Cumulants on the Wiener space

arXiv:0910.1996v1 [math.PR] 12 Oct 2009

by IvanNourdin

∗

and Giovanni Pe ati

†

Université Paris VI and Université Paris Ouest

Abstra t: We ombineinnite-dimensionalintegrationbypartspro edureswithare ursive rela-tiononmoments(reminis entofaformulabyBarbour(1986)),anddedu eexpli itexpressionsfor umulants of fun tionals of a general Gaussian eld. Thesendings yield a ompa t formulafor umulants on a xed Wiener haos, virtually repla ing theusual graph/diagram omputations adoptedinmost of the probabilisti literature.

Key words: Cumulants; Diagram Formulae; Gaussian Pro esses; Malliavin al ulus; Ornstein-Uhlenbe kSemigroup.

2000 Mathemati s Subje t Classi ation: 60F05;60G15; 60H05;60H07.

1 Introdu tion

The integration by parts formula of Malliavin al ulus, ombining derivative operators and anti ipative integrals into a exible tool for omputing and assessing mathemati al expe tations, isa ornerstoneof modernsto hasti analysis. The s opeof itsappli ations, ranging e.g. from density estimates for solutions of sto hasti dierential equations to on entration inequalities,from anti ipative sto hasti al ulus to Greeks omputations inmathemati alnan e, is vividlydes ribed inthe three lassi monographs by Malliavin [10℄, Janson [9℄and Nualart [20℄.

Inre entyears,innite-dimensionalintegrationbypartste hniques havefoundanother fertile ground for appli ations, that is, limit theorems and (more generally) probabilisti approximations. The starting point of this a tive lineof resear h is the paper [21℄, where theauthorsuse Malliavin al ulusinordertorenesome riteriaforasymptoti normality onaxedWiener haos,originallyprovedin[22,24℄(seealso[12℄forsomenon- entral ver-sion of these results). Another importantstep appears in[13℄, where integration by parts on Wiener spa e is ombined with the so- alled Stein's method for probabilisti approxi-mations (see e.g. [6, 25℄), thus yielding expli it upper bounds in the normal and gamma approximations of the lawof fun tionals of Gaussian elds. The te hniques introdu ed in [13℄ have ledto several appli ations and generalizations, for instan e: in[14℄ one an nd appli ationstoEdgeworthexpansions andreversedBerry-Esseeninequalities;[18℄ ontains results formultivariate normalapproximations;[1℄ fo usesonfurther developments inthe

∗

Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, Boîte ourrier 188, 4Pla e Jussieu,75252ParisCedex 5,Fran e. Email: ivan.nourdinupm .fr

†

se ond order Poin aré inequalities; in [19℄, one an nd new expli it expressions for the density of fun tionals of Gaussian eld as well as new on entration inequalities (see also [3℄ for some appli ations in mathemati al statisti s); in [17℄, the results of [13℄ are om-bined withLindeberg-type invarian eprin iples in order todedu e universality results for homogeneoussums (these ndings are further appliedin [15℄ to randommatrix theory).

The aim of this note is to develop yet another striking appli ation of the innite-dimensional integration by parts formula of Malliavin al ulus, namely the omputation of umulants for general fun tionals of a given Gaussian eld. As dis ussed below, our te hniquesmakea ru ialuseofare ursiveformulaformoments(seerelation(2.2)below), whi his the startingpoint ofsome well-known omputationsperformedby Barbour in[2℄ in onne tionwiththe Stein'smethodfor normalapproximations. Assu h,thete hniques developed inthe forth omingse tions an be seen as further rami ations of the ndings of [13℄.

The main a hievement of the present work is a re ursive formula for umulants (see (4.19)), based on a repeated use of integration by parts. Note that umulants of order greaterthantwoarenot linearoperators(forinstan e, these ond umulant oin ideswith the varian e): however, our formula (4.19) implies that umulants of regular fun tionals of Gaussian elds an be always represented as the mathemati al expe tation of some re ursively dened random variable. We shall prove in Se tion 5 that this implies a new ompa t representation for umulants of random variables belonging to a xed Wiener haos. We laim that this result may repla e the lassi diagram omputations adopted inmostoftheprobabilisti literature(seee.g. [4,5,8℄,aswellas[23℄forageneraldis ussion of related ombinatorialresults).

Thepaperisorganizedasfollows. InSe tion2westateandprovesomeusefulre ursive formulae for moments. Se tion 3 ontains basi on epts and results related to Malliavin al ulus. Se tion 4 is devoted to our main statements about umulants onWiener spa e. Finally, in Se tion 5 we spe ialize our results to random variables ontained in a xed Wiener haos.

From nowon, every random obje t is dened on a ommon suitableprobabilityspa e

(Ω, F, P )

.

A knowledgements. Part of this paper has been written while we were visiting the DepartmentofStatisti sofPurdueUniversity, inthe o asionof theworkshopSto hasti Analysisat Purdue, from September 28to O tober 1,2009. We heartily thank Professor Frederi G.Viens forhis warm hospitality and generous support.

2 Moment expansions

nite moments. Sin e we only need Barbour's results in the spe ial ase of polynomial transformations, and for the sake of ompleteness, we hoose to provide a self- ontained presentationinthis simplersetting. See alsoRotar'[27℄forfurtherextensions ofBarbour's ndings.

Denition 2.1 (Cumulants) Let

X

beareal-valuedrandomvariablesu hthat

E|X|

m

_<

∞

for some integer

m > 1

, and dene

φ

X

(t) = E(e

itX

₎

,

t ∈ R

, to be the hara teristi

fun tion of

X

. Then, for

j = 1, ..., m

, the

j

th umulant of

X

, denoted by

κ

j

(X)

, is given by

κ

j

(X) = (−i)

j

d

j

dt

j

log φ

X

(t)|

t=0

.

(2.1)

For instan e,

κ

1

(X) = E(X)

,

κ

2

(X) = E(X

2

_{) − E(X)}

2

_{= Var(X)}

,

κ

3

(X) = E(X

3

_{) −}

3E(X

2

_{)E(X) + 2E(X)}

3

,and so on.

The following relationis exploitedthroughout the paper. Proposition 2.2 Fix

m = 0, 1, 2...

