Functional Analysis For Probability And Stochastic Processes.
Functional Analysis for Probability and Stochastic Processes.An IntroductionThis text is designed both for students of probability and stochastic processes and forstudents of functional analysis. For the reader not familiar with functional analysis adetailed introduction to necessary notions and facts is provided. However, this is not astraight textbook in functional analysis; rather, it presents some chosen parts offunctional analysis that help understand ideas from probability and stochasticprocesses. The subjects range from basic Hilbert and Banach spaces, through weaktopologies and Banach algebras, to the theory of semigroups of bounded linearoperators. Numerous standard and non-standard examples and exercises make thebook suitable for both a textbook for a course and for self-study.adam bobrowsk i is a Professor of Mathematics at Lublin University ofTechnology.
Functional Analysis for Probability andStochastic ProcessesAn IntroductionA. BOBROWSKI
cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São PauloCambridge University PressThe Edinburgh Building, Cambridge cb2 2ru, UKPublished in the United States of America by Cambridge University Press, New Yorkwww.cambridge.orgInformation on this title: www.cambridge.org/9780521831666 Cambridge University Press 2005This publication is in copyright. Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press.First published in print format 2005isbn-13isbn-10978-0-511-13039-7 eBook (NetLibrary)0-511-13039-2 eBook (NetLibrary)isbn-13isbn-10978-0-521-83166-6 hardback0-521-83166-0 hardbackisbn-13isbn-10978-0-521-53937-1 paperback0-521-53937-4 paperbackCambridge University Press has no responsibility for the persistence or accuracy of urlsfor external or third-party internet websites referred to in this publication, and does notguarantee that any content on such websites is, or will remain, accurate or appropriate.
To the most enthusiastic writer ever – my son Radek.
ContentsPrefacepage xi11.11.21.31.41.51.6Preliminaries, notations and conventionsElements of topologyMeasure theoryFunctions of bounded variation. Riemann–Stieltjes integralSequences of independent random variablesConvex functions. Hölder and Minkowski inequalitiesThe Cauchy equation1131723293322.12.22.3Basic notions in functional analysisLinear spacesBanach spacesThe space of bounded linear operators3737446333.13.23.33.43.53.63.7Conditional expectationProjections in Hilbert spacesDefinition and existence of conditional expectationProperties and examplesThe Radon–Nikodym TheoremExamples of discrete martingalesConvergence of self-adjoint operators. and of martingales8080879110110310611244.14.2Brownian motion and Hilbert spacesGaussian families & the definition of Brownian motionComplete orthonormal sequences in a Hilbert space121123127vii
viiiContents4.34.4Construction and basic properties of Brownian motionStochastic integrals13313955.15.25.35.45.55.65.75.8Dual spaces and convergence of probability measuresThe Hahn–Banach TheoremForm of linear functionals in specific Banach spacesThe dual of an operatorWeak and weak topologiesThe Central Limit TheoremWeak convergence in metric spacesCompactness everywhereNotes on other modes of 46.56.66.7The Gelfand transform and its applicationsBanach algebrasThe Gelfand transformExamples of Gelfand transformExamples of explicit calculations of Gelfand transformDense subalgebras of C(S)Inverting the abstract Fourier transformThe Factorization 7.77.8Semigroups of operators and Lévy processesThe Banach–Steinhaus TheoremCalculus of Banach space valued functionsClosed operatorsSemigroups of operatorsBrownian motion and Poisson process semigroupsMore convolution semigroupsThe telegraph process semigroupConvolution semigroups of measures on Markov processes and semigroups of operatorsSemigroups of operators related to Markov processesThe Hille–Yosida TheoremGenerators of stochastic processesApproximation al notes363363
Contents9.29.3ixSolutions and hints to exercisesSome commonly used notations366383ReferencesIndex385390
