Stein's method on Wiener chaos

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Stein’s method on Wiener chaos

Ivan Nourdin, Giovanni Peccati

To cite this version:

Ivan Nourdin, Giovanni Peccati. Stein’s method on Wiener chaos. 2008. �hal-00199024v5�

hal-00199024, version 5 - 10 May 2008

byIvanNourdin

∗

and GiovanniPeati

†

University of Paris VI

Revisedversion: May10,2008

Abstrat: We ombine Malliavin alulus with Stein's method, in order to derive expliit bounds

in theGaussian and Gamma approximationsof random variables in axed Wiener haos of a gen-

eral Gaussianproess. Ourapproahgeneralizes,renesanduniestheentral andnon-entrallimit

theorems formultiple Wiener-Itintegralsreentlyproved(inseveralpapers,from 2005to 2007)by

Nourdin,Nualart,Ortiz-Latorre,Peati andTudor. Weapplyourtehniques toproveBerry-Esséen

bounds in the Breuer-Major CLT for subordinated funtionals of frational Brownian motion. By

usingthewell-knownMehler'sformulaforOrnstein-Uhlenbeksemigroups,wealsoreoveratehnial

resultreentlyprovedbyChatterjee,onerningtheGaussian approximationoffuntionals ofnite-

dimensional Gaussianvetors.

Key words: Berry-Esséenbounds;Breuer-MajorCLT;FrationalBrownianmotion;Gamma

approximation;Malliavinalulus;Multiplestohastiintegrals;Normalapproximation;Stein's

method.

2000 Mathematis SubjetClassiation: 60F05;60G15;60H05; 60H07.

1 Introdution and overview

1.1 Motivations

Let

Z

^{be arandom variable whose lawisabsolutely ontinuouswith respetto theLebesgue}

measure(for instane,

Z

^{is a standard Gaussian random variable or a Gamma random vari-}

able). Supposethat

{ Z _n : n > 1 }

^{isasequeneofrandomvariablesonvergingin}distribution towards

Z

^{,that is:}

for all

z ∈ R

,

P (Z _n 6 z) −→ P(Z 6 z)

^as

n → ∞

^{. (1.1)}

It is sometimes possible to assoiate an expliit uniform bound with the onvergene (1.1),

providingaglobaldesriptionoftheerroronemakeswhenreplaing

P (Z _n 6 z)

^with

P (Z 6 z)

for axed

n > 1

^{. Oneof the most elebrated results inthis diretion isthe following Berry-}

Esséen Theorem (see e.g. Feller [17℄for aproof), thatwe reordhere for futurereferene:

Theorem 1.1 (Berry-Esséen) Let

(U _j ) _j>1

^{beasequene of}independentandidentiallydis- tributed random variables, suh that

E( | U j | ³ ) = ρ < ∞

^,

E(U j ) = 0

^and

E(U _j ² ) = σ ²

^{. Then,}

∗

LaboratoiredeProbabilitésetModèlesAléatoires,UniversitéPierreetMarieCurie,Boîteourrier188,4

PlaeJussieu,75252Paris Cedex5,Frane,ivan.nourdinupm.fr

†

LaboratoiredeStatistiqueThéoriqueetAppliquée,UniversitéPierreetMarieCurie,8èmeétage,bâtiment

A,175rueduChevaleret,75013Paris,Frane,giovanni.peatigmail.om

by setting

Z _n = _σ √ ¹ n

P _n

j=1 U _j , n > 1,

^{one has that}

Z _n −→ ^Law Z ∼ N (0, 1)

^{, as}

n → ∞

^{, and}

moreover:

sup

z ∈ R | P (Z _n 6 z) − P(Z 6 z) | 6 3ρ

σ ³ √ n .

^(1.2)

The aim of this paper is to show that one an ombine Malliavin alulus (see e.g. [35 ℄)

andStein'smethod (seee.g. [11℄),inordertoobtainboundsanalogous to(1.2),wheneverthe

randomvariables

Z _n

^{in(1.1)an be} representedasfuntionals ofa given Gaussian eld. Our results are general, in the sense that (i) they do not rely on any spei assumption on the

underlyingGaussianeld,(ii)theydonot requirethatthevariables

Z n

^{havethespeiform}

ofpartialsums,and (iii)they allow todeal(atleastintheaseofGaussian approximations)

withseveraldierent notions of distane between probabilitymeasures. Assuggested bythe

title, a prominent role will be played by random variables belonging to a Wiener haos of

order

q

⁽

q > 2

^{), that is, random variables having the form of a multiple stohasti Wiener-}

It integral of order

q

^{(see Setion 2 below for preise} denitions). It will be shown that our results provide substantial renements of the entral and non-entral limit theorems for

multiple stohastiintegrals, reently proved in [33 ℄ and [37℄. Among other appliations and

examples, wewill provideexpliitBerry-EsséenboundsintheBreuer-Major CLT(see[5℄) for

elds subordinatedto a frational Brownian motion.

Conerning point (iii), we shall note that, as a by-produt of the exibility of Stein's

method,we will indeedestablishboundsfor Gaussian approximations relatedtoa numberof

distanes of the type

d _H (X, Y ) = sup {| E(h(X)) − E(h(Y )) | : h ∈ H } ,

^(1.3)

where

H

^{is some suitable lass of funtions. For instane: by taking}

H = { h : k h k L 6 1 } ,

where

k·k _L

^{denotestheusualLipshitzseminorm,oneobtainsthe}Wasserstein(orKantorovih- Wasserstein) distane; by taking

H = { h : k h k BL 6 1 } ,

^where

k·k BL = k·k L + k·k _∞

^{, one}

obtains the Fortet-Mourier (or bounded Wasserstein) distane; by taking

H

^{equal to the}

olletionofallindiators

1 B

^{ofBorelsets,oneobtains thetotalvariationdistane;bytaking}

H

^{equal to the lass of all indiators funtions}

1 ( −∞ ,z]

^,

z ∈ R

, one has the Kolmogorov distane, whih is the one taken into aount in the Berry-Esséen bound (1.2). In what

follows, we shall sometimes denote by

d _W (., .)

^,

d _FM (., .)

^,

d _TV (., .)

^and

d _Kol (., .)

^, respetively, the Wasserstein, Fortet-Mourier, total variation and Kolmogorov distanes. Observe that

d _W (., .) > d _FM (., .)

^and

d _TV (., .) > d _Kol (., .)

^{. Also, the topologies indued by}

d _W

^,

d _TV

^and

d _Kol

^{are stronger than the topology of onvergene in} distribution, while one an show that

d FM

^{metrizestheonvergenein}distribution(seee.g. [16,Ch. 11℄fortheseandfurtherresults involvingdistanes onspaes of probabilitymeasures).

