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Probability and Its Applications Published in association with the Applied Probability Trust Editors: J. Gani, C.C. Heyde, P. Jagers, T.G. Kurtz Probability and Its Applications Anderson:Continuous-TimeMarkovChains. Azencott/Dacunha-Castelle:SeriesofIrregularObservations. Bass:DiffusionsandEllipticOperators. Bass:ProbabilisticTechniquesinAnalysis. Chen:Eigenvalues,Inequalities,andErgodicTheory Choi:ARMAModelIdentification. Daley/Vere-Jones:AnIntroductiontotheTheoryofPointProcesses. VolumeI:ElementaryTheoryandMethods,SecondEdition. delaPenã/Gine´:Decoupling:FromDependencetoIndependence. DelMoral:Feynman-KacFormulae:GenealogicalandInteractingParticleSystemswith Applications. Durrett:ProbabilityModelsforDNASequenceEvolution. Galambos/Simonelli:Bonferroni-typeInequalitieswithApplications. Gani(Editor):TheCraftofProbabilisticModelling. Grandell:AspectsofRiskTheory. Gut:StoppedRandomWalks. Guyon:RandomFieldsonaNetwork. Kallenberg:FoundationsofModernProbability,SecondEdition. Last/Brandt:MarkedPointProcessesontheRealLine. Leadbetter/Lindgren/Rootzeń:ExtremesandRelatedPropertiesofRandomSequences andProcesses. Molchanov:TheoryofRandomSets. Nualart:TheMalliavinCalculusandRelatedTopics. Rachev/Ru¨schendorf:MassTransportationProblems.VolumeI:Theory. Rachev/Ru¨schendofr:MassTransportationProblems.VolumeII:Applications. Resnick:ExtremeValues,RegularVariationandPointProcesses. Shedler:RegenerationandNetworksofQueues. Silvestrov:LimitTheoremsforRandomlyStoppedStochasticProcesses. Thorisson:Coupling,Stationarity,andRegeneration. Todorovic:AnIntroductiontoStochasticProcessesandTheirApplications. Ilya Molchanov Theory of Random Sets With 33 Figures IlyaMolchanov DepartmentofMathematicalStatisticsandActuarialScience,UniversityofBerne, CH-3012Berne,Switzerland SeriesEditors J.Gani C.C.Heyde StochasticAnalysisGroupCMA StochasticAnalysisGroup,CMA AustralianNationalUniversity AustralianNationalUniversity CanberraACT0200 CanberraACT0200 Australia Australia P.Jagers T.G.Kurtz MathematicalStatistics DepartmentofMathematics ChalmersUniversityofTechnology UniversityofWisconsin SE-41296Go¨teborg 480LincolnDrive Sweden Madison,WI53706 USA Mathematics Subject Classification (2000): 60-02, 60D05, 06B35, 26A15, 26A24, 28B20, 31C15, 43A05, 43A35,49J53, 52A22,52A30, 54C60, 54C65,54F05, 60E07,60G70, 60F05, 60H25,60J99, 62M30,90C15,91B72,93E20 BritishLibraryCataloguinginPublicationData Molchanov,IlyaS.,1962– Theoryofrandomsets.—(Probabilityanditsapplications) 1. Randomsets 2. Stochasticgeometry I. Title 519.2 ISBN185233892X LibraryofCongressCataloging-in-PublicationData CIPdataavailable. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublish- ers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicencesissuedbythe CopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbesentto thepublishers. ISBN-10:1-85233-892-X ISBN-13:978-185223-892-3 SpringerScience+BusinessMedia springeronline.com ©Springer-VerlagLondonLimited2005 PrintedintheUnitedStatesofAmerica Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsence ofaspecificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandtherefore freeforgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinforma- tion contained in this book and cannot accept any legal responsibility or liability for any errors or omissionsthatmaybemade. Typesetting:Camera-readybyauthor. 12/3830-543210Printedonacid-freepaper SPIN10997451 To mymother Preface History The studies of random geometrical objects go back to the famous Buffon needle problem.Similar to theideasof GeometricProbabilitiesthatcan be tracedbackto the first results in probability theory, the concept of a random set was mentioned forthefirsttimetogetherwiththemathematicalfoundationsofProbabilityTheory. A.N.Kolmogorov[321,p.46]wrotein1933: LetG beameasurableregionoftheplanewhoseshapedependsonchance; inotherwords,letusassigntoeveryelementaryeventξ ofafieldofprob- abilityadefinitemeasurableplaneregionG.Weshalldenoteby J thearea oftheregionG andbyP(x,y)theprobabilitythatthepoint(x,y)belongs totheregionG.Then (cid:1)(cid:1) E(J)= P(x,y)dxdy. One can notice that this is the formulationof Robbins’ theorem and P(x,y) is the coveragefunctionoftherandomsetG. Thefurtherprogressinthetheoryofrandomsetsreliedonthedevelopmentsin thefollowingareas: • studiesofrandomelementsinabstractspaces,forexamplegroupsandalgebras, seeGrenander[210]; • thegeneraltheoryofstochasticprocesses,seeDellacherie[131]; • advancesin image analysis and microscopy