PrefaceThis book is an expanded version of lecture notes for the graduate course “AnIntroduction to Methods of Functional Analysis in Probability and StochasticProcesses” that I gave for students of the University of Houston, Rice University,and a few friends of mine in Fall, 2000 and Spring, 2001. It was quite anexperience to teach this course, for its attendees consisted of, on the one hand,a group of students with a good background in functional analysis having limitedknowledge of probability and, on the other hand, a group of statisticians withouta functional analysis background. Therefore, in presenting the required notionsfrom functional analysis, I had to be complete enough for the latter group whileconcise enough so that the former would not drop the course from boredom.Similarly, for the probability theory, I needed to start almost from scratch for theformer group while presenting the material in a light that would be interestingfor the latter group. This was fun. Incidentally, the students adjusted to thischallenging situation much better than I.In preparing these notes for publication, I made an effort to make the presentation self-contained and accessible to a wide circle of readers. I have added anumber of exercises and disposed of some. I have also expanded some sectionsthat I did not have time to cover in detail during the course. I believe the bookin this form should serve first year graduate, or some advanced undergraduatestudents, well. It may be used for a two-semester course, or even a one-semestercourse if some background is taken for granted. It must be made clear, however,that this book is not a textbook in probability. Neither may it be viewed as atextbook in functional analysis. There are simply too many important subjectsin these vast theories that are not mentioned here. Instead, the book is intendedfor those who would like to see some aspects of probability from the perspective of functional analysis. It may also serve as a (slightly long) introductionto such excellent and comprehensive expositions of probability and stochasticprocesses as Stroock’s, Revuz’s and Yor’s, Kallenberg’s or Feller’s.xi
xiiPrefaceIt should also be said that, despite its substantial probabilistic content, thebook is not structured around typical probabilistic problems and methods. Onthe contrary, the structure is determined by notions that are functional analyticin origin. As it may be seen from the very chapters’ titles, while the body isprobabilistic, the skeleton is functional analytic.Most of the material presented in this book is fairly standard, and the book ismeant to be a textbook and not a research monograph. Therefore, I made littleor no effort to trace the source from which I had learned a particular theoremor argument. I want to stress, however, that I have learned this material fromother mathematicians, great and small, in particular by reading their books. Thebibliography gives the list of these books, and I hope it is complete. See alsothe bibliographical notes to each chapter. Some examples, however, especiallytowards the end of the monograph, fit more into the category of “research”.A word concerning prerequisites: to follow the arguments presented in thebook the reader should have a good knowledge of measure theory and someexperience in solving ordinary differential equations. Some knowledge of abstract algebra and topology would not hurt either. I sketch the needed materialin the introductory Chapter 1. I do not think, though, that the reader should startby reading through this chapter. The experience of going through prerequisitesbefore diving into the book may prove to be like the one of paying a large billfor a meal before even tasting it. Rather, I would suggest browsing throughChapter 1 to become acquainted with basic notation and some important examples, then jumping directly to Chapter 2 and referring back to Chapter 1 whenneeded.I would like to thank Dr. M. Papadakis, Dr. C. A. Shaw, A. Renwick and F. J.Foss (both PhDs soon) for their undivided attention during the course, efforts tounderstand Polish-English, patience in endless discussions about the twentiethcentury history of mathematics, and valuable impact on the course, includinghow-to-solve-it-easier ideas. Furthermore, I would like to express my gratitudeto the Department of Mathematics at UH for allowing me to teach this course.The final chapters of this book were written while I held a special one-yearposition at the Institute of Mathematics of the Polish Academy of Sciences,Warsaw, Poland.A final note: if the reader dislikes this book, he/she should blame F. J.Foss who nearly pushed me to teach this course. If the reader likes it, her/hiswarmest thanks should be sent to me at both addresses: bobrowscy@op.pland a.bobrowski@pollub.pl. Seriously, I would like to thank Fritz Foss for hisencouragement, for valuable feedback and for editing parts of this book. Allthe remaining errors are protected by my copyright.
