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1、ACanonicalRepresentationforAggregatedMarkovProcessesByBretLargetDuquesneUniversityAbstractAdeterministicfunctionofaMarkovprocessiscalledanaggregatedMarkovprocess.Wegivenecessaryandsucientconditionsfortheequivalenceofcontinuous-timeaggregatedMarkovprocesses.Forbothdiscrete-a
2、ndcontinuous-time,weshowthatanyaggregatedMarkovprocesswhichsatisesmildregularityconditionscanbedirectlyconvertedtoacanonicalrepresentationwhichisuniqueforeachclassofequivalentmodels,andfurthermore,isaminimalparameteriza-tionofallthatcanbeidentiedabouttheunderlyingMarkovproce
3、ss.HiddenMarkovmodelsonnitestatespacesmaybeframedasaggregatedMarkovprocessesbyexpandingthestatespaceandthusalsohavecanonicalrepresentations.TheauthorcompletedthemajorpartofthisworkaspartofhisPh.D.dissertationattheUniversityofCaliforniaatBerkeley.AMS1991subjectclassications.P
4、rimary60J27,Secondary60J10Keywordsandphrases.aggregatedMarkovprocess,canonicalrepresentation,hiddenMarkovmodel,identiability,equivalence,minimalparameterization1IntroductionConsideracontinuous-timenitestatespaceMarkovprocess,fX(t);t0gTwithstatespaceS=f1;2;:::;ng,initialdist
5、ributionp,andstationarytransitionmatrixP(t)=Qte,inwhichindividualstateshavebeencollectedtogetherintoseveralaggregates,partitioningthestatespace.Supposethatthestateisnotdirectlyobservable,butthatadeterministicfunctionoftheunderlyingMarkovprocessY(t)=f(X(t))whichmapsthestatesp
6、aceStotheaggregatesetA=f;:::;gcanbeobserved.ThisobservablestochasticprocessofaggregatesfY(t);t0g1miscalledanaggregatedMarkovprocess(AMP).AMPsareasubclassofhiddenMarkovmodels(HMMs),whichmoregenerallyallowtheobservedY(t)toberandomlydeterminedgivenX(t).WhentheHMMtakesvaluesona
7、niteset,expandingthestatespacetobethepairs(X(t);Y(t))anddeningftobetheprojectionfunctionf(x;y)=yallowsthisclassofHMMstobetreatedasAMPsonanitestatespace.FredkinandRice[4]andKienker[8]areamongseveralauthorstostudyAMPsasamodelforionchannelkinetics.ResearchersinmanyeldsapplyHMMs
8、tomodelsequencesofdependentrandomvariables.Somevariedexamplesincludemolecul