, and suppose that

E|X|

m+1

_{< ∞}

. Then

E(X

m+1

) =

m

X

s=0



m

s



κ

s+1

(X)E(X

m−s

).

(2.2)

Proof. ByLeibniz rule, one has that

E(X

m+1

_{) = (−i)}

m+1

d

m+1

dt

m+1

φ

X

(t)|

t=0

= (−i)

m+1

d

m

dt

m



d

dt

log φ

X

(t)



φ

X

(t)









t=0

=

"

_m

X

s=0

(−i)

s+1



m

s



d

s+1

dt

s+1

(log φ

X

(t)) × (−i)

m−s

d

m−s

dt

m−s

φ

X

(t)

#







t=0

,

with

d

0

dt

0

equalto the identity operator. This yields the desired on lusion.

Finally,observe that (2.2) an be rewritten as

E(X

m+1

) =

m

X

s=0

κ

s+1

(X)

s!

m(m − 1) · · · (m − s + 1)E(X

m−s

_),

expansion

E(Xf (X)) =

∞

X

s=0

κ

s+1

(X)

s!

E



d

s

dx

s

f (X)



.

holds for some innitelydierentiable fun tion

f

whi h is not ne essarilya polynomial.

3 Malliavin operators and Gaussian analysis

Weshall nowpresent the basi elements of Gaussian analysis and Malliavin al ulusthat are used inthis paper. The reader isreferred to[9, 10, 20℄ for any unexplained denition orresult.

Let

H

be a real separable Hilbert spa e. For any

q > 1

, let

H

⊗q

be the

q

th tensor power of

H

and denote by

H

⊙q

the asso iated

q

th symmetri tensor power. We write

X = {X(h), h ∈ H}

to indi ate an isonormal Gaussian pro ess over

H

, dened on some

probability spa e

(Ω, F, P )

. This means that

X

is a entered Gaussian family, whose ovarian eisgivenbytherelation

E [X(h)X(g)] = hh, gi

H

. Wealsoassumethat

F = σ(X)

, that is,

F

is generated by

X

.

For every

q > 1

, let

H

q

be the

q

th Wiener haos of

X

, dened as the losed linear subspa e of

L

2

_{(Ω, F, P )}

generated by the family

{H

q

(X(h)), h ∈ H, khk

H

= 1}

, where

H

q

is the

q

th Hermite polynomialgiven by

H

q

(x) = (−1)

q

e

x2

2

d

q

dx

q

e

−

x2

₂

_.

We write by onvention

H

0

= R

. For any

q > 1

, the mapping

I

q

(h

⊗q

_{) = q!H}

q

(X(h))

an

be extended to a linear isometry between the symmetri tensor produ t

H

⊙q

(equipped with the modied norm

√

q! k·k

H

⊗q

) and the

q

th Wiener haos

H

q

. For

q = 0

, we write

I

0

(c) = c

,

c ∈ R

. It is well-known that

L

2

_{(Ω) := L}

2

_{(Ω, F, P )}

an be de omposed into the innite orthogonal sum of the spa es

H

q

. It follows that any square integrable random variable

F ∈ L

2

_(Ω)

admitsthe following (Wiener-It) haoti expansion

F =

∞

X

q=0

I

q

(f

q

),

(3.3)

where

f

0

= E[F ]

, and the

f

q

∈ H

⊙q

,

q > 1

, are uniquely determined by

F

. For every

q > 0

, we denote by

J

q

the orthogonal proje tion operator on the

q

th Wiener haos. In

parti ular, if

F ∈ L

2

_(Ω)

isas in (3.3),then

J

q

F = I

q

(f

q

)

for every

q > 0

.

Let

{e

k

, k > 1}

be a omplete orthonormalsystem in

H

. Given

f ∈ H

⊙p

and

g ∈ H

⊙q

, forevery

r = 0, . . . , p ∧ q

,the ontra tion of

f

and

g

oforder

r

isthe elementof

H

Noti ethatthedenitionof

f ⊗

r

g

doesnotdependonthe parti ular hoi eof

{e

k

, k > 1}

, and that

f ⊗

r

g

is not ne essarily symmetri ; we denote its symmetrization by

f e

⊗

r

g ∈

H

⊙(p+q−2r)

. Moreover,

f ⊗

0

g = f ⊗ g

equalsthe tensorprodu tof

f

and

g

while, for

p = q

,

f ⊗

q

g = hf, gi

H

⊗q

.

It an also be shown that the following multipli ation formula holds: 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).

(3.5)

We now introdu e some basi elements of the Malliavin al ulus with respe t to the isonormalGaussian pro ess

X

. Let

S

be the set of all ylindri al randomvariablesof the form

F = g (X(φ

1

), . . . , X(φ

n

)) ,

(3.6)

where

n > 1

,

g : R

n

_{→ R}

is an innitely dierentiable fun tion su h that its partial derivativeshave polynomial growth, and

φ

i

∈ H

,

i = 1, . . . , n

. The Malliavin derivativeof

F

with respe t to

X

is the element of

L

2

_{(Ω, H)}

dened as

DF =

n

X

i=1

∂g

∂x

i

(X(φ

1

), . . . , X(φ

n

)) φ

i

.

Inparti ular,

DX(h) = h

forevery

h ∈ H

. Byiteration,one an denethe

m

thderivative

D

m

_F

, whi h is an element of

L

2

_{(Ω, H}

⊙m

₎

, for every

m > 2

. For

m > 1

and

p > 1

,

D

m,p

denotes the losureof

S

with respe t tothe norm

k · k

m,p

,dened by the relation

kF k

p

m,p

= E [|F |

p

] +

m

X

i=1

E kD

i

F k

p

H

⊗i

.

One also writes

D

∞

₌

T

m>1

T

p>1

D

m,p

. The Malliavin derivative

D

obeys the following hain rule. If

ϕ : R

n

_{→ R}

is ontinuously dierentiable with bounded partial derivatives

and if

F = (F

1

, . . . , F

n

)

is ave tor of elementsof

D

1,2

, then

ϕ(F ) ∈ D

1,2

and

D ϕ(F ) =

n

X

i=1

∂ϕ

∂x

i

(F )DF

i

.

