Selected Works of R. M. Dudley

by Gine, Evarist; Koltchinskii, Vladimir; Norvaisa, Rimas

• ISBN13: 9781441958204
• ISBN10: 1441958207
• Format: Hardcover
• Copyright: 2010-10-02
• Publisher: Springer Verlag

Summary

For almost fifty years, Richard M. Dudley has been extremely influential in the development of several areas of Probability. His work on Gaussian processes led to the understanding of the basic fact that their sample boundedness and continuity should be characterized in terms of proper measures of complexity of their parameter spaces equipped with the intrinsic covariance metric. His sufficient condition for sample continuity in terms of metric entropy is widely used and was proved by X. Fernique to be necessary for stationary Gaussian processes, whereas its more subtle versions (majorizing measures) were proved by M. Talagrand to be necessary in general.Together with V. N. Vapnik and A. Y. Cervonenkis, R. M. Dudley is a founder of the modern theory of empirical processes in general spaces. His work on uniform central limit theorems (under bracketing entropy conditions and for Vapnik-Cervonenkis classes), greatly extends classical results that go back to A. N. Kolmogorov and M. D. Donsker, and became the starting point of a new line of research, continued in the work of Dudley and others, that developed empirical processes into one of the major tools in mathematical statistics and statistical learning theory.As a consequence of Dudley's early work on weak convergence of probability measures on non-separable metric spaces, the Skorohod topology on the space of regulated right-continuous functions can be replaced, in the study of weak convergence of the empirical distribution function, by the supremum norm. In a further recent step Dudley replaces this norm by the stronger p-variation norms, which then allows replacing compact differentiability of many statistical functionals by Fréchet differentiability in the delta method.Richard M. Dudley has also made important contributions to mathematical statistics, the theory of weak convergence, relativistic Markov processes, differentiability of nonlinear operators and several other areas of mathematics. Professor Dudley has been the adviser to thirty PhD's and is a Professor of Mathematics at the Massachusetts Institute of Technology.

Author Biography

Richard M. Dudley has also made important contributions to mathematical statistics the theory of weak convergence, relativistic Markov processes, differentiability of nonlinear operators and several other area of mathematics. Professor Dudley has been the adviser to thirty PhD's and is a Professor of Mathematics at the Massachusetts Institute of Technology.

Table of Contents

Convergence in Law
Introduction	p. 3
Weak convergence of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math., 10:109-126, 1966. Reprinted with permission of the Illinois Journal of Mathematics, published by the University of Illinois at Urbana-Champaign	p. 5
Measures on non-separable metric spaces. Illinois J. Math., 11:449-453, 1967. Reprinted with permission of the Illinois Journal of Mathematics, published by the University of Illinois at Urbana-Champaign	p. 23
Distances of probability measures and random variables. Ann. Math. Statist., 39:1563-1572, 1968. Reprinted with permission of the Institute of Mathematical Statistics	p. 28
An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions. In A. Beck, R. Dudley, M. Hahn, J. Kuelbs, M. Marcus, editors, Probability in Banach Spaces V (Medford, Mass., 1984), volume 1153 of Lecture Notes in Math., pages 141-178. Springer, Berlin, 1985. Reprinted with permission of Springer Science+Business Media	p. 38
Markov Processes
Introduction	p. 79
Lorentz-invariant Markov processes in relativistic phase space. Ark. Mat., 6:241-268, 1966. Reprinted with permission of Springer Netherlands	p. 81
A note on Lorentz-invariant Markov processes. Ark. Mat., 6:575-581, 1967. Reprinted with permission of Springer Netherlands	p. 109
Asymptotics of some relativistic Markov processes. Proc. Nat. Acad. Sci. U.S.A., 70:3551-3555, 1973.	p. 116
Gaussian Processes
Introduction	p. 123
The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis, 1:290-330, 1967. Reprinted with permission of Elsevier Inc. An electronic version is available at doi: 10.1016/0022-1236(67)90017-1	p. 125
On seminorms and probabilities, and abstract Wiener spaces. Ann. of Math. 93:390-408, 1971. Corrections ibid. 104:391, 1976. Reprinted with permission of Princeton University and the Institute for Advanced Studies	p. 166
Sample functions of the Gaussian process. Ann. Probab, 1:66-103, 1973. Reprinted with permission of the Institute of Mathematical Statistics	p. 187
On the lower tail of Gaussian seminorms. Ann. Probab., 7:319-342, 1979. Reprinted with permission of the Institute of Mathematical Statistics	p. 225
Empirical Processes
Introduction	p. 251
Central limit theorems for empirical measures. Ann. Probab., 6:899-929, 1978. Correction ibid. 7:909-911, 1979. Reprinted with permission of the Institute of Mathematical Statistics	p. 253
Empirical and Poisson processes on classes of sets or functions too large for central limit theorems. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 61:355-368, 1982. Reprinted with permission of Springer Science+Business Media	p. 287
Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 62:509-552, 1983. Reprinted with permission of Springer Science+Business Media	p. 301
Universal Donsker classes and metric entropy. Ann. Probab., 15:1306-1326, 1987. Reprinted with permission of the Institute of Mathematical Statistics	p. 345
Nonlinear functionals and p-variation
Introduction	p. 369
Fréchet differentiability, p-variation and uniform Donsker classes. Ann. Probab., 20:1968-1982, 1992. Reprinted with permission of the Institute of Mathematical Statistics	p. 371
The order of the remainder in derivatives of composition and inverse operators for p-variation norms. Ann. Statist., 22:1-20, 1994. Reprinted with permission of the Institute of Mathematical Statistics	p. 386
Empirical processes and p-variation. In D. Pollard, E. Torgersen, and G. L. Yang, editors, Festschrift for Lucien LeCam, pages 219-233. Springer, New York, 1997. Reprinted with permission of Springer Science+Business Media	p. 406
Miscellanea
Introduction	p. 423
Pathological topologies and random walks on abelian groups. Proc. Amer. Math. Soc., 15:231-238, 1964. Reprinted with permission of the American Mathematical Society	p. 425
Metric entropy of some classes of sets with differentiable boundaries. J. Approximation Theory, 10:227-236, 1974. Reprinted with permission of Elsevier Inc. An electronic, version is available at doi:10.1016/0021-9045(74)90120-8	p. 433
Wiener functional as Itô integrals, Ann. Probab, 5:140-141, 1977. Reprinted with permission of the Institute of Mathematical Statistics	p. 443
A metric entropy bound is not sufficient for learnability. IEEE Trans. Inform. Theory, 40:883-885, 1994. Reprinted with permission of the Institute of Electrical and Electronics Engineers, Inc	p. 445
Asymptotic normality with small relative errors of posterior probabilities of half-spaces. Ann. Statist., 30:1311-1344, 2002. Reprinted with permission of the Institute of Mathematical Statistics	p. 448
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