1.2 Stein's method

We shall now give a short aount of Stein's method, whih is basially a set of tehniques

allowingtoevaluatedistanesofthetype(1.3)bymeansofdierentialoperators. Thistheory

has been initiated by Stein in the path-breaking paper [48℄, and then further developed in

the monograph [49 ℄. The readerisreferredto [11 ℄,[43 ℄ and [44℄for detailedsurveys ofreent

resultsandappliations. ThepaperbyChatterjee[8 ℄providesfurtherinsightsintotheexisting

literature. In what follows, we will apply Stein's method to two types of approximations,

namely Gaussian and (entered) Gamma. We shall denote by

N (0, 1)

^{a standard Gaussian}

random variable. Theentered Gammarandom variableswe areinterested inhavethe form

F (ν ) ^Law = 2G(ν/2) − ν, ν > 0,

^(1.4)

where

G(ν/2)

^{has a Gamma law with parameter}

ν/2

^{. This means that}

G(ν/2)

^{is a (a.s.}

stritly positive)random variablewithdensity

g(x) = ^x

ν 2 −1 e ^−x

Γ(ν/2) 1 (0, ∞ ) (x),

^where

Γ

^istheusual

Gamma funtion. We hoose this parametrization inorder to failitate the onnetion with

our previous paper [33℄ (observe in partiular that, if

ν > 1

^{is an integer, then}

F (ν)

^{has a}

entered

χ ²

distribution with

ν

^{degrees of freedom).}

StandardGaussian distribution. Let

Z ∼ N (0, 1)

^{. Consider a} real-valued funtion

h : R → R

suhthattheexpetation

E(h(Z))

^iswell-dened. TheSteinequationassoiatedwith

h

^and

Z

^{islassially given by}

h(x) − E(h(Z)) = f ^′ (x) − xf (x), x ∈ R .

^(1.5)

A solution to (1.5) is a funtion

f

^{whih is Lebesgue} a.e.-dierentiable, and suh that there exists a version of

f ^′

^{verifying (1.5) for every}

x ∈ R

. The following result is basially due to Stein [48, 49℄. Theproof of point (i) (whose ontent is usually referredas Stein's lemma)

involves astandard use ofthe Fubini theorem(see e.g. [47 ℄ or [11 ,Lemma 2.1℄). Point (ii)is

proved e.g. in [11 , Lemma 2.2℄; point (iii) an be obtained by ombining e.g. thearguments

in[49, p. 25℄and [9, Lemma 5.1℄; a proofof point (iv)is ontained in[49 , Lemma 3,p. 25℄;

point (v) isproved in[8 , Lemma 4.3℄.

Lemma 1.2 (i) Let

W

^{be a random variable. Then,}

W ^Law = Z ∼ N (0, 1)

^{if, andonly if,}

E[f ^′ (W ) − W f (W )] = 0,

^(1.6)

for every ontinuous and pieewise ontinuously dierentiable funtion

f

^{verifying the}

relation

E | f ^′ (Z) | < ∞

^.

(ii) If

h(x) = 1 ( −∞ ,z] (x)

^,

z ∈ R

, then (1.5)admitsa solution

f

^{whih isbounded by}

√ 2π/4

^,

pieewise ontinuously dierentiable andsuh that

k f ^′ k ∞ 6 1

^.

(iii) If

h

^{is bounded by 1/2, then (1.5) admits a solution}

f

^{whih is bounded by}

p π/2

^,

Lebesgue a.e. dierentiable and suh that

k f ^′ k ∞ 6 2

^.

(iv) If

h

^{is bounded and absolutely ontinuous (then, in partiular, Lebesgue-a.e. dieren-}

tiable), then (1.5) has a solution

f

^{whih is bounded and twie} dierentiable, and suh that

k f k ∞ 6 p

π/2 k h − E(h(Z)) k ∞

^,

k f ^′ k ∞ 6 2 k h − E(h(Z )) k ∞

^and

k f ^′′ k ∞ 6 2 k h ^′ k ∞

^.

(v) If

h

^{isabsolutely ontinuous withbounded} derivative, then (1.5) has a solution

f

^whih

is twie dierentiable andsuh that

k f ^′ k ∞ 6 k h ^′ k ∞

^and

k f ^′′ k ∞ 6 2 k h ^′ k ∞

^.

We alsoreall the relation:

2d _TV (X, Y ) = sup {| E(u(X)) − E(u(Y )) | : k u k ∞ 6 1 } .

^(1.7)

Notethat point (ii) and (iii) (via(1.7)) imply the following boundson theKolmogorov and

totalvariation distanebetween

Z

^{andan arbitraryrandom variable}

Y

^:

d _Kol (Y, Z) 6 sup

f ∈F Kol

| E(f ^′ (Y ) − Y f (Y )) |

^(1.8)

d TV (Y, Z) 6 sup

f ∈ F TV

| E(f ^′ (Y ) − Y f (Y )) |

^(1.9)

where

F Kol

^and

F TV

^are,respetively,thelassof pieewiseontinuouslydierentiablefun- tions that are bounded by

√ 2π/4

^{and suh that their derivative is bounded by 1, and the}

lassof pieewise ontinuously dierentiable funtions that are bounded by

p π/2

^{and suh}

thattheir derivative is bounded by 2.

Analogously,byusing (iv)and(v) alongwith therelation

k h k L = k h ^′ k ∞

^,oneobtains

d _FM (Y, Z) 6 sup

f ∈F FM

| E(f ^′ (Y ) − Y f (Y )) | ,

^(1.10)

d _W (Y, Z) 6 sup

f ∈ F W

| E(f ^′ (Y ) − Y f (Y )) | ,

^(1.11)

where:

F FM

^{is the lass of twie} dierentiable funtions that are bounded by

√ 2π

^{, whose}

rst derivative is bounded by 4, and whose seond derivative is bounded by 2;

F W

^{is the}

lassoftwiedierentiablefuntions,whoserstderivativeisboundedby1andwhoseseond

derivative isbounded by2.

Centered Gamma distribution. Let

F (ν )

^{be as in (1.4). Consider a} real-valued funtion

h : R → R

suhthattheexpetation

E[h(F (ν))]

^{exists. TheSteinequationassoiatedwith}

h

and

F (ν)

^is:

h(x) − E[h(F (ν))] = 2(x + ν)f ^′ (x) − xf (x), x ∈ ( − ν, + ∞ ).

