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- multiple integrals
- commutative probability spaces
- joint moments
- probability theory
- moment
- nth multiple
- theorem become
- random variable

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1 2 3Aur´elien Deya , Salim Noreddine and Ivan Nourdin

Abstract: In 2005, Nualart and Peccati [13] proved the so-called Fourth Moment Theorem

asserting that, for a sequence of normalized multiple Wiener-Itˆo integrals to converge to the

standard Gaussian law, it is necessary and suﬃcient that its fourth moment tends to 3. A few

years later, Kemp et al. [9] extended this theorem to a sequence of normalized multiple Wigner

integrals, in the context of the free Brownian motion. The q-Brownian motion, q ∈ (−1,1],

introduced by the physicists Frisch and Bourret [6] in 1970 and mathematically studied by

Boz˙ejko and Speicher [2] in 1991, interpolates between the classical Brownian motion (q = 1)

and the free Brownian motion (q = 0), and is one of the nicest examples of non-commutative

processes. The question we shall solve in this paper is the following: what does the Fourth

Moment Theorem become when dealing with a q-Brownian motion?

Keywords: Central limit theorems; q-Brownian motion; non-commutative probability space;

multiple integrals.

AMS subject classiﬁcations: 46L54; 60H05; 60F05

1. Introduction and main results

The q-Brownian motion was introduced in 1970 by the physicists Frisch and Bourret [6] as

an intermediate model between two standard theoretical axiomatics (see also [7] for another

physical interpretation). From a probabilistic point of view, it may be seen as a smooth and

natural interpolation between two of the most fundamental processes in probability theory: on

the one hand, the classical Brownian motion (W ) deﬁned on a classical probability spacet t≥0

(Ω,F,P); on the other hand, the free Brownian motion (S ) at the core of Voiculescu’s freet t≥0

probability theory and closely related to the study of large random matrices (see [15]).

The mathematical construction of the q-Brownian motion is due to Boz˙ejko and Speicher [2],

anditheavilyreliesonthetheoryofnon-commutative probability spaces. Thus, beforedescribing

our results and for the sake of clarity, let us ﬁrst introduce some of the central concepts of this

theory (see [10] for a systematic presentation).

∗A W -probability space (or a non-commutative probability space) is a von Neumann algebra

A (that is, an algebra of bounded operators on a complex separable Hilbert space, closed under

adjoint and convergence in the weak operator topology) equipped with a trace ϕ, that is, a

unital linear functional (meaning preserving the identity) which is weakly continuous, positive

∗(meaning ϕ(X) ≥ 0 whenever X is a non-negative element of A; i.e. whenever X = YY for

∗some Y ∈ A), faithful (meaning that if ϕ(YY ) = 0 then Y = 0), and tracial (meaning that

ϕ(XY) =ϕ(YX) for all X,Y ∈A, even though in general XY =YX).

1 ´Institut Elie Cartan, Universit´e de Lorraine, BP 70239, 54506 Vandoeuvre-l`es-Nancy, France. Email:

Aurelien.Deya@iecn.u-nancy.fr

2Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e Paris 6, Boˆıte courrier 188, 4 Place

Jussieu, 75252 Paris Cedex 5, France. Email: salim.noreddine@polytechnique.org

3 ´Institut Elie Cartan, Universit´e de Lorraine, BP 70239, 54506 Vandoeuvre-l`es-Nancy, France. Email:

inourdin@gmail.com. Supported in part by the two following (french) ANR grants: ‘Exploration des

Chemins Rugueux’ [ANR-09-BLAN-0114] and ‘Malliavin, Stein and Stochastic Equations with Irregular

Coeﬃcients’ [ANR-10-BLAN-0121].

