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Conditional Measures and Applications Second Edition © 2005 by Taylor & Francis Group, LLC PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Central Florida New Brunswick, New Jersey Orlando, Florida EDITORIAL BOARD M. S. Baouendi Anil Nerode University of California, Cornell University San Diego Donald Passman Jane Cronin University of Wisconsin, Rutgers University Madison Jack K. Hale Fred S. Roberts Georgia Institute of Technology Rutgers University S. Kobayashi David L. Russell University of California, Virginia Polytechnic Institute Berkeley and State University Marvin Marcus Walter Schempp University of California, Universität Siegen Santa Barbara Mark Teply W. S. Massey University of Wisconsin, Yale University Milwaukee © 2005 by Taylor & Francis Group, LLC MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles A. N. Michel and D. Liu, Qualitative Analysis and Synthesis of Recurrent Neural Networks (2002) J. R. Weeks, The Shape of Space, Second Edition (2002) M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces (2002) V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Second Edition (2002) T. Albu, Cogalois Theory (2003) A. Bezdek, Discrete Geometry (2003) M. J. Corless and A. E. Frazho, Linear Systems and Control: An Operator Perspective (2003) I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions (2003) G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative (2003) A. Kelarev, Graph Algebras and Automata (2003) A. H. Siddiqi, Applied Functional Analysis: Numerical Methods, Wavelet Methods, and Image Processing (2004) F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line (2004) G. S. Ladde and M. Sambandham, Stochastic versus Deterministic Systems of Differential Equations (2004) B. J. Gardner and R. Wiegandt, Radical Theory of Rings (2004) J. Haluska, The Mathematical Theory of Tone Systems (2004) C. Menini and F. Van Oystaeyen, Abstract Algebra: A Comprehensive Treatment (2004) E. Hansen and G. W. Walster, Global Optimization Using Interval Analysis, Second Edition, Revised and Expanded (2004) M. M. Rao, Measure Theory and Integration, Second Edition, Revised and Expanded (2004) W. J. Wickless, A First Graduate Course in Abstract Algebra (2004) R. P. Agarwal, M. Bohner, and W-T Li, Nonoscillation and Oscillation Theory for Functional Differential Equations (2004) J. Galambos and I. Simonelli, Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions (2004) Walter Ferrer and Alvaro Rittatore, Actions and Invariants of Algebraic Groups (2005) Christof Eck, Jiri Jarusek, and Miroslav Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems (2005) M. M. Rao, Conditional Measures and Applications, Second Edition (2005) © 2005 by Taylor & Francis Group, LLC Conditional Measures and Applications Second Edition M. M. Rao University of California Riverside, USA Boca Raton London New York Singapore © 2005 by Taylor & Francis Group, LLC Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-57444-593-6 (Hardcover) International Standard Book Number-13: 978-1-57444-593-0 (Hardcover) Library of Congress Card Number 2005041909 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Rao, M. M. (Malempati Madhusudana), 1929- Conditional measures and applications / M.M. Rao.--2nd ed., rev. and expanded. p. cm. Includes bibliographical references and index. ISBN 1-57444-593-6 (alk. paper) 1. Probability measures. I. Title. QA273.6.R36 2005 519.2--dc22 2005041909 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group and the CRC Press Web site at is the Academic Division of T&F Informa plc. http://www.crcpress.com © 2005 by Taylor & Francis Group, LLC To the memory of Professors A. N. Kolmogorov and S. Bochner whose fundamental contributions form a basis of the following work and to friends P. A. Meyer and V. V. Sazonov who helped expand the subject © 2005 by Taylor & Francis Group, LLC Preface to the Second Edition This is a revised and expanded version of the previous edition. Sev- eral typographical errors, slips, and many missing references have now been corrected. Specific changes and details of new inclusions will be explained. Also a sharper treatment of the computational or algorithmic point of view is discussed for a better understanding of the subject. It may be recalled that conditional expectations and the resulting (conditional) probabilities have been successfully introduced by A. N. Kolmogorov in the most general form in 1933 as an application of the then newly established Radon-Nikody´m theorem on abstract measure spaces. Prior to this only special cases, mostly for discrete random variables (or those that take at most countably many values), were in use especially for (Markov or other finite) chains, where explicit calculations are easily performed, using the elementary definition of conditional probability by ratios. In the continuous case (or if the conditioning is based on random variables taking possibly an uncount- able set of values), the ratio definition and related procedures lead to indefinite forms and typically some approximations are used in the literature. The problem is analogous to that confronting in elementary differential calculus in higher dimensions leading to directional derivatives. The earliest concern for the resulting problem was raised by E. Borel in a communication to Kolmogorov who did not have a satisfactory solution. (It was thought that the problem was not well ix © 2005 by Taylor & Francis Group, LLC x Preface to the Second Edition posed.) The difficulty is made quite explicit in the present book, by exposing a set of examples for an ergodic stationary Gaussian process with smooth paths, based on some nontrivial calculations for different types of methods, including those of directional derivatives, based on the work of M. Kac and D. Slepian (1959), ending up with numerous (uncountably many) distinctly different answers for the same question. In fact, the evaluationisa problemof differentiationtheory of measures and is nontrivial. So far no direct and usable method of evaluating general conditional probabilities/expectations is available. To clarify the deeply entrenched calculating problem, one may look for Constructive Analysis, along the lines advocated by the well-known mathematician L. E. J. Brouwer, starting in 1907, and reconsidered more recently by Errett Bishop (1967) as follows. In his “constructive manifesto”, mathematical methods are broadly classified into two classes – one an “idealistic type,” and the other a “constructive