Notealsothatarandomvariable

F

asin(3.3)isin

D

1,2

ifandonlyif

P

∞

q=1

qkJ

q

F k

2

_L

2

_(Ω)

< ∞

and, in this ase,

E (kDF k

We denote by

δ

the adjoint of the operator

D

, also alled the divergen e operator. A random element

u ∈ L

2

_{(Ω, H)}

belongs to the domain of

δ

, noted

Domδ

, if and only if it veries

|EhDF, ui

H

| 6 c

u

kF k

L

2

(Ω)

forany

F ∈ D

1,2

, where

c

u

isa onstantdependingonly

on

u

. If

u ∈ Domδ

, then the random variable

δ(u)

is dened by the duality relationship

( alled integration by parts formula)

E(F δ(u)) = EhDF, ui

H

,

(3.8)

whi hholds for every

F ∈ D

1,2

.

The family

(P

t

, t > 0)

of operators isdened through the proje tion operators

J

q

as

P

t

=

∞

X

q=0

e

−qt

J

q

,

(3.9)

andis alledtheOrnstein-Uhlenbe ksemigroup. Assumethatthepro ess

X

′

,whi hstands foranindependent opyof

X

,issu hthat

X

and

X

′

aredenedontheprodu tprobability spa e

(Ω × Ω

′

_{, F ⊗ F}

′

_{, P × P}

′

₎

. Given a random variable

F ∈ D

1,2

, we an regard it as a measurable mapping from

R

H

to

R

, determined

P ◦ X

−1

-almost surely. Then, for any

t > 0

, we have the so- alled Mehler'sformula:

P

t

F = E

′

F (e

−t

X +

√

1 − e

−2t

_X

′

₎

_,

(3.10) where

E

′

denotes the mathemati al expe tation with respe t to the probability

P

′

. By meansof thisformula, itisimmediatetoprovethat

P

t

isa ontra tionoperatoron

L

p

_(Ω)

,

for all

p > 1

.

Theoperator

L

isdenedas

L =

P

∞

q=0

−qJ

q

,andit anbeproventobetheinnitesimal generator of the Ornstein-Uhlenbe k semigroup

(P

t

)

t>0

. The domainof

L

is

DomL = {F ∈ L

2

(Ω) :

∞

X

q=1

q

2

_kJ

q

F k

2

_L

2

_(Ω)

< ∞} = D

2,2

.

There is an important relation between the operators

D

,

δ

and

L

. A random variable

F

belongs to

D

2,2

if and onlyif

F ∈ Dom (δD)

(i.e.

F ∈ D

1,2

and

DF ∈ Domδ

) and, in this

ase,

δDF = −LF.

(3.11) For any

F ∈ L

2

_(Ω)

, we dene

L

−1

_{F =}

P

∞

q=0

−

1

q

J

q

(F )

. The operator

L

−1

is alled the pseudo-inverse of

L

. Indeed, for any

F ∈ L

2

_(Ω)

, we havethat

L

−1

_{F ∈ DomL = D}

2,2

, and

LL

−1

_{F = F − E(F ).}

(3.12)

Lemma 3.1 Suppose that

F ∈ D

1,2

and

G ∈ L

2

_(Ω)

. Then,

L

−1

_{G ∈ D}

2,2

and we have:

E(F G) = E(F )E(G) + E(hDF, −DL

−1

_Gi

H

).

(3.13)

Proof. By (3.11)and (3.12),

E(F G) − E(F )E(G) = E(F (G − E(G))) = E(F × LL

−1

G) = E(F δ(−DL

−1

G)),

and the result is obtained by usingthe integration by parts formula(3.8). Remark 3.2 Observethat

hDF, −DL

−1

_Gi

H

is not ne essarilysquare-integrable, albeitit is by onstru tion in

L

1

_(Ω)

. On the other hand,we have

hDF, −DL

−1

Gi

H

=

Z

∞

0

e

−t

_{hDF, P}

t

DGi

H

dt,

(3.14)

see indeed identity (3.46) in[13℄.

The next result isa onsequen e of the previous Lemma 3.1. Proposition 3.3 Fix

p > 2

, and assume that

F ∈ L

4p−4

_{(Ω) ∩ D}

1,4

. Then,

F

p

_{∈ D}

1,2

and

DF

p

_{= pF}

p−1

_DF

. Moreover, for any

G ∈ L

2

_(Ω)

,

E(F

p

G) = E(F

p

)E(G) + pE(F

p−1

_{hDF, −DL}

−1

_Gi

H

).

(3.15)

4 A re ursive formula for umulants

The aim of this se tion is to dedu e from formula(2.2) are ursive relationfor umulants of su iently regular fun tionals of the isonormal pro ess

X

. To do this, we need to (re ursively) introdu e some further notation.

Denition 4.1 Let

F ∈ D

1,2

. We dene

Γ

0

(F ) = F

and

Γ

1

(F ) = hDF, −DL

−1

_{F i}

H

. If,

for

j > 1

,the randomvariable

Γ

j

(F )

is awell-dened elementof

L

2

_(Ω)

, we set

Γ

j+1

(F ) =

hDF, −DL

−1

_Γ

j

(F )i

H

. Observethatthedenitionof

Γ

j+1

(F )

iswellgiven,sin e(asalready

observed in general) the square-integrability of

Γ

j

(F )

implies that

L

−1

_Γ

j

(F ) ∈ Dom L =

D

2,2

_{⊂ D}

1,2

.

The followingstatementprovidessu ient onditions on

F

,ensuring that the random variable

Γ

j

(F )

iswell dened.

Lemma 4.2 1. Fix an integers

j > 1

, and let

F, G ∈ D

j,2

j

. Then

hDF, −DL

−1

_Gi

H

∈

D

j−1,2

j−1

, where we set by onvention (but onsistently!)

D

0,1

_{= L}

1

_(Ω)

. 2. Fixaninteger

j > 1

,andlet

F ∈ D

j,2

j

. Then,forall

k = 1, . . . , j

, wehavethat

Γ

k

(F )

isa well-denedelement of

D

j−k,2

j−k

; in parti ular, one has that

Γ

k

(F ) ∈ L

1

_(Ω)

3. If

F ∈ D

∞

(in parti ularif

F

equals a nitesum of multipleintegrals), then

Γ

j

(F ) ∈

D

∞

for every

j > 0

.

