Multivariate central limit theorems for Rademacher functionals with applications Kai Krokowski and Christoph Thäle† ∗ 7 1 0 Abstract 2 n Quantitativemultivariatecentrallimittheoremsforgeneralfunctionalsofpossiblynon-symmetricand a non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus J withthesmartpathmethodfornormalapproximation. Inparticular,adiscretemultivariatesecond-order 5 Poincaréinequalityisdeveloped. Asafirstapplication,thenormalapproximationofvectorsofsubgraph 2 counting statisticsintheErdo˝s-Rényirandomgraphisconsidered. Inthis context, wefurtherspecialize tothenormalapproximationofvectorsofvertexdegrees.Inasecondapplicationweproveaquantitative ] multivariatecentrallimittheoremforvectorsofintrinsicvolumesinducedbyrandomcubicalcomplexes. R P Keywords. DiscreteMalliavin calculus, intrinsicvolume, multivariate central limittheorem, smartpath . h method,subgraphcount,randomgraph,randomcubicalcomplex,vertexdegree t MSC.Primary60F05;Secondary05C80,60C05,60D05,60H07. a m [ 1 Introduction 1 v SupposethatX =(Xk)k NisaRademachersequence,thatis,asequenceofindependentrandomvariables 5 satisfying, for all k N∈, P(X = 1) = p and P(X = 1) = q = 1 p for some p (0,1). Further, k k k k k k 6 fixadimensionpara∈meterd NandletF = F (X),...−,F = F (X)be−drandomvariabl∈esdependingon 3 ∈ 1 1 d d possibly infinite many members of the Rademacher sequence X. We shall referto such random variables 7 0 as Rademacher functionals in what follows. The goal of this paper is to derive handy conditions under . which the random vector F = (F ,...,F ) consisting of d Rademacher functionals is close in distribution 1 1 d 0 toad-dimensionalGaussianrandomvector. Inourpaperthedistributionalclosenesswillbemeasuredby 7 meansofamultivariateprobabilitymetricbasedonfourtimespartiallydifferentiabletestfunctions. Wewill 1 providetwoversionsofsucharesult. OneisinthespiritoftheMalliavin-Steinmethodandexpressesthe v: distributionalclosenessintermsofso-calleddiscreteMalliavinoperators. Thesecondoneisamultivariate i discretesecond-orderPoincaréinequality,aboundwhichonlyinvolvesthefirst-andsecond-orderdiscrete X Malliavin derivatives of the Rademacher functionals F ,...,F , or, more precisely, their moments up to 1 d r orderfour. Moreformally,ifF = F(X)isaRademacherfunctional,thediscreteMalliavinderivativeD Fin a k directionk ∈ N is definedas DkF = √pkqk(Fk+−Fk−), where Fk± is the Rademacherfunctional for which the kth coordinate X of the Rademacher sequence X is conditioned to be 1. The second-order discrete k derivative is iteratively given by DkDℓF for k,ℓ N. Such a bound is part±icularlyattractive for concrete ∈ applicationsasdemonstratedinthepresenttext. Letusdescribethepurposeandthecontentofourpaperinsomemoredetail. (i) First of all, our aim is to provide a multivariate quantitative central limit theorem for vectors of RademacherfunctionalsbybringingtogetherthediscreteMalliavincalculusofvariationswiththeso- calledsmart-pathmethodfornormalapproximation. Thisleadstoalimittheoreminthespiritofthe Mallavin-Steinmethod andgeneralizesanearlierresultfrom[9], wherethe underlyingRademacher