^(1.12)

The following statement ollets some slight variations around results proved by Stein [49 ℄,

Diaonis and Zabell [15 ℄, Luk [26℄, Shoutens [46℄ and Pikett [41 ℄. It is theGamma oun-

terpart of Lemma 1.2. Theproofis detailedinSetion 7.

Lemma 1.3 (i) Let

W

^{be a} real-valued random variable (not neessarily with values in

( − ν, + ∞ )

^{) whose law admits a density with respet to the Lebesgue measure. Then,}

W ^Law = F(ν)

^{if, andonly if,}

E[2(W + ν) ₊ f ^′ (W ) − W f(W )] = 0,

^(1.13)

where

a ₊ := max(a, 0)

^{, for every smooth funtion}

f

^{suh that the mapping}

x 7→ 2(x + ν ) + f ^′ (x) − xf (x)

^isbounded.

(ii) If

| h(x) | 6 c exp(ax)

^forevery

x > − ν

^andforsome

c > 0

^and

a < 1/2

^,andif

h

^istwie

dierentiable, then(1.12) hasa solution

f

^{whihisbounded on}

( − ν, + ∞ )

^, dierentiable and suh that

k f k ∞ 6 2 k h ^′ k ∞

^and

k f ^′ k ∞ 6 k h ^′′ k ∞

^.

(iii) Suppose that

ν > 1

^{is an integer. If}

| h(x) | 6 c exp(ax)

^{for every}

x > − ν

^{and for}

some

c > 0

^and

a < 1/2

^{, and if}

h

^{is twie} dierentiable with bounded derivatives, then (1.12) hasa solution

f

^{whihisbounded on}

( − ν, + ∞ )

^, dierentiable andsuh that

k f k ∞ 6 p

2π/ν k h k ∞

^and

k f ^′ k ∞ 6 p

2π/ν k h ^′ k ∞

^.

Now dene

H 1 = { h ∈ C _b ² : k h k ∞ 6 1, k h ^′ k ∞ 6 1 } ,

^(1.14)

H 2 = { h ∈ C _b ² : k h k ∞ 6 1, k h ^′ k ∞ 6 1, k h ^′′ k ∞ 6 1 } ,

^(1.15)

H 1,ν = H 1 ∩ C _b ² (ν)

^(1.16)

H 2,ν = H 2 ∩ C _b ² (ν)

^(1.17)

where

C _b ²

^{denotes the lass of twie} dierentiable funtions (with support in

R

) and with boundedderivatives, and

C _b ² (ν)

^{denotes thesubsetof}

C _b ²

^{omposedoffuntionswithsupport}

in

( − ν, + ∞ )

^{. Note that point (ii) in the previous statement implies that, by adopting the}

notation (1.3) and for every

ν > 0

^{and every real random variable}

Y

^{(not neessarily with}

supportin

( − ν, + ∞ )

^),

d _{H 2,ν} (Y, F (ν)) 6 sup

f ∈F 2,ν

| E[2(Y + ν)f ^′ (Y ) − Y f (Y )] |

^(1.18)

where

F 2,ν

^{is the lass of} dierentiable funtions with support in

( − ν, + ∞ )

^{, bounded by 2}

andwhose rstderivatives arebounded by 1. Analogously,point (iii) impliesthat, for every

integer

ν > 1

^,

d _{H 1,ν} (Y, F (ν)) 6 sup

f ∈ F 1,ν

| E[2(Y + ν)f ^′ (Y ) − Y f (Y )] | ,

^(1.19)

where

F 1,ν

^{is the lass of} dierentiable funtions with support in

( − ν, + ∞ )

^{, bounded by}

p 2π/ν

^{and whose rst} derivatives are also bounded by

p 2π/ν

^{. A little inspetion shows}

thatthefollowingestimates also hold: for every

ν > 0

^{andevery random variable}

Y

^,

d _{H 2} (Y, F (ν)) 6 sup

f ∈ F 2

| E[2(Y + ν) + f ^′ (Y ) − Y f (Y )] |

^(1.20)

where

F 2

^{is the lass of funtions (dened on}

R

) that are ontinuous and dierentiable on

R \{ ν }

^{, bounded by}

max { 2, 2/ν }

^{, and whose rst} derivatives are bounded by

max { 1, 1/ν + 2/ν ² }

^. Analogously,for everyinteger

ν > 1

^,

d _{H 1} (Y, F (ν)) 6 sup

f ∈ F 1

| E[2(Y + ν) ₊ f ^′ (Y ) − Y f (Y )] | ,

^(1.21)

where

F 1

^{is the lass of funtions (on}

R

) that are ontinuous and dierentiable on

R \{ ν }

^,

boundedby

max { p

2π/ν, 2/ν }

^,andwhoserstderivativesareboundedby

max { p

2π/ν, 1/ν + 2/ν ² }

^.

Now, the ruial issue is how to estimatethe right-hand side of (1.8)(1.11) and (1.18)

(1.21) for a given hoie of

Y

^{. Sine Stein's initial} ontribution [48 ℄, an impressive panoply of tehniques has been developed in this diretion (see again [10 ℄ or [43 ℄ for a survey; here,

we shallquote e.g.: exhangeable pairs [49℄,diusion generators [3, 19℄, size-bias transforms

[20 ℄, zero-bias transforms [21 ℄, loal dependeny graphs [10℄ and graphial-geometri rules

[8℄). Starting from the next setion, we will show that, when working within theframework

of funtionals of Gaussian elds, one an veryeetively estimate expressions suh as(1.8)

(1.11),(1.18) and(1.19) byusingtehniquesofMalliavinalulus. Interestingly,aentralrole

isplayed byan innite dimensional version of the same integration by parts formula that is

atthevery heart ofStein's haraterization oftheGaussian distribution.

1.3 The basi approah (with some examples)

Let

H

^{be a realseparable Hilbertspaeand, for}

q > 1

^{, let}

H ^{⊗ q}

^(resp.

H ^{⊙ q}

^{) be the}

q

^thtensor

produt (resp.

q

^{thsymmetri tensorprodut) of}

H

^{. We write}

X = { X(h) : h ∈ H }

^(1.22)

to indiate a entered isonormal Gaussian proess on

H

^{. For every}

q > 1

^{, we denote by}

I _q

the isometry between

H ^{⊙ q}

^{(equipped with the norm}

√

q! k · k H ^⊗q

^{) and the}

q

^{th Wiener haos}

of

X

^{. Note that, if}

H

^{is a}

σ

^{-nite measure spae withno atoms, theneah random variable}

I q (h)

^,

h ∈ H ^{⊙ q}

^{, has the form of a multiple Wiener-It integral of order}

q

^{. We denote by}

L ² (X) = L ² (Ω, σ(X), P )

^{the spae of square integrable funtionals of}

X

^{, and by}

D ^1,2

the domain of the Malliavin derivative operator

D

^{(see the forthoming Setion 2 for preise}

denitions). Reall that, for every

F ∈ D ^1,2

, the derivative

DF

^{is a random element with}

values in

H

^.

