Cumulants on the Wiener Space
18
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe et accède à tout notre catalogue !
Découvre YouScribe et accède à tout notre catalogue !
18
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Publié par
Langue
English
Cumulants on the Wiener Space by Ivan Nourdin ? and Giovanni Peccati † Université Paris VI and Université Paris Ouest Abstract: We combine infinite-dimensional integration by parts procedures with a recursive rela- tion on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual graph/diagram computations adopted in most of the probabilistic literature. Key words: Cumulants; Diagram Formulae; Gaussian Processes; Malliavin calculus; Ornstein- Uhlenbeck Semigroup. 2000 Mathematics Subject Classification: 60F05; 60G15; 60H05; 60H07. 1 Introduction The integration by parts formula of Malliavin calculus, combining derivative operators and anticipative integrals into a flexible tool for computing and assessing mathematical expectations, is a cornerstone of modern stochastic analysis. The scope of its applications, ranging e.g. from density estimates for solutions of stochastic di?erential equations to concentration inequalities, from anticipative stochastic calculus to Greeks computations in mathematical finance, is vividly described in the three classic monographs by Malliavin [10], Janson [9] and Nualart [20]. In recent years, infinite-dimensional integration by parts techniques have found another fertile ground for applications, that is, limit theorems and (more generally) probabilistic approximations.
- berry-esseen inequalities
- isonormal gaussian process
- combinatorial results
- computations adopted
- polynomial growth
- random variable
- gaussian field
Publié par
Langue
English
∗ †
∗
†
aristhetheWieneralSpacetob(moreyforIvwanMoNourdin1tslineandxedGioearsvyieldinganniePmeccaticourrierulanNanUniversit?groundParispVI[21],andsomeUniversit?edParistheseOuestpartsAbstract:xi-WnormalehniquescomforbineBerry-Esseinnite-dimensionalcusesinUnivtegrationarisonbAyPpartsoprois,ceduresximationswithofaisrecursivuseeinrela-normalittionorigion24]momenvtsortan(reminiscenintisofda[6,formbulaximationsbGaussianyinBarberalourone(1986)),andandcondeduceapproexplicitelopmenexpressionstforariscumPlaceulanrance.tsdal'X,ofterrefunctionalsdeofAarance.generalaGaussianhereld.applications,ThesetheoremsndingsprobabilisticyieldTheaincompactactivformresearculapapforthecumvinulanultstoonforaonxedcWproinenalsoernon-cencsionhaos,Anothervstepi[13],rtuballWienerybinedreplacingStein'stheprobabilisticusual(seegraph/diagramthcomputationsuppadoptediningammamosttheoffunctionalstheTheprobabilistictroliterature.haKeytowandords:inCumndulanorthts;eDiagraminequalities;Fresultsormariateulae;[1]GaussianfurtherCuminoratoireProProbabilit?scesses;Al?Malliaivin?calculus;,Ornstein-,Uhlen75252b5,ecivan.nourdin@upmc.keSemigroupersit?.2000D?fense,MathematicsenSubR?publique,jectandClassication:Univ60F05;VI,60G15;giovanni.peccati60H05;found60H07.n1tInfertiletroforductionthatThelimitinandtegrationgenerally)bapproy.partsstartingformoulatofthisMalliaevinofcalculus,hcomthebiningerderivwhereativauthorseMalliaopceratorscandusanorderticipatirenevcriteriaeasymptoticinytegralsainWienertohaos,anallyevxiible[22,to(seeol[12]forsomecomputingtralander-assessingofmathematicalresults).expimpectations,tisappaincornerstonewhereoftegrationmoydeonrnspacestocomcwithhasticso-calledanalysis.methoTheforscopapproemationsofe.g.its25]),applications,usrangingexplicite.g.erfoundsrtheoandmapprodensitofylaestimatesofforofsolutionselds.oftecstoincducedhastic[13]dviledesevrenapplicationstialgeneralizations,equationsinstance:to[14]concencantrationapplicationsinequalities,Edgewfromexpansionsanrevticipativrsedeensto[18]ctainshasticforcalculusultivtonormalGreeksximations;computationsfoinonmathematicaldevnanctse,theisLabvividlydedeetscribd?lesedainotheres,threeersitclassicPmonographsVIbBo?tey188Malli4aJussieu,vinP[10],CedexJaFnEmail:sofrnEquip[9]MoandUnivNualartP[20].OuestInNanrelacen200tvyueears,lainnite-dimensional92000interre,tegrationLSTb,yersit?partsaristecFhniquesEmail:ha@gmail.comve(Ω,F,P)
obwithreplacequasi-sureisanalysis;ts.[16]28dealsthiswithbinatoriinnite-dimensionalFinallysecondworderpPrecursivoincar?elonginginequalities;5,inst[19]is,tainedoneocanondhisnewrelatieablexplshallicitcumexpressionsWforoftheadensitasysomeofresultsfunctionalsabofecializeGaussianFeldprobabilitaserwofellhasticasheartilynewort.c2.2)oncencumtrationspinequalities[(seevalsoin[3]compactforrandomsomeWieneapplicationsthisincomputationsmathematicalliteraturestatistics);ellindiscussion[17],paptheSectionresultsandofform[13]tainsarevincom-ourbinedtswith5Lindebrandomerg-tdypon,eoninariatevartariancewrittenprinciplesDepartmeninyorderwtfromo1,deducerederiunivyersalitexpansioyourresultsell-knoforolvinghomogeneousalreadysumsw(thesesomendingsyareandfurtherdenedappliedWinv[15]5toarandomtationmtsatrariablesiaxctheory).claimThemaaimclassicofinthisprobabilistinotee.g.isasto[23]deveeloprelatedyresults).etisanotherws.strikingwapplicationtofvtherecursivinnite-fordimensional3inconceptstegratiotonSectionbotedystatemenpartscumformWienerulainofeMalliaresultsvinariablescalculus,anamelycthenocomputationeryofiscumcommonulanspacetskforts.generalthisfunctionalsbofwavisitinggivofenUnivGaussianineld.ofAsStodiscussedatbbeloOctobw,WourProfessortecVienshniquesarmmakgenerouseMomenaThecrucialtuse(Propofaarecursivrecursivinetsformts.ulathisforseedmomenastscase)(seemrelationulae(2.2)BarbbLemmaelo2w),elywhicrandomhariable.isetheprostartingepSectionointhattimpliesofnewsomerepresenwforell-knoulanwnofcomputationsvpberformedtobxedyrBarbhaos.oureinthat[2]resultinyconnecttheidiagramonadoptedwithmostthetheStein'scmetho(seed[4,for8],normalwapproasximations.forAsgsucneralh,ofthecomtecalhniquesThedevereloporganizededfolloinInthe2forthcomingesectionsacanebproeeseenusefulasefurtherulaeramicationsmomenofSectiontheconndingsbasicofand[13].relatedTheMalliamaincalculus.ac4hievdevementotmainoftstheoutpresenulantonwspace.ork,isSectionawrecursivspeourformtoulavforconcuminulaxenWienertshaos.(seerom(4.19)),wbasedevorandomnjectadenedrepaeatedsuitableuseyincase,relationultivmAof.inctegrationnbwledgemenyPparts.ofNotepapthathascumeenulanwhiletseofereotherdterStatisticsgreaterPurduethanersitt,wtheoccasionarethenotorkshoplinearcopAnalysiseratorsPurdue,(forSepteminstance,erthetosecondercum2009.ulanetthankcoincidesFwithG.theforvwariance):hospitalithoandwsuppev2er,tournsformstartinguoinlofaanalysis(4.19)ositionimpliesisthatwcumwnulanetsonofvregularmomenfuncandtiulanonalsAsofdiscussed,Gaussianresulteldsthecan(asbelleaalwecialaofysrerepresenarktedformasbtheA.D.mathematicalourexp2,ectation1oCorollaryfsomemX E|X| <
itX∞ m > 1 φ (t) = E(e ) t ∈ RX
X j = 1,...,m j X κ (X)j
jdjκ (X) = (−i) logφ (t)| .j X t=0jdt
2 2 3κ (X) = E(X) κ (X) = E(X )− E(X) = Var(X) κ (X) = E(X )−1 2 3
2 33E(X )E(X)+2E(X)
m+1m = 0,1,2... E|X| <∞
mX mm+1 m−sE(X ) = κ (X)E(X ).s+1
s
s=0
m+1dm+1 m+1E(X ) = (−i) φ (t)|X t=0m+1dt
m d dm+1 = (−i) logφ (t) φ (t)X Xm dt dt
t=0" # m s+1 m−sX m d d s+1 m−s= (−i) (logφ (t))×(−i) φ (t) ,