sequence hasbeenassumedtobehomogeneous andsymmetric, meaningthat p = q = 1/2forall k k k Ninabovenotation. ∈ ∗RuhrUniversityBochum,FacultyofMathematics,NA3/28,D-44780Bochum,Germany.E-mail:[email protected] †RuhrUniversityBochum,FacultyofMathematics,NA3/68,D-44780Bochum,Germany.E-mail:[email protected] 1 (ii) Fromthisresult,afurtheraimofthistextistodevelopadiscretemultivariatesecond-orderPoincaré inequality,thatis,aboundforthemultivariatenormalapproximationthatonlyinvolvesthefirst-and second-orderdiscreteMalliavinderivatives,or,moreprecisely,itsmomentsuptoorderfour. Sucha resultcanberegardedasthemultivariateanalogueofthemaintheoremobtainedin[10]. (iii) Finally, we want todemonstrate the flexibility and applicabilityof our discretemultivariate second- orderPoincaréinequalitybymeansofexamplesfromthetheoryofrandomgraphsandrandomtopol- ogy. First,wearegoingtoprovideaboundoforderO(n 1)forthemultivariatenormalapproximation − ofavectorofsubgraphcountsintheclassicalErdo˝s-Rényirandomgraph.Thisgeneralizes(inadiffer- entprobabilitymetric)aresultofReinertandRöllin[18],wherevectorsofthenumberofedges,2-stars andtriangleshavebeenconsidered,andaddsarateofconvergencetotherelatedcentrallimittheorem in the paperof Janson andNowicki [6]. Moreover, for the same modelwe also providea multivari- atecentrallimittheoremfortherandomvectorofvertexdegreeswitharateofconvergenceoforder O(n 1/2). ThiscanbeseenasaversionoftheresultofGoldsteinandRinott[4]andisthemultivariate − analogueofarelatedBerry-EsseenboundprovedbyGoldstein[3]andKrokowski,Reichenbachsand Thäle [10]. Second, we consider the vector of intrinsic volumes determined by different models of random cubical complexes in Rd and derive bounds of orderO(n d/2) on the error in their normal − approximation.ThisconstitutesamultivariateextensionofthecentrallimittheoremprovidedbyWer- manandWright[20]andisinline with recentdevelopmentsinthe activefieldof randomtopology, see[1,2,7,11]aswellasthereferencescitedtherein. Our results continue a recentline of researchconcerning limit theorems for Rademacher functionals. The field has been opened by Nourdin, Peccati and Reinert [12], who proved first limit theorems for a class of smooth probabilitymetrics. Later, Krokowski, Reichenbachs andThäle [9, 10] considered Berry-Esseen boundsandprovidedafirstunivariatediscretesecond-orderPoincaréinequality. Zheng[21]hasobtained arefinedboundfortheWassersteindistanceandalsoprovedalmostsurecentrallimittheorems. Moreover, Privault and Torrisi [16] as well as Krokowski [8] also derived bounds for the Poisson approximation of Rademacherfunctionals. This text is organized as follows. In Section 2 we briefly recall the basis of discrete Malliavin calculus in order to keep the paper reasonably self-contained. A first quantitative multivariate central limit theorem for functionals of a possibly non-symmetric and non-homogeneous infinte Rademacher sequence based on the discrete Malliavin-Stein method is presented in Section 3.1, while Section 3.2 contains the discrete multivariate second-order Poincaré inequality. The applications to subgraph and vertex degree counts in theErdo˝s-Rényirandomgraphandtotheintrinsicvolumesofrandomcubicalcomplexesarediscussedin thefinalSection4. 