1

62

∗InaW -probabilityspace(A,ϕ),werefertotheself-adjointelementsofthealgebraasrandom

variables. Any random variableX has a law: this is the unique compactly supported probability

measure μ onR with the same moments as X; in other words, μ is such that

Z

Q(x)dμ(x) =ϕ(Q(X)), (1)

R

for any real polynomial Q. Thus, and as in the classical probability theory, the focus is more

on the laws (which, in this context, is equivalent to the sequence of moments) of the random

variables than on the underlying space (A,ϕ) itself. For instance, we say that a sequence

{X } of random variables such that X ∈ (A ,ϕ ) converges to X ∈ (A,ϕ) if, for everyk k≥1 k k k

r rpositive integer r, one has ϕ (X ) → ϕ(X ) as k → ∞. In the same way, we consider herek k

(i)that any family{X } of random variables on (A,ϕ) is ‘characterized’ by the set of all of itsi∈I

(i)(i ) (i )1 rjoint moments ϕ(X ···X ) (i ,...,i ∈ I, r∈N), and we say that {X } converges to1 r i∈Ik

(i){X } (when k→∞) if the convergence of the joint moments holds true (see [10, Lecture 4]i∈I

for further details on non-commutative random systems).

It turns out that a rather sophisticated combinatorial machinery is hidden behind most of

these objects, see [10]. This leads in particular to the notion of a crossing/non crossing pairing,

which is a central tool in the theory.

Deﬁnition1.1. 1. Letr be an even integer. A pairing of{1,...,r} is any partition of{1,...,r}

intor/2 disjoint subsets, each of cardinality 2. We denote byP ({1,...,r}) the set of all pairings2

of {1,...,r}.

2. When π ∈P ({1,...,r}), a crossing in π is any set of the form {{x ,y },{x ,y }} with2 1 1 2 2

{x ,y} ∈ π and x < x < y < y . The number of such crossings is denoted by Cr(π). Thei i 1 2 1 2

subset of all non-crossing pairings in P ({1,...,r}) (i.e., the subset of all π ∈ P ({1,...,r})2 2

satisfying Cr(π) = 0) is denoted by NC ({1,...,r}).2

By means of the objects given in Deﬁnition 1.1, it is simple to compute the joint moments

related to the classical Brownian motion W or to the free Brownian motion S, and this actually

leads to the so-called Wick formula. Namely, for every t ,...,t ≥ 0, one has1 r

X Y

E W ···W = (t ∧t ), (2)t t i jr1

π∈P ({1,...,r}){i,j}∈π2

X Y

ϕ S ···S = (t ∧t ). (3)t t i j1 r

π∈NC ({1,...,r}){i,j}∈π2

It is possible to go smoothly from (2) to (3) by using the q-Brownian motion, which is one of

the nicest examples of non-commutative processes.

∗Deﬁnition 1.2. Fix q∈ (−1,1). A q-Brownian motion on some W -probability space (A,ϕ) is

a collection {X} of random variables on (A,ϕ) satisfying that, for every integer r ≥1 andt t≥0

every t ,...,t ≥ 0,1 r

X Y

Cr(π)ϕ X ···X = q (t ∧t ). (4)t t i j1 r

π∈P ({1,...,r}) {i,j}∈π2

The existence of such a process, far from being trivial, is ensured by the following result.

∗Theorem 1.3 (Boz˙ejko, Speicher). For every q ∈ (−1,1), there exists a W -probability space

(q)

(A ,ϕ ) and a q-Brownian motion {X } built on it.q q t≥0t

As is immediately seen, formula (4) allows us to recover (2) by choosing q = 0 (we adopt

0the usual convention 0 = 1). On the other hand, although the classical Brownian motion3

∗W cannot be identiﬁed with a process living on some W -probability space (the laws of its

(q)marginals being not compactly supported), it can legitimately be considered as the limit ofX

−when q → 1 . (This extension procedure can even be made rigorous by considering a larger

(q)class of non-commutative probability spaces.) As such, the family {X } of q-Brownian0≤q<1

motions with a parameter q between 0 and 1 happens to be a ‘smooth’ interpolation between S

and W.

