Conditional measures and applications

492 Pages·2005·2.267 MB·English

by  M.M. Rao

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. Reasonable efforts
have been made to publish reliable data and information, but the author and the publisher cannot assume
responsibility for the validity of all materials or for the consequences of their use.
No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic,
mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and
recording, or in any information storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com
(http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive,
Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration
for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate
system of payment has been arranged.
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 algorith-
mic 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 elemen-
tary 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 gen-
eral 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 construc-
tive 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 re-
sults 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 prob-
lems. 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 con-
structive 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 mea-
sures therefore, a lack of unique computational methods/algorithms
will typically give answers depending on the procedures used (“direc-
tional” procedures) and lead to subjective and tentative solutions. (See
the dilemma in H. Cram´er 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 appli-
cations 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 prob-
lem), and then satisfactory results, including uniqueness when avail-
able, 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 deriva-
tive 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 condi-
tional expectations both as vector integrals and in special cases using
a kernel representation for Gaussian processes when conditional proba-
bilities 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 mea-
sures(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