High Dimensional Probability II, 2000
Progress in Probability Series, Vol. 47

High dimensional probability, in the sense that encompasses the topics rep­ resented in this volume, began about thirty years ago with research in two related areas: limit theorems for sums of independent Banach space valued random vectors and general Gaussian processes. An important feature in these past research studies has been the fact that they highlighted the es­ sential probabilistic nature of the problems considered. In part, this was because, by working on a general Banach space, one had to discard the extra, and often extraneous, structure imposed by random variables taking values in a Euclidean space, or by processes being indexed by sets in R or Rd. Doing this led to striking advances, particularly in Gaussian process theory. It also led to the creation or introduction of powerful new tools, such as randomization, decoupling, moment and exponential inequalities, chaining, isoperimetry and concentration of measure, which apply to areas well beyond those for which they were created. The general theory of em­ pirical processes, with its vast applications in statistics, the study of local times of Markov processes, certain problems in harmonic analysis, and the general theory of stochastic processes are just several of the broad areas in which Gaussian process techniques and techniques from probability in Banach spaces have made a substantial impact. Parallel to this work on probability in Banach spaces, classical proba­ bility and empirical process theory were enriched by the development of powerful results in strong approximations.

Moment Bounds for Self-Normalized Martingales.- Exponential and Moment Inequalities for U-Statistics.- A Multiplicative Inequality for Concentration Functions of n-Fold Convolutions.- On Exact Maximal Khinchine Inequalities.- Strong Exponential Integrability of Martingales with Increments Bounded by a Sequence of Numbers.- On Uniform Laws of Large Numbers for Smoothed Empirical Measures.- Weak Convergence of Smoothed Empirical Processes: Beyond Donsker Classes.- Limit Theorems for Smoothed Empirical Processes.- Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes.- Continuité de certaines fonctions aléatoires gaussiennes à valeurs dans lp, 1?pof Cross Validation for Spline Smoothing.- Rademacher Processes and Bounding the Risk of Function Learning.- Bootstrapping Empirical Distributions under Auxiliary Information.- On the Characteristic Function of the Matrix von Mises-Fisher Distribution with Application to SO(N)—Deconvolution.- Testing for Ellipsoidal Symmetry of a Multivariate Distribution.