A quantitative fourth moment theorem in free probability theory
Guillaume Cébron



Enhancement of the magnetoelectric effect in multiferroic CoFe_2O_4/PZT bilayer by induced uniaxial magnetic anisotropy
    Alex Aubert1,
Vincent Loyau1,
Frédéric Mazaleyrat1, and 
Martino LoBue1
1SATIE UMR 8029 CNRS, ENS Paris-Saclay,
Cachan, 94235 FRANCE

DOI : 10.1109/TMAG.2017.2696162

    December 30, 2023
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   A quantitative "fourth moment theorem" for any self-adjoint element in a homogeneous Wigner chaos is provided: the Wasserstein distance is controlled by the distance from the fourth moment to two. The proof uses the free counterpart of the Stein discrepancy. On the way, the free analogue of the WSH inequality is established.
      
    
      


§ INTRODUCTION



 §.§ A quantitative fourth moment theorem

Let 𝒮 be a family of centered jointly semicircular variables in some tracial W^*-probability space (𝒜,τ) such that the Hilbert space in L^2(𝒜,τ) generated by 𝒮 is separable. For any integer n≥ 0, let us denote by 𝒫_n the Wigner chaos of order n, that is to say the (complex) Hilbert space in L^2(𝒜,τ) generated by the set
{1_𝒜}∪{S_1⋯ S_k:1≤ k≤ n,S_1,…, S_k ∈𝒮}. Let us denote by ℋ_n the homogeneous Wigner chaos of order n, defined by

    ℋ_n:=𝒫_n⊖𝒫_n-1  (=𝒫_n∩𝒫_n-1^⊥).

In this article, the following quantitative bound of the 2-Wasserstein distance W_2 from an element of the  n-th homogeneous chaos to the semicircular variable is provided.

Let n≥ 2. Let us consider a self-adjoint element F=F^* of the n-th homogeneous chaos ℋ_n such that τ(F^2)=1, and let S be a standard semicircular variable. We have 
    W_2(F,S) ≤  n^3/4(τ(F^4)-2)^1/4.


In <cit.>, Kemp, Nourdin, Peccati, and Speicher showed that, for a sequence of such variables in a fixed homogeneous chaos, convergence of the fourth moment controls convergence in distribution towards the semicircular law. However, they provided a quantitative bound in terms of the fourth moment only in the special case of Wigner chaos of order two: they proved that, for F in the 2-th homogeneous chaos,

    d_𝒞_2(F,S)  ≤1/2√(3/4)(τ(F^4)-2)^1/2

for an ad hoc distance d_𝒞_2 which is in fact weaker than the Wassertein distance W_2 (see Lemma <ref>). Whether a similar fourth moment bound holds for chaoses of higher orders, as in the commutative setting (see Nualart and Peccati <cit.> and Nourdin and Peccati <cit.>), is a question which has first been investigated by Bourguin and Campese in <cit.>. They provided the following bound

    d_𝒞_2(F,S)  ≤√(C_n)(τ(F^4)-2)^1/2

for variables in a homogeneous Wigner chaos of order n which are fully symmetric (see Section <ref>), and for a constant C_n which grows asymptotically linearly with n.


Theorem <ref> completes their result: the assumption of full symmetry is removed, and the distance d_𝒞_2 is replaced by the Wasserstein distance W_2. The proof of Theorem <ref> uses the notions of free Stein kernel and free Stein discrepancy introduced by Fathi and Nelson in <cit.> (see Section <ref>), and the free Malliavin calculus as defined by Biane and Speicher in <cit.> (see Section <ref>).

One can consult <cit.> and the reference therein to get a larger overview about the classical and free results related to the "fourth moment theorem".





 §.§ Notation
A non-commutative probability space (𝒜,τ) is a unital *-algebra with a linear functional τ:𝒜→ℂ such that τ(1_𝒜)=1 and τ(A^*A)≥ 0 for all A∈𝒜. Let (𝒜,τ) be a tracial W^*-probability space, that is to say a non-commutative probability space such that 𝒜 is a von Neumann algebra, and τ is a faithful normal tracial state. For all A_1,…,A_n∈𝒜,  we denote by W^*(A_1,…,A_n) the von Neumann subalgebra of 𝒜 generated by A_1,…,A_n.

Let 𝒜_1 and 𝒜_2 be two von Neumann subalgebras of 𝒜. These algebras are called free if, for all n∈ℕ, and all indices i_1 ≠ i_2 ≠…≠ i_n, whenever A_j ∈𝒜_i_j and τ(A_j) = 0 for all 1≤ j ≤ n, we have τ(A_1⋯ A_n) = 0. We say that a family A_1,…, A_n ∈𝒜 is free from another family B_1,…, B_m ∈𝒜 if the algebras W^*(A_1,…, A_n) and W^*(B_1,…, B_n) are free.

