Gaussian approximation of functionals: Malliavin calculus and Stein s method
20 pages
English

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Gaussian approximation of functionals: Malliavin calculus and Stein's method

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20 pages
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Niveau: Supérieur, Doctorat, Bac+8
Gaussian approximation of functionals: Malliavin calculus and Stein's method Gesine Reinert Abstract. Combining Malliavin calculus and Stein's method has recently lead to a new framework for normal and for chi-square approximation, and for second-order Poincaré inequalities. Applications include functionals of Gaussian random fields as well as func- tionals of infinite Poisson and Rademacher sequences. Here we present the framework and an extension which leads to invariance principles for multilinear homogeneous sums of general centered independent random variables. In the spirit of Stein's method, the key is weak dependence, which can be quantified by the influence function of a variable. In particular we see that Wiener chaos is universal, in the sense that the normal approx- imations of any homogeneous sum can be completely characterised by first switching to its Wiener chaos counterpart. Then the Malliavin-Stein framework yields simple upper bounds and convergence criteria to the normal (and to the chi-square) distribution. These results are joint work with Ivan Nourdin and Giovanni Peccati. 2010 Mathematics Subject Classification. Primary 60F05, 60F17; secondary 60G15, 60G50, 60H07. Keywords. Central limit theorems, isonormal Gaussian processes, linear functionals, second-order Poincaré inequalities, Malliavin calculus, Stein's method, universality. 1. Introduction Over the last few years a very fruitful interplay between Malliavin calculus and Stein's method has been developed, yielding not only a new angle on limit theo- rems, but also new results, notably invariance theorems and a result on the uni- versality of Wiener chaos.

  • exchangeable ran- dom variables

  • gaussian approximation

  • stein's method

  • has been extended

  • thus

  • pair couplings

  • normal distribution

  • random variable

  • any homogeneous


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Gaussianapproximationoffunctionals:MalliavincalculusandStein’smethodGesineReinertAbstract.CombiningMalliavincalculusandStein’smethodhasrecentlyleadtoanewframeworkfornormalandforchi-squareapproximation,andforsecond-orderPoincaréinequalities.ApplicationsincludefunctionalsofGaussianrandomfieldsaswellasfunc-tionalsofinfinitePoissonandRademachersequences.Herewepresenttheframeworkandanextensionwhichleadstoinvarianceprinciplesformultilinearhomogeneoussumsofgeneralcenteredindependentrandomvariables.InthespiritofStein’smethod,thekeyisweakdependence,whichcanbequantifiedbytheinfluencefunctionofavariable.InparticularweseethatWienerchaosisuniversal,inthesensethatthenormalapprox-imationsofanyhomogeneoussumcanbecompletelycharacterisedbyfirstswitchingtoitsWienerchaoscounterpart.ThentheMalliavin-Steinframeworkyieldssimpleupperboundsandconvergencecriteriatothenormal(andtothechi-square)distribution.TheseresultsarejointworkwithIvanNourdinandGiovanniPeccati.2010MathematicsSubjectClassification.Primary60F05,60F17;secondary60G15,60G50,60H07.Keywords.Centrallimittheorems,isonormalGaussianprocesses,linearfunctionals,second-orderPoincaréinequalities,Malliavincalculus,Stein’smethod,universality.1.IntroductionOverthelastfewyearsaveryfruitfulinterplaybetweenMalliavincalculusandStein’smethodhasbeendeveloped,yieldingnotonlyanewangleonlimittheo-rems,butalsonewresults,notablyinvariancetheoremsandaresultontheuni-versalityofWienerchaos.Theseresultsallowtoderiveboundsondistributionaldistancesforfairlygeneraluni-ormultivariatereal-valuedfunctionalsofrandomfields,andtheyincludemanywell-studiednormalapproximationsasspecialcases.Belowareafewexamplestoillustratetherangeofobjectswhichhavebeenconsidered.Theseobjectsareoftenphrasedintermsofsymmetricreal-valuedfunctionsfonRd,thatis,f(iσ(1),...,iσ(d))=f(i1,...,id)foranypermutationσon{1,...,d}andany(i1,...,id)Rd.Oftenwealsoassumethatfvanishesondiagonals,thatis,f(i1,...,id)=0wheneverthereexistk6=jsuchthatik=ij.Example1.1.Multilinearhomogeneoussums.FixintegersN,d>2.LetX={Xi:i>1}beacollectionofcenteredindependentrandomvariables;andletf:{1,...,N}dRbeasymmetricfunctionvanishingondiagonals.ThenQd(X)=f(i1,...,id)Xi1XidX1i1,...,idN
2GesineReinertiscalledthemultilinearhomogeneoussum,oforderd,basedonfandonthefirstNelementsofX.InSection5weshallgiveboundstothestandardnormaldistribution,inWassersteindistance,forsuchsums.Example1.2.FunctionalsofRademachersequences.LetX={Xn:n>1}beaRademachersequence,thatis,asequenceofi.i.d.randomvariableswithvaluesin{−1,1}suchthatP(X1=1)=P(X1=1)=1/2.Assumethatfn:NnRvanishesondiagonals,andputQ(X)=fn(i1,...,in)Xi1Xin.XXn0i1,...,inWeareinterestedinassessinghowfarawayQ(X)isfromanormallydistributedrandomvariable.Notethathere,incontrasttoExample1.1,thesumispotentiallyoveraninfinitenumberofsummands.Aneasyexampleisthenumberof(weighted)d-runsinaninfiniteBernoullisequence;see[35].AsavariantitispossibletoobtainsimilarresultswhenreplacingtheRademachersequencebyasequenceofcompensatedPoissonvariables;see[41].Example1.3.AfunctionalofaGaussianprocesRs.LetBbeacenteredGaussianprocesswithstationaryincrementssuchthatR|ρ(x)|dx<,whereρ(uv):=E(Bu+1Bu)(Bv+1Bv).Letf:RRbeanon-constantfunctionwhichistwicecontinuouslydifferentiable,letZ∼N(0,1)beastandardnormalvariable,assumethatE|f(Z)|<andE[f′′(Z)4]<.Fixa<binRand,foranyT>0,considerFT=f(Bu+1Bu)E[f(Z)]du.1ZbT�TTaInSection4weshallassessitsdistributionaldistancetothestandardnormal.Example1.4.Functionsoffunctionsofindependentnormals.Anexam-plewhichisconsideredin[10]istoletX=(X1,...,Xn),whereeachXi=ui(Z),withZstandardnormal,anduibeingareal-valued,smoothfunction,foreachi.Letg:RRbetwicecontinuouslydifferentiableandputW=g(X).ForexamplewecouldconsiderW=ijai,jXiXj;orW=ReTr(f(A(X))),whereA(x)isaPcomplexn×nmatrix.Undera“symmetricinteractionrule”ong,[10]derivesanormalapproximationforW,withbounds,usingStein’smethod.Someapplicationswherefunctionalsofrandomfieldsplayanimportantroleincludepowerandbipowervariationsofstochasticintegralsasin[6]and[17],thecorrelationstructureofMexicanneedlets(whichareaclassofwaveletsystems)andmodelsforcosmologicaldataanalysis(see[23]and[26]);andtheestimationofHurstindices,see[55].Thecommonthemebehindtheseexamplesandapplicationsarepossiblynon-linearfunctionalsofindependentrandomelementswhichadmitachaosdecompo-sition,see(8),whichlookslikeF=E(F)+n!fn(i1,...,in)Xi1Xin,(1)XXn>1i1<i2<...<in
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