Stein's method and normal approximation of Poisson functionals

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Stein’s method and normal approximation of Poisson

functionals

Giovanni Peccati, Josep Lluís Solé, Murad Taqqu, Frederic Utzet

To cite this version:

Giovanni Peccati, Josep Lluís Solé, Murad Taqqu, Frederic Utzet. Stein’s method and normal

approx-imation of Poisson functionals. 2008. �hal-00308627�

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by G.Pe ati

∗

,J.L. Solé

†

,M.S.Taqqu

‡

and F.Utzet

§

Thisversion: July31,2008

Abstra t: We ombineStein'smethodwithaversionofMalliavin al ulusonthePoissonspa e. As aresult,weobtainexpli itBerry-EsséenboundsinCentralLimitTheorems(CLTs)involvingmultiple Wiener-Itintegrals with respe t to a generalPoisson measure. We provideseveral appli ations to CLTsrelatedtoOrnstein-Uhlenbe kLévypro esses.

Keywords: Malliavin al ulus; Normalapproximation;Ornstein-Uhlenbe kpro ess;Poisson measure; Stein's method;Wasserstein distan e.

2000 Mathemati s Subje tClassi ation: 60F05;60G51;60G60; 60H05

1 Introdu tion

Ina re ent seriesofpapers, NourdinandPe ati[13 ,14℄,and Nourdin,Pe atiandRéveilla [15 ℄,have shownthatone anee tively ombineMalliavin al ulusonaGaussian spa e(see e.g. [16℄)andStein'smethod(seee.g. [3,26℄)inordertoobtainexpli itboundsforthenormal andnon-normal approximationof smooth fun tionals of Gaussian elds.

The aim of the present paperis to extend theanalysis initiated in [13℄ to theframework of the normal approximation (in the Wasserstein distan e) of regular fun tionals of Poisson measures dened on abstra t Borelspa es. Asinthe Gaussian ase, the main ingredients of ouranalysis arethefollowing:

1. A set of Stein dierential equations, relating thenormal approximation inthe W asser-stein distan e to rstorder dierential operators.

2. A (Hilbertspa e-valued) derivativeoperator

D

,a ting onreal-valued square-integrable random variables.

3. An integration byparts formula, involving theadjoint operator of

D

.

4. Apathwiserepresentation of

D

whi h,inthePoisson ase,involvesstandarddieren e operators.

Asaby-produ tofour analysis,weobtainsubstantial generalizationsoftheCentralLimit Theorems (CLTs)forfun tionalsofPoissonmeasures(forinstan e,forsequen esofsingleand double Wiener-It integrals) re ently proved in [21, 22 , 23℄ (see also [4, 20 ℄ for appli ations

∗

LaboratoiredeStatistiqueThéoriqueetAppliquée,UniversitéPierreetMarieCurie,175rueduChevaleret, 75013Paris,Fran e,E-mail: giovanni.pe atigmail. om

†

DepartamentdeMatemàtiques,Fa ultatdeCièn ies,UniversitatAutónomadeBar elona,08193Bellaterra (Bar elona),Spain;E-mail: jllsolemat.uab.es

‡

Boston University, Departement of Mathemati s, 111 Cummington Road, Boston (MA), USA. E-mail: muradmath.bu.edu.

§

DepartamentdeMatemàtiques,Fa ultatdeCièn ies,UniversitatAutónomadeBar elona,08193Bellaterra (Bar elona),Spain;E-mail: utzetmat.uab.es

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of thepresent paper(see Theorem 5.1below) is a CLT for sequen es of multiple Wiener-It integrals ofarbitrary(xed) orderwithrespe tto ageneral Poissonmeasure. Our onditions areexpressedinterms of ontra tion operators, and an be seenasa Poisson ounterpart to theCLTs onWienerspa eproved byNualartand Pe ati [18 ℄ andNualartandOrtiz-Latorre [17 ℄. The readeris referredto De reusefond and Savy[5℄ for other appli ations ofStein-type te hniquesandMalliavinoperatorstotheassessmentofRubinsteindistan eson onguration spa es.

The remainder of the paper is organized as follows. In Se tion 2, we dis uss some pre-liminaries, involvingmultipli ation formulae, Malliavinoperators, limit theorems and Stein's method. In Se tion 3,we derive a general inequality on erning theGaussian approximation of regular fun tionals of Poisson measures. Se tion 4 is devoted to upper bounds for the Wasserstein distan e, and Se tion 5 to CLTs for multiple Wiener-It integrals of arbitrary order. Se tion 6deals withsumsofa singleandadoubleintegral. InSe tion 7,weapplyour results to non-linear fun tionals ofOrnstein-Uhlenbe kLévypro esses.

2 Preliminaries 2.1 Poisson measures

Throughoutthe paper,

(Z, Z, µ)

indi atesameasurespa esu hthat

Z

isaBorelspa eand

µ

isa

σ

-nitenon-atomi Borelmeasure. Wedenethe lass

Z

µ

as

Z

µ

= {B ∈ Z : µ (B) < ∞}

. The symbol

N = { b

b

N (B) : B ∈ Z

µ

}

indi ates a ompensated Poisson random measure on

(Z, Z)

with ontrol

µ.

This means that

N

b

is a olle tion of random variables dened on someprobability spa e

(Ω, F, P)

,indexedbytheelementsof

Z

µ

,and su hthat: (i) for every

B, C ∈ Z

µ

su hthat

B ∩ C = ∅

,

N (B)

b

and

N (C)

b

areindependent, (ii)for every

B ∈ Z

µ

,

b

N (B)

law

= P (B) − µ (B)

,

where

P

(B)

isaPoissonrandomvariablewithparameter

µ (B)

. Notethatproperties (i)-(ii) imply, in parti ular, that

N

b

is an independently s attered (or ompletely random) measure (see e.g. [23 ℄).

Remark 2.1 A ording e.g. to [25 , Se tion 1℄, and due to the assumptions on the spa e

(Z, Z, µ)

,it isalways possible to dene

(Ω, F, P)

insu h awaythat

Ω =







ω =

n

X

j=0

δ

z

j

, n ∈ N ∪ {∞}, z

j

∈ Z







(2.1)

where

δ

z

denotes theDira massat

z

,and

N

b

isthe ompensated anoni almapping given by

b

N (B)(ω) = ω(B) − µ(B), B ∈ Z

µ

.

(2.2)

Also, in this ase one an take

F

to be the

P

- ompletion of the

σ

-eld generated by the mapping(2.2). Notethat, undertheseassumptions, onehasthat

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the stru tureof

Ω

and

N

b

,are impli itlysatised. In parti ular,

F

isthe

P

- ompletion ofthe

σ

-eld generated bythe mapping in(2.2).

Foreverydeterministi fun tion

h ∈ L

2 _{(Z, Z, µ) = L}

2 _(µ)

wewrite

N (h) =

b

R

Z

h (z) b

N (dz)

to indi ate the Wiener-It integral of

h

with respe t to

N

b

(see e.g. [9℄ or [27℄). We re all that, for every

h ∈ L

2 _(µ)

, the random variable

N (h)

b

has an innitely divisible law, with Lévy-Khin hine exponent (seeagain [27 ℄)givenby

ψ (h, λ) = log E

h

e

iλ b

N (h)

i

=

Z

h

e

iλh(z)

_{− 1 − iλh (z)}

i

µ(dz),

_{λ ∈ R.}

(2.4)

Re allalsotheisometri relation: forevery

g, h ∈ L

2 _(µ)

,

E[ b

N (g) b

N (h)] =

R

Z

h (z) g (z) µ (dz)

. Fix

n > 2.