(q)

Deﬁnition1.4. Letq∈ (−1,1). For everyt≥ 0, the distribution ofX is called the (centered)t

q-Gaussian law with variance t. We denote it by G (0,t). Otherwise stated, a given probabilityq

measureν onR is distributed according toG (0,t) if it is compactly supported and if its momentsq

are given by

Z Z X

2k+1 2k k Cr(π)x dν(x) = 0 and x dν(x) =t q . (5)

R R

π∈P ({1,...,2k})2

The probability measure ν ∼G (0,1) is absolutely continuous with respect to the Lebesgue mea-q q

−2 2√ √sure; its density is supported on , and is given, within this interval, by

1−q 1−q

∞Yp1 2cosθn n 2iθ 2 √ν (dx) = 1−qsinθ (1−q )|1−q e | , where x = with θ∈ [0,π].q

π 1−q

n=1

By convention, we also set G (0,t) as being the probability measure whose density with respect1

to the Lebesgue measure is given by

2x1 −

2t√ e , x∈R,

2πt

that is, G (0,t) =N(0,t).1

(q)For every q∈ (−1,1), the process X shares many similarities with the classical (resp. free)

Brownian motion. For instance, it also appears as a limit process of some generalized random

walks (see [3, Theorem 0]). For this reason, one sometimes considers this ‘q-deformation’ of S

and W (see [2]). Also, and similarly to the free Brownian motion case, the q-Brownian motion

appears as the limit of some particular sequences of random matrices (see [14]).

In the seminal paper [13], Nualart and Peccati highlighted a powerful convergence criterion

for the normal approximation of sequences of multiple integrals with respect to the classical

Brownian motion. From now on, we will refer to it as the Fourth Moment Theorem. A few

years later, it was extended by Kemp et al. [9] for the free Brownian motion S and its multiple

Wigner integrals.

The question we shall solve in this paper is the following: what does the Fourth Moment

Theorem become when dealing more generally with a q-Brownian motion? Before stating our

main result and in order to put it into perspective, let us be more speciﬁc with the two afore-

mentioned versions of the Fourth Moment Theorem that are already known (that is, in the

W Sclassical and free Brownian motion cases). We let I (resp. I ) denote the nth multiplen n

integrals with respect to W (resp. S), as they are constructed in [12] (resp. [1]). The following

two theorems are, respectively, the versions of the Fourth Moment Theorem in the classical case

(q = 1) and in the free case (q = 0).

Theorem 1.5 (Nualart, Peccati). Fix n ≥ 2 and let {f } be a sequence of real-valuedk k≥1

2 nsymmetric functions in L (R ) satisfying+

W 2 2E[I (f ) ] =n!kf k n → 1 as k→∞.2k kn L (R )

+

Then, the following two assertions are equivalent as k→∞:4

W 4(i) E I (f ) → 3.kn

W(ii) The sequence I (f ) converges in law to N(0,1) =G (0,1).k 1n

Theorem 1.6 (Kemp, Nourdin, Peccati, Speicher). Fix n≥ 2 and let{f } be a sequence ofk k≥1

2 nmirror-symmetricfunctionsinL (R )(thatis, eachf issuchthatf (t ,...,t ) =f (t ,...,t )k k 1 n k n 1+

for almost all t ,...,t ≥ 0), satisfying1 n

S 2 2ϕ I (f ) =kf k → 1 as k→∞.2 nk kn L (R )

+

Then, the following two assertions are equivalent as k→∞:

S 4(i) ϕ I (f ) → 2.kn

S(ii) The sequence I (f ) converges in law to S(0,1) =G (0,1).k 0n

(q) ∗Now, ﬁx a parameter q ∈ [0,1], and consider a q-Brownian motion X on some W -

probability space (A ,ϕ ). (Note that in the three forthcoming statements, we extend the deﬁ-q q

(q) (1)nition of X to q = 1 by naturally setting X :=W and by replacing (A ,ϕ ) by (Ω,F,P).)1 1

(q)As in the classical and free cases, to each n≥ 0 we may associate with X a natural notion of

(q)Xnth multiple integralI , see Donati-Martin [5] or Section 2.1 for the details. We are now in an

position to state the main result of the present paper, which is a suitable interpolation between

Theorem 1.5 and Theorem 1.6, but in a somehow unexpected way (see indeed the comment

following its statement).

