Theory of random sets PDF
Preview Theory of random sets
Probability Theory and Stochastic Modelling 87 Ilya Molchanov Theory of Random Sets Second Edition Probability Theory and Stochastic Modelling Volume 87 Editors-in-chief PeterW.Glynn,Stanford,CA,USA AndreasE.Kyprianou,Bath,UK YvesLeJan,Orsay,France AdvisoryBoard SørenAsmussen,Aarhus,Denmark MartinHairer,Coventry,UK PeterJagers,Gothenburg,Sweden IoannisKaratzas,NewYork,NY,USA FrankP.Kelly,Cambridge,UK BerntØksendal,Oslo,Norway GeorgePapanicolaou,Stanford,CA,USA EtiennePardoux,Marseille,France EdwinPerkins,Vancouver,Canada HalilMeteSoner,Zürich,Switzerland TheProbabilityTheoryandStochasticModellingseriesisamergerandcontin- uationofSpringer’stwowellestablishedseriesStochasticModellingandApplied ProbabilityandProbabilityandItsApplicationsseries.Itpublishesresearchmono- graphsthatmakeasignificantcontributiontoprobabilitytheoryoranapplications domain in which advanced probability methods are fundamental. Books in this seriesareexpectedtofollowrigorousmathematicalstandards,whilealsodisplaying the expository quality necessary to make them useful and accessible to advanced studentsaswellasresearchers.Theseriescoversallaspectsofmodernprobability theoryincluding (cid:129) Gaussianprocesses (cid:129) Markovprocesses (cid:129) Randomfields,pointprocessesandrandomsets (cid:129) Randommatrices (cid:129) Statisticalmechanicsandrandommedia (cid:129) Stochasticanalysis aswellasapplicationsthatinclude(butarenotrestrictedto): (cid:129) Branchingprocessesandothermodelsofpopulationgrowth (cid:129) Communicationsandprocessingnetworks (cid:129) Computationalmethodsinprobabilityandstochasticprocesses,includingsimu- lation (cid:129) Geneticsandotherstochasticmodelsinbiologyandthelifesciences (cid:129) Informationtheory,signalprocessing,andimagesynthesis (cid:129) Mathematicaleconomicsandfinance (cid:129) Statisticalmethods(e.g.empiricalprocesses,MCMC) (cid:129) Statisticsforstochasticprocesses (cid:129) Stochasticcontrol (cid:129) Stochasticmodelsinoperationsresearchandstochasticoptimization (cid:129) Stochasticmodelsinthephysicalsciences Moreinformationaboutthisseriesathttp://www.springer.com/series/13205 Ilya Molchanov Theory of Random Sets Second Edition 123 IlyaMolchanov InstituteofMathematicalStatistics andActuarialScience UniversityofBern Bern,Switzerland ISSN2199-3130 ISSN2199-3149 (electronic) ProbabilityTheoryandStochasticModelling ISBN978-1-4471-7347-2 ISBN978-1-4471-7349-6 (eBook) DOI10.1007/978-1-4471-7349-6 LibraryofCongressControlNumber:2017949364 MathematicsSubjectClassification(2010):Primary:60D05;Secondary:26E25,28B20,52A22,49J53, 54C65,60B05,60E07,60F15,60G55,60G57,62M30 Originallypublishedintheseries:ProbabilityandItsApplications ©Springer-VerlagLondonLtd.2005,2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagLondonLtd. Theregisteredcompanyaddressis:236Gray’sInnRoad,LondonWC1X8HB,UnitedKingdom To mymother Preface SomeHistory The study of random geometrical objects goes back to the famous Buffon needle problem.Similar to the ideas of Geometric Probability,which can be traced back to the very origins of probability, the conceptof a random set was mentioned for the first time together with the mathematical foundations of Probability Theory. A.N.Kolmogorov[493,p.46]wrotein1933(translatedfromGerman): LetGbeameasurableregionoftheplanewhoseshapedependsonchance;inotherwords, letusassigntoeveryelementaryevent(cid:2)ofafieldofprobabilityadefinitemeasurableplane regionG.WeshalldenotebyJtheareaoftheregionGandbyP.x;y/theprobabilitythat thepoint.x;y/belongstotheregionG.Then “ E.J/D P.x;y/dxdy: OnemightobservethatthisisaformulationofRobbins’theoremandP.x;y/isthe coveragefunctionoftherandomsetG. Further progress in the theory of random sets relied on developments in the followingareas: (cid:129) studies of random elements in general topologicalspaces, in groups and semi- groups,see,e.g.,Grenander[326]; (cid:129) thegeneraltheoryofstochasticprocesses,seeDellacherie[220],andthetheory ofcapacities,seeChoquet[172]; (cid:129) set-valuedanalysisandmultifunctions,seeCastaingandValadier[158]; (cid:129) advances in image analysis and microscopy that required a satisfactory mathematical theory of distributions for binary images (or random