We denote by

L

2 _(µ

n

₎

the spa e of real valued fun tions on

Z

n

that are square-integrablewithrespe tto

µ

n

,andwewrite

L

2 s

(µ

n

)

toindi atethesubspa eof

L

2 _(µ

n

₎

omposed of symmetri fun tions. For every

f ∈ L

2 s

(µ

n

)

, we denote by

I

n

(f )

the multiple Wiener-It integral of order

n

, of

f

with respe t to

N

b

. Observe that, for every

m, n > 2

,

f ∈ L

2 s

(µ

n

)

and

g ∈ L

2 s

(µ

m

)

,one hasthe isometri formula(see e.g. [31℄):

EI

n

(f )I

m

(g)

= n!hf, gi

L

2 _(µ

n

₎

1 _(n=m)

.

(2.5)

Now x

n > 2

and

f ∈ L

2 _(µ

n

₎

(not ne essarily symmetri ) and denote by

f

˜

the anoni al symmetrization of

f

: for futureusewe stress that, asa onsequen e of Jenseninequality,

k ˜

f k

L

2 _(µ

n

₎

6 kfk

_L

2 _(µ

n

₎

.

(2.6)

TheHilbertspa eofrandomvariablesofthetype

I

n

(f )

,where

n > 1

and

f ∈ L

2 s

(µ

n

)

is alled the

n

th Wiener haos asso iated with

N

b

. We also use the following onventional notation:

I

1 (f ) = b

N (f )

,

f ∈ L

2 _(µ)

;

I

n

(f ) = I

n

( ˜

f )

,

f ∈ L

2 _(µ

n

₎

,

n > 2

(this onvention extends the denition of

I

n

(f )

to non-symmetri fun tions

f

);

I

0 (c) = c

,

c ∈ R

. The following proposition, whose ontentisknownasthe haoti representation property of

N

b

,isoneofthe ru ial results usedinthis paper. Seee.g. [19 ℄ or [32 ℄for a proof.

Proposition 2.3 (Chaoti de omposition) Everyrandomvariable

F ∈ L

2 _{(F, P) = L}

2 _(P)

admitsa (unique) haoti de omposition of the type

F = E(F ) +

∞

X

n>1

I

n

(f

n

),

(2.7)

where the series onverges in

L

2

and,for ea h

n > 1

, the kernel

f

n

isan element of

L

2 s

(µ

n

)

.

2.2 Contra tions, stars and produ ts

Wenowre allausefulversionofthemultipli ationformula formultiplePoissonintegrals. To this end, we dene, for

q, p > 1

,

f ∈ L

2 s

(µ

p

)

,

g ∈ L

2 s

(µ

q

)

,

r = 0, ..., q ∧ p

and

l = 1, ..., r

, the ( ontra tion) kernel on

Z

p+q−r−l

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f g

from

p + q

to

p + q − r − l

as follows:

r

variables are identied and, among these,

l

are integrated out. This ontra tion kernel isformallydened asfollows:

f ⋆

l

_r

g(γ

1 , . . . , γ

r−l

, t

1 , . . . , t

p−r

, s

1 , . . . , s

q−r

)

(2.8)

=

Z

l

f (z

1 , . . . , z

l

, γ

1 , . . . , γ

r−l

, t

1 , . . . , t

p−r

)g(z

1 , . . . , z

l

, γ

1 , . . . , γ

r−l

, s

1 , . . . , s

q−r

)µ

l

(dz

1 ...dz

l

)

, and,for

l = 0

,

f ⋆

0 _r

g(γ

1 , . . . , γ

r

, t

1 , . . . , t

p−r

, s

1 , . . . , s

q−r

) = f (γ

1 , . . . , γ

r

, t

1 , . . . , t

p−r

)g(γ

1 , . . . , γ

r

, s

1 , . . . , s

q−r

),

(2.9) sothat

f ⋆

0

0 g(t

1 , . . . , t

p

, s

1 , . . . , s

q

) = f (t

1 , . . . , t

p

)g(s

1 , . . . , s

q

)

. For example,if

p = q = 2

,

f ⋆

0 ₁

g (γ, t, s) = f (γ, t) g (γ, s)

,

f ⋆

1

1 g (t, s) =

Z

f (z, t) g (z, s) µ (dz)

(2.10)

f ⋆

1 ₂

g (γ) =

Z

f (z, γ) g (z, γ) µ (dz)

(2.11)

f ⋆

2 ₂

g =

Z

f (z

1 , z

2 ) g (z

1 , z

2 ) µ (dz

1 ) µ (dz

2 ) .

(2.12)

Thequite suggestive `star-type'notation isstandard, and hasbeen rstused by Kabanovin [8℄ (butseealsoSurgailis [31℄). Plainly,forsome hoi eof

f, g, r, l

the ontra tion

f ⋆

l

r

g

may not exist,in the sense that its denition involves integrals thatare not well-dened. Onthe positiveside,the ontra tions ofthe following three typesarewell-dened (although possibly innite)for every

q > 2

and every kernel

f ∈ L

2 s

(µ

q

)

: (a)

f ⋆

0 r

f (z

1 , ...., z

2q−r

)

,where

r = 0, ...., q

,asobtained from(2.9),bysetting

g = f

; (b)

f ⋆

l

q

f (z

1 , ..., z

q−l

) =

R

_Z

l

f

2 (z

1 , ..., z

q−l

, ·)dµ

l

,forevery

l = 1, ..., q

; ( )

f ⋆

r

f

,for

r = 1, ...., q − 1

.

Inparti ular, a ontra tion of the type

f ⋆

l

q

f

,where

l = 1, ..., q − 1

mayequal

+∞

at some point

(z

1 , ..., z

q−l

)

. Thefollowing (elementary)statement ensuresthatanykernel of thetype

f ⋆

r

g

is square-integrable.

Lemma 2.4 Let

p, q > 1

, and let

f ∈ L

2 s

(µ

q

)

and

g ∈ L

2 s

(µ

p

)

. Fix

r = 0, ..., q ∧ p

. Then,

f ⋆

r

_r

_{g ∈ L}

2 (µ

p+q−2r

)

.

Proof. Just use equation (2.8) in the ase

l = r

, and dedu e the on lusion by a standard useoftheCau hy-S hwarz inequality.

2

We also re ord the following identity (whi h is easily veried bya standard Fubini argu-ment), valid for every

q > 1

,every

p = 1, ..., q

:

Z

2q−p

(f ⋆

0 _p

f )

2 dµ

2q−p

=

Z

p

(f ⋆

q−p

_q

f )

2 dµ

p

,

(2.13) for every

f ∈ L

2 s

(µ

q

)

. The forth oming produ tformulafor two Poissonmultiple integrals is proved e.g. in[8℄ and[31 ℄.

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Proposition 2.5 (Produ t formula) Let

f ∈ L

2 s

(µ

p

)

and

g ∈ L

2 s

(µ

q

)

,

p, q > 1

, and sup-pose moreover that

f ⋆

l

r

g ∈ L

2 (µ

p+q−r−l

)

for every

r = 0, ..., p ∧ q

and

l = 1, ..., r

su h that

l 6= r

. Then,

I

p

(f )I

q

(g) =

p∧q

X

r=0

r!

p

r

q

r

_X

r

l=0

r

l

I

q+p−r−l

(

f ⋆

^

l

r

g),

(2.14)

where the tilde

e

stands for symmetrization,that is,

^

f ⋆

l

r

g (x

1 , ..., x

q+p−r−l

) =

1 (q + p − r − l)!