2 Discrete Malliavin calculus Inthissectionwe brieflyrecallthe basisof discreteMalliavincalculus. We refertothe monograph [15] as wellastothepapers[9,10,12]fordetails,proofsandfurtherreferences. Rademacher sequences. Let p := (pk)k N be a sequence of success probabilities 0 < pk < 1 and put q := (qk)k N with qk := 1 pk. Furthe∈rmore, let (Ω, ,P) be the following probability space: Ω := 1,+1 N∈, := power( −1,+1 ) N, where power( )Fdenotes the power set of the argumentset, and ⊗ a{P−s:e=quen}∞kc=e1(oFpfkiδn+d1e+peqnkdδe−n1{t)−rwanitdhoδm}±1vabreiianbgletshedeufinniet-dmoan·ss(DΩi,rac,Pm)ebaysuXre(aωt)±:1=. ωWe, fleotrXeve:=ry(kXk)kN∈Nanbde N F k k ∈ ω := (ωk)k N Ω. We refer to such a sequence X as (possibly non-symmetric and non-homogeneous infinite) Rad∈ema∈chersequence. We also definethe standardizedsequenceY := (Yk)k N byputtingYk := h(Voamr(oXgek)n)e−o1u/s2(aXnkd−syEm[mXke]t)ri=cR(a2d√empkaqckh)e−r1(sXeqku−enpcke,+thqakt)ifso,rifepve=ryqk ∈=N1/.2,Nfoortealtlhka∈t YNk .= Xk, iff X is a k k ∈ Discretemultiplestochasticintegralsandchaosdecomposition. Letusdenotebyκthecountingmeasure on N. We put ℓ2(N) n := L2(Nn, (N) n,κ n) for every n N and referto the elements of thatspace ⊗ ⊗ ⊗ P ∈ 2 as kernels. Let ℓ2(N) n denote the subset of ℓ2(N) n consisting of symmetric kernels and let ℓ2(N) n ◦ ⊗ 0 ⊗ be the subset of kernels vanishing on diagonals, that is, vanishing on the complement of the set ∆ := n (i ,...,i ) Nn :i =i forj = k . Wethenputℓ2(N) n :=ℓ2(N) n ℓ2(N) n. { 1 n ∈ j 6 k 6 } 0 ◦ ◦ ∩ 0 ⊗ Forn Nand f ℓ2(N) n,wedefinethediscretemultiplestochasticintegralofordernof f by ∈ ∈ 0 ◦ J (f):= ∑ f(i ,...,i )Y ... Y = ∑ f(i ,...,i )Y ... Y n 1 n i1· · in 1 n i1· · in (i1,...,in)∈Nn (i1,...,in)∈∆n =n! ∑ f(i ,...,i )Y ... Y . 1 n i1· · in 1≤i1<...<in<∞ Inaddition,weputℓ2(N) 0 :=Rand J (c):=c,foreveryc R. ⊗ 0 ∈ ItisanimportantfactthateveryF L2(Ω)admitsadecompositionoftheform ∈ ∞ F =E[F]+ ∑ J (f ) (1) n n n=1 withuniquelydeterminedkernels f ℓ2(N) n. n ∈ 0 ◦ DiscreteMalliavinderivative. Foreveryω = (ω ,ω ,...) Ωandk Nwedefinethetwosequences 1 2 ωk byputting ωk := (ω ,...,ω , 1,ω ,...). Furtherm∈ore,foreve∈ry F L1(Ω), ω Ω and k N let±Fk±(ω):= F(ω±k ). For1suchankF−1th±edisckr+e1teMalliavinderivativeisdefined∈byDF:=(D∈kF)k Nwit∈h ± ∈ DkF :=√pkqk(Fk+−Fk−), k ∈N. (2) Notethatitimmediatelyfollowsfrom(2)that,foreveryk N,D FisindependentofX . Inthefollowing k k ∈ westateaproductformulaforthediscreteMailliavinderivative.IfF,G L1(Ω),then ∈ X D (FG)= (D F)G+F(D G) k (D F)(D G), k N. (3) k k k k k − √p q ∈ k k For m N let us further define the iterated discrete Malliavin derivative of order m of F by DmF := ∈ (Dkm1,...,kmF)k1,...,km∈N with Dkm1,...,kmF := Dkm(Dkm1−,..1.,km 1F),foreveryk1,...,km ∈ N,where D0F := F. Given F L2(Ω) with chaos representation F = E[F]+∑−∞ J (f ) as in (1) and m N, we will say that F ∈ n=1 n n ∈ ∈ dom(Dm),providedthat ∞ n! E[ DmF 2 ] = ∑ n! f 2 < ∞. k kℓ2(N)⊗m n=m(n−m)! k nkℓ2(N)⊗n If F dom(D) with chaos decomposition (1), D F can be P-almost surely be identified with the random k ∈ variablegivenby ∞ D F = ∑ nJ (f ( ,k)), k n 1 n − · n=1 where f ( ,k)standsforthekernel f withoneofitsvariablesfixedtobek (whichoneisirrelevant,since n n · thekernelsaresymmetric). Discretedivergence. Wewillnowdefinethe discretedivergenceoperatorδ anditsdomaindom(δ). Let ∑fn∞∈ Jℓ20(N(f)◦(n−,1k⊗))ℓfo2(rNev)e,rfyork evNer.yFnor∈sucNh,ua,nwdecsoanystihdaetruthedsoemqu(eδn),ceifu := (uk)k∈N given by uk := n=1 n−1 n · ∈ ∈ ∞ ∑ n!kfn1∆nk2ℓ2(N)⊗n < ∞, n=1 e 3 where f denotesthecanonicalsymmetrizationof f . Foru dom(δ),thediscretedivergenceoperatorδis n n ∈ thendefinedby e ∞ δ(u) := ∑ Jn(fn1∆n). n=1 e One can interpret δ as the operator that is adjoint to the discrete Malliavin derivative. Namely, if F ∈ dom(D)andu dom(δ),then ∈ E[Fδ(u)]=E[hDF,uiℓ2(N)]. (4) DiscreteOrnstein-Uhlenbeckoperatoranditsinverse. Next,wedefinethediscreteOrnstein-Uhlenbeck operator L and its (pseudo-)inverse L 1. Given F L2(Ω), again with chaos representation F = E[F]+ − ∑∞ J (f )asabove,wesaythatF dom(L),if ∈ n=1 n n ∈ ∞ ∑ n2n! f 2 <∞. k nkℓ2(N)⊗n n=1 ForF dom(L),thediscreteOrnstein-UhlenbeckoperatorListhendefinedby ∈ ∞ LF := ∑ nJ (f ). n n − n=1 ForcentredF L2(Ω),its(pseudo-)inverseisgivenasfollows: ∈ ∞ 1 L−1F := ∑ Jn(fn). − n n=1 Discrete Ornstein-Uhlenbecksemigroup. Finally, we introduce the semigroup associated with the dis- creteOrnstein-UhlenbeckoperatorL. ThediscreteOrnstein-Uhlenbecksemigroup(P) isdefinedby t t 0 ≥ ∞ PtF :=E[F]+ ∑ e−ntJn(fn), t 0. ≥ n=1 The process associated with the discrete Ornstein-Uhlenbeck semigroup is given as follows. For every ekx∈poNne,nletitaXllyk∗ dbeistarnibiuntdeedpreannddeonmtcvoapryiaobfleXskw. iFthurmtheearnm1o,rew,hleetre(ZZk)kis∈NindbeepaensedqeunetnocfeXofainnddeXpen,fdoernetvaenrdy k k k∗ k ∈N. Foreveryrealt ≥0,letXt :=(Xkt)k∈N with Xkt := Xk∗1{Zk≤t}+Xk1{Zk>t}, k ∈N. Then, (Xt) is the discrete Ornstein-Uhlenbeck process associated with the Ornstein-Uhlenbeck semi- t 0 group (P) ≥ . The relation of Ornstein-Uhlenbeck semigroup and process is exhibited in the following t t 0 formula,kn≥ownasMehler’sformula. IfF L2(Ω),thenitP-almostsurelyholdsthat ∈ PF =E[F(Xt) X], t 0. (5) t | ≥ Integrationby parts, integratedMehler’s formula and Poincaré inequality. We notice that the discrete Malliavin operators D, δ and L are related by the identity L = δD. Moreover, the following discrete − integrationbypartsformulaisvalid. IfF,G dom(D),then ∈ E[(F−E[F])G]=E[h−DL−1(F−E[F]),DGiℓ2(N)]. (6) Indeed,therelationL= δDandtheadjointnessofDandδin(4)yield − E[(F−E[F])G]=E[LL−1(F−E[F])G]=E[−δDL−1(F−E[F])G]=E[h−DL−1(F−E[F]),DGiℓ2(N)]. 4 Thefollowing identitycanbe seenasanintegratedversionof Mehler’sformula. If m,k ,...,k N and 1 m F dom(Dm)withE[F] =0,thenitP-almostsurelyholdsthat ∈ ∈ ∞ −Dkm1,...,kmL−1F =Z0 e−mtPtDkm1,...,kmFdt. (7) Fromthis,onecanimmediatelydeducethefollowingimportantinequality. Ifm,k ,...,k N,α 1and 1 m F dom(Dm)withE[F] =0,then ∈ ≥ ∈ E[|Dkm1,...,kmL−1F|α] ≤E[|Dkm1,...,kmF|α]. (8) Finally,letusrecalladiscreteversionoftheclassicalPoincaréinequality. ForeveryF L1(Ω),itholdsthat ∈ Var(F) E[ DF 2 ]. (9) ≤ k kℓ2(N) 3 Multivariate central limit theorems 3.1 Adiscrete Malliavin-Stein bound Inthefollowing,wewillproveaboundontheerrorinthemultivariatenormalapproximationofvectorsof generalfunctionalsofpossiblynon-symmetricandnon-homogeneousinfiniteRademachersequences. This way we generalize Theorem 5.1 in [9], where only functionals of symmetric Rademacher sequences have beenconsidered. The proof proceedsalong the lines of [9], butthere area number of subtletiesarising in the more general case here that were not present before. In particular, in the non-symmetric case a new summandinthe errorboundbecomesvisibleasfurtherdiscussedinRemark3.2below. Tomakethisand otherphenomenatransparent,weincludethefulldetails. ThedistancebetweenthelawofavectorofRademacherfunctionalsandamultivariatenormaldistribution willbemeasuredbytheso-calledd -distancethatisdefinedasfollows. Fixd Nandletn = 1,...,d. For 4 anntimespartiallydifferentiablefunctiong :Rd Rweput ∈ → ∂k M (g):= max g k 1≤i1,...,ik≤d(cid:13)∂xi1...∂xik (cid:13)∞ (cid:13) (cid:13) foreveryk =1,...,n,where ∞denotesthesuprem(cid:13)umnormofth(cid:13)eargumentfunction. Thed4-distance betweenthedistributionsoftkw·okRd-valuedrandomvectorsXandYisdefinedby d (X,Y) :=sup E[g(X)] E[g(Y)] , 4 | − | g where the supremum is running over all four times partially differentiable functions g : Rd R with → boundedpartialderivativesfulfilling M (g),M (g),M (g),M (g) 1. 1 2 3 4 ≤ Theorem 3.1. Fix d N and let F ,...,F be Rademacher functionals with F dom(D), E[F] = 0 and E[ (pq) 1/4DF 4 ∈] < ∞, for eve1ry i = d1,...,d. Define F := (F ,...,F ) aind∈let N := (N ,.i..,N ) be a k − ikℓ4(N) 1 d 1 d centred Gaussian random vectorwith symmetric and positivesemidefinite covariancematrix Σ := (Σ )d . Fur- ij i,j=1 ther,let 1 d A1 := 2 ∑ E[|Σij−hDFj,−DL−1Fiiℓ2(N)|], i,j=1 1 p q d 2 d A2 := E | − | ∑ DFj ,∑ DL−1Fi , 4 √pq | | |− | ℓ2(N) hD (cid:16)j=1 (cid:17) i=1 E i 5 1 d 3 d A3 := E ∑ DFj ,∑ DL−1Fi . 24 pq | | |− | ℓ2(N) j=1 i=1 hD (cid:16) (cid:17) E i Then, d (F,N) A +A +A . 4 1 2 3 ≤ 5 Remark 3.2. A comparison of Theorem 3.1 with Theorem 5.1 in [9] shows that the extension to vectors of general functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences comesatthecostsofanadditionalsummandinthebound,namely p q d 2 d E | − | ∑ DFj ,∑ DL−1Fi . √pq | | |− | ℓ2(N) hD (cid:16)j=1 (cid:17) i=1 E i However,resortingtothecasewheretheunderlyingRademachersequenceissymmetric,i.e.,if p = q = k k 1/2,foreveryk N,thisadditionalsummandvanishesandourboundinTheorem3.1coincideswiththe ∈ onefrom[9]withanimprovementbyafactor1/2ontheconstantinfrontofthethirdterm. The proof of Theorem 3.1 relies on two multivariate integration by parts formulae, a Gaussian one and an approximate one from Malliavin calculus which combines (6) with a multivariate chain rule for the discrete gradient operator. We start by recalling the multivariate Gaussian integration by parts formula fromEquation(A.41)in[19]. Lemma 3.3. Fix d N and let N := (N ,...,N ) be a centred Gaussian random vector with symmetric and positivesemidefinite ∈covariance matrix