Theorem 1.7. Fix n ≥ 1, recall that q ∈ [0,1], and let {f } be a sequence of real-valuedk k≥1

2 nsymmetric functions in L (R ) satisfying+

X(q)X 2 inv(σ) 2 ϕ I (f ) = q kf k → 1 as k→∞,2 nq k kn L (R )

σ∈Sn

where the notation inv(σ) refers to the number of inversions in σ, i.e.,

inv(σ) := Card{1≤i<j≤n : σ(i)>σ(j)}.

Then the following two assertions are equivalent as k→∞:

(q) 2X 4 n(i) ϕ I (f ) → 2+q .q kn

(q)X(ii) The sequence I (f ) converges in law to G (0,1).2k nn q

When, in Theorem 1.7, we consider a value of q which is strictly between 0 and 1, we get

(q)Xthat any suitably-normalized sequence {I (f )} satisfying the fourth moment condition (i)kn

converges in law, see (ii), to a random variable which is expressed by means of the parameter

2nq and not q, as could have been legitimately expected by trying to guess the right statement

with the help of Theorems 1.5 and 1.6. But this phenomenon was of course impossible to predict

by taking a look at the case where q ∈ {0,1} because, for these two values, we precisely have

2nthat q =q .

Two natural questions emerge from Theorem 1.7: (a) what can be said when q∈ (−1,0)? (b)

what happens if the functions f are only mirror-symmetric (as in Theorem 1.6)? Regardingk

(a), it is not diﬃcult to build explicit counterexamples where the equivalence between (i) and√

(ii) in Theorem 1.7 fails. For instance, with q =−1/2, n = 2 and f = f = 21 , we have2k [0,1]

(q) (q) (q)X 2 X 4 4 Xϕ (I (f) ) = 1 and ϕ (I (f) ) = 2 +q , but I (f) is not G 4(0,1)-distributed (sinceq q q2 2 25

√(q)X 3 2ϕ (I (f) ) = 2(1 +q) = 0). See Remark 3.3 for the details. Actually, we do not knowq 2

if this counterexample hides a general phenomenon or not. Does Theorem 1.7 continue to be

true for all q < 0 except (possibly) for some values of q, or is it always false when q < 0? On

the other hand, to answer question (b) is unfortunately out of the scope of this paper. Indeed,

to do so would imply to change almost all our computations (in order to take into account the

lack of full symmetry) and especially those in the proof of Proposition 3.2, which serves us as a

starting point towards Theorem 1.7. We postpone this further analysis to another paper.

Theorem1.7willbeobtainedinSection3asaconsequenceofamoregeneralmultidimensional

version (namely, Theorem 3.1). In fact, we will even prove that (i) and (ii) are both equivalent

to a third assertion that only involves the sequence{f } and not the value of q (provided itk k≥1

belongs to [0,1]). As a consequence, we shall deduce the following transfer principle.

Theorem 1.8 (Transfer principle). Fix n ≥ 1 and let {f } be a sequence of real-valuedk k≥1

2 n 2symmetric functions in L (R ) satisfying kf k → 1 as k→∞. For every q∈ [0,1], setnk 2+ L (R )

+

X

2 inv(σ)σ := q > 0.q

σ∈Sn

Then, the following two assertions are equivalent as k→∞:

(q)X 2(i) The sequence I (f ) converges in law to G 2(0,σ ) for one particular q∈ [0,1].k nn qq

(q)X 2(ii) The sequence I (f ) converges in law to G 2(0,σ ) for all q∈ [0,1].k nn qq

As a nice application of all the previous material, we oﬀer the following theorem. (We will

(q) (q)

prove it in Section 3.) For every q ∈ [0,1], let us denote by H ,H ,... the sequence of0 1

q-Hermite polynomials, determined by the recurrence

(q) (q) (q) (q)(q)H (x) = 1, H (x) =x and xH (x) =H (x)+[n] H ,q0 1 n n+1 n−1

n1−qwhere [n] = (with the convention that [n] = n). These polynomials are related to theq 11−q

(q)q-Brownian motion X through the formula

(q) (q)(q) X X ⊗n 2 2H I (e) =I e , e∈L (R ), kek = 1. (6)2+n 1 n L (R )+

We then have:

Theorem 1.9 (q-version of the Breuer-Major theorem). Fix q ∈ [0,1] and let n ≥ 1. Let