that required a satisfactory mathe- maticaltheoryofdistributionsforbinaryimages(orrandomsets),seeSerra[532]. ThemathematicaltheoryofrandomsetscanbetracedbacktothebookbyMath- eron[381].G.Matheronformulatedtheexactdefinitionofarandomclosedsetand developedtherelevanttechniquesthatenrichedtheconvexgeometryandlaidoutthe foundationsof mathematical morphology.Broadly speaking, the convex geometry contributionconcernedpropertiesoffunctionalsofrandomsets,whilethemorpho- logicalpartconcentratedonoperationswiththesetsthemselves. VIII Preface The relationship between random sets and convex geometry later on has been thoroughly explored within the stochastic geometry literature, see, e.g. Weil and Wieacker[607]. Withinthestochastic geometry,randomsetsrepresentonetypeof objects along with point processes, random tessellations, etc., see Stoyan, Kendall andMecke[544].Themaintechniquesstemfromconvexandintegralgeometry,see Schneider[520]andSchneiderandWeil[523]. ThemathematicalmorphologypartofG.Matheron’sbookgaverisetonumerous applications in image processing (Dougherty [146]) and abstract studies of opera- tionswithsets,oftenintheframeworkofthelatticetheory(Heijmans[228]). Since 1975 when G. Matheron’s book [381] was published, the theory of ran- domsetshasenjoyedsubstantialdevelopments.D.G.Kendall’sseminalpaper[295] already contained the first steps into many directions such as lattices, weak con- vergence,spectral representation,infinite divisibility.Many of these conceptshave beenelaboratedlateroninconnectiontotherelevantideasinpuremathematics.This mademanyoftheconceptsandnotationusedin[295]obsolete,sothatwewillfollow the modernterminologythatfits betterinto the system developedby G. Matheron; mostofhisnotationwastakenasthebasisforthecurrenttext. Themoderndirectionsinrandomsetstheoryconcern • relationshipstothetheoriesofsemigroupsandcontinuouslattices; • propertiesofcapacities; • limittheoremsforMinkowskisumsandrelevanttechniquesfromprobabilitiesin Banachspaces; • limittheoremsforunionsofrandomsets, whicharerelatedtothetheoryofex- tremevalues; • stochasticoptimisationideasinrelationtorandomsetsthatappearasepigraphs ofrandomfunctions; • studiesofpropertiesoflevelsetsandexcursionsofstochasticprocesses. Thesedirectionsconstitutethemaincoreofthisbookwhichaimstocasttherandom sets theory in the conventionalprobabilistic frameworkthat involvesdistributional properties,limittheoremsandtherelevantanalyticaltools. Centraltopicsofthebook Thewholestorytoldinthisbookconcentratesonseveralimportantconceptsinthe theoryofrandomsets. Thefirstconceptisthecapacityfunctionalthatdeterminesthedistributionofa random closed set in a locally compact Hausdorff separable space. It is related to positivedefinitefunctionsonsemigroupsandlattices.Unlikeprobabilitymeasures, the capacity functional is non-additive. The studies of non-additive measures are abundant,especially,inviewofapplicationstogametheory,wherethenon-additive measure determinesthe gain attained by a coalition of players. The capacity functional can be used to characterise the weak convergenceof random sets and some propertiesoftheirdistributions.Inparticular,thisconcernsunionsofrandomclosed sets, where the regular variation property of the capacity functional is of primary Preface IX importance.Itispossibletoconsiderrandomcapacitiesthatunifytheconceptsofa randomclosed set and a randomupper semicontinuousfunction.However,the ca- pacityfunctionaldoesnothelptodealwithanumberofotherissues,forinstanceto definetheexpectationofarandomclosedset. Here the leading role is taken over by the concept of a selection, which is a (single-valued)randomelementthatalmostsurely belongsto a randomset. In this frameworkit is convenientto view a randomclosed set as a multifunction(or set- valued function) on a probability space and use the well-developed machinery of set-valuedanalysis.Itispossibletofinda countablefamilyofselectionsthatcom- pletelyfillstherandomclosedsetandiscalleditsCastaingrepresentation.Bytaking expectationsofintegrableselections,onedefinestheselectionexpectationofaran- domclosedset.However,thefamiliesofallselectionsareveryrichevenforsimple randomsets. Fortunately,itispossibletoovercomethisdifficultybyusingtheconceptofthe supportfunction.Theselectionexpectationofarandomsetdefinedofanon-atomic probability space is always convex and can be alternatively defined by taking the expectationof the supportfunction.The Minkowski sum of random sets is defined as the set of sums of all their points or all their selections and can