Multivariate central limit theorems for Rademacher functionals with applications

0.28 MB·

by  Kai Krokowski

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