Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference PDF
Preview Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference
Progress in Probability Volume 30 Series Editors Thomas Liggett Charles Newman Loren Pitt Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference Richard M. Dudley Marjorie G. Hahn James Kuelbs Editors Springer Science+Business Media, LLC Richard M. Dudley Marjorie G. Hahn Depl. of Mathematics Department of Mathematics MIT Tufts University Cambridge, MA 02139 Medford, MA 02178 James Kuelbs Depl. of Mathematics University of Wisconsin Madison, WI 53706 Library of Congress Cataloging-in-Publication Data Probability in Banach spaces, 8 : proceedings of the eighth international conference I edited by Richard M. Dudley, Marjorie G. Rahn, James Kuelbs. p. cm. ._- (Progress in probability ; 30) "From a two-week NSF-sponsored session '" held at Bowdoin College in the summer of 1991 "--Pref. Includes bibliographical references. ISBN 978-1-4612-6728-7 ISBN 978-1-4612-0367-4 (eBook) DOI 10.1007/978-1-4612-0367-4 1. Probabilities--Congresses. 2. Banach spaces--Congresses. 1. Dudley, R. M. (Richard M.) II. Rahn, Marjorie G. III. Kuelbs, James. IV. Title: Probability in Banach spaces, eight. V. Series. QA273.43.P773 1992 92-17649 519.2--dc20 CIP Printed on acid-free paper © 1992 Springer Science+Business Media New York Originally published by Birkhliuser Boston in 1992 Softcover reprint of the hardcover 1s t edition 1992 Copyright is not claimed for works of D.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrleval system, or transmitted, in any form or by any means, electronic, mechanical, photo copying, recording, Of otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Springer Science+Business Media, LLC for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, D.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC ISBN 978-1-4612-6728-7 Camera-ready copy prepared by the Authors in TeX. 9 8 7 6 5 432 1 CONTENTS Contents ...................................................................... v Preface ....................................................................... ix Matching Theorems An exposition of Talagrand's mini-course on matching theorems Marjorie C. Hahn and Yongzhao Shao ..................................... 3 The Ajtai-Komlos-Tusnady matching theorem for general measures Michel Talagrand ......................................................... 39 Some generalizations of the Euclidean two-sample matching problem Joe Yukich ............................................................... 55 Inequalities and Approximations Sharp bounds on the Lp norm of a randomly stopped multilinear form with an application to Wald's equation Victor de La Peria ......................................................... 69 On Hoffmann-Jorgensen's inequality for U-processes Evarist Cine and Joel Zinn ............................................... 80 The Poisson counting argument: A heuristic for understanding what makes a Poissonized sum large Marjorie C. Hahn and Michael J. Klass ................................... 92 On the lower tail of Gaussian measures on lp Wenbo V. Li ............................................................ 106 Conditional versions of the Strassen-Dudley Theorem Ditlev Monrad and Walter Philipp ....................................... 116 An approach to inequalities for the distributions of infinite-dimensional martingales losi! Pinelis ............................................................. 128 Stochastic Processes Random integral representations for classes of limit distributions similar to Levy class Lo. III Zbigniew J. Jurek ........................................................ 137 Asymptotic dependence of stable self-similar processes of Chentsov-type Piotr S. Kokoszka and Murad S. Taqqu .................................. 152 Distributions of stable processes on spaces of measurable functions Rimas Norvaisa ......................................................... 166 Harmonizability, V-boundedness, and stationary dilation of Banach space-valued processes Philip H. Richard ........................................................ 189 Weak Convergence and Large Deviations Asymptotic behavior of self-normalized trimmed sums: Nonnormallimits III Marjorie C. Hahn and Daniel C. Weiner ................................ 209 On large deviations of Gaussian measures in Banach spaces Marek Slaby ............................................................. 228 Mosco convergence and large deviations Sandy Zabell ............................................................ 245 Strong Limit Theorems and Approximations A functional LIL approach to pointwise Bahadur-Kiefer theorems Paul Deheuvels and David M. Mason .................................... 255 The Glivenko-Cantelli theorem in a Banach space setting Vladimir Dobric ......................................................... 267 Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics Evarist Cine and Joel Zinn .............................................. 273 Self-normalized bounded laws of the iterated logarithm in Banach spaces A nant Codbole .......................................................... 292 Rates of clustering for weakly convergent Gaussian vectors and some applications Victor Goodman and Jim K uelbs ........................................ 304 On the almost sure summability of B-valued random variables Bernard Heinkel ......................................................... 325 On the rate" of Clustering in Strassen's LIL for Brownian Motion Michel Talagrand ........................................................ 339 Local Times of Stochastic Processes A central limit theorem for the renormalized self-intersection local time of a stationary process Simeon M. Berman ...................................................... 351 Moment generating functions for local times of symmetric Markov processes and random walks Michael B. Marcus and Jay Rosen ....................................... 364 Empirical Processes and Applications Partial-sum processes with random lattice-points and indexed by Vapnik Cervonenkis classes of sets in arbitrary sample spaces Miguel A. Arcones, Peter Caenssler, and Klaus Ziegler ................... 