X

σ

f ⋆

l

_r

g x

_σ(1)

, ..., x

_{σ(q+p−r−l)}

,

where

σ

runs over all

(q + p − r − l)!

permutations of the set

{1, ..., q + p − r − l}

.

Remark2.6 (Multiple integrals and Lévy pro esses). In the multiple Wiener-It integrals introdu ed in this se tion, the integrators are ompensated Poisson measures of the type

b

N (dz)

,denedon someabstra tBorelspa e. Itiswell-knownthatone analso buildsimilar obje ts inthe framework ofLévypro esses indexedbythe realline. Suppose indeedthatwe are given a adlagLévypro ess

X = {X

t

, t ≥ 0}

(that is,

X

hasstationary andindependent in rements,

X

is ontinuous in probability and

X

0 = 0

), dened on a omplete probability spa e

(Ω, F, P )

,andwith Lévytriplet

(γ, σ

2 _{, ν)}

where

γ ∈ R, σ ≥ 0

and

ν

is aLévymeasure on

R

(see e.g. [27 ℄). Thepro ess

X

admits aLévy-It representation

X

t

= γt + σW

t

+

Z Z

(0,t]×{|x|>1}

x dN (s, x) + lim

ε↓0

Z Z

(0,t]×{ε<|x|≤1}

x d e

N (s, x),

(2.15)

where: (i)

{W

t

, t ≥ 0}

isa standard Brownian motion, (ii)

N (B) = #{t : (t, ∆X

t

) ∈ B},

B ∈ B((0, ∞) × R

0 ),

is the jump measure asso iated with

X

(where

R

0 = R − {0}

,

∆X

t

= X

t

− X

t−

and

#A

denotesthe ardinal ofa set

A

),and(iii)

d e

_{N (t, x) = dN (t, x) − dt dν(x)}

isthe ompensated jump measure. The onvergen e in(2.15) isa.s. uniform (in

t

) on every bounded interval. Following It [7 ℄, the pro ess

X

an be extended to an independently s atteredrandommeasure

M

on

(R

+

× R, B(R

+

× R))

asfollows. First, onsiderthemeasure

dµ

∗

(t, x) = σ

2 dt dδ

0 (x) + x

2 dt dν(x),

where

δ

0

is theDira measureat point

0

,and

dt

isthe Lebesgue measureon

R

. This meansthat, for

E ∈ B(R

+

× R)

,

µ

∗

(E) = σ

2 Z

E(0)

dt +

Z Z

E

′

x

2 dt dν(x),

where

E(0) = {t ∈ R

+

: (t, 0) ∈ E}

and

E

′

_{= E − {(t, 0) ∈ E}}

; this measure is ontinuous (see It [7 ℄,p. 256). Now, for

E ∈ B(R

+

× R)

with

µ

∗

_{(E) < ∞,}

dene

M (E) = σ

Z

E(0)

dW

t

+ lim

n

ZZ

{(t,x)∈E:1/n<|x|<n}

x d e

N (t, x),

(7)

(with onvergen ein

L

2 _(Ω)

),thatis,

M

isa enteredindependentlys atteredrandommeasure su hthat

E

M (E

1 )M (E

2 )] = µ

∗

(E

1 ∩ E

2 ),

for

E

1 , E

2 ∈ B(R

+

× R)

with

µ

∗

_(E

1 ) < ∞

and

µ

∗

_(E

2 ) < ∞

. Now write

L

2 n

= L

2 ((R

+

×

R)

n

_{, B(R}

+

× R)

n

, µ

∗⊗n

).

For every

f ∈ L

2 n

,one an nowdene a multiple sto hasti integral

I

n

(f )

, with integrator

M

, by using the same pro edure as in the Poisson ase (the rst onstru tion ofthistypeisduetoItsee[7℄). Sin e

M

isdenedonaprodu tspa e, inthis s enario we an separate the time and thejump sizes. If theLévy pro ess has no Brownian omponents, then

M

has the representation

M (dz) = M (ds, dx) = x e

N (ds, dx)

, that is,

M

is obtained by integrating a fa tor

x

with respe t to the underlying ompensated Poisson measure ofintensity

dt ν(dx)

. Therefore,inorder todene multipleintegrals of order

n

with respe tto

M

,oneshould onsiderkernelsthataresquare-integrablewithrespe tsthemeasure

(x

2 _{dt ν(dx))}

⊗n

. The produ tformula ofmultiple integrals inthis dierent ontext wouldbe analogoustothe onegiveninProposition2.5,but inthedenitions ofthe ontra tionkernels one hasto to take into a ount thesupplementary fa tor

x

(see e.g. [10℄ for more details on thispoint).

2.3 Four Malliavin-type operators

In this se tion, we introdu e four operators of the Malliavin-type, that are involved in the estimatesofthesubsequentse tions. Ea hoftheseoperatorsisdenedintermsofthe haoti expansions of the elements inits domain, thus impli itly exploiting thefa t that the haoti representationproperty(2.7)indu esanisomorphism between

L

2 _(P)

andthesymmetri Fo k spa e anoni ally asso iated with

L

2 _(µ)

. The reader is referred to [19℄ or [28℄ for more details on the onstru tion of these operators, as well as for further relations with haoti expansionsandFo kspa es. One ru ialfa tinour analysisisthatthederivativeoperator

D

(tobeformallyintrodu ed below)admitsaneat hara terizationintermsofausualdieren e operator (see the forth oming Lemma 2.7).

Forlater referen e,we re allthatthespa e

L

2 _{(P; L}

2 _{(µ)) ≃ L}

2 _{(Ω × Z, F ⊗ Z, P ⊗ µ)}

isthe spa eof themeasurable randomfun tions

u : Ω × Z → R

su h that

E

Z

u

2 _z

µ(dz)

< ∞.

Inwhatfollows,given

f ∈ L

2 s

(µ

q

)

(

q > 2

) and

z ∈ Z

,wewrite

f (z, ·)

to indi atethefun tion on

Z

q−1

given by

(z

1 , ..., z

q−1

) → f(z, z

1 , ..., z

q−1

)

.

i) Thederivative operator

D

. The derivative operator, denoted by

D

, transforms random variablesintorandomfun tions. Formally,thedomainof

D

,written

domD

,isthesetofthose random variables

F ∈ L

2 _(P)

admitting a haoti de omposition (2.7) su h that

X

n>1

nn!kf

n

k

2 _L

2 _(µ

n

₎

< ∞.

(2.16)

If

F

veries(2.16) (that is, if

F ∈ domD

),thentherandomfun tion

z → D

z

F

isgiven by

D

z

F =

X

n>1

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For instan e, if

F = I

1 (f )

, then

D

z

F

is the non-random fun tion

z → f(z)

. If

F = I

2 (f )

, then

D

z

F

is the randomfun tion

z → 2I

1 (f (z, ·)).

(2.18)

By exploiting the isometri properties of multiple integrals, and thanks to (2.16), one sees immediately that

DF ∈ L

2 _{(P; L}

2 _(µ))

, for every

F ∈ domD

. Now re all that, in this paper, the underlyingprobabilityspa e

(Ω, F, P)

issu h that

Ω

isthe olle tionofdis rete measures givenby(2.1). Fix

z ∈ Z

;givenarandomvariable

F : Ω → R

,we dene

F

z

tobetherandom variable obtained by adding the Dira mass

δ

z

to the argument of

F

. Formally, for every

ω ∈ Ω

,we set

F

z

(ω) = F (ω + δ

z

).