∗{G} be a q-Gaussian centered stationary family of random variables on some W -probabilityl l∈N

space (A,ϕ), meaning that there exists ρ :Z→R such that, for every integer r ≥ 1 and every

l ,...,l ≥ 1, one has1 r

X Y

Cr(π)ϕ G ...G = q ρ(l −l ).l l a br1

π∈P ({1,...,r}) {a,b}∈π2

P nAssume further that ρ(0) = 1 (this just means that G ∼G (0,1) for every l) and |ρ(l)| isl q l∈Z

ﬁnite. Then, as k→∞,

[kt] s 2X X X n1 f.d.d. (q )(q) inv(σ) n√ H (G ) −−−→ q ρ(l) X , (7)ln t k t≥0l=0 σ∈S l∈Znt≥0

2

n(q )where ‘f.d.d.’ stands for the convergence in law of all ﬁnite-dimensional distributions and X

2nis a q -Brownian motion.

66

The rest of the paper is divided into two sections. In Section 2, we recall and prove some

useful results relative to the so-called q-Gaussian chaos, which is nothing but a generalization

of both the Wiener and Wigner chaoses. Notably, therein we extend the formula (4) to the case

of multiple integrals with respect to the q-Brownian motion (Theorem 2.7). Once endowed with

this preliminary material, we devote Section 3 to the proofs of Theorems 1.7, 1.8 and 1.9.

2 nFor the rest of the paper, we assume that L (R ) stands for the set of all real-valued square-+

nintegrable functions onR .+

2. q-Brownian chaos and product formulae

(q)Throughout this section, we ﬁx a parameter q∈ (−1,1), as well as a q-Brownian motion X

∗on some W -probability space (A ,ϕ ). As a ﬁrst step towards Theorem 1.7, our aim is toq q

(q)generalize the formula (4) to the case of multiple integrals with respect to X .

2.1. Multiple integrals. For every integer n≥ 1, the collection of all random variables of the

type Z

(q) (q) (q)X 2 nI (f) = f(t ,...,t )dX ···dX , f ∈L (R ),1 nn t t +1 nnR

+

(q)is called thenthq-Gaussian chaos associated withX , and has been deﬁned by Donati-Martin

[5] along the same lines as the classical Wiener chaos (see, e.g., [12]), namely:

(q) (q) (q) (q) (q)X- ﬁrst deﬁne I (f) = (X −X )···(X −X ) when f has the forma an 1 nb b1 n

f(t ,...,t ) =1 (t )×···×1 (t ), (8)1 n (a ,b ) 1 (a ,b ) n1 1 n n

where the intervals (a ,b ), i = 1,...,n, are pairwise disjoint;i i

(q)X- extend linearly the deﬁnition of I (f) to the class E of simple functions vanishing on di-n

agonals, that is, to functions f that are ﬁnite linear combinations of indicators of the type

(8);

2 m 2 n- observe that, for all simple functions f ∈L (R ) and g∈L (R ) vanishing on diagonals,+ +

(q) (q) (q) (q)X X X ∗ XhI (f),I (g)i 2 =ϕ I (f) I (g) =δ hf,gi , (9)q m,n qm n L (A ,ϕ ) m nq q

2 nwhere the sesquilinear formh.,.i is deﬁned for all f,g∈L (R ) byq +

ZX

inv(σ)hf,gi := q f(t ,...,t )g(t ,...,t )dt ···dt (10)q σ(1) σ(n) 1 n 1 n

nR

+σ∈Sn

and where δ stands for the Kronecker symbol;m,n

2 n- exploit the fact that the form h.,.i is strictly positive on L (R ) (see [2, Proposition 1])q +

(q)Xin order to extend I (f) to functions f in the completion F of E with respect to h.,.i .q qn P2 inv(σ) 2Observe ﬁnally that, owing to the estimate kfk ≤ q kfk , one can rely onn2q σ∈Sn L (R )

+

2 nthe inclusion L (R )⊂F for every q∈ (−1,1) and every n≥ 1.q+

2 n 2 nOfcourse,relation(9)continuestoholdforeverypairf ∈L (R )andg∈L (R ). Moreover,+ +