Proof. Without loss of generality, we assume during all the proof that

H

has the form

L

2

_{(A, A , µ)}

, where

(A, A )

is a measurable spa e and

µ

is a

σ

-nite measure with no atoms.

1. Let

k ∈ {0, . . . , j − 1}

. UsingLeibniz rule for

D

(see e.g. [20, Exer i e 1.2.13℄),one

has that

− D

k

hDF, −DL

−1

Gi

H

= D

k

Z

A

D

a

F D

a

L

−1

G µ(da)

(4.16)

=

k

X

l=0



k

l

 Z

A

D

k−l

(D

a

F ) e

⊗D

l

(D

a

L

−1

G) µ(da),

with

⊗

e

the usual symmetri tensor produ t. Notethat, todedu e (4.16), itissu ient to onsider randomvariables

F, G

that havethe form(3.6) (forwhi hthe formulaisevident, sin e itbasi allyboilsdown tothe originalLeibnizrule fordierential al ulus), andthen toapply astandard approximation argument. From (4.16), one therefore dedu es that

kD

k

hDF, −DL

−1

Gi

H

k

H

⊗k

6

k

X

l=0



k

l



Z

A

D

k−l

(D

a

F ) e

⊗D

l

(D

a

L

−1

G) µ(da)

H

⊗k

6

k

X

l=0



k

l

 Z

A

kD

k−l

_(D

a

F )k

H

⊗(k−l)

kD

l

_(D

a

L

−1

G)k

H

⊗l

µ(da)

6

k

X

l=0



k

l

sZ

A

kD

k−l

_(D

a

F )k

2

_H

⊗(k−l)

µ(da)

sZ

A

kD

l

_(D

a

L

−1

G)k

2

H

⊗l

µ(da)

=

k

X

l=0



k

l



kD

k−l+1

F k

H

⊗(k−l+1)

kD

l+1

L

−1

Gk

_H

⊗(l+1)

.

(4.17) By mimi king the argumentsused in the proof of Proposition 3.1in[16℄(see also(3.14)), it ispossibleto prove that

−D

l+1

L

−1

G =

Z

∞

0

Consequently, for any real

p > 1

,

E



_kD

l+1

L

−1

_Gk

p

_H

⊗(l+1)



6

E

Z

∞

0

e

−(l+1)t

_kP

t

D

l+1

Gk

H

⊗(l+1)

dt



p



6

1

(l + 1)

p−1

Z

∞

0

e

−(l+1)t

E



_kP

t

D

l+1

Gk

p

_H

⊗(l+1)



dt

6

1

(l + 1)

p−1

E



kD

l+1

Gk

p

H

⊗(l+1)

 Z

∞

0

e

−(l+1)t

dt

=

1

(l + 1)

p

E



kD

l+1

Gk

p

H

⊗(l+1)



,

(4.18)

where, to get the last inequality, we used the ontra tion property of

P

t

on

L

p

_(Ω)

. Fi-nally, by ombining (4.17) with (4.18) through the Cau hy-S hwarz inequality on the one hand, and by the assumptions on

F

and

G

on the other hand, we immediately get that

kD

k

_{hDF, −DL}

−1

_Gi

H

k

H

⊗k

belongs to

L

2

j−1

(Ω)

for all

k = 0, . . . , j − 1

, yielding the

an-noun ed result.

2. Fix

j > 1

and

F ∈ D

j,2

j

. The proof is a hieved by re ursion on

k

. For

k = 1

, the desired property is true, due to Point 1 applied to

G = F

. Now, assume that the desired property is true for

k

(

6

j − 1

), and let us prove that it also holds for

k + 1

. Indeed, we

have that

Γ

k+1

(F ) = hDF, −DL

−1

_Γ

k

(F )i

H

, that

F ∈ D

j,2

j

⊂ D

j−k,2

j−k

(assumption on

F

) andthat

Γ

k

(F ) ∈ D

j−k,2

j−k

(re urren eassumption). Point1yieldsthe desired on lusion. 3. The proof isimmediatelyobtained by a repeated appli ation of Point 2.

✷

We will now provide a new hara terization of umulants for fun tionals of Gaussian pro esses: itisthefundamentaltoolyieldingthemainresultsofthe paper. Notethat,due toLemma 4.2(Point2), the following statement appliesinparti ular torandom variables in

D

m,2

m

(

m > 2

).

Theorem 4.3 Fix an integer

m > 2

, and suppose that: (i) the random variable

F

is an element of

L

4m−4

_{(Ω) ∩ D}

1,4

, (ii) for every

j 6 m − 1

, the random variable

Γ

j

(F )

is in

L

2

_(Ω)

. Then, for every

s 6 m

,

κ

s+1

(F ) = s!E[Γ

s

(F )].

(4.19)

Proof. The proof is a hieved by re ursion on

s

. First observe that

κ

1

(F ) = E(F ) =

E[Γ

0

(F )]

, so that the laim is proved for

s = 0

. Now suppose that (4.19) holds for every

s = 0, ..., l

,where

l 6 m − 1

. A ording to (2.2),wehave that

E(F

l+1

) =

l−1

X

s=0



l

s



κ

s+1

(F )E(F

l−s

) + κ

l+1

(F ).

(4.20)

On the otherhand, by applying(3.15) tothe ase

p = l

and

G = F

, we dedu e that

By the re urren e assumption, and by applying again (3.15) to the ase

p = l − 1

and

G = Γ

1

(F )

, one dedu es therefore that

E(F

l+1

) = E(F

l

)κ

1

(F ) + lE(F

l−1

Γ

1

(F ))

= κ

1

(F )E(F

l

) + κ

2

(F )lE(F

l−1

) + l(l − 1)E(F

l−2

Γ

2

(F )).

Iterating this pro edure yields eventually

E(F

l+1

) =

l−1

X

s=0



l

s



κ

s+1

(F )E(F

l−s

) + l!E[Γ

l

(F )],

so that one dedu es from (4.20) that relation (4.19) holds for

s = l + 1

. The proof is on luded.

Remark 4.4 Suppose that

F = I

q

(f )

,where

q > 2

and

f ∈ H

⊙q

. Then,

L

−1

_{F = −q}

−1

_F

, and onsequently

E[Γ

s

(F )] = E[hDF, −DL

−1

Γ

s−1

(F )i

H

] = E[F Γ

s−1

(F )]

(4.21)

= E[h−DL

−1

F, DΓ

s−1

(F )i

H

] =

1

q

E[hDF, DΓ

s−1

(F )i

H

].