(2.19)

Aversionofthe following result,whose ontent isoneofthestaplesofour analysis,isproved e.g. in[19, Theorem6.2℄ (seealso [28, Proposition5.1℄). One shouldalso notethattheproof given in[19℄ appliestotheframeworkof aRadon ontrolmeasure

µ

: however, thearguments extend immediately to the ase of a Borel ontrol measure onsidered in the present paper, and wedo not reprodu e them here(see, however, theremarksbelow). It relates

F

z

to

D

z

F

andprovidesa representation of

D

asa dieren e operator.

Lemma 2.7 For every

F ∈ domD

,

D

z

F = F

z

− F, a.e.−µ(dz).

(2.20)

Remark2.8 1. IntheIt-typeframeworkdetailed inRemark2.6, we ouldalternatively usethe denitionoftheMalliavinderivative introdu ed in[28℄, thatis,

D

_(t,x)

F =

F

(t,x)

− F

x

, a.e.−x

2 _dtν(dx).

F = I

1 (f )

,formula(2.20)isjusta onsequen eofthe straight-forward relation

F

z

− F =

Z

f (x)( b

N (dx) + δ

z

(dx)) −

Z

f (x) b

N (dx) =

Z

f (x)δ

z

(dx) = f (z).

3. To have an intuition of theproof of (2.20) in the general ase, onsider for instan e a random variableof the type

F = b

N (A) × b

N (B)

,where thesets

A, B ∈ Z

µ

aredisjoint. Then, onehas that

F = I

2 (f )

,where

f (x, y) = 2

−1

_{1

A

(x)1

B

(y) + 1

A

(y)1

B

(x)}

. Using (2.18), we havethat

D

z

F = b

N (B)1

A

(z) + b

N (A)1

B

(z), z ∈ Z,

(2.21)

and

(9)

Nowx

z ∈ Z

. There arethreepossible ases: (i)

z /

∈ A

and

z /

∈ B

,(ii)

z ∈ A

,and (iii)

z ∈ B

. If

z

isasin(i),then

F

z

= F

. If

z

is asin(ii)(resp. asin(iii)),then

F

z

− F = { b

N + δ

z

}(A) × b

N (B) − b

N (A) b

N (B) = b

N (B)

(resp.

F

z

− F = b

N (A) × { b

N + δ

z

}(B) − b

N (A) b

N (B) = b

N (A) ).

Asa onsequen e,onededu esthat

F

z

− F = b

N (B)1

A

(z) + b

N (A)1

B

(z)

,sothatrelation (2.20) is obtained from(2.21).

4. Observe also that Lemma 2.7 yields that, if

F, G ∈ domD

aresu h that

F G ∈ domD

, then

D(F G) = F DG + GDF + DGDF,

(2.22)

(see [19, Lemma 6.1℄ fora detailed proof of thisfa t).

ii) TheSkorohodintegral

δ

. Observe that, due to the haoti representation property of

N

b

, every randomfun tion

u ∈ L

2 _{(P, L}

2 _(µ))

admitsa (unique)representation ofthetype

u

z

=

X

n>0

I

n

(f

n

(z, ·)), z ∈ Z,

(2.23)

where, for every

z

, the kernel

f

n

(z, ·)

is an element of

L

2 s

(µ

n

)

. The domain of the Skorohod integral operator, denoted by

domδ

, is dened as the olle tions of those

u ∈ L

2 _{(P, L}

2 _(µ))

su hthat the haoti expansion (2.23) veriesthe ondition

X

n>0

(n + 1)!kf

n

k

2 _L

2 _(µ

n+1

₎

< ∞.

(2.24)

If

u ∈ domδ

,thenthe random variable

δ(u)

isdened as

δ(u) =

X

n>0

I

n+1

( ˜

f

n

),

(2.25)

where

f

˜

n

stands for the anoni al symmetrization of

f

(as a fun tion in

n + 1

variables). For instan e, if

u(z) = f (z)

is a deterministi element of

L

2 _(µ)

, then

δ(u) = I

1 (f )

. If

u(z) = I

1 (f (z, ·))

, with

f ∈ L

2 s

(µ

2 )

,then

δ(u) = I

2 (f )

(we stress thatwe have assumed

f

to besymmetri ). Thefollowing lassi result, proved e.g. in[19℄,providesa hara terization of

δ

asthe adjoint of thederivative

D

.

Lemma 2.9 (Integration by parts formula) For every

G ∈ domD

and every

u ∈ domδ

, onehas that

E[Gδ(u)] = E[hDG, ui

L

2 _(µ)

],

(2.26) where

hDG, ui

L

2 _(µ)

=

Z

D

z

G × u(z)µ(dz).

(10)

iii) TheOrnstein-Uhlenbe k generator

L

. The domain of the Ornstein-Uhlenbe k generator (see [16 , Chapter 1℄), written

domL

, is given by those

F ∈ L

2 _(P)

su h that their haoti expansion (2.7) veries

X

n>1

n

2 _n!kf

n

k

2 L

2 _(µ

n

₎

< ∞.

If

F ∈ domL

,then the randomvariable

LF

isgiven by

LF = −

X

n>1

nI

n

(f

n

).

(2.27)

Note that

E(LF ) = 0

, by denition. The following result is a dire t onsequen e of the denitions of

D

,

δ

and

L

.

Lemma 2.10 For every

F ∈ domL

, one has that

F ∈ domD

and

DF ∈ domδ

. Moreover,

δDF = −LF.

(2.28)

Proof. The rst part of the statement is easily proved by applying thedenitions of

domD

and

domδ

given above. In view of Proposition 2.3, it is now enough to prove (2.28) for a random variable of the type

F = I

q

(f )

,

q > 1

. In this ase, one has immediately that

D

z

F = qI

q−1

(f (z, ·))

,sothat

δDF = qI

q

(f ) = −LF

.

2

iv)The inverse of

L

. Thedomain of

L

−1

,denotedby

L

2

0 (P)

,is thespa eof entered random variables in

L

2 _(P)

. If

F ∈ L

2

0 (P)

and

F =

P

n>1

I

n

(f

n

)

(and thus is entered) then

L

−1

_{F = −}

X

n>1

1 n

I

n

(f

n

).

(2.29)

2.4 Normalapproximationinthe Wassersteindistan e, viaStein'smethod Weshallnowgivea shorta ount of Stein'smethod,asapplied to normalapproximationsin the Wasserstein distan e. We denote by

Lip(1)

the lass of real-valued Lips hitz fun tions, from

R

to

R

,withLips hitz onstantlessorequaltoone,thatis,fun tions

h

thatareabsolutely ontinuousandsatisfytherelation

kh

′

_k

∞

6

1

. Giventworeal-valuedrandomvariables

U

and

Y

,theWasserstein distan e between the lawsof

U

and

Y

,written

d

W

(U, Y )

isdened as

d

W

(U, Y ) =

sup

h∈Lip(1)

|E[h(U)] − E[h(Y )]|.

(2.30)

We re all that the topology indu ed by

d

W

on the lass of probability measures over

R

is stri tly stronger than the topology of weak onvergen e (see e.g. [6℄). We will denote by

N

_{(0, 1)}

thelawof a entered standard Gaussian randomvariable.