Σ :=1(Σ )d d . Furthermore, let g : Rd R be a partially differentiable ij i,j=1 → functionwithboundedpartialderivativesandE[ Ng(N) ] <∞,foreveryi =1,...,d. Then,foreveryi =1,...,d, i | | d ∂ E[Ng(N)]= ∑Σ E g(N) . i ij ∂x j=1 h j i Thefollowinglemmacontainsamultivariatechainruleforthediscretegradientoperator,whichisagener- alizationofProposition2.1in[16]tothed-dimensionalcase. AlsonotethatitnotonlygeneralizesLemma 5.1in[9]tothecasewheretheunderlyingRademachersequenceisnon-symmetricandnon-homogeneous, but also improves on the constants in the bound for the remainder term. For these reasons, we include a detailedproof. Lemma3.4. LetFbearandomvectorofRademacherfunctionalsasinTheorem3.1. Furthermore,let f : Rd R beathricepartiallydifferentiablefunction. Then,foreveryk N, → ∈ d ∂ X d ∂2 ∂2 Dkf(F) = ∑ ∂x f(F)DkFi− 4√pkq ∑ ∂x ∂x f(F+k )+ ∂x ∂x f(F−k ) (DkFi)(DkFj)+Rk(F) (10) i=1 i k k i,j=1 i j i j (cid:16) (cid:17) withF±k := (F1)±k ,...,(Fd)±k andaremaindertermRk(F)thatfulfils (cid:0) (cid:1) 5 d ∂3 Rk(F)≤ 12pkqk i,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|(DkFi)(DkFj)(DkFℓ)| (11) (cid:13) (cid:13) foreveryk N. (cid:13) (cid:13) ∈ Proof. Fixk Nandobservethat ∈ Dkf(F)= √pkqk(f(F+k )− f(F−k ))= √pkqk(f(F+k )− f(F))−√pkqk(f(F−k )− f(F)). (12) Now,aTaylorseriesexpansionof f atFyieldsthat,foreveryx:=(x ,...,x ) Rd, 1 d ∈ d ∂ 1 d ∂2 f(x) f(F) = ∑ f(F)(x F)+ ∑ f(F)(x F)(x F) − ∂x i− i 2 ∂x ∂x i− i j− j i=1 i i,j=1 i j 1 d ∂3 + 6i,j∑,ℓ=1∂xi∂xj∂xℓ f(F+θ(x−F))(xi−Fi)(xj−Fj)(xℓ−Fℓ) 6 wwaityh,istofmoleloθw:=sfrθo(mx,(F1)2)∈th(0a,t,1f)o.rBeyvreer-ywkritinNg,eachofthequantities f(F+k )− f(F)and f(F−k )− f(F)inthis ∈ d ∂ 1 d ∂2 D f(F) =√p q ∑ f(F)((F)+ F)+ √p q ∑ f(F)((F)+ F)((F)+ F) k k k ∂x i k − i 2 k k ∂x ∂x i k − i j k − j i=1 i i,j=1 i j d ∂ +R1(F,F+k )−√pkqk ∑ ∂x f(F)((Fi)−k −Fi) i=1 i 1 d ∂2 − 2√pkqk ∑ ∂x ∂x f(F)((Fi)−k −Fi)((Fj)−k −Fj)−R2(F,F−k ) i,j=1 i j d ∂ 1 d ∂2 = ∑ ∂x f(F)DkFi+ 2√pkqk ∑ ∂x ∂x f(F)(((Fi)+k −Fi)((Fj)+k −Fj)−((Fi)−k −Fi)((Fj)−k −Fj)) i=1 i i,j=1 i j +R1(F,F+k )−R2(F,F−k ), (13) wherefromtheidentities 1 Fk+−F =(Fk+−Fk−)1{Xk=−1} = √pkqk(DkF)1{Xk=−1} (14) and 1 Fk−−F = (Fk−−Fk+)1{Xk=+1} =−√pkqk(DkF)1{Xk=+1} (15) itfollowsthat,foreveryk N, ∈ 1 d ∂3 |R1(F,F+k )| ≤ 6√pkqki,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|((Fi)+k −Fi)((Fj)+k −Fj)((Fℓ)+k −Fℓ)| (cid:13) (cid:13) 1 d (cid:13) ∂3 (cid:13) = 6pkqk i,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|(DkFi)(DkFj)(DkFℓ)|1{Xk=−1} (16) (cid:13) (cid:13) (cid:13) (cid:13) and 1 d ∂3 |R2(F,F−k )| ≤ 6√pkqki,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|((Fi)−k −Fi)((Fj)−k −Fj)((Fℓ)−k −Fℓ)| (cid:13) (cid:13) 1 d (cid:13) ∂3 (cid:13) = 6pkqk i,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|(DkFi)(DkFj)(DkFℓ)|1{Xk=+1} . (17) (cid:13) (cid:13) Again,byvirtueof(14)and(15),forev(cid:13)eryk N,t(cid:13)hesecondsummandontherighthandsideof(13)can ∈ berewrittenas 1 d ∂2 2√pkqk ∑ ∂x ∂x f(F)(((Fi)+k −Fi)((Fj)+k −Fj)−((Fi)−k −Fi)((Fj)−k −Fj)) i,j=1 i j 1 d ∂2 = ∑ f(F)(D F)(D F)(1 1 ) 2√pkqk i,j=1∂xi∂xj k i k j {Xk=−1}− {Xk=+1} X d ∂2 = k ∑ f(F)(D F)(D F). (18) −2√p q ∂x ∂x k i k j k k i,j=1 i j 7 AnotherTaylorseriesexpansionof ∂x∂i∂2xjf atF+k andF−k ,respectively,yieldsthat,foreveryi,j,k ∈N, ∂2 ∂2 d ∂3 ∂xi∂xj f(F) = ∂xi∂xj f(F+k )+ℓ∑=1∂xℓ∂xi∂xj f(F+k +θ1(F−F+k ))(Fℓ−(Fℓ)+k ) and ∂2 ∂2 d ∂3 ∂xi∂xj f(F) = ∂xi∂xj f(F−k )+ℓ∑=1∂xℓ∂xi∂xj f(F−k +θ2(F−F−k ))(Fℓ−(Fℓ)−k ), whereθ1 := θ1(F,F+k )∈ (0,1)andθ2 :=θ2(F,F−k ) ∈ (0,1).Thisaddsupto ∂2 1 ∂2 ∂2 1 d ∂3 ∂xi∂xj f(F) = 2(cid:16)∂xi∂xj f(F+k )+ ∂xi∂xj f(F−k )(cid:17)+ 2 ℓ∑=1∂xℓ∂xi∂xj f(F+k +θ1(F−F+k ))(Fℓ−(Fℓ)+k ) 1 d ∂3 + 2 ℓ∑=1∂xℓ∂xi∂xj f(F−k +θ2(F−F−k ))(Fℓ−(Fℓ)−k ) foreveryi,j,k N,andthus,itfollowsfrom(18)thatforeveryk N ∈ ∈ 1 d ∂2 2√pkqk ∑ ∂x ∂x f(F)(((Fi)+k −Fi)((Fj)+k −Fj)−((Fi)−k −Fi)((Fj)−k −Fj)) i,j=1 i j X d ∂2 ∂2 = −4√pkq ∑ ∂x ∂x f(F+k )+ ∂x ∂x f(F−k ) (DkFi)(DkFj)−R3(F,F+k )−R4(F,F−k ), (19) k k i,j=1(cid:16) i j i j (cid:17) wherebythefactthat X 1foreveryk Nandanotherapplicationof(14)and(15)itholdsthat | k| ≤ ∈ 1 d ∂3 |R3(F,F+k )| ≤ 4√pkqk i,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|(DkFj)(DkFℓ)((Fi)+k −Fi)| (cid:13) (cid:13) 1 d (cid:13) ∂3 (cid:13) = 4pkqk i,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|(DkFi)(DkFj)(DkFℓ)|1{Xk=−1} (20) (cid:13) (cid:13) (cid:13) (cid:13) and 1 d ∂3 |R4(F,F−k )| ≤ 4√pkqk i,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|(DkFj)(DkFℓ)((Fi)−k −Fi)| (cid:13) (cid:13) 1 d (cid:13) ∂3 (cid:13) = 4pkqk i,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|(DkFi)(DkFj)(DkFℓ)|1{Xk=+1} . (21) (cid:13) (cid:13) Combining(13)and(19)finallyyieldsthat(cid:13)foreveryk (cid:13)N ∈ d ∂ X d ∂2 ∂2 Dkf(F)= ∑ ∂x f(F)DkFi− 4√pkq ∑ ∂x ∂x f(F+k )+ ∂x ∂x f(F−k ) (DkFi)(DkFj) i=1 i k k i,j=1 i j i j (cid:16) (cid:17) +R1(F,F+k )−R2(F,F−k )−R3(F,F+k )−R4(F,F−k ), wherebecauseof(16),(17),(20)and(21)wehavethat 5 d ∂3 |R1(F,F+k )|+|R2(F,F−k )|+|R3(F,F+k )|+|R4(F,F−k )| ≤ 12pkqk i,j∑,ℓ=1(cid:13)∂xi∂xj∂xℓ f(cid:13)∞|(DkFi)(DkFj)(DkFℓ)|. (cid:13) (cid:13) (cid:13) (cid:13) Theproofisthuscomplete. 8 Letusnowturntothealreadyannouncedmultivariateapproximateintegrationbypartsformula. Thenext resultnot only generalizesLemma 5.2 in [9] to the case in which the underlying Rademacher sequence is allowedtobenon-symmetricandnon-homogeneous, butalsoimprovestheconstantsintheboundforthe remainder term. We emphasize that Lemma 3.5 is the first instance where the additional boundary term discussedinRemark3.2showsup. Lemma3.5. LetFbeavectorofRademacherfunctionalsasinTheorem3.1. Furthermore,let f :Rd Rbeathrice → partiallydifferentiablefunctionwithboundedpartialderivatives. Then,foreveryi =1,...,d, d ∂ E[Fif(F)]= ∑E ∂x f(F)hDFj,−DL−1Fiiℓ2(N) +E[hR(F),−DL−1Fiiℓ2(N)] j=1 h j i witharemainderR(F)thatsatisfiestheestimate 1 p q d 2 |E[hR(F),−DL−1Fiiℓ2(N)]| ≤ 2M2(f)E |√−pq| ∑|DFj| ,|−DL−1Fi| ℓ2(N) j=1 hD (cid:16) (cid:17) E i (22) 5 1 d 3 + M3(f)E ∑ DFj , DL−1Fi . 