(q)Xthe above sketched construction implies that I (f) is self-adjoint if and only if f is mirrorn

∗ ∗symmetric, i.e., f =f where f (t ,...,t ) :=f(t ,...,t ).1 n n 1

Let us now report one of the main results of [5], namely the generalization of the product

formulaformultipleWiener-Itˆointegralstotheq-Brownianmotioncase. Inthesequel, weadopt

the following notation.7

2 nNotation. With every f ∈ L (R ) and every p ∈ {1,...,n}, we associate the function+

(p) 2 nf ∈L (R ) through the formulaq +

X

(p) α(σ)f (t ,...,t ,s ,...,s ) := q f(t ,...,s ,...,s ,...,t ),1 n−p p 1 1 p 1 n−pq

σ:{1,...,p}→{1,...,n}ց

where, in the right-hand-side, σ is decreasing (this fact is written in symbols as σ ց), s is ati

the place σ(i), and

pX p(p+1)

α(σ) := (n+1−σ(i))− .

2

i=1

[p] 2 nBesides, we deﬁne another function f ∈L (R ) byq +

X

[p] β(σ)f (s ,...,s ,t ,...,t ) := q f(t ,...,s ,...t ),1 p 1 n−p 1 i n−pq

σ:{1,...,p}→{1,...,n}

where, in the right-hand-side, s is at the place σ(i) andi

pX p(p+1)

β(σ) := σ(i)− +inv(σ).

2

i=1

(See Theorem 1.7 for the deﬁnition of inv(σ).)

We now introduce the central concept of contractions.

2 nDeﬁnition 2.1. Fix n,m ≥ 1 as well as p ∈ {1,...,min(m,n)}. Let f ∈ L (R ) and g ∈+

2 mL (R ).+

p m+n−2p21. The pth contraction f ⌢g∈L (R ) of f and g is deﬁned by the formula+

p

f ⌢g(t ,...,t )1 m+n−2p

Z

= f(t ,...,t ,s ,...,s )g(s ,...,s ,t ,...,t )ds ...ds .1 n−p p 1 1 p n−p+1 m+n−2p 1 p

p

R

+

p m+n−2p22. The pth q-contraction f ⌢ g∈L (R ) of f and g is deﬁned by the formulaq +

p p(p) [p]f ⌢ g =f ⌢g .q q q

0 0

3. We also set f ⌢ g =f ⌢g =f⊗g.q

Thesecontractionsappearnaturallyintheproductformulaformultipleintegralswithrespect

to the q-Brownian motion, that we state now.

2 n 2 mTheorem 2.2 (Donati-Martin). Let f ∈L (R ) and g∈L (R ) with n,m≥ 1. Then+ +

min(n,m)X (q) (q) (q) pX X XI (f)I (g) = I f ⌢ g . (11)qn m n+m−2p

p=08

2.2. Respecting pairings. As in [9], the notion of a respecting pairing will play a prominent

role in our study.

Deﬁnition 2.3. Let n ,...,n be positive integers and n = n +...+n . The set {1,...,n}1 r 1 r

is then partitioned accordingly as {1,...,n} = B ∪ B ∪ ...∪ B , where B = {1,...,n },1 2 r 1 1

B ={n +1,...,n +n }, ..., B ={n +...+n +1,...,n}. We denote this partition by2 1 1 2 r 1 r1

n ⊗···⊗n , and we will refer to the sets B as the blocks of n ⊗···⊗n .1 r i 1 r

Then, we say that a pairing π∈P ({1,...,n}) respects n ⊗···⊗n if every pair{l,m}∈π is2 1 r

such that l∈B and m∈B with i =j. In the sequel, the subset of such respecting pairings ini j

P ({1,...,n}) will be denoted as C (n ⊗···⊗n ).2 2 1 r

n n2 1 2 rFinally, given π ∈ C (n ⊗...⊗n ) and functions f ∈ L (R ),...,f ∈ L (R ), we deﬁne2 1 r 1 r+ +

the pairing integral

Z Z

f ⊗···⊗f := dt ···dt1 r 1 n

nπ R+

Y

×f (t ,...,t )f (t ,...,t )···f (t ,...,t ) δ(t −t ), (12)1 1 n 2 n +1 n +n r n +...+n +1 n i j1 1 1 2 1 r−1

{i,j}∈π

where δ stands for a Dirac mass at 0.