1Preliminaries, notations and conventionsFinite measures and various classes of functions, including random variables, are examples of elements of natural Banach spaces and thesespaces are central objects of functional analysis. Before studying Banach spaces in Chapter 2, we need to introduce/recall here the basictopological, measure-theoretic and probabilistic notions, and examplesthat will be used throughout the book. Seen from a different perspective,Chapter 1 is a big “tool-box” for the material to be covered later.1.1 Elements of topology1.1.1 Basics of topology We assume that the reader is familiar withbasic notions of topology. To set notation and refresh our memory, let usrecall that a pair (S, U) where S is a set and U is a collection of subsetsof S is said to be a topological space if the empty set and S belong toU, and unions and finite intersections of elements of U belong to U. Thefamily U is then said to be the topology in S, and its members are calledopen sets. Their complements are said to be closed. Sometimes, whenU is clear from the context, we say that the set S itself is a topologicalspace. Note that all statements concerning open sets may be translatedinto statements concerning closed sets. For example, we may equivalentlydefine a topological space to be a pair (S, C) where C is a collection ofsets such that the empty set and S belong to C, and intersections andfinite unions of elements of C belong to C.An open set containing a point s S is said to be a neighborhood ofs. A topological space (S, U) is said to be Hausdorff if for all p1 , p2 S,there exists A1 , A2 U such that pi Ai , i 1, 2 and A1 A2 .Unless otherwise stated, we assume that all topological spaces consideredin this book are Hausdorff.1
2Preliminaries, notations and conventionsThe closure, cl(A), of a set A S is defined to be the smallest closedset that contains A. In other words, cl(A) is the intersection of all closedsets that contain A. In particular, A cl(A). A is said to be dense inS iff cl(A) S.A family V is said to be a base of topology U if every element of Uis a union of elements of V. A family V is said to be a subbase of U ifthe family of finite intersections of elements of V is a base of U.If (S, U) and (S , U ) are two topological spaces, then a map f : S S is said to be continuous if for any open set A in U its inverse imagef 1 (A ) is open in S.Let S be a set and let (S , U ) be a topological space, and let {ft , t T}be a family of maps from S to S (here T is an abstract indexing set).Note that we may introduce a topology in S such that all maps ft arecontinuous, a trivial example being the topology consisting of all subsetsof S. Moreover, an elementary argument shows that intersections of finiteor infinite numbers of topologies in S is a topology. Thus, there existsthe smallest topology (in the sense of inclusion) under which the ftare continuous. This topology is said to be generated by the family{ft , t T}.1.1.2 Exercise Prove that the family V composed of sets of the formft 1 (A ), t T, A U is a subbase of the topology generated by ft , t T.1.1.3 Compact sets A subset K of a topological space (S, U) is said tobe compact if every open cover of K contains a finite subcover. This means that if V is a collection of open sets such that K B V B,then there exists a finite collection of sets B1 , . . . , Bn V such that nK 1 1 Bi . If S is compact itself, we say that the space (S, U) iscompact (the reader may have noticed that this notion depends as muchon S as it does on U). Equivalently, S is compact if, for any family Ct , t T of closed subsets of S such that t T Ct , there exists na finite collection Ct1 , . . . , Ctn of its members such that i 1 Cti .A set K is said to be relatively compact iff its closure is compact.A topological space (S, U) is said to be locally compact if for everypoint p S there exist an open set A and a compact set K, such thats A K. The Bolzano–Weierstrass Theorem says that a subsetof Rn is compact iff it is closed and bounded. In particular, Rn is locallycompact.
1.2 Measure theory31.1.4 Metric spaces Let X be an abstract space. A map d : X X R is said to be a metric iff for all x, y, z X(a) d(x, y) d(y, x),(b) d(x, y) d(x, z) d(z, y),(c) d(x, y) 0 iff x y.A sequence xn of elements of X is said to converge to x X iflimn d(xn , x) 0. We call x the limit of the sequence (xn )n 1 andwrite limn xn x. A sequence is said to be convergent if it converges to some x. Otherwise it is said to be divergent.An open ball B(x, r) with radius r and center x is defined as the setof all y X such that d(x, y) r. A closed ball with radius r and centerx is defined similarly as the set of y such d(x, y) r. A natural way tomake a metric space into a topological space is to take all open balls asthe base of the topology in X. It turns out that under this definition asubset A of a metric space is closed iff it contains the limits of sequenceswith elements in A. Moreover, A is compact iff every sequence of itselements contains a converging subsequence and its limit belongs to theset A. (If S is a topological space, this last condition is necessary butnot sufficient for A to be compact.)A function f : X Y that maps a metric space X into a normedspace Y is continuous at x X if for any sequence xn converging tox, limn f (xn ) exists and equals f (x) (xn converges in X, f (xn ) converges in Y). f is called continuous if it is continuous at every x X(this definition agrees with the definition of continuity given in 1.1.1).1.2 Measure theory1.2.1 Measure spaces and measurable functions Although we assumethat the reader is familiar with the rudiments of measure theory aspresented, for example, in [103], let us recall the basic notions. A familyF of subsets of an abstract set Ω is said to be a σ-algebra if it contains Ωand complements and countable unions of its elements. The pair (Ω, F)is then said to be a measurable space. A family F is said to be analgebra or a field if it contains Ω, complements and finite unions of itselements.A function µ that maps a family F of subsets of Ω into R such thatµ( n NAn ) n 1µ(An )(1.1)