12 pq | | |− | ℓ2(N) j=1 hD (cid:16) (cid:17) E i Proof. Fixi =1,...,d.Bytheintegrationbypartsformula(6)wehavethat E[Fif(F)] =E[hDf(F),−DL−1Fiiℓ2(N)]. (23) Here, we implicitly used the fact that f(F) dom(D), which can be verified as follows. At first, by the meanvaluetheoremitholdsthatforeveryk∈ N ∈ d ∂ |Dkf(F)| = √pkqk|f(F+k )− f(F−k )| = √pkqk ∑ ∂x f(F−k +θ(F+k −F−k ))((Fi)+k −(Fi)−k ) (cid:12)i=1 i (cid:12) d (cid:12) d (cid:12) (cid:12) (cid:12) ≤ √pkqkM1(f)∑|(Fi)+k −(Fi)−k | = M1(f)∑|DkFi|, i=1 i=1 whereθ (0,1).Thus,anapplicationoftheCauchy-Schwarzinequalityyieldsthat ∈ ∞ ∞ d 2 ∞ d E[ Df(F) 2 ] =E ∑(D f(F))2 (M (f))2E ∑ ∑ D F d(M (f))2E ∑ ∑(D F)2 k kℓ2(N) k ≤ 1 | k i| ≤ 1 k i hk=1 i hk=1(cid:16)i=1 (cid:17) i hk=1i=1 i d = d(M (f))2∑E[ DF 2 ] (24) 1 k ikℓ2(N) i=1 andfinitenessoftherighthandsidein(24)followsfromtheassumptionsthat,foreveryi =1,...,d, ∂ f is ∂xi boundedandF dom(D).Now,byplugging(10)into(23)weimmediatelyget i ∈ d ∂ E[Fif(F)]= ∑E ∂x f(F)hDFj,−DL−1Fiiℓ2(N) j=1 h j i d X ∂2 ∂2 −j,∑ℓ=1EhD4√pq(cid:16)∂xj∂xℓ f(F+)+ ∂xj∂xℓ f(F−)(cid:17)(DFj)(DFℓ),−DL−1FiEℓ2(N)i +E[hR1(F),−DL−1Fiiℓ2(N)] (25) withF+ :=(F+k )k N andF− :=(F−k )k NaswellasaremainderR1(F)whichby(11)fulfilstheestimate ∈ ∈ 5 1 d 3 |E[hR1(F),−DL−1Fiiℓ2(N)]| ≤ 12M3(f)E pq ∑|DFj| ,|−DL−1Fi| ℓ2(N) . (26) hD (cid:16)j=1 (cid:17) E i 9 Asaconsequence,weonlyneedtofurtherboundthesecondtermin(25). ByvirtueoftheCauchy-Schwarz inequalityand(8)weseethat,foreveryj,ℓ N, ∈ ∞ X ∂2 ∂2 Ehk∑=1(cid:12)4√pkkqk(cid:16)∂xj∂xℓ f(F+k )+ ∂xj∂xℓ f(F−k )(cid:17)(DkFj)(DkFℓ)(−DkL−1Fi)(cid:12)i (cid:12) ∞ (cid:12) 1M(cid:12)2(f)E ∑ 1 (DkFj)(DkFℓ)( DkL−1Fi) (cid:12) ≤ 2 √p q | − | hk=1 k k i ∞ ∞ 1 1 1/2 1/2 ≤ 2M2(f) E ∑ p q (DkFj)2(DkFℓ)2 E ∑(DkL−1Fi)2 (cid:16) hk=1 k k i(cid:17) (cid:16) hk=1 i(cid:17) ∞ ∞ ∞ 1 1 1/4 1 1/4 1/2 M2(f) E ∑ (DkFj)4 E ∑ (DkFℓ)4 E ∑(DkFi)2 ≤ 2 p q p q (cid:16) hk=1 k k i(cid:17) (cid:16) hk=1 k k i(cid:17) (cid:16) hk=1 i(cid:17) 1 = 2M2(f)(E[k(pq)−1/4DFjk4ℓ4(N)])1/4(E[k(pq)−1/4DFℓk4ℓ4(N)])1/4(E[kDFik2ℓ2(N)])1/2 andfinitenessofthisexpressionfollowsfromtheassumptionsthatFi ∈dom(D)andk(pq)−1/4DFikℓ2(N) < ∞foreveryi =1,...,d. Thus,anexchangeofexpectationandsummationisvalidduetotheFubini-Tonelli theorem,andthe independenceof Xk and (∂x∂j∂2xℓ f(F+k )+ ∂x∂j∂2xℓf(F−k ))(DkFj)(DkFℓ)(−DkL−1Fi), forevery k N,yieldsthat,foreveryj,ℓ N, ∈ ∈ X ∂2 ∂2 E 4√pq ∂xj∂xℓ f(F+)+ ∂xj∂xℓ f(F−) (DFj)(DFℓ),−DL−1Fi ℓ2(N) hD (cid:16) (cid:17) E i ∞ p q ∂2 ∂2 =k∑=14√k−pkqkkEh(cid:16)∂xj∂xℓ f(F+k )+ ∂xj∂xℓ f(F−k )(cid:17)(DkFj)(DkFℓ)(−DkL−1Fi)i p q ∂2 ∂2 =E 4√−pq ∂xj∂xℓ f(F+)+ ∂xj∂xℓ f(F−) (DFj)(DFℓ),−DL−1Fi ℓ2(N) . (27) hD (cid:16) (cid:17) E i Byplugging(27)into(25)wethenget d ∂ E[Fif(F)]= ∑E ∂x f(F)hDFj,−DL−1Fiiℓ2(N) +E[hR1(F),−DL−1Fiiℓ2(N)] j=1 h j i −E[hR2(F),−DL−1Fiiℓ2(N)] (28) witharemainderterm p q ∞ ∂2 ∂2 R2(F) := 4√−pq j,∑ℓ=1(cid:16)∂xj∂xℓ f(F+)+ ∂xj∂xℓ f(F−)(cid:17)(DFj)(DFℓ) satisfying 1 p q d 2 |E[hR2(F),−DL−1Fiiℓ2(N)]| ≤ 2M2(f)E |√−pq| ∑|DFj| ,|−DL−1Fi| ℓ2(N) . (29) j=1 hD (cid:16) (cid:17) E i Finally, the assertionfollows from(28) uponputting R(F) := R (F) R (F) andusingthe boundsin(26) 1 2 − and(29). Remark3.6. InthesymmetriccasewheretheunderlyingRademachersequencesatisfies p = q =1/2for k k everyk N,theboundfortheremaindertermin(22)simplifiesto ∈ 5 d 3 |E[hR(F),−DL−1Fiiℓ2(N)]| ≤ 3M3(f)E ∑|DFj| ,|−DL−1Fi| ℓ2(N) , hD(cid:16)j=1 (cid:17) E i since p q =0,foreveryk N. k− k ∈ 10