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STOCHASTICDOMINATIONFORITERATEDCONVOLUTIONSANDCATALYTICMAJORIZATIONGUILLAUMEAUBRUNANDIONNECHITAAbstract.Westudyhowiteratedconvolutionsofprobabilitymeasurescompareunderstochasticdomination.Wegivenecessaryandsufficientconditionsfortheexistenceofanintegernsuchthatnisstochasticallydominatedbyνnfortwogivenprobabilitymeasuresandν.AsaconsequenceweobtainasimilartheoremonthemajorizationorderforvectorsinRd.Inparticularweproveresultsaboutcatalysisinquantuminformationtheory.DominationstochastiquepourlesconvolutionsitéréesetcatalysequantiqueRésumé.Nousétudionscommentlesconvolutionsitéréesdesmesuresdeprobabilitéssecompar-entpourladominationstochastique.Nousdonnonsdesconditionsnécessairesetsuffisantespourl’existenced’unentierntelquensoitstochastiquementdominéeparνn,étantdonnéesdeuxmesuresdeprobabilitésetν.NousobtenonsencorollaireunthéorèmesimilairepourdesvecteursdeRdetlarelationdeSchur-domination.Plusspécifiquement,nousdémontronsdesrésultatssurlacatalyseenthéoriequantiquedel’information.IntroductionandnotationsThisworkisacontinuationof[1],wherewestudythephenomenonofcatalyticmajorizationinquantuminformationtheory.AprobabilisticapproachtothisquestioninvolvesstochasticdominationwhichweintroduceinSection1anditsbehaviorwithrespecttotheconvolutionofmeasures.WegiveinSection2aconditiononmeasuresandνfortheexistenceofanintegernsuchthatnisstochasticallydominatedbyνn.WegatherfurthertopologicalandgeometricalaspectsinSection3.Finally,weapplytheseresultstoouroriginalproblemofcatalyticmajorization.InSection4weintroducethebackgroundforquantumcatalyticmajorizationandwestateourresults.Section5containstheproofsandinSection6weconsideraninfinitedimensionalversionofcatalysis.Weintroducenowsomenotationandrecallbasicfactsaboutprobabilitymeasures.WewriteP(R)forthesetofprobabilitymeasuresonR.WedenotebyδxtheDiracmassatpointx.IfP(R),wewritesuppforthesupportof.Wewriterespectivelymin[−∞,+)andmax(−∞,+]forminsuppandmaxsupp.Wealsowrite(a,b)and[a,b]asashortcutfor((a,b))and([a,b]).Theconvolutionoftwomeasuresandνisdenotedν.RecallthatifXandYareindependentrandomvariablesofrespectivelawsandν,thelawofX+Yisgivenbyν.Theresultsofthispaperarestatedforconvolutionsofmeasures,theyadmitimmediatetranslationsinthelanguageofsumsofindependentrandomvariables.ForλR,thefunctioneλisdefinedbyeλ(x)=exp(λx).1.StochasticdominationAnaturalwayofcomparingtwoprobabilitymeasuresisgivenbythefollowingrelation1991MathematicsSubjectClassification.Primary60E15;Secondary94A05.Keywordsandphrases.Stochasticdomination,iteratedconvolutions,largedeviations,majorization,catalysis.ResearchwassupportedinpartbytheEuropeanNetworkPhenomenainHighDimensions,FP6MarieCurieActions,MCRN-511953.1