sets), see Serra[790]. The mathematicaltheoryof randomsets can be traced backto Matheron[581] andKendall[454].Theprincipalnewfeatureisthatrandomsetsmayhavedifferent shapes and the development of this idea is crucial in the study of random sets. vii viii Preface G. Matheronformulatedthe verydefinitionof a randomclosed set and developed the relevantprobabilisticand geometric techniques.D.G. Kendall’s seminalpaper [454]onrandomsetsalreadycontainedthefirststepsintomanyfurtherdirections suchaslattices,weakconvergence,spectralrepresentation,infinitedivisibility.Most oftheseaspectswereelaboratedlater oninconnectionwith relevantideasinpure mathematicsandclassicalprobabilitytheory.Thishasmademanyoftheconcepts and the notation used in [454] obsolete, so we will follow instead the modern terminologythatfitsbetterintothesystemdevelopedbyG.Matheron;mostofhis notationwastakenasthebasisforthismonograph. The relationship between random sets and convex geometry later on has been thoroughly explored within the stochastic geometry literature, mostly in the sta- tionary setting, see, e.g., Schneider and Weil [780]. Within stochastic geometry, random sets represent one type of object along with point processes and random tessellations, see Chiu, Stoyan, Kendall and Mecke [169]. The mathematical morphologypartofG.Matheron’sbookgaverisetonumerousapplicationsinimage processing (Dougherty [239] and Serra [790]) and abstract studies of operations withsets,oftenintheframeworkoflatticetheory(Heijmans[355]). Since1975,whenG.Matheron’sbook[581]waspublished,thetheoryofrandom setshasenjoyedsubstantialdevelopmentsconcerning (cid:129) relationshipstothetheoriesofsemigroupsandcontinuouslattices; (cid:129) propertiesofcapacities; (cid:129) limittheoremsforMinkowskisumsbasedupontechniquesfromprobabilitiesin Banachspaces; (cid:129) limit theorems for unions of random sets in relation to the theory of extreme values; (cid:129) stochasticoptimisationideasinrelationtorandomsetsthatappearasepigraphs ofrandomfunctions; (cid:129) propertiesoflevelsetsandexcursionsofstochasticprocesses. These developments constitute the core of this book, which aims to cast the theory of random sets into the conventionalprobabilistic framework that involves distributionalproperties,limittheoremsandrelatedanalyticaltools. Central TopicsoftheBook This book concentrates on several basic concepts in the theory of random sets. The first is the capacity functional that determines the distribution of a random closed set in a locally compact Hausdorff separable space. Unlike probability measures, the capacity functional is non-additive. The studies of non-additive set functionsareabundant,especially,inviewofgametheoryapplicationstodescribe the gainattained bya coalitionof players,in statistics as belief functionsin order to modelsituations where the underlyingprobability measure is uncertain, and in mathematicalfinance,wherenon-additivesetfunctionsareessentialtoassessrisk. Preface ix The capacity functional can be used to characterise the weak convergence of randomsets and some propertiesof their distributions.In particular,this concerns unionsof randomclosedsets, wherethe regularvariationpropertyofthe capacity functionalisofprimaryimportance.However,thecapacityfunctionaldoesnothelp to deal with a number of other issues, for instance to define the expectation of a randomclosedset. Here the leading role is taken over by the concept of a selection, which is a (single-valued)randomelementthatalmostsurelybelongstoarandomset.Inthis framework,itisconvenienttoviewarandomclosedsetasamultifunction(orset- valued function) on a probability space and use the well-developed machinery of set-valuedanalysis,see,e.g.,HuandPapageorgiou[402].Bytakingexpectationsof integrableselections,onedefinestheselectionexpectationofarandomclosedset. Theselectionexpectationofarandomsetdefinedonanon-atomicprobabilityspace isalwaysconvexandcanbealternativelydefinedastheconvexsetwhosesupport functionequalstheexpectedsupportfunctionofarandomset.TheMinkowskisum of random sets is introduced as the set of sums of all their points (or all their selections)andcanbeequivalentlydefinedusingthearithmeticsumofthesupport functions. Therefore, limit theorems for Minkowski sums of random