Several appli ations of formula (4.19) (and, impli itly, of (4.21)) are detailed in the next se tion.

5 Cumulants on Wiener haos

5.1 General statement

Wenow fo us onthe omputation of umulants asso iatedto randomvariables belonging toaxedWiener haos, thatis,randomvariableshavingtheformofamultipleWiener-It integral. Ourmainndingsare olle tedinthe followingstatement,providingaquite om-pa trepresentationfor umulantsasso iatedwith multipleintegralsofarbitraryorders. In the forth omingSe tion5.2,wewill ompareour resultswiththe lassi diagramformulae that are ustomarily used inthe probabilisti literature.

Theorem 5.1 Let

q > 2

, and assume that

F = I

q

(f )

, where

f ∈ H

⊙q

. Denote by

κ

s

(F )

,

s > 1

, the umulants of

F

. We have

κ

1

(F ) = 0

,

κ

2

(F ) = q!kfk

2

H

⊗q

and, for every

s > 3

,

κ

s

(F ) = q!(s −1)!

X

c

q

(r

1

, . . . , r

s−2

)

(...((f e

_⊗

r

1

f ) e

⊗

r

2

f ) . . . e

⊗

r

s−3

f ) e

⊗

r

s−2

f, f



H

⊗q

,

(5.22)

where the sum

P

runs over all olle tions of integers

r

1

, . . . , r

s−2

su h that:

(i)

1 6 r

1

, . . . , r

s−2

6

q

;

(ii)

r

1

+ . . . + r

s−2

=

(iii)

r

1

< q

,

r

1

+ r

2

<

3q

2

,

. . .

,

r

1

+ . . . + r

s−3

<

(s−2)q

2

; (iv)

r

2

6

2q − 2r

1

,

. . .

,

r

s−2

6

(s − 2)q − 2r

1

− . . . − 2r

s−3

;

and where the ombinatorial onstants

c

q

(r

1

, . . . , r

s−2

)

are re ursively dened by the rela-tions

c

q

(r) = q(r − 1)!



q − 1

r − 1



2

,

and, for

a > 2

,

c

q

(r

1

, . . . , r

a

) = q(r

a

− 1)!



aq − 2r

1

− . . . − 2r

a−1

− 1

r

a

− 1



q − 1

r

a

− 1



c

q

(r

1

, . . . , r

a−1

).

Remark 5.2 1. If

sq

is odd, then

κ

s

(F ) = 0

,see indeed ondition

(ii)

.

2. If

q = 2

and

F = I

2

(f )

,

f ∈ H

⊙2

,thentheonlypossibleintegers

r

1

, . . . , r

s−2

verifying

(i) − (iv)

in the previous statement are

r

1

= . . . = r

s−2

= 1

. On the other hand,

we immediately ompute that

c

2

(1) = 2

,

c

2

(1, 1) = 4

,

c

2

(1, 1, 1) = 8

, and so on. Therefore,

κ

s

(F ) = 2

s−1

(s − 1)!

(...(f ⊗

1

f ) . . . f ) ⊗

1

f, f



H

⊗2

,

andwe re overthe lassi alexpression ofthe umulantsof adoubleintegral(asused e.g. in[7℄ or[14℄).

3. If

q > 2

and

F = I

q

(f )

,

f ∈ H

⊙q

,then (5.22) for

s = 4

reads

κ

4

(I

q

(f )) = 6q!

q−1

X

r=1

c

q

(r, q − r)

(f e

_⊗

r

f ) e

⊗

q−r

f, f



H

⊗q

=

3

q

q−1

X

r=1

rr!

2



q

r



4

(2q − 2r)!

(f e

_⊗

r

f ) ⊗

q−r

f, f



H

⊗q

=

3

q

q−1

X

r=1

rr!

2



q

r



4

(2q − 2r)!

f e

_⊗

r

f, f ⊗

r

f



H

⊗(2q−2r)

=

3

q

q−1

X

r=1

rr!

2



q

r



4

(2q − 2r)!kf e

⊗

r

f k

2

H

⊗(2q−2r)

,

(5.23)

and we re over the expression for

κ

4

(F )

dedu ed in [17, Se tion 3.1℄ by a dierent route. Formula (5.23) should be ompared with the following identity, rst estab-lished in[22,p. 183℄: forevery

q > 2

and every

f ∈ H

both symmetrized and non-symmetrized ontra tions.

Proof of Theorem 5.1. Let us rst show that the following formula (5.25) is in order: for

any

s > 2

,

Γ

s−1

(F ) =

q

X

r

1

=1

. . .

[(s−1)q−2r

_X

1

−...−2r

s−2

]∧q

r

s−1

=1

c

q

(r

1

, . . . , r

s−1

)1

{r

1

<q}

. . . 1

_{r

₁

_+...+r

_s−2

_<

(s−1)q

2

}

×I

sq−2r

1

−...−2r

s−1

(...(f e

⊗

r

1

f ) e

⊗

r

2

f ) . . . f ) e

⊗

r

s−1

f

.

(5.25) We shall prove (5.25) by indu tion, assuming without loss of generality that

H

has the form

L

2

_{(A, A , µ)}

,where

(A, A )

isa measurablespa eand

µ

isa

σ

-nitemeasure without atoms. When

s = 2

, identity (5.25) simplyreads

Γ

1

(F ) =

q

X

r=1

c

q

(r)I

2q−2r

(f e

⊗

r

f ).

(5.26)

Let usprove (5.26) by means of the multipli ation formula(3.5), see also[21℄ for similar omputations. Wehave

Γ

1

(F ) = hDF, −DL

−1

F i

H

=

1

q

kDF k

2

H

= q

Z

A

I

q−1

f (·, a)

2

µ(da)

= q

q−1

X

r=0

r!



q − 1

r



2

I

2q−2−2r

Z

A

f (·, a)e

⊗

r

f (·, a)µ(da)



= q

q−1

X

r=0

r!



q − 1

r



2

I

2q−2−2r

f e

⊗

r+1

f

= q

q

X

r=1

(r − 1)!



q − 1

r − 1



2

I

2q−2r

f e

⊗

r

f

,

Γ

s

(F ) = hDF, −DL

−1

Γ

s−1

F i

H

=

q

X

r

1

=1

. . .