We start by observing that, thanks to (1.6), for every

h ∈ H

^{suh that}

k h k ^H = 1

^{and for}

everysmooth funtion

f

^,wehave

E[X(h)f (X(h))] = E[f ^′ (X(h))]

^{. Ourpointisthatthis last}

relation is a very partiular ase of the following orollary of the elebrated integration by

partsformulaofMalliavin alulus: for every

Y ∈ D ^1,2

withzero mean,

E[Y f (Y )] = E[ h DY, − DL ^{− 1} Y i ^H f ^′ (Y )],

^(1.23)

where thelinear operator

L ^{− 1}

^{is the inverse of the generator of the}Ornstein-Uhlebek semi- group,noted

L

^{. The readerisreferredtoSetion 2andSetion 3for denitionsandfor afull}

disussion of this point; here, we shall note that

L

^{is an} innite-dimensional version of the generator assoiated with Ornstein-Uhlenbek diusions (see [35 , Setion 1.4℄ for a proof of

thisfat),anobjetwhihisalsoruialintheBarbour-Götzegeneratorapproah toStein's

method [3,19℄.

Itfollowsthat,forevery

Y ∈ D ^1,2

withzeromean,theexpressionsappearingontheright- hand side of (1.8)(1.11) (or (1.18)(1.21)) an be assessed by rst replaing

Y f (Y )

^with

h DY, − DL ^{− 1} Y i ^H f ^′ (Y )

^insidetheexpetation,andthenbyevaluatingthe

L ²

^{distanebetween}

1

^(resp.

2Y + 2ν

^{)andtheinnerprodut}

h DY, − DL ^{− 1} Y i ^H

^{. Ingeneral,these}omputationsare arriedout byrstresorting to therepresentation of

h DY, − DL ^{− 1} Y i ^H

^{asa(possiblyinnite)}

series of multiple stohasti integrals. We will see that, when

Y = I _q (g)

^{, for}

q > 2

^and

some

g ∈ H ^{⊙ q}

^,then

h DY, − DL ^{− 1} Y i ^H = q ^{− 1} k DY k ² H

^{. Inpartiular, byusing this last relation}

one an dedue bounds involving quantities that are intimately related to the entral and

non-entral limit theorems reently proved in[37℄,[36 ℄ and[33 ℄.

Remark 1.4 1. Theruialequality

E[I _q (g)f (I _q (g))] = E[q ^{− 1} k DI _q (g) k ² H f ^′ (I _q (g))]

^,inthe

ase where

f

^{is a omplex} exponential, has been rst used in [36℄, in order to give renements (as well asalternate proofs) of the main CLTs in [37℄ and [39 ℄. The same

relation has been later applied in [33℄, where a haraterization of non-entral limit

theoremsformultipleintegralsisprovided. Notethatneither[33℄nor[36 ℄areonerned

with Stein's method or, more generally, with bounds on distanes between probability

measures.

2. We will see that formula (1.23) ontains as a speial ase a result reently proved by

Chatterjee [9 ,Lemma 5.3℄,intheontextof limittheoremsfor linearstatistisofeigen-

values of random matries. Theonnetion between thetwo results an be established

by means of the well-known Mehler's formula (see e.g. [28, Setion 8.5, Ch. I℄ or

[35 , Setion 1.4℄), providing a mixture-type representation of the innite-dimensional

Ornstein-Uhlenbek semigroup. See Remarks 3.6 and 3.12 below for a preise disus-

sion of this point. See e.g. [31℄ for a detailed presentation of the innite-dimensional

Ornstein-Uhlebek semigroup.

3. We stress that therandom variable

h DY, − DL ^{− 1} Y i ^H

^{appearing in (1.23) is ingeneral}

not measurable with respet to

σ(Y )

^{. For instane, if}

X

^{is taken to be the Gaussian}

spae generated by a standard Brownian motion

{ W _t : t > 0 }

^and

Y = I ₂ (h)

^with

h ∈ L ² _s ([0, 1] ² )

^,then

D t Y = 2 R 1

0 h(u, t)dW u

^,

t ∈ [0, 1]

^,and

h DY, − DL ^{− 1} Y i L ² ([0,1]) = 2 I ₂ (h ⊗ 1 h) + 2 k h k ² _L ² _([0,1] ² ₎

whih is, ingeneral, not measurable withrespetto

σ(Y )

^(thesymbol

h ⊗ 1 h

^indiates

a ontrationkernel, anobjet thatwill be dened inSetion 2).

4. Note that(1.23) also impliestherelation

E[Y f (Y )] = E[τ (Y )f ^′ (Y )],

^(1.24)

where

τ (Y ) = E[ h DY, − DL ^{− 1} Y i ^H | Y ]

^{. Some general results for the existeneof a real-}

valued funtion

τ

^{satisfying (1.24) are ontained e.g. in [6℄. Note that, in general, it}

is very hard to nd an analyti expression for

τ (Y )

^{, espeially when}

Y

^{is a random}

variablewitha very omplex struture,suh ase.g. a multipleWiener-It integral. On

the other hand,we will seethat, in manyases, the random variable

h DY, − DL ^{− 1} Y i ^H

is remarkably tratable and expliit. See the forthoming Setion 6,whih is based on

[49 , Leture VI℄, for a general disussions of equations of the type (1.24). See Remark

3.10 belowfor aonnetion withGoldsteinand Reinert'szerobias transform[20℄.

5. The reader is referredto [42 ℄ for appliations of integration by parts tehniques to the

Stein-typeestimationofdriftsofGaussianproesses. See[23℄foraSteinharaterization

ofBrownianmotionsonmanifoldsbymeansofintegrationbypartsformulae. See[13 ℄for

aonnetionbetween Stein'smethodandalgebrasofoperatorsonongurationspaes.

Before proeeding to a formal disussion, and in order to motivate the reader, we shall

provide two examples of the kind of results that we will obtain in the subsequent setions.

The rst statement involves double Wiener-It integrals, that is, random variables living in

the seondhaosof

X

^{. The proof isgiven inSetion 7.}

Theorem 1.5 Let

(Z _n ) _n>1

^{be a sequene belonging to the seond Wiener haos of}

X

^.

