• 1. Donsker's theorem and inequalities--
• 2. Gaussian processes, sample continuity--
• 3. Definition of Donsker classes--
• 4. Vapnik-Cervonenkis combinatorics--
• 5. Measurability--
• 6. Limit theorems for VC-type classes--
• 7. Metric entropy with bracketing--
• 8. Approximation of functions and sets--
• 9. Two samples and the bootstrap--
• 10. Uniform and universal limit theorems--
• 11. Classes too large to be Donsker-- Appendix A. Differentiating under an integral sign-- Appendix B. Multinomial distributions-- Appendix C. Measures on nonseparable metric spaces-- Appendix D. An extension of Lusin's theorem-- Appendix E. Bochner and Pettis integrals-- Appendix F. Non-existence of some linear forms-- Appendix G. Separation of analytic sets-- Appendix H. Young-Orlicz spaces--
• Appendix I. Versions of isonormal processes.
• (source: Nielsen Book Data)

In this new edition of a classic work on empirical processes the author, an acknowledged expert, gives a thorough treatment of the subject with the addition of several proved theorems not included in the first edition, including the Bretagnolle-Massart theorem giving constants in the Komlos-Major-Tusnady rate of convergence for the classical empirical process, Massart's form of the Dvoretzky-Kiefer-Wolfowitz inequality with precise constant, Talagrand's generic chaining approach to boundedness of Gaussian processes, a characterization of uniform Glivenko-Cantelli classes of functions, Gine and Zinn's characterization of uniform Donsker classes, and the Bousquet-Koltchinskii-Panchenko theorem that the convex hull of a uniform Donsker class is uniform Donsker. The book will be an essential reference for mathematicians working in infinite-dimensional central limit theorems, mathematical statisticians, and computer scientists working in computer learning theory. Problems are included at the end of each chapter so the book can also be used as an advanced text.
(source: Nielsen Book Data)

• Preface
• 1 Introduction and Overview
• 2 Definitions and Basic Properties of Extended Riemann-Stieltjes Integrals
• 3 Phi-variation and p-variation; Inequalities for Integrals
• 4 Banach Algebras
• 5 Derivatives and Analyticity in Normed Spaces
• 6 Nemytskii Operators on Some Function Spaces
• 7 Nemytskii Oerators on Lp Spaces
• 8 Two-Function Composition
• 9 Product Integration
• 10 Nonlinear Differential and Integral Equations
• 11 Fourier Series
• 12 Stochastic Processes and Phi-Variation
• Appendix Nonatomic Measure Spaces
• References
• Subject Index
• Author Index
• Index of Notation.

Concrete Functional Calculus focuses primarily on differentiability of some nonlinear operators on functions or pairs of functions. This includes composition of two functions, and the product integral, taking a matrix- or operator-valued coefficient function into a solution of a system of linear differential equations with the given coefficients. In this book existence and uniqueness of solutions are proved under suitable assumptions for nonlinear integral equations with respect to possibly discontinuous functions having unbounded variation. Key features and topics: Extensive usage of p-variat.

• 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.- Frechet 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 Ito Integrals.- A Metric Entropy Bound is Not Sufficient for Learnability.- Asymptotic Normality with Small Relative Errors of Posterior Probabilities of Half-Spaces.
• (source: Nielsen Book Data)

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 Frechet 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.
(source: Nielsen Book Data)

• 1. Foundations: set theory--
• 2. General topology--
• 3. Measures--
• 4. Integration--
• 5. Lp spaces: introduction to functional analysis--
• 6. Convex sets and duality of normed spaces--
• 7. Measure, topology, and differentiation--
• 8. Introduction to probability theory--
• 9. Convergence of laws and central limit theorems--
• 10. Conditional expectations and martingales--
• 11. Convergence of laws on separable metric spaces--
• 12. Stochastic processes--
• 13. Measurability: Borel isomorphism and analytic sets-- Appendixes: A. Axiomatic set theory-- B. Complex numbers, vector spaces, and Taylor's theorem with remainder-- C. The problem of measure-- D. Rearranging sums of nonnegative terms-- E. Pathologies of compact nonmetric spaces-- Indices.
• (source: Nielsen Book Data)

This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.
(source: Nielsen Book Data)

• 1. Foundations: set theory--
• 2. General topology--
• 3. Measures--
• 4. Integration--
• 5. Lp spaces: introduction to functional analysis--
• 6. Convex sets and duality of normed spaces--
• 7. Measure, topology, and differentiation--
• 8. Introduction to probability theory--
• 9. Convergence of laws and central limit theorems--
• 10. Conditional expectations and martingales--
• 11. Convergence of laws on separable metric spaces--
• 12. Stochastic processes--
• 13. Measurability: Borel isomorphism and analytic sets-- Appendixes: A. Axiomatic set theory-- B. Complex numbers, vector spaces, and Taylor's theorem with remainder-- C. The problem of measure-- D. Rearranging sums of nonnegative terms-- E. Pathologies of compact nonmetric spaces-- Indices.
• (source: Nielsen Book Data)