Nowlet

X ∼ N (0, 1)

. Considerareal-valuedfun tion

h : R → R

su hthattheexpe tation

E[h(X)]

iswell-dened. TheStein equationasso iated with

h

and

X

is lassi allygiven by

h(x) − E[h(X)] = f

′

(x) − xf(x),

x ∈ R.

(2.31)

Asolution to (2.31)is afun tion

f

dependingon

h

whi h isLebesgue a.e.-dierentiable, and su h that there exists a version of

f

′

verifying (2.31) for every

x ∈ R

. The following result isbasi ally due to Stein [29, 30℄. The proofof point (i) (whose ontent isusually referredas Stein's lemma)involvesastandarduseoftheFubinitheorem(see e.g. [3 ,Lemma 2.1℄). Point (ii) isproved e.g. in [2,Lemma 4.3℄.

(11)

Lemma 2.11 (i) Let

W

be a random variable. Then,

W

Law

= X ∼ N (0, 1)

if, andonlyif,

E[f

′

_{(W ) − W f (W )] = 0,}

(2.32) for every ontinuous and pie ewise ontinuously dierentiable fun tion

f

verifying the relation

E|f

′

_{(X)| < ∞}

.

(ii) If

h

isabsolutely ontinuouswithbounded derivative,then(2.31)has asolution

f

h

whi h is twi e dierentiable andsu h that

kf

′

h

k

∞

6 kh

′

k

∞

and

kf

′′

h

k

∞

6 2kh

′

k

∞

.

Let

F

W

denotethe lassoftwi edierentiablefun tions,whoserstderivativeisbounded by 1 and whose se ond derivative is bounded by2. If

h ∈ Lip(1)

and thus

kh

′

_k

∞

6

1

,then the solution

f

h

,appearing inLemma 2.11(ii), satises

kf

′

h

k

∞

6

1

and

kf

′′

h

k

∞

6

2

, andhen e

f

h

∈ F

W

.

Now onsidera Gaussian random variable

X ∼ N (0, 1)

, and let

Y

be a generi random variablesu h that

EY

2 _{< ∞}

. Byintegrating both sidesof(2.31)withrespe ttothelawof

Y

andbyusing(2.30), one seesimmediately thatPoint (ii)of Lemma 2.11impliesthat

d

W

(Y, X) 6 sup

f ∈F

W

|E(f

′

(Y ) − Y f(Y ))|.

(2.33)

Note that the square-integrability of

Y

implies that the quantity

E[Y f(Y )]

is well dened (re all that

f

isLips hitz). In the subsequent se tions, we will show thatone an ee tively bound thequantityappearing on the RHS of(2.33) bymeans of theoperators introdu ed in Se tion 2.3.

3 A general inequality involving Poisson fun tionals

The following estimate, involvingthenormal approximationof smooth fun tionals of

N

b

,will be ru ialfor the rest ofthepaper. Weusethenotation introdu ed intheprevious se tion. Theorem 3.1 Let

F ∈ domD

be su h that

E(F ) = 0

. Let

X ∼ N (0, 1)

. Then,

d

W

(F, X) 6 E

|1 − hDF, −DL

−1

F i

L

2 _(µ)

|

+

Z

E |D

z

F |

2 _|D

z

L

−1

F |

µ(dz)

(3.1)

6 q

E (1 − hDF, −DL

−1

_{F i}

L

2 _(µ)

)

2 +

Z

E |D

z

F |

2 _|D

z

L

−1

F |

µ(dz),

(3.2)

where we used the standard notation

hDF, −DL

−1

F i

L

2 _(µ)

= −

Z

[D

z

F × D

z

L

−1

F ] µ(dz).

Moreover, if

F

has the form

F = I

q

(f )

, where

q > 1

and

f ∈ L

2 s

(µ

q

)

, then

hDF, −DL

−1

F i

L

2 _(µ)

= q

−1

kDF k

2 L

2 _(µ)

(3.3)

Z

E |D

z

F |

2 |D

z

L

−1

F |

µ(dz) = q

−1

Z

E |D

z

F |

3 _µ(dz).

(3.4)

(12)

fun tion

f

su h that

kf

′

_k

∞

6

1

and

kf

′′

_k

∞

6

2

(that is, for every

f ∈ F

W

), the quantity

|E[f

′

_{(F )−F f (F )]|}

issmallerthantheRHSof(3.2). Toseethis,x

f ∈ F

W

andobservethat, by using (2.19), for every

ω

and every

z

one hasthat

D

z

f (F )(ω) = f (F )

z

(ω) − f(F )(ω) =

f (F

z

)(ω) − f(F )(ω)

. Nowusetwi eLemma 2.7, ombinedwithastandard Taylorexpansion, and write

D

z

f (F ) = f (F )

z

− f(F ) = f(F

z

) − f(F )

(3.5)

= f

′

(F )(F

z

− F ) + R(F

z

− F ) = f

′

(F )(D

z

F ) + R(D

z

F ),

where (dueto the fa t that

kf

′′

_k

∞

6

2

) the mapping

y → R(y)

is su h that

|R(y)| 6 y

2

for every

y ∈ R

. We an therefore apply (in order) Lemma 2.10 and Lemma 2.9 (in the ase

u = DL

−1

F

and

G = f (F )

)and infer that

E[F f (F )] = E[LL

−1

F f (F )] = E[−δ(DL

−1

F )f (F )] = E[hDf(F ), −DL

−1

F i

L

2 _(µ)

].

A ording to (3.5), onehasthat

E[hDf(F ), −DL

−1

F i

L

2 _(µ)

] = E[f

′

(F )hDF, −DL

−1

F i

_L

2 _(µ)

] + E[hR(DF ), −DL

−1

F i

_L

2 _(µ)

].

Itfollows that

|E[f

′

(F ) − F f (F )]| 6 |E[f

′

(F )(1 − hDF, −DL

−1

F i

L

2 _(µ)

)]| + |E[hR(DF ), −DL

−1

F i

_L

2 _(µ)

]|.

Bythefa tthat

kf

′

_k

∞

6

1

andbyCau hy-S hwarz,

|E[f

′

(F )(1 − hDF, −DL

−1

F i

L

2 _(µ)

)]| 6 E[|1 − hDF, −DL

−1

F i

_L

2 _(µ)

|]

6 q

E (1 − hDF, −DL

−1

_{F i}

L

2 _(µ)

)

2 .

Ontheotherhand,one sees immediately that

|E[hR(DF ), −DL

−1

F i

L

2 _(µ)

]| 6

Z

E |R(D

z

F )D

z

L

−1

F |

µ(dz)

6 Z

Z

E |D

z

F |

2 |D

z

L

−1

F |

µ(dz),

Relations(3.3) and(3.4) areimmediate onsequen esofthedenitionof

L

−1

givenin(2.29). Theproofis omplete.

2

Remark3.2 Notethat,ingeneral,theboundsinformulae(3.1)(3.2) anbe innite. Inthe forth omingSe tions47,wewillexhibit several examplesofrandom variablesin

domD

su h thattheboundsinthestatement of Theorem3.1are nite.

