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 ofmathematics.
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.- Weak Convergence of Probabilities on Nonseparable Metric Spaces and Empirical Measures on Euclidean Spaces.- Measures on Non-Separable Metric Spaces.- Distances of Probability Measures and Random Variables.- An Extended Wichura Theorem, Definitions of Donsker Class, and Weighted Empirical Distributions.- Markov Processes.- Lorentz-invariant Markov processes in relativistic phase space.- A note on Lorentz-invariant Markov processes.- Asymptotics of Some Relativistic Markov Processes.- Gaussian Processes.- The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes.- On seminorms and probabilities, and abstract Wiener spaces.- Sample Functions of the Gaussian Process.- On the Lower Tail of Gaussian Seminorms.- Empirical Processes.- Special Invited Paper.- Empirical and Poisson Processes on Classes of Sets or Functions Too Large for Central Limit Theorems.- Invariance Principles for Sums of Banach Space Valued Random Elements and Empirical Processes.- Universal Donsker Classes and Metric Entropy.- Nonlinear functionals and p-variation.- Fréchet Differentiability, p-Variation and Uniform Donsker Classes.- The Order of the Remainder in Derivatives of Composition and Inverse Operators for p-Variation Norms.- Empirical Processes and p-variation.- Miscellanea.- Pathological Topologies and Random Walks on Abelian Groups.- Metric Entropy of Some Classes of Sets with Differentiable Boundaries.- Wiener Functionals as Itô Integrals.- A Metric Entropy Bound is Not Sufficient for Learnability.- Asymptotic Normality with Small Relative Errors of Posterior Probabilities of Half-Spaces.
Review :
From the reviews:
"This book is comprised of a number of Professor R. M. Dudley's best works ... . Each part begins with an introduction followed by a number of papers written by Dudley and his co-researchers. ... The book also presents a biographical sketch of Dudley and provides his bibliography (1961-2009). ... This is a comprehensive collection of work within mathematical statistics and probability theory and will be of special interest to graduate students and researchers in probability, statistics and related fields." (Technometrics, Vol. 53 (2), May, 2011)
