying the Vapnik–Cervonenkis comdinatorial conditions defined later in this section Donsker the Vapnik-Cervonenkis combinatorial conditions defined later in this section [Durst and Dudley (1981) and Dudley (1984) Chapter 11]. The use of such classes allows a variety of extensions of the Roy–Watson results to general (multidimensional) sample spaces [Pollard (1979) and Moore and Subblebine (1981)]. Vapnik and Cervonenkis (1974) indicated application of their families of sets to classification (pattern recognition) problems. More recently, the classes have been applied to tree-structured classifiacation [Breiman, Friedman, Olshen and Stone (1984), Chapter 12].

Wiener Functionals as Ito Integrals

Almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable Banach space. It is then shown that empirical processes readily fit into this general framework. Thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes.

The Order of the Remainder in Derivatives of Composition and Inverse Operators for

Let \({\eta}=\left({\eta_j}\right)\) be a sequence of independent Gaussian, means 0, Varriance 1, random variable in all of this paper. We shall then study the distribution of