High Dimensional Probability

1. Front Matter

   Pages i-viii

   PDF

2. Weak Convergence of the Row Sums of a Triangular Array of Empirical Processes

   • Miguel A. Arcones

   Pages 1-25

3. Self-Normalized Large Deviations in Vector Spaces

   • Amir Dembo, Qi-Man Shao

   Pages 27-32

4. Consistency of M-Estimators and One-Sided Bracketing

   • Richard M. Dudley

   Pages 33-58

5. Small Deviation Probabilities of Sums of Independent Random Variables

   • T. Dunker, M. A. Lifshits, W. Linde

   Pages 59-74

6. Strong Approximations to the Local Empirical Process

   • Uwe Einmahl, David M. Mason

   Pages 75-92

7. On Random Measure Processes with Application to Smoothed Empirical Processes

   • Peter Gaenssler, Daniel Rost, Klaus Ziegler

   Pages 93-102

8. A Consequence For Random Polynomials of a Result of De La Peña and Montgomery-Smith

   • Evarist Giné

   Pages 103-110

9. Distinctions Between the Regular and Empirical Central Limit Theorems for Exchangeable Random Variables

   • Marjorie G. Hahn, Gang Zhang

   Pages 111-143

10. Laws of Large Numbers and Continuity of Processes

    • Bernhard Heinkel

    Pages 145-149

11. Convergence in Law of Random Elements and Random Sets

    • Jørgen Hoffmann-Jørgensen

    Pages 151-189

12. Asymptotics of Spectral Projections of Some Random Matrices Approximating Integral Operators

    • Vladimir I. Koltchinskii

    Pages 191-227

13. A Short Proof of the Gaussian Isoperimetric Inequality

    • Michel Ledoux

    Pages 229-232

14. Some Shift Inequalities for Gaussian Measures

    • Wenbo V. Li, James Kuelbs

    Pages 233-243

15. A Central Limit Theorem for the Sock-Sorting Problem

    • Wenbo V. Li, Geoffrey Pritchard

    Pages 245-248

16. Oscillations of Gaussian Stein’s Elements

    • Mikhail Lifshits, Michel Weber

    Pages 249-261

17. A Sufficient Condition for the Continuity of High Order Gaussian Chaos Processes

    • Michael B. Marcus

    Pages 263-276

18. On Wald’s Equation and First Exit Times for Randomly Stopped Processes with Independent Increments

    • Victor H. de la Peña

    Pages 277-286

19. The Best Doob-Type Bounds for the Maximum of Brownian Paths

    • Goran Peškir

    Pages 287-296

20. Optimal Tail Comparison Based on Comparison of Moments

    • Iosif Pinelis

    Pages 297-314

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.

• Estimator
• Gaussian measure
• Probability
• Random variable
• Stochastic processes
• mixing
• random measure
• statistics
• stochastic process

• Institut für Mathematische Stochastik, Universität Freiburg, Freiburg, Germany

  Ernst Eberlein

• Department of Mathematics, Tufts University, Medford, USA

  Marjorie Hahn

• Equipe d’analyse, Tour 46, Université Paris VI, Paris Cedex 05, France

  Michel Talagrand

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