Remark 3.3 Let

G

be anisonormalGaussian pro ess(see [16 ℄)oversome separableHilbert spa e

H

,and supposethat

F ∈ L

2 _(σ(G))

is entered and dierentiable inthesense of Malli-avin. Then, the Malliavin derivative of

F

, noted

DF

, isa

H

-valued random element, and in [13 ℄itisproved thatone hasthe following upperboundon theWassersteindistan e between the lawof

F

andthe law of

X ∼ N (0, 1)

:

d

W

(F, X) 6 E|1 − hDF, −DL

−1

F i

H

|,

where

L

−1

(13)

Now x

h ∈ L

2 _(µ)

. It is lear that the random variable

I

1 (h) = b

N (h)

is an element of

domD

. One has that

D

z

N (h) = h(z)

b

; moreover

−L

−1

_{N (h) = b}

_b

_{N (h)}

. Thanks to these relations, one dedu es immediately from Theorem 3.1 the following renement of Part A of Theorem3 in[23 ℄.

Corollary 3.4 Let

h ∈ L

2 _(µ)

andlet

X ∼ N (0, 1)

. Then,the followingbound holds:

d

W

( b

N (h), X) 6

1 − khk

2 L

2 _(µ)

+

Z

|h(z)|

3 _µ(dz).

(3.6)

Asa onsequen e, if

µ(Z) = ∞

andif a sequen e

{h

k

} ⊂ L

2 _{(µ) ∩ L}

3 _(µ)

veries, as

k → ∞

,

kh

k

L

2 _(µ)

→ 1

and

kh

k

L

3 _(µ)

→ 0,

(3.7)

onehas the CLT

b

N (h

k

)

−→ X,

law

(3.8)

andthe inequality(3.6) provides an expli it upper bound in the Wasserstein distan e.

In the following se tion, we will use Theorem 3.1 in order to prove general bounds for the normalapproximation ofmultiple integralsof arbitraryorder. Asa preparation, we now present two simple appli ationsof Corollary3.4.

Example 3.5 Consider a entered Poisson measure

N

b

on

Z = R

+

, with ontrol measure

µ

equal to the Lebesgue measure. Then, the random variable

k

−1/2

_{N ([0, k]) = b}

_b

_{N (h}

k

)

, where

h

k

= k

−1/2

1 [0,k]

,isan elementof therstWiener haosasso iatedwith

N

b

. Plainly,sin ethe random variables

N ((i − 1, i])

b

(

i = 1, ..., k

) are i.i.d. entered Poisson withunitary varian e, astandard appli ation of theCentral Limit Theorem yields that, as

k → ∞

,

N (h

b

k

)

law

→ X ∼

N

_{(0, 1)}

. Moreover,

N (h

b

k

) ∈ domD

for every

k

,and

D b

N(h

k

) = h

k

. Sin e

kh

k

2 L

2 _(µ)

= 1

and

Z

|h

k

|

3 _{µ(dz) =}

1 k

1/2

,

onededu es fromCorollary 3.4that

d

W

( b

N (h

k

), X) 6

1 k

1/2

,

whi h is onsistent withthe usual Berry-Esséenestimates.

2

Example 3.6 Fix

λ > 0

. We onsiderthe Ornstein-Uhlenbe k Lévypro ess given by

Y

_t

λ

=

√

2λ

Z

t

−∞

Z

R

u exp (−λ (t − x)) b

N (du, dx)

,

t > 0

, (3.9)

where

N

b

isa enteredPoissonmeasureover

R × R

,with ontrolmeasuregivenby

µ(du, dx) =

ν (du) dx

,where

ν (·)

ispositive,non-atomi andnormalized insu hawaythat

R

u

2 _{dν = 1}

.

(14)

We assume also that

R

|u|

3 dν < ∞.

Note that

Y

λ

t

is a stationary moving average Lévy pro ess. A ording to [23 , Theorem 4℄,on hasthat, as

T → ∞

,

A

λ

_T

=

p

1 2T /λ

Z

T

0 Y

_t

λ

dt

law

_{→ X ∼ N (0, 1) .}

We shallshowthat Corollary 3.4impliestheexisten e ofa nite onstant

q

λ

> 0

,depending uniquely on

λ

,su hthat, for every

T > 0

d

W

(A

λ

T

, X) 6

q

λ

T

1/2

.

(3.10)

Tosee this,rstuseaFubinitheoreminorderto represent

A

λ

T

asanintegral withrespe t to

b

N

,that is, write

A

λ

_T

=

Z

T

−∞

Z

R

u

"

2λ

2T /λ

1

2 Z

T

x∨0

exp (−λ (t − x)) dt

#

b

N (du, dx) := b

N (h

λ

_T

).

Clearly,

A

λ

T

∈ domD

and

DA

λ

T

= h

λ

T

. By using the omputations ontained in [23 , Proof of Theorem 4℄, one sees immediately that there exists a onstant

β

λ

, depending uniquely on

λ

andsu h that

Z

R×R

|h

λ

T

(u, x)|

3 ν(du)dx 6

β

λ

T

1/2

.

Sin eonehasalsothat

|kh

λ

T

k

2 L

2 _(µ)

− 1| = O(1/T )

,theestimate(3.10) isimmediatelydedu ed from Corollary 3.4.

2

4 Bounds on the Wasserstein distan e for multiple integrals of arbitrary order

Inthisse tionweestablishgeneralupperboundsformultipleWiener-Itintegralsofarbitrary order

q > 2

, with respe t to the ompensated Poisson measure

N

b

. Our te hniques hinge on theforth omingTheorem4.2, whi husestheprodu tformula(2.14)and theinequality(3.2).

4.1 The operators

G

q

p

and

G

b

q

p

Fix

q > 2

andlet

f ∈ L

2 s

(µ

q

)

. Theoperator

G

q

p

transformsthefun tion

f

,of

q

variables,into a fun tion

G

q

p

f

of

p

variables,where

p

an be aslargeas

2q

. When

p = 0

,we set

G

q

₀

_{f = q!kfk}

2 _L

2 _(µ

q

₎

and,for every

p = 1, ..., 2q

,we dene thefun tion

(z

1 , ..., z

p

) → G

q

p

f (z

1 , ..., z

p

)

, from

Z

p

into

R

,asfollows:

G

q

_p

f (z

1 , ..., z

p

) =

q

X

r=0

r

X

l=0

1 {2q−r−l=p}

r!

q

r

2 r

l

^

f ⋆

l

r

f (z

1 , ..., z

p

),

(4.1)

wherethe `star' ontra tions have been dened informulae (2.8) and(2.9), and thetilde

e

indi ates a symmetrization. Thenotation (4.1) is mainly introdu ed inorder to give a more

(15)

ompa t representation of the RHS of (2.14) when

g = f

. Indeed, supposethat

f ∈ L

2 s

(µ

q

)

(

q > 2

), and that

f ⋆

l

r

f ∈ L

2 (µ

2q−r−l

)

for every

r = 0, ..., q

and

l = 1, ..., r

su h that

l 6= r

; then,byusing(2.14) and (4.1),one dedu es that

I

q

(f )

2 =

2q

X

p=0

I

p

(G

q

p

f ),

(4.2) where

I

0 (G

q

0 f ) = G

q

0 f = q!kfk

2 L

2 _(µ

q

₎

. Note that the advantage of (4.2) (over (2.14)) is that the square

I

q

(f )

2

is now represented as an orthogonal sum of multiple integrals. As before, given

f ∈ L

2 s

(µ

q

)

(

q > 2

) and

z ∈ Z

, we write

f (z, ·)

to indi ate thefun tion on

Z

q−1

given by

(z

1 , ..., z

q−1

) → f(z, z

1 , ..., z

q−1

)

. To simplify thepresentation of the forth oming results, wenowintrodu e afurther assumption.

