Law of large numbers for non-additive measures

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1、Lawoflargenumbersfornon-additivemeasuresYannR´ebill´e∗AbstractOuraimistogiveforsomeclassesnon-additivemeasuressomelimittheorems.Forbalancedgamesweobtainaweakandstronglawoflargenumbersforboundedrandomvariables,asharperconclusionisobtainwithexactgames.Weprovideanextensiontoupperenveloppemeasures.Keywo

2、rds:Lawoflargenumbers,exactgames,multi-priormodel.AMSClassification:28C15,91A121IntroductionNon-additivemeasuresarenowadaysstudiedindifferentfieldofexpertise.Theyareoriginallyknownascapacitiesinpotentialtheory([4]),gameswithtransfer-ableutilitiyincooperativegametheory,fuzzymeasuresinartificialintelligen

3、ceorarXiv:0801.0984v1[math.PR]7Jan2008impreciseprobabilitiesinstatistics.SinceKolmogorov’s([8])axiomatictreatiseonprobability,themeasuretheoreticapproachbecamethestandardframework.σ-additivemeasuresturnedouttobetheappropriateobjectstomodelrandomphenomenon.Amajorrequirementforagoodprobabilitytheoryis

4、tobeabletogiveafrequentistjustificationtoprobabilitynumbersvialimitfrequenciesorequivalentlylawsoflargenumbersshouldhold.Weaddressthisquestionfornon-additivemeasures.Animportantclassofgameswhichcontainsomeverymildadditivityconditionsarebalancedgames([2],[12])andwithmorestructure,ex-actgames([13]).Thi

5、sgamesareparticularlyimportantsincetheyintroducethecore,akeyconcepttounderstandthegeometryofagame.AnaturalapproachistointroducetheusualMarkov’sconditionstoobtainweakandstronglawoflargenumbersforbalancedandexactgames.Ourresultscanbesharpenedthroughupperintegrals.Thereinterestforupperintegralsrelyonth

6、epossibilitytodeal∗Universit´eParisI,CERMSEM,106-112boulevarddel’Hopital,75647ParisCedex13,France.E-mailaddress:yann.rebille@noos.fr12withsetofmeasures.Thisgivesamoreflexibletreatementofuncertaintyindeci-sionmakingtheoryasinthemulti-priormodelofGilboa-Schmeidler(1989)([7],see[3]forσ-measures).Ourappr

7、oachismoreelementaryanddeparturesfromtheexistingtopologicalresults,wherepowerfulanalyticalmethodsareused,seeMarinacci([10])forcompactspacesandMaccheroni-Marinacci([9])forpolishspaces.2DefinitionLet(Ω,A