type.” A recent letter in the AMS Notices (May 2004) by G. van der Geer, points out the deeply held views on these ap- proaches between L. E. J. Brouwer and D. Hilbert which resulted in seriousprofessional conflicts. Tryingtosoftenthisview, butsupporting Brouwer in a pragmatic way, E. Bishop (1967) in his book emphasizes the constructive approach while trying to (re)prove many results of the idealistic approach. His motto is: “Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give a constructive proof.” As an example, he devoted a considerable part of the book to classical mathematics from this cherished viewpoint. This included a version of the Radon- Nikody´m theorem. The result obtained by him (and a slight improve- ment worked out by D. Bridges (1985) in the revision) in this form is quiteinadequateforconditionalmeasures studiedinthisbookaswellas in the current developments of Probability Theory. The Kac-Slepian illustration (or paradox) noted above shows that the Bishop-Bridges work does not, as yet, help resolve our difficulties. It is essential that we compare the constructive attempts with the (unique) existence results obtained by the idealistic approach to appreciate the underlying practical and philosophical underpinnings, a distinction which has not been clearly addressed or even recognized in the literature. Thus the constructive approach is to approximate by a suitable com- © 2005 by Taylor & Francis Group, LLC Preface to the Second Edition xi putable (preferably algorithmic) procedure to use it in practical problems. Note that results based on some form of the axiom of choice (e.g., the existence of maximal ideals in algebra studies or the general Hahn-Banach theorem) are generally beyond the uniqueness assertions as well as computational procedures. Further developments in constructive analysis may help in future. It should perhaps also be noted that computational methods willbe of real use when they are accompa- nied by the uniqueness assertions. In applications of conditional measures therefore, a lack of unique computational methods/algorithms will typically give answers depending on the procedures used (“directional” procedures) and lead to subjective and tentative solutions. (See the dilemma in H. Cramér and M. R. Leadbetter (1967, p.222).) A similar situation confronts in “Bayesian analysis” (which would use a constructive approach); and the “Neyman-Pearson methods” used in likelihood ratios (or the Radon-Nikody´m (or RN) derivatives as gen- eralized by U. Grenander) have to follow the above noted idealistic approach because of its relation with the RN theorem. It is important that this distinction be recognized. As already noted, all these applications clearly need some appropriate algorithms to implement. The hoped for resolution (if ever it is found) may be regarded as a basic justification to advocate subjective probabilities, for Bayes and related practices, in obtaining some solutions for problems of practical import. However, in the present treatments the idealistic approach predom- inates (e.g., for work in Markov processes where transition probability functions are typically assumed as part of the formulation of the problem), and then satisfactory results, including uniqueness when available, are obtained in that context. The presentation of material in this edition is sharpened to make this dichotomy visible and indicates the matter at almost every turn to focus on what is known to exist and what is constructible. This point has not been made explicit in the existing works, as far as I know, and even in such a classic treatise as Doob’s (1953, p. 612), in calculating a Radon-Nikody´m density using a martingale theorem, it is stated briefly that “the result (the derivative in question) depends on the (difference) net used, except in trivial cases.” This important point should have been highlighted at the very beginning of its occurrence. Thus the “idealistic approach” naturally dominates in our work. Bishop (1967, p.233) also observes that the © 2005 by Taylor & Francis Group, LLC xii Preface to the Second Edition “Birkhoff ergodic theorem” is beyond the “constructive approach” as ofthetimeofwritinghisbook,orevennow, andthesameistrueofmar- tingale convergence (noted on p. 360 of his book). With this detailed explanation of these (philosophical) problems, some of the additions and modifications for the present expanded version can be explained as follows. The first six chapters contain essentially the same material as the first edition modulo the simple corrections and minor additions. Some elaborations and the basic frame work of conditioning is given in these chapters. It forms the First part of the book that everybody should study. The details are explained in the preface of the first edition (ap- pended). The remaining five chapters, considerably expanded, divide into two parts, with Chapters 7 and 8 as the Second, and Chapters 9, 10, 11 forming the Third part. These last two parts contain new material and demand a more serious and advanced treatment of the subject. Thus these parts can be conveniently designated as follows. I: Basic Framework; II: Abstract Foundations; and III: Advanced Applications. The last two parts will be described in more detail. Chapter 7 has a new section on integral representations of conditional expectations both as vector integrals and in special cases using a kernel representation for Gaussian processes when conditional probabilities are evaluated using the “ratio definition” of densities, prevalent in current applications. It is pointed out that this type of evaluation does not necessarily give the asserted Radon-Nikody´m density. The reader may compare this work, which involves many specialized prop- erties, with the Kac-Slepian paradox to appreciate the difficulties. The remaining chapters continue the idealistic approach, and the work con- centrates on the existence problem for different types but with little on constructive methods. In fact Section 8.3 now contains a detailed treatment of the theory of projective limits of regular conditional measures(cf., Rao and Sazonov,1993),including anovelapplicationto the Weil-Mackey-Bruhat formula. A discussion of the dichotomy of absolute continuity and singularity of probability measures is found in Section 9.3. Also some additional arguments and results are pre- sented to enhance the previous applications in Chapter 10. The final chapter has undergone a major expansion and elaboration, as it deals © 2005 by Taylor & Francis Group, LLC