AssumptionA. Fortherestofthe paper,whenever onsidering akernel

f ∈ L

2 s

(µ

q

)

,wewill impli itly assume thatevery ontra tion of thetype

(z

1 , ..., z

2q−r−l

) −→ |f| ⋆

l

r

|f|(z

1 , ..., z

2q−r−l

)

is well-dened and nite for every

r = 1, ..., q

, every

l = 1, ..., r

and every

(z

1 , ..., z

2q−r−l

) ∈

Z

2q−r−l

.

Assumption A ensures, in parti ular, that the following relations are true for every

r =

0, ...., q − 1

,every

l = 0, ..., r

and every

(z

1 , ..., z

2(q−1)−r−l

) ∈ Z

2(q−1)−r−l

:

Z

[f (z, ·) ⋆

l

r

f (z, ·)]µ(dz)={f ⋆

l+1

r+1

f },

and

Z

[

_{f (z, ·) ⋆}

^

l

r

f (z, ·)]µ(dz)=

^

f ⋆

l+1

_r+1

f

(notethatthesymmetrizationontheLHSofthese ondequalitydoesnotinvolvethevariable

z

).

The notation

G

q−1

p

f (z, ·)

(

p = 0, ..., 2(q − 1)

) stands for the a tion of the operator

G

q−1

p

(deneda ordingto (4.1))on

f (z, ·)

,that is,

G

q−1

_p

_{f (z, ·)(z}

1 , ..., z

p

)

(4.3)

=

q−1

X

r=0

r

X

l=0

1 {2(q−1)−r−l=p}

r!

q − 1

r

2 r

l

^

f (z, ·) ⋆

l

r

f (z, ·)(z

1 , ..., z

p

).

For instan e, if

f ∈ L

2 s

(µ

3 )

,then

G

2 ₁

_{f (z, ·)(a) = 4 × f(z, ·) ⋆}

1 ₂

_{f (z, ·)(a) = 4}

Z

f (z, a, u)

2 µ(du).

Notealso that,for a xed

z ∈ Z

,thequantity

G

q−1

0 f (z, ·)

isgiven bythefollowing onstant

G

q−1

₀

_{f (z, ·) = (q − 1)!}

Z

q−1

f

2 _{(z, ·)dµ}

q−1

.

(4.4)

Finally,for every

q > 2

and every

f ∈ L

2 s

(µ

q

)

,we set

b

G

q

₀

f =

Z

G

q−1

₀

_{f (z, ·)µ(dz) = q}

−1

G

q

₀

_{f = (q − 1)!kfk}

2 _L

2 _(µ

q

₎

,

(16)

and,for

p = 1, ..., 2(q − 1)

,we dene the fun tion

(z

1 , ..., z

p

) → b

G

q

p

f (z

1 , ..., z

p

)

,from

Z

p

into

R

,asfollows:

b

G

q

_p

_{f (·) =}

Z

G

q−1

_p

_{f (z, ·)µ(dz),}

(4.5)

or,more expli itly,

b

G

q

_p

f (z

1 , ..., z

p

)

=

Z

q−1

X

r=0

r

X

l=0

1 {2(q−1)−r−l=p}

r!

q − 1

r

2 r

l

^

f (z, ·) ⋆

l

r

f (z, ·)(z

1 , ..., z

p

)µ(dz)

(4.6)

=

q

X

t=1

t∧(q−1)

_X

s=1

1 {2q−t−s=p}

(t − 1)!

q − 1

t − 1

2 t − 1

s − 1

^

f ⋆

s

t

f (z

1 , ..., z

p

),

(4.7)

wherein(4.7)we have usedthe hangeofvariables

t = r + 1

and

s = l + 1

,aswellasthefa t that

p > 1

. We stressthat thesymmetrizationin(4.6) doesnotinvolve thevariable

z

.

4.2 Bounds on the Wasserstein distan e

When

q > 2

and

µ(Z) = ∞

, we shall fo us on kernels

f ∈ L

2 s

(µ

q

)

verifying the following te hni al ondition: for every

p = 1, ..., 2(q − 1)

,

Z

"sZ

Z

p

{G

q−1

p

f (z, ·)}

2 dµ

p

#

µ(dz) < ∞.

(4.8)

Aswill be ome lear from the subsequent dis ussion, therequirement (4.8) is usedto justify a Fubini argument.

Remark 4.1 1. When

q = 2

, one dedu es from (4.3) that

G

1

1 f (z, ·)(x) = f(z, x)

2

and

G

1 ₂

_{f (z, ·)(x, y) = f(z, x)f(z, y)}

. Itfollowsthat, inthis ase, ondition(4.8)isveriedif, and only if,the following relation holds:

Z

sZ

Z

f (z, a)

4 _{µ(da) µ(dz) < ∞.}

(4.9)

Indeed, sin e

f

issquare-integrable, theadditional relation

Z

sZ

Z

2 f (z, a)

2 _{f (z, b)}

2 _{µ(da)µ(db) µ(dz) < ∞}

(4.10)

is always satised,sin e

Z

sZ

Z

2 f (z, a)

2 _{f (z, b)}

2 _{µ(da)µ(db)µ(dz) = kfk}

2 L

2 _(µ

2 ₎

.

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(stronger) ondition: forevery

p = 1, ..., 2(q−1)

andevery

(r, l)

su hthat

2(q−1)−r−l =

p

Z

"sZ

Z

p

{f(z, ·) ⋆

l

r

f (z, ·)}

2 dµ

p

#

µ(dz) < ∞.

(4.11) 3. When

µ(Z) < ∞

and

Z

p

{G

q−1

p

f (z, ·)}

2 dµ

p

µ(dz) < ∞,

ondition (4.8) is automati ally satised (to see this, just apply the Cau hy-S hwarz inequality).

4. Arguing asin the previous point, a su ient ondition for (4.8) to be satised,is that the support of the symmetri fun tion

f

is ontained ina set of the type

A × · · · × A

where

A

is su hthat

µ(A) < ∞

.

Theorem 4.2 (Wasserstein bounds on a xed haos) Fix

q > 2

and let

X ∼ N (0, 1)

. Let

f ∈ L

2 s

(µ

q

)

be su h that:

(i) whenever

µ(Z) = ∞

, ondition (4.8) issatised for every

p = 1, ..., 2(q − 1)

;

(ii) for

dµ

-almost every

z ∈ Z

, every

r = 1, ..., q − 1

and every

l = 0, ..., r − 1

, the kernel

f (z, ·) ⋆

l

r

f (z, ·)

isan element of

L

2 s

(µ

2(q−1)−r−l

)

.

Denote by

I

q

(f )

the multiple Wiener-It integral, of order

q

, of

f

with respe t to

N

b

. Then, the followingbound holds:

d

W

(I

q

(f ), X) 6

v

u

t(1 − q!kfk

2 L

2 _(µ

q

₎

)

2 + q

2 2(q−1)

_X

p=1

p!

Z

p

n

b

G

q

p

f

o

2 dµ

p

(4.12)

+ q

2 q

_{(q − 1)!kfk}

2 _L

2 _(µ

q

₎

×

v

u

t

2(q−1)

X

p=0

p!

Z

p

G

q−1

p

f (z, ·)

2 dµ

p

µ(dz),

(4.13)

where the notations

G

b

q

p

f

and

G

q−1

p

f (z, ·)

are dened, respe tively, in (4.5)(4.7) and (4.3). Moreover, the bound appearing on the RHS of (4.12)(4.13) an be assessed by means of the followingestimate:

v

u

t(1 − q!kfk

2 L

2 _(µ

q

₎

)

2 + q

2 2(q−1)

_X

p=1

p!