1. Assume that

E(Z _n ² ) → 1

^and

E(Z _n ⁴ ) → 3

^as

n → ∞

^{. Then}

Z _n −→ ^Law Z ∼ N (0, 1)

^as

n → ∞

^{. Moreover, we have:}

d _TV (Z _n , Z) 6 2 r 1

6

E(Z _n ⁴ ) − 3

+ 3 + E(Z _n ² ) 2

E(Z _n ² ) − 1 .

2. Fix

ν > 0

^{andassume that}

E(Z _n ² ) → 2ν

^and

E(Z _n ⁴ ) − 12E(Z _n ³ ) → 12ν ² − 48ν

^as

n → ∞

^.

Then,as

n → ∞

^,

Z _n −→ ^Law F(ν)

^{, where}

F (ν)

^{hasa entered Gamma}distributionofparameter

ν

^{. Moreover, we have:}

d _{H 2} (Z _n , F (ν)) 6 max { 1, 1/ν, 2/ν ² }

s 1 6

E(Z _n ⁴ ) − 12E(Z _n ³ ) − 12ν ² +48ν +

8 − 6ν + E(Z _n ² ) 2

E(Z _n ² ) − 2ν ,

where

H 2

^{isdened by (1.15).}

ever, these operators will appear in the general statements presented in Setion 3). For

instane, whenappliedto theasewhere

X

^{istheisonormalproessgenerated byafrational}

Brownianmotion,therstpointofTheorem1.5anbeusedtoderivethefollowingboundfor

theKolmogorovdistaneintheBreuer-MajorCLTassoiatedwithquadratitransformations:

Theorem 1.6 Let

B

^{be a frational Brownian motionwithHurst index}

H ∈ (0, ³ / 4 )

^{. We set}

σ ² _H = 1

2 X

t ∈ Z

| t + 1 | ^2H + | t − 1 | ^2H − 2 | t | ^2H 2

< ∞ ,

and

Z _n = 1 σ _H √ n

n − 1

X

k=0

n ^2H (B _(k+1)/n − B _k/n ) ² − 1

, n > 1.

Then, as

n → ∞

^,

Z _n −→ ^Law Z ∼ N (0, 1)

^{. Moreover, there exists a onstant}

c _H

^(depending

uniquely on

H

^{)suh that, forany}

n > 1

^:

d _Kol (Z n , Z ) 6 c H

n ^{1 2 ∧ ( 3 2 − 2H)} .

^(1.25)

Notethatboth Theorem1.5and1.6will be signiantly generalizedinSetion 3and Setion

4(see,inpartiular, theforthoming Theorems 3.1, 3.11and4.1).

Remark1.7 1. When

H = ¹ / 2

^,then

B

^{isastandardBrownian motion(and thereforehas}

independent inrements), and we reover from theprevious resultthe rate

n ^{− 1/2}

^,that

ouldbealsoobtainedbyapplyingtheBerry-EsséenTheorem1.1. Thisrateisstillvalid

for

H < ¹ / 2

^{. But,for}

H > ¹ / 2

^{,therateweanproveinthe}Breuer-Major CLTis

n ^{2H − 2 3}

^.

2. Totheauthorsknowledge,Theorem1.6anditsgeneralizationsaretherstBerry-Esséen

boundseverestablished for the Breuer-Major CLT.

3. Tokeepthelengthofthispaperwithinlimits,wedonotderivetheexpliitexpressionof

someoftheonstants(suhasthequantity

c _H

^{informula(1.25))omposingourbounds.}

As will beome lear later on, the exat value of these quantities an be dedued by a

areful bookkeeping of the bounding onstants appearing at the dierent stagesof the

proofs.

1.4 Plan

Therestofthe paperisorganizedasfollows. InSetion 2wereallsome notionsof Malliavin

alulus. In Setion3 westate anddisuss our mainboundsinStein-typeestimates forfun-

tionalsofGaussianelds. Setion4ontainsanappliationtotheBreuer-MajorCLT. Setion

5 deals withGamma-type approximations. Setion 6 providesa unieddisussionof approx-

imations bymeans ofabsolutelyontinuous distributions. Proofsandfurther renementsare

olleted inSetion 7.

The reader isreferred to [25℄ or [35℄ for any unexplainednotion disussed inthis setion. As

in (1.22), we denote by

X = { X(h) : h ∈ H }

^{an isonormal Gaussian proess over}

H

^{. By}

denition,

X

^{is a entered Gaussian family indexed by the elements of}

H

^{and suh that, for}

every

h, g ∈ H

^,

E

X(h)X(g)

= h h, g i ^H .

^(2.26)

As before, we use the notation

L ² (X) = L ² (Ω, σ(X), P )

^{. It is well-known (see again [35 ,}

Ch. 1℄or [25℄)thatanyrandom variable

F

^{belonging to}

L ² (X)

^{admits thefollowing haoti}

expansion:

F = X ∞

q=0

I q (f q ),

^(2.27)

where

I ₀ (f ₀ ) := E[F]

^{,theseriesonvergesin}

L ²

^{andthesymmetrikernels}

f _q ∈ H ^{⊙ q}

^,

q > 1

^,are

uniquelydeterminedby

F

^{. Asalreadydisussed,inthepartiularasewhere}

H = L ² (A, A , µ)

^,

where

(A, A )

^{isa measurablespaeand}

µ

^isa

σ

^{-nite andnon-atomimeasure, one hasthat}

H ^{⊙ q} = L ² _s (A ^q , A ^{⊗ q} , µ ^{⊗ q} )

^{is the spae of symmetri and square integrable funtions on}

A ^q

^.

Moreover, for every

f ∈ H ^{⊙ q}

^{, the random variable}

I _q (f )

^{oinides withthemultiple Wiener-}

It integral (of order

q

^{) of}

f

^{with respet to}

X

^{(see [35, Ch. 1℄). Observe that a random}

variable of the type

I _q (f )

^{, with}

f ∈ H ^{⊙ q}

^{, has nite moments of all orders (see e.g. [25, Ch.}

VI℄). See again [35 , Ch. 1℄or [45 ℄ for a onnetionbetween multiple Wiener-Itand Hermite

polynomials. For every

q > 0

^{, we write}

J _q

^{to indiate the orthogonalprojetion operator on}

the

q

^{th Wiener haos assoiated with}

X

^{, so that, if}

F ∈ L ² (Ω, F , P )

^{is as in (2.27), then}

J q F = I q (f q )

^{for every}

q > 0

^.

