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.

Inhalt
Weak Convergence of the Row Sums of a Triangular Array of Empirical Processes.- Self-Normalized Large Deviations in Vector Spaces.- Consistency of M-Estimators and One-Sided Bracketing.- Small Deviation Probabilities of Sums of Independent Random Variables.- Strong Approximations to the Local Empirical Process.- On Random Measure Processes with Application to Smoothed Empirical Processes.- A Consequence for Random Polynomials of a Result of De La Peña and Montgomery-Smith.- Distinctions Between the Regular and Empirical Central Limit Theorems for Exchangeable Random Variables.- Laws of Large Numbers and Continuity of Processes.- Convergence in Law of Random Elements and Random Sets.- Asymptotics of Spectral Projections of Some Random Matrices Approximating Integral Operators.- A Short Proof of the Gaussian Isoperimetric Inequality.- Some Shift Inequalities for Gaussian Measures.- A Central Limit Theorem for the Sock-Sorting Problem.- Oscillations of Gaussian Stein's Elements.- A Sufficient Condition for the Continuity of High Order Gaussian Chaos Processes.- On Wald's Equation and First Exit Times for Randomly Stopped Processes with Independent Increments.- The Best Doob-type Bounds for the Maximum of Brownian Paths.- Optimal Tail Comparison Based on Comparison of Moments.- The Bootstrap of Empirical Processes for ?-Mixing Sequences.