For instance, consider the following pairing

π :={(1,7),(2,4),(3,6),(5,8)}

as an element of C (3⊗2⊗3). Then it is readily checked that2

Z Z

f ⊗f ⊗f = f (t ,t ,t )f (t ,t )f (t ,t ,t )dt dt dt dt .1 2 3 1 1 2 3 2 2 4 3 3 1 4 1 2 3 4

4π R+

2 nLemma 2.4. Let f,g∈L (R ) and recall the deﬁnition (10) of hf,gi . We haveq+

ZX

inv(σ) ∗hf,gi = q f⊗g (13)q

P (σ)2σ∈Sn

where, for every σ∈S , the pairing P (σ)∈C (n⊗n) is explicitly given byn 2 2

P (σ) :={(n+1−i,n+σ(i)), 1≤i≤n}.2

Proof. With each σ ∈S , we may associate σe ∈S given by σe(i) = n+1−σ(n+1−i). Wen n

then have, by the deﬁnition (12),

Z Z

∗ ∗f⊗g = f(s ,...,s )g (s ,...,s )ds ···ds1 n −1 −1 1 nn+1−eσ (1) n+1−eσ (n)

nP (eσ) R2 +

Z

= f(s ,...,s )g(s ,...,s )ds ···ds−1 −11 n 1 nn+1−eσ (n) n+1−eσ (1)nR

+

Z

= f(t ,...,t )g(t ,...,t )dt ···dt .−1 −1 1 nσ(1) σ(n) σ(n+1−eσ (n)) σ(n+1−eσ (1))

nR

+

−1 −1Now, we observe that σ(n+1−σe (i)) = n+1−σe(σe (i)) = n+1−i, so that σ(n+1−

−1σe (n+1−i)) =i for any i. We deduce that

Z Z

∗f⊗g = f(t ,...,t )g(t ,...,t )dt ···dt .1 n 1 nσ(1) σ(n)

nP (eσ) R2 +

69

Thus, since it is further readily checked that inv(σe) = inv(σ) and that σ →σe is an involution,

we get

Z ZX X

inv(σ) ∗ inv(σ) ∗q f⊗g = q f⊗g

P (σ) P (eσ)2 2σ∈S σ∈Sn n

ZX

inv(σ)= q f(t ,...,t )g(t ,...,t )dt ···dt1 n 1 nσ(1) σ(n)

nR+σ∈Sn

= hf,gi .q

2.3. Joint moments of multiple integrals. Let us ﬁnally turn to the main concern of this

(q) (q)X 1 X rsection, that is, to the extension of (4) for multiple integralsI (f ),...,I (f ). To achieven n1 r

this goal, we focus on the following construction procedure.

Fix some positive integers n ,...,n with r ≥ 3, as well as p∈ {1,...,min(n ,n )}. Then,1 r 1 2

′givenπ ∈C ((n +n −2p)⊗n ⊗···⊗n ),σ :{1,...,p}→{1,...,n }ցandσ :{1,...,p}→2 1 2 3 r 1 1 2

′{1,...,n }, we construct a pairing π = F(σ ,σ ,π ) ∈ C (n ⊗n ⊗...⊗n ) as follows (see2 1 2 2 1 2 r

Figure 1 for an illustration):

1) In π, the ﬁrst two blocks {1,...,n } and {n +1,...,n +n } are connected via exactly p1 1 1 2

pairs given by

(σ (i),n +σ (i)), 1≤i≤p .1 1 2

2) The interactions between the n +n −2p remaining points in {1,...,n +n } and the set1 2 1 2

{n +n +1,...,n +...,n }, as well as the interactions within{n +n +1,...,n +...,n },1 2 1 r 1 2 1 r

′are governed along π .