[(s−1)q−2r

_X

1

−...−2r

s−2

]∧q

r

s−1

=1

qc

q

(r

1

, . . . , r

s−1

)1

{r

1

<q}

. . . 1

{r

1

+...+r

s−2

<

(s−1)q

₂

}

×1

{r

1

+...+r

s−1

<

sq

₂

}

I

q−1

(f ), I

sq−2r

1

−...−2r

s−1

−1

(...(f e

⊗

r

1

f ) e

⊗

r

2

f ) . . . f ) e

⊗

r

s−1

f



H

=

q

X

r

1

=1

. . .

[(s−1)q−2r

_X

1

−...−2r

s−2

]∧q

r

s−1

=1

[sq−2r

1

−...−2r

_X

s−1

]∧q

r

s

=1

c

q

(r

1

, . . . , r

s−1

) × q(r

s

− 1)!

×



sq − 2r

1

− . . . − 2r

s−1

− 1

r

s

− 1



q − 1

r

s

− 1



1

{r

1

<q}

. . . 1

_{r

₁

_+...+r

_s−2

_<

(s−1)q

2

}

×1

{r

1

+...+r

s−1

<

sq

₂

}

I

(s+1)q−2r

1

−...−2r

s

(...(f e

⊗

r

1

f ) e

⊗

r

2

f ) . . . f ) e

⊗

r

s

f

,

whi h is the desired formula for

Γ

s

(F )

. The proof of (5.25) for all

s > 1

is thus nished. Now, letus take the expe tation onboth sides of (5.25). Weget

κ

s

(F ) = (s − 1)!E[Γ

s−1

(F )]

= (s − 1)!

q

X

r

1

=1

. . .

[(s−1)q−2r

_X

1

−...−2r

s−2

]∧q

r

s−1

=1

c

q

(r

1

, . . . , r

s−1

)1

{r

1

<q}

. . . 1

_{r

₁

_+...+r

_s−2

_<

(s−1)q

2

}

×1

{r

1

+...+r

s−1

=

sq

₂

}

× (...(f e

⊗

r

1

f ) e

⊗

r

2

f ) . . . f ) e

⊗

r

s−1

f.

Observe that, if

2r

1

+ . . . + 2r

s−1

= sq

and

r

s−1

6

(s − 1)q − 2r

1

− . . . − 2r

s−2

then

2r

s−1

= q + (s − 1)q − 2r

1

− . . . − 2r

s−2

>

q + r

s−1

,so that

r

s−1

>

q

. Therefore,

κ

s

(F ) = (s − 1)!

q

X

r

1

=1

. . .

[(s−2)q−2r

_X

1

−...−2r

s−3

]∧q

r

s−2

=1

c

q

(r

1

, . . . , r

s−2

, q)1

{r

1

<q}

. . . 1

_{r

₁

_+...+r

_s−3

_<

(s−2)q

2

}

×1

{r

1

+...+r

s−2

=

(s−2)q

₂

}

(...(f e

_⊗

r

1

f ) e

⊗

r

2

f ) . . . f ) e

⊗

r

s−2

f, f



H

⊗q

,

whi his the announ ed result, sin e

c

q

(r

1

, . . . , r

s−2

, q) = q!c

q

(r

1

, . . . , r

s−2

)

.

✷

5.2 Combinatorial expression of umulants

Wenow provide a lassi ombinatorialrepresentation of umulantsof the type

κ

s

(F )

, in the ase where: (i)

s > 2

, (ii)

q > 2

, (iii)

sq

is even, (iv)

F = I

q

(f )

(with

f ∈ H

⊙q

) and (v)

H

= L

2

_{(A, A , µ)}

Denition 5.3 For

s > 2

, onsidertheset

[s] = {1, ..., s}

oftherst

s

integers. For

q > 2

, wedenote by

K(s, q)

the lass ofallnon-oriented graphs

γ

over

[s]

satisfyingthe following properties:

-

γ

doesnot ontainedges of thetype

(j, j)

,

j = 1, ..., s

,that is,noedges of

γ

onne t avertex with itself. One an equivalently say that

γ

has no loops.

- Multiple edges are allowed, that is, an edge an appear

k > 2

times into

γ

; in this ase, we say that

k

is the multipli ity of the edge. Also, if

i, j

are onne ted by an edge of multipli ity

k

, wesay that

i

and

j

are onne ted

k

times.

- Every vertex appears in exa tly

q

edges ( ounting multipli ities). -

γ

is onne ted.

If

sq

is odd, then

K(s, q)

is empty. If

sq

is even, then ea h

γ ∈ K(s, q)

ontains

ex-a tly

sq/2

edges ( ounting multipli ities). For instan e: an element of

K(3, 2)

is

γ =

{(1, 2), (2, 3), (3, 1)}

; an element of

K(3, 4)

is

γ = {(1, 2), (1, 2), (2, 3), (2, 3), (1, 3), (1, 3)}

(note that ea hedge has multipli ity 2).

Denition 5.4 Given

q, s > 2

su hthat

sq

iseven, and

γ ∈ K(s, q)

,wedenethe onstant

w(γ)

asfollows.

(a) Forevery

j = 1, ..., s

, onsider ageneri ve tor

L(j) = (l(j, 1), ..., l(j, q))

of

q

distin t obje ts. Write

{L(j)}

for the set of the omponents of

L(j)

.

(b) Starting from

γ

, build a mat hing

m(γ)

over

L :=

S

j=1,...,s

{L(j)}

as follows. Enu-meratetheverti es

v

1

, ..., v

sq/2

of

γ

. If

v

1

links

i

1

and

j

1

andhas multipli ity

k

1

,then pi k

k

1

elementsof

L(i

1

)

and

k

1

elementsof

L(j

1

)

and build amat hing between the two

k

1

-sets. If

v

2

links

i

2

and

j

2

and has multipli ity

k

2

, then pi k

k

2

elements of

L(i

2

)

(amongthose notalready hosen attheprevious step, whenever

i

2

equals

i

1

or

j

1

) and

k

2

elements of

L(j

2

)

(among those not already hosen at the previous step, whenever

j

2

equals

i

1

or

j

1

) and build a mat hing between the two

k

2

-sets. Repeat the operation up tothe step

sq/2

.