Let

{ e _k , k ≥ 1 }

^{be aomplete} orthonormal systemin

H

^{. Given}

f ∈ H ^{⊙ p}

^and

g ∈ H ^{⊙ q}

^,for

every

r = 0, . . . , p ∧ q

^,the

r

^{thontration of}

f

^and

g

^{is theelement of}

H ^{⊗ (p+q − 2r)}

^{dened as}

f ⊗ r g =

X ∞

i 1 ,...,i r =1

h f, e _{i 1} ⊗ . . . ⊗ e _{i r} i H ^⊗r ⊗ h g, e _{i 1} ⊗ . . . ⊗ e _{i r} i H ^⊗r .

^(2.28)

Notethat, inthe partiular asewhere

H = L ² (A, A , µ)

^(with

µ

non-atomi), one hasthat

f ⊗ r g = Z

A ^r

f (t ₁ , . . . , t _{p − r} , s ₁ , . . . , s _r ) g(t _{p − r+1} , . . . , t _{p+q − 2r} , s ₁ , . . . , s _r )dµ(s ₁ ) . . . dµ(s _r ).

Moreover,

f ⊗ 0 g = f ⊗ g

^{equals the tensor produt of}

f

^and

g

^{while, for}

p = q

^,

f ⊗ p g = h f, g i H ^⊗p

^{. Notethat, ingeneral (and exeptfor trivial ases), theontration}

f ⊗ r g

^{is not a}

symmetrielement of

H ^{⊗ (p+q − 2r)}

^{. The anonial} symmetrization of

f ⊗ r g

^{is written}

f ⊗ e r g

^.

We alsohave the usefulmultipliation formula: if

f ∈ H ^{⊙ p}

^and

g ∈ H ^{⊙ q}

^,then

I _p (f )I _q (g) =

p ∧ q

X

r=0

r!

p r

q r

I _{p+q − 2r} (f ⊗ e r g).

^(2.29)

Let

S

^{be the setof all smooth ylindrial random variables of theform}

F = g X(φ ₁ ), . . . , X(φ _n )

where

n > 1

^,

g : R ⁿ → R

is a smooth funtion with ompat support and

φ _i ∈ H

^{. The}

Malliavinderivative of

F

^withrespetto

X

^{isthe element of}

L ² (Ω, H )

^{dened as}

DF =

X n

i=1

∂g

∂x _i X(φ ₁ ), . . . , X(φ _n ) φ _i .

In partiular,

DX(h) = h

^{for every}

h ∈ H

^{. By iteration, one an dene the}

m

^{th derivative}

D ^m F

^{(whihisanelementof}

L ² (Ω, H ^{⊗ m} )

^)forevery

m > 2

^{. Asusual,for}

m > 1

^,

D ^m,2

denotes the losureof

S

^{withrespetto the norm}

k · k m,2

^{,dened bytherelation}

k F k ² m,2 = E F ²

+ X m

i=1

E

k D ⁱ F k ² H ^⊗i

.

Note that, if

F 6 = 0

^and

F

^{is equal to a nite sum of multiple Wiener-It integrals, then}

F ∈ D ^m,2

for every

m > 1

^{and the law of}

F

^{admits a density with respet to the Lebesgue}

measure. TheMalliavin derivative

D

^{satisesthe followinghainrule: if}

ϕ : R ⁿ → R

isin

C _b ¹

(thatis,theolletionofontinuouslydierentiablefuntionswitha boundedderivative) and

if

{ F _i } i=1,...,n

^{isavetor of elementsof}

D ^1,2

,then

ϕ(F ₁ , . . . , F _n ) ∈ D ^1,2

and

Dϕ(F ₁ , . . . , F _n ) = X n

i=1

∂ϕ

∂x _i (F ₁ , . . . , F _n )DF _i .

Observe that the previous formula still holds when

ϕ

^{is a Lipshitz funtion and the law of}

(F 1 , . . . , F n )

^{hasa densitywithrespetto theLebesgue measureon}

R ⁿ

(see e.g. Proposition 1.2.3 in[35 ℄).

We denote by

δ

^{the adjoint of the operator}

D

^{, also alled the divergene operator. A}

random element

u ∈ L ² (Ω, H )

^{belongs to the domain of}

δ

^{, noted}

Domδ

^{, if, and only if, it}

satises

| E h DF, u i ^H | 6 c _u k F k L ²

^{for any}

F ∈ S ,

where

c _u

^{isaonstant depending uniquelyon}

u

^{. If}

u ∈ Domδ

^{,thentherandomvariable}

δ(u)

isdened by the duality relationship(ustomarily alledintegration bypartsformula):

E(F δ(u)) = E h DF, u i ^H ,

^(2.30)

whih holds for every

F ∈ D ^1,2

. One sometimes needs the following property: for every

F ∈ D ^1,2

and every

u ∈ Domδ

^{suh that}

F u

^and

F δ(u) + h DF, u i ^H

^aresquareintegrable, one hasthat

F u ∈ Domδ

^and

δ(F u) = F δ(u) − h DF, u i ^H .

^(2.31)

Theoperator

L

^{,atingonsquareintegrablerandomvariablesofthetype(2.27),isdened}

through theprojetion operators

{ J _q } q > 0

^as

L = P _∞

q=0 − qJ _q ,

^{and is alled the} innitesimal generator of the Ornstein-Uhlenbek semigroup. It veries the following ruial property: a

random variable

F

^{isanelementof}

DomL (= D ^2,2 )

^{if,andonlyif,}

F ∈ DomδD

^(i.e.

F ∈ D ^1,2

and

DF ∈ Domδ

^{),and inthisase:}

δDF = − LF.

^{Notethatarandomvariable}

F

^asin(2.27)

is in

D ^1,2

(resp.

D ^2,2

) if,and only if,

X ∞

q=1

q k f _q k ² H ^⊙q < ∞ (

^resp.

X ∞

q=1

q ² k f _q k ² H ^⊙q < ∞ ),

and also

E

k DF k ² H

= P

q > 1 q k f _q k ² H ^⊙q

^{. If}

H = L ² (A, A , µ)

^(with

µ

non-atomi), then the derivativeofarandomvariable

F

^{asin(2.27) anbeidentiedwiththeelementof}

L ² (A × Ω)

given by

D a F = X ∞

q=1

qI q − 1 f q ( · , a)

, a ∈ A.

^(2.32)

We also dene the operator

L ^{− 1}

^{,whih is the inverse of}

L

^{,as follows: for every}

F ∈ L ² (X)

withzero mean, we set

L ^{− 1} F = P

q>1 − ¹ _q J _q (F)

^{. Note that}

L ^{− 1}

^{is an operator with values}

in

D ^2,2

. The following Lemma ontains two statements: therst one (formula (2.33)) is an immediateonsequeneofthedenitionof

L

^{andoftherelation}

δD = − L

^{,whereastheseond}

(formula (2.34))orrespondsto Lemma 2.1in[33℄.