379 Learnability models and Vapnik-Chervonenkis combinatorics Anselm Blumer .......................................................... 390 Nonlinear functionals of empirical measures Richard M. Dudley ...................................................... 403 KAC empirical processes and the bootstrap Chris A. J. Klaassen and Jon A. Wellner ............................... .411 Functional limit theorems for probability forecasts Deborah Nolan .......................................................... 430 Exponential bounds in Vapnik-Cervonenkis classes of index 1 Daphne L. Smith and Richard M. Dudley ................................ 451 Applications to Statistics and Engineering Tail estimates for empirical characteristic functions, with applications to random arrays George Benke and W. James Hendricks .................................. 469 The radial process for confidence sets Rudolf Beran ............................................................ 479 Stochastic search in a Banach space P. Warwick Millar ...................................................... 497 PREFACE Probability limit theorems in infinite-dimensional spaces give conditions un der which convergence holds uniformly over an infinite class of sets or functions. Early results in this direction were the Glivenko-Cantelli, Kolmogorov-Smirnov and Donsker theorems for empirical distribution functions. Already in these cases there is convergence in Banach spaces that are not only infinite-dimensional but nonsep arable. But the theory in such spaces developed slowly until the late 1970's. Meanwhile, work on probability in separable Banach spaces, in relation with the geometry of those spaces, began in the 1950's and developed strongly in the 1960's and 70's. We have in mind here also work on sample continuity and boundedness of Gaussian processes and random methods in harmonic analysis. By the mid-70's a substantial theory was in place, including sharp infinite-dimensional limit theorems under either metric entropy or geometric conditions. Then, modern empirical process theory began to develop, where the collection of half-lines in the line has been replaced by much more general collections of sets in and functions on multidimensional spaces. Many of the main ideas from probability in separable Banach spaces turned out to have one or more useful analogues for empirical processes. Tightness became "asymptotic equicontinuity." Metric entropy remained useful but also was adapted to metric entropy with bracketing, random entropies, and Kolchinskii-Pollard entropy. Even norms themselves were in some situations replaced by measurable majorants, to which the well-developed separable theory then carried over straightforwardly. Also, what had seemed to be disparate parts of the two theories developed unexpected connections. For example, "Type 2" is one very useful geometric concept in separable Banach spaces, while in empirical processes the VC (Vapnik-Cervonenkis) combinatorial condition (non-shattering of large finite sets) is very important. Gilles Pisier proved in 1984 that a class of sets is a VC class if and only if a certain operator is of type 2. In empirical process theory as applied to statistics, one first estimates an un- x PREFACE known probability measure P by an observed empirical measure Pn. Then it is possible to construct a confidence set for P by finding the variability with respect p!! - to a supremum over some class of sets or functions of the difference Pn where p!! is a bootstrap empirical measure, which results from iterating the operation of taking the empirical measure and can be observed by simulation. In the 1980's, probability in separable Banach spaces, and some parts of em pirical process theory that had been closely based on the separable case, such as Gaussian randomization methods, developed into a highly advanced set of tech niques, especially in the work of Gine, Zinn, Ledoux and Talagrand. Thus, Gine and Zinn were able to prove that the uniform asymptotic normality of families of (linear) bootstrap statistics was equivalent to the same property for empirical (non bootstrap) measures. Not only the result itself but its proof were impressive, being based on a number of facts that had been first proved in separable Banach spaces by themselves and others, including Ledoux and Talagrand. And, large classes of functions are known for which the uniform asymptotic normality holds. This book resulted from a two-week NSF-sponsored session on probability in Banach spaces held at Bowdoin College in the summer of 1991. Although the con ference had a rough division between probability in separable Banach spaces in the first week and empirical processes in the second week, there were talks on both top ics in both weeks. It was clear that probability in both separable and non-separable Banach spaces is thriving with new ideas, new techniques, and applications to a multitude of new problems. Central to the program were minicourses on "Matching Theorems and Empir ical Discrepancies" given by Michel Talagrand in the first week and on "Empirical Processes" given by Evarist Gine and Joel Zinn in the second week. We wish to thank the speakers for these minicourses and NSF for its financial support. We ex tend our gratitude to the many people who helped referee the papers submitted for publication in this proceedings. Their numerous constructive suggestions led to im proved clarity and results in the papers accepted. This volume presents Talagrand's PREFACE XI course, and a selection of other papers. These papers indicate the broad range of interests of the participants and the variety of problems that can be attacked with the methods developed in this area. Two examples, indicating this range of appli cations, are the use of probability in Banach space theory for studying local times of Markov processes and relations of empirical process ideas with developments in theoretical computer science (learning theory). The volume should provide the reader with a good view of the present status of probability in Banach spaces in its many different aspects. Richard M. Dudley Marjorie G. Hahn James Kuelbs April, 1992
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