be equivalently formalisedusingthearithmeticsumofthesupportfunctions.Therefore,limittheo- remsforMinkowskisumsofrandomsetscanbederivedfromtheexistingresultsin Banachspaces,sincethefamilyofsupportfunctionscanbeembeddedintoaBanach space.Thesupportfunctionconceptestablishesnumerouslinkstoconvexgeometry ideas.Italsomakesitpossibletostudyset-valuedprocesses,e.g.set-valuedmartin- galesandset-valuedshot-noise. Importantexamplesofrandomclosedsetsappearasepigraphsofrandomlower semicontinuous functions. Viewing the epigraphs as random closed sets makes it possibletoobtainresultsforlowersemicontinuousfunctionsundertheweakestpos- sibleconditions.Inparticular,thisconcernstheconvergenceofminimumvaluesand minimisers,whichisthesubjectofstochasticoptimisationtheory. It is possible to consider the family of closed sets as both a semigroup and a lattice.Therefore,randomclosedsetsaresimplyaspecialcaseofgenerallattice-or semigroup-valuedrandomelements.Theconceptofprobabilitymeasureonalattice isindispensableinthemoderntheoryofrandomsets. Itisconvenienttoworkwithrandomclosed sets,whichisthetypicalsettingin thisbook,althoughinsomeplaceswementionrandomopensetsandrandomBorel sets. Plan Since the conceptofa setis centralformathematics,the bookis highlyinterdisci- plinary and aims to unite a number of mathematical theories and concepts: capac- ities, convex geometry, set-valued analysis, topology, harmonic analysis on semi- groups, continuous lattices, non-additive measures and upper/lower probabilities, limit theorems in Banach spaces, general theory of stochastic processes, extreme values,stochasticoptimisation,pointprocessesandrandommeasures. X Preface Thebookstartswithadefinitionofrandomclosedsets.ThespaceEwhichran- dom sets belong to, is very often assumed to be locally compact Hausdorffwith a countablebase.TheEuclideanspaceRd isagenericexample(apartfromraremo- ments when E is a line). Often we switch to the more general case of E being a Polish space or Banach space (if a linear structure is essential). Then the Choquet theorem concerning the existence of random sets distributions is proved and rela- tionshipswithset-valuedanalysis(ormultifunctions)andlatticesareexplained.The restofChapter1reliesontheconceptofthecapacityfunctional.Firstithighlights relationships between capacity functionalsand properties of random sets, then de- velopssome analytictheory,convergenceconcepts,applicationsto pointprocesses andrandomcapacitiesandfinallyexplainsvariousinterpretationsforcapacitiesthat stemfromgametheory,impreciseprobabilitiesandrobuststatistics. Chapter2concernsexpectationconceptsforrandomclosedsets.Themainpart isdevotedtotheselection(orAumann)expectationthatisbasedontheideaofthe selection.Chapter3continuesthistopicbydealingwithMinkowskisumsofrandom sets.Thedualrepresentationoftheselectionexpectation–asasetofexpectationsof allselectionsandastheexpectationofthesupportfunction–makesitpossibletore- fertolimittheoremsinBanachspacesinordertoprovethecorrespondingresultsfor randomclosedsets. Thegeneralityofpresentationvariesinordertoexplainwhich propertiesofthecarrierspaceEareessentialforparticularresults. Theschemeofunionsforrandomsetsiscloselyrelatedtoextremesofrandom variablesandfurthergeneralisationsforpointwiseextremesofstochasticprocesses. Chapter 4 describes the main results for the union scheme and explains the back- groundideasthatmostlystemfromthestudiesoflattice-valuedrandomelements. Chapter5isdevotedtolinksbetweenrandomsetsandstochasticprocesses.On theonehand,thisconcernsset-valuedprocessesthatdevelopin time,inparticular, set-valuedmartingales.Ontheotherhand,thesubjectmatterconcernsrandomsets interpretations of conventional stochastic processes, where random sets appear as graphs,levelsetsorepigraphs(hypographs). TheAppendicessummarisethenecessarymathematicalbackgroundthatisnor- mallyscatteredbetweenvarioustexts.Thereisanextensivebibliographyandade- tailedsubjectindex. Several areas that are related to random sets are only mentioned in brief. For instance,theseareasincludethetheoryofset-indexedprocesses,whererandomsets appearasstoppingtimes(orstoppingsets),excursionsofrandomfieldsandpotential theoryforMarkovprocessesthatprovidesfurtherexamplesofcapacitiesrelatedto hittingtimesandpathsofstochasticprocesses. Itisplannedthatacompanionvolumetothisbookwillconcernmodelsofran- dom sets (germ-grain models, random fractals, growth processes, etc), convex ge- ometrytechniques,statisticalinferenceforstationaryandcompactrandomsetsand relatedmodellingissuesinimageanalysis.