Lemma 2.1 Fix an integer

q > 2

^{and set}

F = I _q (f )

^{, with}

f ∈ H ^{⊙ q}

^{. Then,}

δDF = qF.

^(2.33)

Moreover, for every integer

s > 0

^,

E F ^s k DF k ² H

= q

s + 1 E F ^s+2

.

^(2.34)

3 Stein's method and integration by parts on Wiener spae

3.1 Gaussian approximations

Our rst result provides expliit bounds for the normal approximation of random variables

thatareMalliavinderivable. Althoughitsproofisquiteeasytoobtain,thefollowingstatement

will beentral for the rest ofthe paper.

Theorem 3.1 Let

Z ∼ N (0, 1)

^,andlet

F ∈ D ^1,2

be suhthat

E(F ) = 0

^{. Then,thefollowing}

bounds are inorder:

d _W (F, Z) 6 E[(1 − h DF, − DL ^{− 1} F i ^H ) ² ] ^1/2 ,

^(3.35)

d _FM (F, Z) 6 4E[(1 − h DF, − DL ^{− 1} F i ^H ) ² ] ^1/2 .

^(3.36)

If,inaddition,the lawof

F

^{isabsolutelyontinuouswithrespettotheLebesgue measure, one}

hasthat

d _Kol (F, Z) 6 E[(1 − h DF, − DL ^{− 1} F i ^H ) ² ] ^1/2 ,

^(3.37)

d _TV (F, Z) 6 2E[(1 − h DF, − DL ^{− 1} F i ^H ) ² ] ^1/2 .

^(3.38)

Proof. Start by observing that one an write

F = LL ^{− 1} F = − δDL ^{− 1} F

^{. Now let}

f

^be

a real dierentiable funtion. By using the integration by parts formula and the fat that

Df(F ) = f ^′ (F )DF

^{(notethat, forthis formulato holdwhen}

f

^isonlya.e. dierentiable, one needs

F

^{to have anabsolutelyontinuous law, see}Proposition 1.2.3 in[35 ℄),we dedue

E(F f (F )) = E[f ^′ (F) h DF, − DL ^{− 1} F i ^H ].

Itfollowsthat

E[f ^′ (F ) − F f (F )] = E(f ^′ (F )(1 − h DF, − DL ^{− 1} F i ^H ))

^{sothatrelations (3.35)}

(3.38) an bededued from(1.8)(1.11) andthe Cauhy-Shwarz inequality.

2

We shall now prove that the bounds appearing in the statement of Theorem 3.1 an be

expliitly omputed, whenever

F

^{belongs to axed Wienerhaos.}

Proposition 3.2 Let

q > 2

^{be an integer, and let}

F = I _q (f )

^{, where}

f ∈ H ^{⊙ q}

^{. Then,}

h DF, − DL ^{− 1} F i ^H = q ^{− 1} k DF k ² H

^{, and}

E[(1 − h DF, − DL ^{− 1} F i ^H ) ² ] = E[(1 − q ^{− 1} k DF k ² H ) ² ]

^(3.39)

= (1 − q! k f k ² H ^⊗q ) ² + q ²

q − 1

X

r=1

(2q − 2r)!(r − 1)! ² q − 1

r − 1 4

k f ⊗ e r f k ² _H ^⊗2(q−r)

^(3.40)

6 (1 − q! k f k ² H ^⊗q ) ² + q ²

q − 1

X

r=1

(2q − 2r)!(r − 1)! ² q − 1

r − 1 4

k f ⊗ r f k ² H ^⊗2(q−r) .

^(3.41)

Proof. The equality

h DF, − DL ^{− 1} F i ^H = q ^{− 1} k DF k ² H

^{is an immediate onsequene of the}

relation

L ^{− 1} I _q (f ) = − q ^{− 1} I _q (f )

^{. Fromthe}multipliationformulaebetweenmultiplestohasti integrals, see(2.29),one dedues that

k D[I _q (f)] k ² H = qq! k f k ² H ^⊗q + q ²

q − 1

X

r=1

(r − 1)!

q − 1 r − 1

2

I _{2(q − r)} f ⊗ e r f

(3.42)

(seealso[36,Lemma 2℄). Wethereforeobtain (3.40)byusingtheorthogonalityandisometri

properties of multiple stohasti integrals. The inequality in (3.41) is just a onsequene of

the relation

k f ⊗ e r f k H ^⊗2(q−r) 6 k f ⊗ r f k H ^⊗2(q−r)

^.

2

The previous result should be ompared with the forthoming Theorem 3.3, where we

ollet the main ndings of [36 ℄ and [37℄. In partiular, the ombination of Proposition 3.2

and Theorem 3.3 shows that, for every (normalized) sequene

{ F _n : n > 1 }

^{living in a xed}

Wiener haos, the bounds given in (3.35)(3.36) are tight with respet to theonvergene

in distributiontowards

Z ∼ N (0, 1)

^{,inthe sense thatthese boundsonvergeto zeroif, and}

only if,

F n

^{onverges in}distribution to

Z

^.

Theorem 3.3 ([36, 37℄) Fix

q > 2

^{, and onsider a sequene}

{ F _n : n > 1 }

^{suh that}

F _n = I _q (f _n )

^,

n > 1

^{, where}

f _n ∈ H ^{⊙ q}

^{. Assume moreover that}

E[F _n ² ] = q! k f _n k ² H ^⊗q → 1

^{. Then, the}

followingfour onditions are equivalent, as

n → ∞

^:

(i)

F _n

^{onverges in} distributionto

Z ∼ N (0, 1)

^;

(ii)

E[F _n ⁴ ] → 3

^;

(iii) for every

r = 1, ..., q − 1

^,

k f _n ⊗ r f _n k H ^⊗2(q−r) → 0

^;

(iv)

k DF _n k ² H → q

ⁱⁿ

L ²

^.