( ) Dene

S

q

asthegroupofallpermutationsof

[q]

. Forevery

σ ∈ S

q

,denethe ve tor

L

σ

(j) = (l

σ

(j, 1), ..., l

σ

(j, q))

, where

l

σ

(j, p) = l(j, σ(p))

,

p = 1, ..., q

.

(d) Dene

S

(s)

q

as the

s

th produ t group of

S

q

, that is,

S

(s)

q

is the olle tion of all

s

-ve tors of the type

σ = (σ

1

, ..., σ

s

)

, where

σ

j

∈ S

q

,

j = 1, ..., s

, endowed with the usual produ t groupstru ture.

(e) For every

σ = (σ

1

, ..., σ

s

) ∈ S

(s)

q

, build a new mat hing

m

σ

(γ)

over

L

by repeating the same operation as at point (b), with the ve tor

L

σ

j

(j)

repla ing

L(j)

for every

(f) Deneanequivalen erelationover

S

(s)

q

by writing

σ ∼

γ

π

whenever

m

σ

(γ) = m

π

(γ)

. Let

S

(s)

q

/ ∼

γ

bethe quotient of

S

(s)

q

with respe t to

∼

γ

. (g) Dene

w(γ)

tobethe ardinalityof

S

(s)

q

/ ∼

γ

. Forinstan e, one an provethat

w(γ) = 2

s−1

for every

s > 2

and every

γ ∈ K(s, 2)

. Also,

w(γ) = q!

forevery

q > 2

and every

γ ∈ K(2, q)

.

Denition 5.5 For

q > 2

,let

f ∈ H

⊙q

,thatis,

f

isasymmetri elementof

L

2

_(A

q

_{, A}

q

_{, µ}

q

₎

.

Fix

γ ∈ K(s, q)

, where

s > 2

and

sq

iseven. Starting from

f

and

γ

,wedene a fun tion

(a

1

, ..., a

sq/2

) 7→ f

γ

(a

1

, ..., a

sq/2

),

of

sq/2

variables, asfollows: (i) juxtapose

s

opiesof

f

, and

(ii) ifthe verti es

j

and

l

arelinkedby

r

edges,thenidentify

r

variablesinthe argument of the

j

th opy of

f

with

r

variablesin the argumentof the

l

th opy. By symmetry, the position of the identied

r

variables is immaterial. Also, by onne tedness, one has ne essarily

r < q

.

The resultingfun tion

f

γ

isa (not ne essarilysymmetri ) elementof

L

1

(A

(sq/2)

, A

(sq/2)

, µ

(sq/2)

).

For instan e, if

γ = {(1, 2), (2, 3), (3, 1)} ∈ K(3, 2)

,then

f

γ

(x, y, z) = f (x, y)f (y, z)f (z, x)

.

If

γ = {(1, 2), (1, 2), (2, 3), (2, 3), (1, 3), (1, 3)} ∈ K(3, 4)

, then

f

γ

(t, u, v, x, y, z) = f (t, u, v, x)f (t, u, y, z)f (y, z, v, x).

We now turn to the main statement of this se tion, relating graphs and umulants. The rst part is lassi (see e.g. [23℄ for a proof), and shows how to use the fun tions

f

γ

to ompute the umulants of the random variable

F = I

q

(f )

. The se ond part of the statement makes use of (5.22), and shows indeed that Theorem 5.1 impli itly provides a ompa t representation of well-known ombinatorialexpressions.

Proposition 5.6 Let

q > 2

and assume that

F = I

q

(f )

, where

f ∈ H

⊙q

. Assume the integer

s > 2

is su h that

sq

is even. Then,

κ

s

(F ) =

X

γ∈K(s,q)

w(γ)

Z

A

(sq/2)

f

γ

(a

1

, ..., a

sq/2

)µ(da

1

) · · · µ(da

sq/2

).

(5.27)

As a onsequen e,by using (5.22), one dedu es the identity

(s − 1)!q!

X

c

q

(r

1

, . . . , r

s−2

)

(...(f e

_⊗

r

1

f ) e

⊗

r

2

f ) . . . f ) e

⊗

r

s−2

f, f



H

⊗q

(5.28)

=

X

γ∈K(s,q)

w(γ)

Z

A

(sq/2)

f

γ

(a

1

, ..., a

sq/2

)µ(da

1

) · · · µ(da

sq/2

).

(5.29) where

P

Remark 5.7 Beingbasedonasum overthewhole set

K(s, q)

,formula(5.27)isof ourse more ompa t than (5.22). However, sin e it does not ontain any hint about how one should enumerate the elements of

K(s, q)

, expression (5.27) is mu h harder to ompute and asses. On the other hand, (5.22) is based on re ursive relations and inner produ ts of ontra tions, sothat one ouldin prin iple ompute

κ

s

(F )

by implementinganexpli it algorithm.

5.3 CLTs on Wiener haos

We on lude the paperbyprovidinganew proof(based onour newformula(5.22)) ofthe following result, rst proved in [22℄ and yielding a ne essary and su ient ondition for CLTs on axed haos.

Theorem 5.8 (See [22℄) Fix aninteger

q > 2

, and let

(F

n

)

n>1

be a sequen e of theform

F

n

= I

q

(f

n

)

, with

f

n

∈ H

⊙q

su h that

E[F

2

n

] = q!kf

n

k

2

H

⊗q

= 1

for all

n > 1

. Then,

as

n → ∞

, we have

F

n

→ N(0, 1)

in law if and only if

kf

n

⊗

e

r

f

n

k

H

⊗(2q−2r)

→ 0

for all

r = 1, . . . , q − 1

.