This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.
(source: Nielsen Book Data)

• 1. Foundations: set theory--
• 2. General topology--
• 3. Measures--
• 4. Integration--
• 5. Lp spaces: introduction to functional analysis--
• 6. Convex sets and duality of normed spaces--
• 7. Measure, topology, and differentiation--
• 8. Introduction to probability theory--
• 9. Convergence of laws and central limit theorems--
• 10. Conditional expectations and martingales--
• 11. Convergence of laws on separable metric spaces--
• 12. Stochastic processes--
• 13. Measurability: Borel isomorphism and analytic sets-- Appendixes: A. Axiomatic set theory-- B. Complex numbers, vector spaces, and Taylor's theorem with remainder-- C. The problem of measure-- D. Rearranging sums of nonnegative terms-- E. Pathologies of compact nonmetric spaces-- Indices.
• (source: Nielsen Book Data)

This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.
(source: Nielsen Book Data)

• Preface.- A survey on differentiability of six operators in relation to probability and statistics.- Product integrals, Young integrals and p-variation.- Differentiability of the composition and inverse operators for regulated and a.e. continuous functions.- Bibliographies on p-variation and phi-variation.- Subject Index.- Author Index.
• (source: Nielsen Book Data)

The book is about differentiability of six operators on functions or pairs of functions: composition (f of g), integration (of f dg), multiplication and convolution of two functions, both varying, and the product integral and inverse operators for one function. The operators are differentiable with respect to p-variation norms with optimal remainder bounds. Thus the functions as arguments of the operators can be nonsmooth, possibly discontinuous, but four of the six operators turn out to be analytic (holomorphic) for some p-variation norms. The reader will need to know basic real analysis, including Riemann and Lebesgue integration. The book is intended for analysts, statisticians and probabilists. Analysts and statisticians have each studied the differentiability of some of the operators from different viewpoints, and this volume seeks to unify and expand their results.
(source: Nielsen Book Data)

• Preface--
• 1. Introduction: Donsker's theorem, metric entropy and inequalities--
• 2. Gaussian measures and processes-- sample continuity--
• 3. Foundations of uniform central limit theorems: Donsker classes--
• 4. Vapnik- ervonenkis combinatorics--
• 5. Measurability--
• 6. Limit theorems for Vapnik- ervonenkis and related classes--
• 7. Metric entropy, with inclusion and bracketing--
• 8. Approximation of functions and sets--
• 9. Sums in general Banach spaces and invariance principles--
• 10. Universal and uniform central limit theorems--
• 11. The two-sample case, the bootstrap, and confidence sets--
• 12. Classes of sets or functions too large for central limit theorems-- Appendices-- Subject index-- Author index-- Index of notation.
• (source: Nielsen Book Data)

This book shows how the central limit theorem for independent, identically distributed random variables with values in general, multidimensional spaces, holds uniformly over some large classes of functions. The author, an acknowledged expert, gives a thorough treatment of the subject, including several topics not found in any previous book, such as the Fernique-Talagrand majorizing measure theorem for Gaussian processes, an extended treatment of Vapnik-Chervonenkis combinatorics, the Ossiander L2 bracketing central limit theorem, the Gine-Zinn bootstrap central limit theorem in probability, the Bronstein theorem on approximation of convex sets, and the Shor theorem on rates of convergence over lower layers. Other results of Talagrand and others are surveyed without proofs in separate sections. Problems are included at the end of each chapter so the book can be used as an advanced text. The book will interest mathematicians working in probability, mathematical statisticians and computer scientists working in computer learning theory.
(source: Nielsen Book Data)

• Foundations: set theory. General topology. Measures. Integration. Lp spaces: introduction to functional analysis. Convex sets and duality of normed spaces. Measure, topology and differentiation. Introduction to probability theory. Convergence of laws and central limit theorems. Conditional expectation and martingales. Convergence of laws on separable metric spaces. Stochastic processes. Measurability: Borel isomorphism and analytic sets. Appendices.
• (source: Nielsen Book Data)

Written by one of the best-known probabilists in the world this text offers a clear and modern presentation of modern probability theory and an exposition of the interplay between the properties of metric spaces and those of probability measures. This text is the first at this level to include discussions of the subadditive ergodic theorems, metrics for convergence in laws and the Borel isomorphism theory. The proofs for the theorems are consistently brief and clear and each chapter concludes with a set of historical notes and references. This book should be of interest to students taking degree courses in real analysis and/or probability theory.
(source: Nielsen Book Data)