The impliations (i)

↔

⁽ⁱⁱ⁾

↔

^{(iii) have been rst proved in}

[

³⁷

]

^{by means of stohasti}

alulus tehniques. The fat that (iv) is equivalent to either one of onditions (i)(iii) is

proved in[36℄. Note thatTheorem 3.1 and Proposition 3.2above provide an alternate proof

of the impliations (iii)

→

^(iv)

→

^{(i). The impliation (ii)}

→

^{(i) an be seen as a drasti}

simpliation of the method of moments and umulants, that is a ustomary toolin order

to prove limit theorems for funtionals of Gaussian elds (see e.g. [5, 7,18, 27 , 50 ℄). In [39 ℄

onean nd amultidimensional versionofTheorem 3.3.

works,suh as:

p

-variationsof stohastiintegrals withrespetto Gaussian proesses[2, 12 ℄, quadrati funtionalsofbivariateGaussian proesses[14℄,self-intersetionloaltimesoffra-

tionalBrownianmotion[24℄,approximationshemesforsalarfrationaldierentialequations

[32 ℄,high-frequenyCLTsforrandomeldsonhomogeneousspaes[29,30,38 ℄,needletsanal-

ysis on the sphere [1 ℄, estimation of self-similarity orders [55℄, power variations of iterated

Brownian motions [34℄. We expetthat the new bounds proved in Theorem 3.1 and Propo-

sition 3.2 will lead to further renements of these results. See Setion 4 and Setion 5 for

appliationsand examples.

As shown in the following statement, the ombination of Proposition 3.2 and Theorem

3.3implies that,onanyxedWienerhaos,theKolmogorov,total variationandWasserstein

distanes metrize theonvergene indistribution towards Gaussian random variables. Other

topologial haraterizations of the set of laws of random variables belonging to a xed sum

of Wiener haoses aredisussed in[25 , Ch. VI℄.

Corollary 3.5 Let the assumptions and notation of Theorem 3.3 prevail. Then, the fat

that

F _n

^{onverges in} distribution to

Z ∼ N (0, 1)

^{is equivalent to either one of the following}

onditions:

(a)

d _Kol (F _n , Z) → 0

^;

(b)

d TV (F n , Z) → 0

^;

()

d _W (F _n , Z ) → 0

^.

Proof. If

F _n ^Law → Z

^{then,byTheorem 3.3, we have neessarilythat}

k DF _n k ² H → q

ⁱⁿ

L ²

^{. The}

desired onlusionfollows immediately fromrelations (3.35), (3.37),(3.38) and (3.39).

2

Notethatthe previous resultisnot trivial, sine thetopologiesindued by

d _Kol

^,

d _TV

^and

d _W

^{are strongerthan onvergene in}distribution.

Remark3.6 (Mehler's formula and Stein's method, I). In [9, Lemma 5.3℄, Chatterjee has

proved the following result (we use a notation whih is slightly dierent from the original

statement). Let

Y = g(V )

^{, where}

V = (V ₁ , ..., V _n )

^{is a vetor of entered i.i.d. standard}

Gaussian random variables, and

g : R ⁿ → R

is a smooth funtion suh that: (i)

g

^{and its}

derivatives have subexponential growth at innity, (ii)

E(g(V )) = 0

^{,and (iii)}

E(g(V ) ² ) = 1

^.

Then,for anyLipshitz funtion

f

^{,one hasthat}

E[Y f (Y )] = E[S(V )f ^′ (Y )],

^(3.43)

where,for every

v = (v 1 , ..., v n ) ∈ R ⁿ

,

S(v) = Z ₁

0

1 2 √

t E

" _n X

i=1

∂g

∂v _i (v) ∂g

∂v _i ( √ tv + √

1 − tV )

#

dt,

^(3.44)

sothat, for instane, for

Z ∼ N (0, 1)

^{andbyusing(1.9),Lemma 1.2(iii),(1.7) and Cauhy-}

Shwarzinequality,

d _TV (Y, Z) 6 2E[(S(V ) − 1) ² ] ^1/2 .

^(3.45)

of generality, we an assume that

V _i = X(h _i )

^{, where}

X

^{is an isonormal proess over some}

Hilbert spae of the type

H = L ² (A, A , µ)

^and

{ h ₁ , ..., h _n }

^{is an} orthonormal system in

H

^{. Sine}

Y = g(V 1 , . . . , V n )

^{, we have}

D a Y = P n i=1

∂g

∂x i (V )h i (a)

^{. On the other hand, sine}

Y

^{is entered and square} integrable, it admits a haoti representation of the form

Y = P

q>1 I _q (ψ _q )

^{. Thisimpliesinpartiularthat}

D _a Y = P _∞

q=1 qI _{q − 1} (ψ _q (a, · ))

^{. Moreover, onehas}

that

− L ^{− 1} Y = P

q>1 1

q I _q (ψ _q )

^{, so that}

− D _a L ^{− 1} Y = P

q>1 I _{q − 1} (ψ _q (a, · ))

^{. Now, let}

T _z

^,

z > 0

^,

denote the (innite dimensional) Ornstein-Uhlenbek semigroup, whose ation on random

variables

F ∈ L ² (X)

^{isgiven by}

T _z (F) = P

q>0 e ^{− qz} J _q (F )

^{. Wean write}

Z ₁

0

1 2 √

t T _ln(1/ ^√ _t) (D _a Y )dt = Z _∞

0

e ^{− z} T _z (D _a Y )dz = X

q> 1

1

q J _{q − 1} (D _a Y )

= X

q>1

I _{q − 1} (ψ _q (a, · )) = − D _a L ^{− 1} Y .

^(3.46)

NowreallthatMehler'sformula(seee.g. [35 ,formula(1.54)℄)impliesthat,foreveryfuntion

f

^withsubexponential growth,

T _z (f(V )) = E

f (e ^{− z} v + p

1 − e ^{− 2z} V ) _v=V , z > 0.

In partiular, by applying this last relation to the partial derivatives

∂g

∂v i

,

i = 1, ..., n

^{, we}

dedue from(3.46) that

Z ₁

0

1 2 √

t T _ln(1/ ^√ _t) (D _a Y )dt = X n

i=1

h _i (a) Z ₁

0

1 2 √

t E ∂g

∂v _i ( √ t v + √

1 − t V ) dt

v=V .

Consequently, (3.43) follows,sine

h DY, − DL ^{− 1} Y i ^H =

* _n X

i=1

∂g

∂v i

(V )h _i , X n

i=1

Z 1

0

1 2 √

t E ∂g

∂v i

( √

t v + √

1 − t V ) dt

v=V h _i +

H

= S(V ).

2

See also Houdré and Pérez-Abreu [22℄ for related omputations in an innite-dimensional

setting.

The following resultonerns nite sumsof multiple integrals.

Proposition 3.7 For

s > 2

^{, x}

s

^integers

2 6 q 1 < . . . < q s

^{. Consider a sequene of the}

form

Z _n = X s

i=1

I _{q i} (f _n ⁱ ), n > 1,

where

f _n ⁱ ∈ H ^{⊙ q i}

^{. Set}

I =

(i, j, r) ∈ { 1, . . . , s } ² × N : 1 6 r 6 q _i ∧ q _j

^and

(r, q _i , q _j ) 6 = (q _i , q _i , q _i ) .