Z

p

n

b

G

q

p

f

o

2 dµ

p

₆

_{|1 − q!kfk}

2 L

2 _(µ

q

₎

|

(4.14)

+ q

q

X

t=1

t∧(q−1)

_X

s=1

1 {26t+s62q−1}

(2q − t − s)!

1/2

(t − 1)!

1/2

×

(4.15)

×

q − 1

t − 1

s − 1

1/2

kf ⋆

s

t

f k

L

2 _(µ

2q−t−s

₎

,

(18)

f ⋆

q−b

_q

_{f ∈ L}

2 (µ

b

_{), ∀b = 1, ..., q − 1,}

(4.16) then,

v

u

t

2(q−1)

_X

p=0

p!

Z

p

G

q−1

p

f (z, ·)

2 dµ

p

2 µ(dz) 6

(4.17)

q

X

b=1

b−1

X

a=0

1 {16a+b62q−1}

(a + b)!

1/2

(q − a − 1)!

1/2

×

(4.18)

×

q − 1

q − 1 − a

q − b

1/2

kf ⋆

a

b

f k

L

2 _(µ

2q−a−b

₎

.

Remark4.3 1. There are ontra tionnorms in(4.18) thatdonot appearintheprevious formula(4.15), and vi eversa. For example,in (4.18) one has

kf ⋆

0 b

f k

L

2 _(µ

2q−b

₎

,where

b = 1, ..., q

(this orresponds to the ase

b ∈ {1, ..., q}

and

a = 0

). By using formula (2.13), these norms anbe omputedasfollows:

kf ⋆

0 b

f k

L

2 _(µ

2q−b

₎

= kf ⋆

q−b

_q

f k

_L

2 _(µ

b

₎

, b = 1, ..., q − 1;

(4.19)

kf ⋆

0 q

f k

L

2 _(µ

q

₎

=

sZ

Z

q

f

4 _dµ

q

_.

(4.20) Westress that

f

2 _{= f ⋆}

0 q

f

,and therefore

f ⋆

0 q

f ∈ L

2 (µ

q

)

if,and only if,

f ∈ L

4 _(µ

q

₎

. 2. One should ompare Theorem 4.2 with Proposition 3.2 in [13℄, whi h provides upper

boundsforthenormalapproximationofmultipleintegrals withrespe tto anisonormal Gaussian pro ess. The boundsin[13 ℄ arealsoexpressedinterms of ontra tions ofthe underlying kernel.

Example 4.4 (Double integrals). We onsider a double integral of the type

I

2 (f )

, where

f ∈ L

2 s

(µ

2 )

satises(4.9)(a ordingto Remark4.1(1),thisimpliesthat(4.8) issatised). We suppose that the following three onditions are satised: (a)

EI

2 (f )

2 _{= 2kfk}

2 L

2 _(µ

2 ₎

= 1

, (b)

f ⋆

1

2 f ∈ L

2 (µ

1 )

and ( )

f ∈ L

4 _(µ

2 ₎

. Sin e

q = 2

here, one has

f (z, ·) ⋆

0

1 f (z, ·)(a) = f(z, a)

2

whi hissquare-integrable,andhen eAssumption(ii)and ondition(4.16)inTheorem4.2are veried. Usingrelations (4.19)(4.20), we an dedu ethefollowing bound ontheWasserstein distan ebetween the lawof

I

2 (f )

andthe lawof

X ∼ N (0, 1)

:

d

W

(I

2 (f ), X)

(4.21)

6 √

_{8kf ⋆}

1 ₁

_{f k}

_L

2 _(µ

2 ₎

+

n

2 +

√

8(1 +

√

6)

o

_{kf ⋆}

1 ₂

_{f k}

L

2 _(µ

2 ₎

+ 4

sZ

Z

2 f

4 _dµ

2 _.

To obtain (4.21), observe rst that, sin e assumption (a) above is in order, then relations (4.12)(4.13) inthestatement ofTheorem 4.2yield that

(19)

(4.15) = 2{2

1/2

kf ⋆

1

1 f k

L

2 _(µ

2 ₎

+ kf ⋆

1 ₂

f k

_L

2 _(µ)

}

and

(4.18) = kf ⋆

1

2 f k

L

2 _(µ)

{3!

1/2

+ 1} + 2

1/2

kfk

2 L

4 _(µ

2 ₎

,

sin e

kf ⋆

1

2 f k

L

2 _(µ)

= kf ⋆

0 ₁

f k

_L

2 _(µ

3 ₎

. A general statement, involving random variables of the type

F = I

1 (g) + I

2 (h)

isgiven inTheorem6.1.

2

Example 4.5 (Triple integrals). We onsider a random variable of the type

I

3 (f )

, where

f ∈ L

2 s

(µ

3 )

veries (4.8) (for instan e, a ording to Remark 4.1(4), we may assume that

f

has support ontained in some re tangle of nite

µ

3

-measure). We shall also suppose that the following three onditions are satised: (a)

EI

3 (f )

2 _{= 3!kfk}

2 L

2 _(µ

3 ₎

= 1

, (b) For every

r = 1, ..., 3

,andevery

l = 1, ..., r∧2

,onehasthat

f ⋆

l

r

f ∈L

2 (µ

6−r−l

)

,and( )

f ∈ L

4 _(µ

3 ₎

. One an he k that all the assumptions in thestatement of Theorem 4.2 (in the ase

q = 3

) are satised. Inviewof(4.19)(4.20),wethereforededu e(exa tlyasinthepreviousexampleand after some tedious bookkeeping!) the following bound on the Wasserstein distan e between the lawof

I

3 (f )

andthelaw of

X ∼ N (0, 1)

:

d

W

(I

3 (f ), X) 6 3(4!

1/2

)kf ⋆

1

1 f k

L

2 _(µ

4 ₎

+ (6 + 2

√

18)(3!

1/2

_{)kf ⋆}

1 ₂

_{f k}

_L

2 _(µ

3 ₎

(4.22)

+6

√

_{2kf ⋆}

2 ₂

_{f k}

_L

2 _(µ

2 ₎

+ {6 +

√

18(4 + 2(4!

1/2

_{))}kf ⋆}

1 ₃

_{f k}

_L

2 _(µ

2 ₎

+{

√

18(2

1/2

+ 5!

1/2

_{) + 6}kf ⋆}

2 ₃

_{f k}

_L

2 _(µ

1 ₎

+{3

3/2

(18

1/2

_)}

sZ

Z

3 f

4 _dµ

3 _.

2

Proof of Theorem 4.2. First observe that, a ording to Theorem3.1, we have that

d

W

(I

q

(f ), X) 6

v

u

tE

"

_{1 −}

1 q

kDI

q

(f )k

2 L

2 _(µ)

2 #

+

1 q

Z

E|D

z

I

q

(f )|

3 µ(dz).

Therest ofthe proof isdivided infour steps:

(S1) Proof of the fa t that

s

E

_{1 −}

1 q

kDI

q

(f )k

2 L

2 _(µ)

2

is less or equal to the quantity appearing at theline (4.12).

(S2) Proof of the fa t that

1 q

R

Z

E|D

z

I

q

(f )|

3 µ(dz)

is lessor equal to thequantity appearing at theline (4.13).

(S3) Proof of theestimate displayed informulae (4.14)(4.15).