High Dimensional Probability

What is high dimensional probability? Under this broad name we collect topics with a common philosophy, where the idea of high dimension plays a key role, either in the problem or in the methods by which it is approached. Let us give a specific example that can be immediately understood, that of Gaussian processes. Roughly speaking, before 1970, the Gaussian processes that were studied were indexed by a subset of Euclidean space, mostly with dimension at most three. Assuming some regularity on the covariance, one tried to take advantage of the structure of the index set. Around 1970 it was understood, in particular by Dudley, Feldman, Gross, and Segal that a more abstract and intrinsic point of view was much more fruitful. The index set was no longer considered as a subset of Euclidean space, but simply as a metric space with the metric canonically induced by the process. This shift in perspective subsequently lead to a considerable clarification of many aspects of Gaussian process theory, and also to its applications in other settings. Front Matter....Pages i-viii Weak Convergence of the Row Sums of a Triangular Array of Empirical Processes....Pages 1-25 Self-Normalized Large Deviations in Vector Spaces....Pages 27-32 Consistency of M -Estimators and One-Sided Bracketing....Pages 33-58 Small Deviation Probabilities of Sums of Independent Random Variables....Pages 59-74 Strong Approximations to the Local Empirical Process....Pages 75-92 On Random Measure Processes with Application to Smoothed Empirical Processes....Pages 93-102 A Consequence For Random Polynomials of a Result of De La Peña and Montgomery-Smith....Pages 103-110 Distinctions Between the Regular and Empirical Central Limit Theorems for Exchangeable Random Variables....Pages 111-143 Laws of Large Numbers and Continuity of Processes....Pages 145-149 Convergence in Law of Random Elements and Random Sets....Pages 151-189 Asymptotics of Spectral Projections of Some Random Matrices Approximating Integral Operators....Pages 191-227 A Short Proof of the Gaussian Isoperimetric Inequality....Pages 229-232 Some Shift Inequalities for Gaussian Measures....Pages 233-243 A Central Limit Theorem for the Sock-Sorting Problem....Pages 245-248 Oscillations of Gaussian Stein’s Elements....Pages 249-261 A Sufficient Condition for the Continuity of High Order Gaussian Chaos Processes....Pages 263-276 On Wald’s Equation and First Exit Times for Randomly Stopped Processes with Independent Increments....Pages 277-286 The Best Doob-Type Bounds for the Maximum of Brownian Paths....Pages 287-296 Optimal Tail Comparison Based on Comparison of Moments....Pages 297-314 The Bootstrap of Empirical Processes for α-Mixing Sequences....Pages 315-330 Back Matter....Pages 332-335 What is high dimensional probability? Under this broad term one finds a collection of topics associated by the fact that �n plays a key role in each, whether the idea of high dimension �n is expressed in the problem or in the methods by which it is approached. For example, the study of probability in Banach spaces gave impetus to a number of methods whose importance has gone far beyond the original goal of extending limit laws to the vector valued case. Familiar applications are in the areas of empirical processes, the use of majorizing measures to study regularity of stochastic processes, and the theory of concentration of measure. Many of the new ideas, results and directions of this newly evolving field were explored on a broad front at the Conference on High Dimensional Probability held at Oberwolfach in August 1996. The papers in this volume are marked by vitality and diversity and will give researchers and graduate students in probability or statistics much to whet their interest What is high dimensional probability? Under this broad term one finds a collection of topics associated by the fact that ñ plays a key role in each, whether the idea of high dimension ñ is expressed in the problem or in the methods by which it is approached. For example, the study of probability in Banach spaces gave impetus to a number of methods whose importance has gone far beyond the original goal of extending limit laws to the vector valued case. Familiar applications are in the areas of empirical processes, the use of majorizing measures to study regularity of stochastic processes, and the theory of concentration of measure. Many of the new ideas, results and directions of this newly evolving field were explored on a broad front at the Conference on High Dimensional Probability held at Oberwolfach in August 1996. The papers in this volume are marked by vitality and diversity and will give researchers and graduate students in probability or statistics much to whet their interest