4Preliminaries, notations and conventionsfor all pairwise-disjoint elements An , n N of F such that the union n N An belongs to F is called a measure. In most cases F is a σalgebra but there are important situations where it is not, see e.g. 1.2.8below. If F is a σ-algebra, the triple (Ω, F, µ) is called a measure space.Property (1.1) is termed countable additivity. If F is an algebraand µ(S) , (1.1) is equivalent tolim µ(An ) 0 whenever An F, An An 1 ,n An .(1.2)n 1The reader should prove it.The smallest σ-algebra containing a given class F of subsets of a set isdenoted σ(F). If Ω is a topological space, then B(Ω) denotes the smallestσ-algebra containing open sets, called the Borel σ-algebra. A measureµ on a measurable space (Ω, F) is said to be finite (or bounded) ifµ(Ω) . It is said to be σ-finite if there exist measurable subsets Ωn , n N, of Ω such that µ(Ωn ) and Ω n N Ωn .A measure space (Ω, F, µ) is said to be complete if for any set A Ωand any measurable B conditions A B and µ(B) 0 imply that Ais measurable (and µ(A) 0, too). When Ω and F are clear from thecontext, we often say that the measure µ itself is complete. In Exercise1.2.10 we provide a procedure that may be used to construct a completemeasure from an arbitrary measure. Exercises 1.2.4 and 1.2.5 prove thatproperties of complete measure spaces are different from those of measure spaces that are not complete.A map f from a measurable space (Ω, F) to a measurable space(Ω , F ) is said to be F measurable, or just measurable iff for anyset A F the inverse image f 1 (A) belongs to F. If, additionally, allinverse images of measurable sets belong to a sub-σ-algebra G of F, thenwe say that f is G measurable, or more precisely G/F measurable.If f is a measurable function from (Ω, F) to (Ω , F ) thenσf {A F A f 1 (B) where B F }is a sub-σ-algebra of F. σf is called the σ-algebra generated by f . Ofcourse, f is G measurable if σf G.The σ-algebra of Lebesgue measurable subsets of a measurable subsetA Rn is denoted Mn (A) or M(A) if n is clear from the context, andthe Lebesgue measure in this space is denoted lebn , or simply leb. A standard result says that M : M(Rn ) is the smallest complete σ-algebracontaining B(Rn ). In considering the measures on Rn we will alwaysassume that they are defined on the σ-algebra of Lebesgue measurable
1.2 Measure theory5sets, or Borel sets. The interval [0, 1) with the family of its Lebesguesubsets and the Lebesgue measure restricted to these subsets is oftenreferred to as the standard probability space. An n-dimensionalrandom vector (or simply n-vector) is a measurable map from a probability space (Ω, F, P) to the measurable space (Rn , B(Rn )). A complexvalued random variable is simply a two dimensional random vector; we tend to use the former name if we want to consider complexproducts of two-dimensional random vectors. Recall that any random nvector X is of the form X (X1 , ., Xn ) where Xi are random variablesXi : Ω R.1.2.2 Exercise Let A be an open set in Rn . Show that A is union ofall balls contained in A with rational radii and centers in points withrational coordinates. Conclude that B(R) is the σ-algebra generated byopen (resp. closed) intervals. The same result is true for intervals of theform (a, b] and [a, b). Formulate and prove an analog in Rn .1.2.3 Exercise Suppose that Ω and Ω are topological spaces. If a mapf : Ω Ω is continuous, then f is measurable with respect to Borelσ-fields in Ω and Ω . More generally, suppose that f maps a measurablespace (Ω, F) into a measurable space (Ω, F ), and that G is a class ofmeasurable subsets of Ω such σ(G ) F . If inverse images of elementsof G are measurable, then f is measurable.1.2.4 Exercise Suppose that (Ω, F, µ) is a measure space, and f mapsΩ into R. Equip R with the σ-algebra of Borel sets and prove that fis measurable iff sets of the form {ω f (ω) t}, t R belong to F.(Equivalently: sets of the form {ω f (ω) t}, t R belong to F.) Proveby example that a similar statement is not necessarily true if Borel setsare replaced by Lebesgue measurable sets.1.2.5 Exercise Let (Ω, F, µ) be a complete measure space, and f bea map f : Ω R. Equip R with the algebra of Lebesgue measurablesets and prove that f is me
Functional Analysis for Probability and Stochastic Processes. An Introduction This text is designed both for students of probability and stochastic processes and for students of functional analysis. For the reader not familiar with functional analysis a detailed introduction to necessary notions and facts is provided. However, this is not a
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