Proof. Observe that

κ

1

(F

n

) = 0

and

κ

2

(F

n

) = 1

. For

κ

s

(F

n

)

,

s > 3

, we onsider the expression (5.22). Let

r

1

, . . . , r

s−2

be some integers su h that

(i)



(iv)

in Theorem 5.1 are satised. Using Cau hy-S hwarz inequality and then su essively

kg e

⊗

r

hk

H

⊗(p+q−2r)

6

kg ⊗

r

hk

H

⊗(p+q−2r)

6

kgk

H

⊗p

khk

H

⊗q

whenever

g ∈ H

⊙p

,

h ∈ H

⊙q

and

r = 1, . . . , p ∧ q

, weget that



h(...(f

n

⊗

e

r

1

f

n

) e

⊗

r

2

f

n

) . . . f

n

) e

⊗

r

s−2

f

n

, f

n

i

H

⊗q





6

_k(...(f

_n

_⊗

e

_r

₁

f

n

) e

⊗

r

2

f

n

) . . . f

n

) e

⊗

r

s−2

f

n

k

H

⊗q

kf

n

k

H

⊗q

6

_kf

_n

_⊗

e

_r

₁

f

n

k

H

⊗

_(2q−2r1)

kf

_n

k

s−2

H

⊗q

= (q!)

1−

s

2

kf

_n

⊗

e

_r

1

f

n

k

H

⊗

_(2q−2r1)

.

(5.30)

Assume now that

kf

n

⊗

e

r

f

n

k

H

⊗(2q−2r)

→ 0

for all

r = 1, . . . , q − 1

, and x aninteger

s > 3

. By ombining (5.22) with (5.30),we get that

κ

s

(F

n

) → 0

as

n → ∞

. Hen e, applying the method of moments or umulants, we get that

F

n

→ N(0, 1)

in law. Conversely, assume

that

F

n

→ N ∼ N(0, 1)

in law. Sin e the sequen e

(F

n

)

lives inside the

q

th haos, and

be ause

E[F

2

n

] = 1

for all

n

, we have that, for every

p > 1

,

sup

n>1

E[|F

n

|

p

] < ∞

(see e.g. Janson [9, Ch. V℄). This implies immediatelythat

κ

4

(F

n

) = E[F

4

n

] − 3 → E[N

4

] − 3 = 0

.

Hen e,identity(5.23)allowsto on ludethat

kf

n

⊗

e

r

f

n

k

H

⊗(2q−2r)

→ 0

forall

r = 1, . . . , q −1

.

✷

Referen es

limittheorem. Probab.Theory Rel. Fields 72(2),289-303.

[3℄ J.-C. Breton, I. Nourdin and G. Pe ati (2009). Exa t onden e intervals for the Hurst parameter of a fra tional Brownian motion. Ele tron. J. Statist. 3, 416-425 (ele troni )

[4℄ P. Breuer et P. Major (1983). Central limit theorems for non-linear fun tionals of Gaussianelds. J. Mult. Anal.13, 425-441.

[5℄ D. Chambers et E. Slud (1989). Central limit theorems for nonlinear fun tionals of stationary Gaussianpro esses. Probab. Theory Rel. Fields 80, 323-349.

[6℄ L.H.Y.ChenandQ.-M.Shao(2005).Stein'smethodfornormalapproximation.In: An Introdu tion to Stein's Method (A.D. Barbour and L.H.Y. Chen, eds), Le ture Notes Series No.4, Institute for Mathemati al S ien es, National University of Singapore, Singapore University Press and World S ienti 2005,1-59.

[7℄ R.FoxetM.S. Taqqu(1987).Multiplesto hasti integralswithdependentintegrators. J. Mult. Anal.21(1),105-127.

[8℄ L. Giraitis and D. Surgailis (1985). CLT and other limit theorems for fun tionals of Gaussianpro esses. Zeits hrift für Wahrs h. verw. Gebiete 70, 191-212.

[9℄ S. Janson(1997). GaussianHilbert Spa es. Cambridge University Press, Cambridge. [10℄ P. Malliavin (1997). Sto hasti Analysis. Springer-Verlag, Berlin, Heidelberg, New

York.

[11℄ D. Marinu i (2007). A Central Limit Theorem and Higher Order Results for the Angular Bispe trum. Probab. Theory Rel. Fields 141, 389-409.

[12℄ I. NourdinandG.Pe ati(2007).Non- entral onvergen e of multipleintegrals.Ann. Probab., 37(4),1412-1426.

[13℄ I. Nourdin and G. Pe ati (2009). Stein's method on Wiener haos. Probab. Theory Rel. Fields145, no. 1,75-118.

[14℄ I.NourdinandG.Pe ati(2008).Stein'smethodandexa tBerry-Esséenasymptoti s for fun tionalsof Gaussian elds. Ann. Probab., to appear.

[15℄ I. Nourdin and G. Pe ati (2008). Universal Gaussian u tuations of non-Hermitian matrix ensembles. Preprint.

sums: universality of GaussianWiener haos. Preprint.

[18℄ I. Nourdin, G. Pe ati and A. Réveilla (2008). Multivariate normal approximation using Stein'smethodand Malliavin al ulus.Ann. Inst. H. Poin aré Probab. Statist., to appear.

[19℄ I. Nourdin and F.G. Viens (2008). Density estimates and on entration inequalities with Malliavin al ulus.Ele tron. J. Probab., toappear

[20℄ D. Nualart (2006). The Malliavin al ulus and related topi s of Probability and Its Appli ations. Springer-Verlag, Berlin,se ond edition.

[21℄ D.NualartandS.Ortiz-Latorre(2008).Centrallimittheoremsformultiplesto hasti integrals and Malliavin al ulus.Sto h. Pro . Appl. 118 (4), 614-628.

[22℄ D. Nualart and G. Pe ati (2005). Central limit theorems for sequen es of multiple sto hasti integrals. Ann. Probab. 33 (1), 177-193.

[23℄ G. Pe ati and M.S. Taqqu (2008). Moments, umulants and diagram formulae for non-linear fun tionalsof random measures (Survey). Preprint.

[24℄ G.Pe atiandC.A. Tudor(2005).Gaussianlimitsforve tor-valuedmultiple sto has-ti integrals.Séminaire de Probabilités XXXVIII, LNM1857. Springer-Verlag,Berlin Heidelberg New York, pp. 247-262.

[25℄ G. Reinert (2005). Three general approa hes toStein's method. In: An Introdu tion to Stein's Method (A.D. Barbour and L.H.Y. Chen, eds), Le ture Notes Series No.4, InstituteforMathemati alS ien es, NationalUniversityofSingapore,Singapore Uni-versity Press and World S ienti 2005, 183-221.

[26℄ G.-C. Rotaand J.Shen (2000).On the ombinatori sof umulants.J. Comb. Theory Series A 91, 283-304.