High dimensional probability II

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. Front Matter....Pages i-xi Front Matter....Pages 1-1 Moment Bounds for Self-Normalized Martingales....Pages 3-11 Exponential and Moment Inequalities for U -Statistics....Pages 13-38 A Multiplicative Inequality for Concentration Functions of n -Fold Convolutions....Pages 39-47 On Exact Maximal Khinchine Inequalities....Pages 49-63 Strong Exponential Integrability of Martingales with Increments Bounded by a Sequence of Numbers....Pages 65-76 Front Matter....Pages 77-77 On Uniform Laws of Large Numbers for Smoothed Empirical Measures....Pages 79-87 Weak Convergence of Smoothed Empirical Processes: Beyond Donsker Classes....Pages 89-105 Limit Theorems for Smoothed Empirical Processes....Pages 107-113 Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes....Pages 115-133 Front Matter....Pages 135-135 Continuité de certaines fonctions aléatoires gaussiennes à valeurs dans l p , 1 ≤ p < ∞....Pages 137-161 A Note on the Gaussian Correlation Conjecture....Pages 163-171 Probability Estimates for Lower Levels of Certain Gaussian Processes with Stationary Increments....Pages 173-179 Front Matter....Pages 181-181 Asymptotic Independence of the Local Empirical Process Indexed by Functions....Pages 183-205 The Azéma-Yor Embedding in Brownian Motion with Drift....Pages 207-221 A New Way to Obtain Estimates in the Invariance Principle....Pages 223-245 Front Matter....Pages 247-247 On the Law of the Iterated Logarithm for Local Times of Recurrent Random Walks....Pages 249-259 A General Compact Law of the Iterated Logarithm in Banach Spaces....Pages 261-278 Front Matter....Pages 279-279 Dominating Points for General Sequences of Random Vectors....Pages 281-291 Large Deviations for Local Empirical Measures....Pages 293-312 Front Matter....Pages 313-313 An Example Concerning Tightness of Sums of B-Valued Random Variables....Pages 315-327 Front Matter....Pages 313-313 Images and Level Sets of Additive Random Walks....Pages 329-345 Front Matter....Pages 347-347 Lee-Yang Models, Selfdecomposability and Negative-Definite Functions....Pages 349-357 When an Isotropic Random Vector is Pseudo-Isotropic....Pages 359-366 Support Fragmentation for Multiplicative Cascade Measures....Pages 367-376 On Simulating Fractional Brownian Motion....Pages 377-387 Front Matter....Pages 389-389 On Robust Recursive Nonparametric Curve Estimation....Pages 391-403 Variable Kernel Estimates: on the Impossibility of Tuning the Parameters....Pages 405-424 Almost Sure Asymptotic Optimality of Cross Validation for Spline Smoothing....Pages 425-441 Rademacher Processes and Bounding the Risk of Function Learning....Pages 443-457 Front Matter....Pages 459-459 Bootstrapping Empirical Distributions under Auxiliary Information....Pages 461-476 On the Characteristic Function of the Matrix von Mises—Fisher Distribution with Application to SO ( N )—Deconvolution....Pages 477-492 Testing for Ellipsoidal Symmetry of a Multivariate Distribution....Pages 493-510 Back Matter....Pages 511-512