sets can be derivedfrom the existing results for randomelements in functionalspaces. These toolsmakeitpossibletoexploreset-valuedmartingales. Importantexamplesofrandomclosedsetsappearasepigraphsofrandomlower semicontinuous functions. Viewing the epigraphs as random closed sets makes it possible to obtain results for lower semicontinuous functions under the weakest possibleconditions.Inparticular,thisconcernstheconvergenceofminimumvalues andminimisers,whichisasubjectofstochasticoptimisationtheory. It is possible to consider the family of closed sets as both a semigroup and a lattice.Therefore,theresultsonlattice-orsemigroup-valuedrandomelementsare veryusefulinthetheoryofrandomsets. Plan Since the concept of a set is central for mathematics, the book is highly inter- disciplinary and relies on tools from a number of mathematical theories and concepts: capacities, convex geometry, set-valued analysis, topology, harmonic analysisonsemigroups,continuouslattices,non-additivemeasuresandupper/lower probabilities, limit theorems in Banach spaces, the general theory of stochastic processes, extreme values, stochastic optimisation, point processes and random measures. The bookstarts with the definitionofa randomclosed set. The spaceE which randomsetsbelongtoisveryoftenassumedtobelocallycompactHausdorffwith a countable base. The Euclidean space Rd is a generic example. Often we switch to the more general case of E being a Polish space or Banach space (if a linear structureisessential).Itisconvenienttoworkwithrandomclosedsets,whichisthe x Preface typicalsettinginthisbook,althoughinsomeplaceswementionrandomopensets andrandomBorelsets.Choquet’stheoremconcerningtheexistenceofrandomset distributionsisprovedandrelationshipswithset-valuedanalysis(ormultifunctions) andlatticesareexplained.TherestofChap.1reliesontheconceptofthecapacity functional.Ithighlightsrelationshipsbetweencapacityfunctionalsandpropertiesof randomsets,developssomeanalytictheory,convergenceconcepts,applicationsto pointprocessesandrandomcapacitiesandfinallysurveysvariousinterpretationsfor capacitiesthatstemfromgametheory,impreciseprobabilitiesandrobuststatistics. Specialattentionisdevotedtothecaseofrandomconvexcompactsets(orconvex bodiesifthecarrierspaceisEuclidean). Chapter2concernsexpectationconceptsforrandomclosedsets.Themainpartis devotedtotheselection(orAumann)expectationbasedontheideaofanintegrable selection.Chapter3continuesthistopicbydealingwithMinkowskisumsofrandom sets.Thedualrepresentationoftheselectionexpectation—asthesetofexpectations ofallselectionsandastheexpectationofthesupportfunction—makesitpossibleto refertolimittheoremsinBanachspacesinordertoderivethecorrespondingresults forrandomclosedsets. The study of unions for random sets is closely related to extremes of random variablesandfurthergeneralisationsforpointwiseextremesofstochasticprocesses. Chapter4describesthemainresultsfortheunionsofrandomsetsandexplainsthe backgroundideasthat are related to the studiesof lattice-valuedrandomelements andregularvariationonabstractspaces. Chapter 5 is devoted to links between random sets and stochastic processes. This concerns set-valued processes that develop in time, in particular, set-valued martingales.Furthermore,thisrelatestorandomsetsinterpretationsofconventional stochastic processes, where random sets appear as graphs, level sets or epigraphs (hypographs).Severalareasrelatedtorandomsetsandstochasticprocessesareonly mentionedinbrief,forinstance,thetheoryofset-indexedprocesses,whererandom sets appear as stopping times (or stopping sets), excursions of random fields, and potentialtheoryforMarkovprocessesthatprovidesfurtherexamplesofcapacities relatedtohittingtimesandpathsofstochasticprocesses. The Appendices summarise the necessary mathematical background; it stems fromvariouspartsofmathematicsandisnormallyscatteredbetweenvarioustexts. SecondEdition Theperiodbetweenthefirstandsecondeditionswitnessedtheappearanceofseveral booksonstochasticgeometryandrandomsetsauthoredbyNguyen[651],Schneider and Weil [780], Chiu, Stoyan, Kendalland Mecke [169], on randommeasures by Kallenberg[444],PoissonpointprocessesbyLastandPenrose[526],andonnon- additivemeasuresbyGrabisch[321]andCuzzolin[196].
The list of books you might like