This construction is clearly a one-to-one procedure. That is, given a pairingπ∈C (n ⊗n ⊗2 1 2

···⊗n )suchthattheﬁrsttwoblocks{1,...,n }and{n +1,...,n +n }arelinkedby(exactly)r 1 1 1 2

p pairs with p∈{1,...,min(n ,n )}, there exists a unique σ :{1,...,p}→{1,...,n }ց, a1 2 1 1

′uniqueσ :{1,...,p}→{1,...,n } and a unique pairingπ ∈C ((n +n −2p)⊗n ⊗···⊗n )2 2 2 1 2 3 r

p′such that π = F(σ ,σ ,π ). Besides, by the very deﬁnition of the q-contraction f ⌢ g, the1 2 q

following result is easily checked:

′Lemma 2.5. Fix p∈{1,...,min(n ,n )} and π ∈ C ((n +n −2p)⊗n ⊗···⊗n ). Then,1 2 2 1 2 3 r

n n2 1 2 rfor all functions f ∈L (R ),...,f ∈L (R ), one has1 r+ +

Z ZXp α(σ )+β(σ )1 2f ⌢ f ⊗f ⊗···⊗f = q f ⊗f ⊗···⊗f . (14)1 q 2 3 r 1 2 r

′ ′π F(σ ,σ ,π )1 2σ :{1,...,p}→{1,...,n }ց1 1

σ :{1,...,p}→{1,...,n }2 2

Our second ingredient for the generalization of (4) lies in the following computation of the

′crossings in F(σ ,σ ,π ):1 2

′Lemma 2.6. Fix p ∈ {1,...,min(n ,n )}, π ∈ C ((n + n − 2p)⊗ n ⊗···⊗ n ), σ :1 2 2 1 2 3 r 1

{1,...,p}→{1,...,n }ց and σ :{1,...,p}→{1,...,n }. Then1 2 2

′ ′Cr(F(σ ,σ ,π )) =α(σ )+β(σ )+Cr(π ). (15)1 2 1 2

′ ′Proof. Set π := F(σ ,σ ,π ). The diﬀerence D := Cr(π)− Cr(π ) is given by the number of1 2

crossings in π that involve at least one of the pairs{(σ (i),n +σ (i)), 1≤i≤p}. In order to1 1 2

compute this quantity, consider the following iterative procedure:10

(a)

• • • • • • • • • • • •

σ (3) σ (2) σ (1) 6+σ (2) 6+σ (3) 6+σ (1)1 1 1 2 2 2

(b)

• • • • • • • • • • • • • •

(c)

• • • • • • • • • • • • • • • • • • • •

Figure 1. (a) Pairs constructed from σ : {1,2,3} → {1,...,6} ց and1

′σ :{1,2,3}→{1,...,6}; (b) A pairingπ ∈ C (6⊗4⊗4); (c) The resulting2 2

′pairing π = F(σ ,σ ,π )∈ C (6⊗6⊗4⊗4).1 2 2

- Step 1: Compute the number of crossings that involve the pair (σ (1),n +σ (1)). Since σ1 1 2 1

is decreasing and π ∈ C (n ⊗...⊗n ), this is just the number of points between σ (1) and2 1 r 1

n +σ (1), i.e., (n +σ (1))−σ (1)−1.1 2 1 2 1

- Step 2: Compute the number of crossings that involve the pair (σ (2),n +σ (2)), leaving1 1 2

asidethepossiblecrossingsbetween(σ (2),n +σ (2))and(σ (1),n +σ (1))(theyhavealready1 1 2 1 1 2

been taken into account in Step 1). Since σ (1)>σ (2), this leads to (n +σ (2))−σ (2)−2−1 1 1 2 1

1 crossings.{σ (1)<σ (2)}2 2

...

- Step l: Compute the number of crossings that involve the pair (σ (l),n +σ (l)), leaving aside1 1 2

the possible crossings between (σ (l),n +σ (l)) and (σ (1),n +σ (1)),...,(σ (l− 1),n +1 1 2 1 1 2 1 1

σ (l−1)) (they have already been taken into account in the previous steps). This yields (n +2 1P

l−1σ (l))−σ (l)−l− 1 crossings.2 1 {σ (j)<σ (l)}j=1 2 2

...