In this article, H_1⊗ H_2 denotes the algebraic tensor product of H_1 and H_2 if H_1 and H_2 are only vector spaces, and H_1⊗ H_2 denotes the Hilbert space tensor product of H_1 and H_2 whenever H_1 and H_2 are two Hilbert spaces.

We denote by L^2(𝒜,τ) the Hilbert space given by the completion of 𝒜 with respect to the inner product ⟨ A,B⟩_L^2(𝒜,τ)=τ(A^*B) (remark that if 𝒜 is a subspace of bounded operators, L^2(𝒜,τ) can be identified with a space of operators affiliated with 𝒜). If ℬ is a W^*-subalgebra of 𝒜, we denote by L^2(ℬ,τ) the Hilbert space generated by ℬ in L^2(𝒜,τ). Moreover, for all A ∈𝒜, we denote by τ(B|A) the orthogonal projection of an element B∈ L^2(𝒜,τ) onto L^2(W^*(A),τ) and call it the conditional expectation of B given A.

Finally, given a self-adjoint element A=A^* of 𝒜, there exists a unique probability measure μ on ℝ, called the distribution of A, such that, for all bounded Borel function h on ℝ, we have

    τ(h(A))=∫_ℝhμ.





 §.§ Organisation of the paper


In Section <ref>, the free entropy, the free Stein kernel and the free Stein discrepancy are introduced. Proposition <ref> states that the Wasserstein distance is bounded by the Stein discrepancy, and Proposition <ref> is the free analogue of the WSH inequality, that is to say an improvement of the free Talagrand inequality involving the Wasserstein distance, the Stein discrepancy and the free entropy.

In Section <ref>, the Wigner chaoses and the Malliavin derivative are introduced. Corollary <ref> exhibits a Stein kernel for any element of the Wigner chaos, and Proposition <ref> states that the free Stein discrepancy of an element in a homogeneous chaos is controlled by the distance from the fourth moment to 2. Finally, our quantitative "fourth moment theorem" is summed up in Theorem <ref>.



§ FUNCTIONAL INEQUALITIES INVOLVING THE STEIN DISCREPANCY

The noncommutative derivative, or free difference quotient, is the linear map ∂ from the set of polynomial ℂ[X] to its tensor product ℂ[X]⊗ℂ[X] defined by ∂ X^n=∑_k=1^n X^k-1⊗ X^n-k. A self-adjoint element S of (𝒜,τ) is called a standard semicircular variable if the distribution of S has density √(4-t^2)/2π on [-2,2]. Equivalently, a self-adjoint element S of (𝒜,τ) is a standard semicircular variable if it satisfies the following Schwinger-Dyson equation: for all polynomials P∈ℂ[X], we have

    ⟨ S,P(S) ⟩_L^2(𝒜,τ)=⟨ 1_𝒜⊗ 1_𝒜,∂ P(S) ⟩_L^2(𝒜,τ)⊗ L^2(𝒜,τ).

In the two following sections, we present two different ways of deforming this equation. One leads to the definition of the free Fisher information, due to Voiculescu <cit.>. The other leads to the definition of the free Stein discrepancy, due to Fathi and Nelson <cit.>.


 §.§ Free Fisher information and free entropy

Let F be a self-adjoint element F of (𝒜,τ). The conjugate variable of F, if it exists, is the element ξ∈ L^2(W^*(F),τ) such that, for all P∈ℂ[X],

    ⟨ξ,P(F) ⟩_L^2(𝒜,τ)=⟨ 1_𝒜⊗ 1_𝒜,∂ P(F) ⟩_L^2(𝒜,τ)⊗ L^2(𝒜,τ).

The free Fisher information of F relative to S is the constant

    Φ^*(F|S):=ξ-F^2_L^2(𝒜,τ)

if there exists a conjugate variable of F, and +∞ if not. Even if Φ^*(F|S)=+∞, it is possible to perturb F in order to get a finite free Fisher information. Without loss of generality, let us assume that there exists a standard semicircular variable S such that F and S are free in (𝒜,τ), and set
F_t:=e^-tF+√(1-e^-2t)S.
For all t>0, 
the free Fisher information of F_t is finite. The free entropy of F relative to S is the quantity

    χ^*(F|S):=∫_0^∞Φ^*(F_t|S) t  ∈ [0,+∞].

Denoting by μ the distribution of F, we have also the explicit formula

    χ^*(F|S)=1/2∫_ℝx^2μ(x)-∬_ℝ^2log |x-y|μ(x)μ(x)-3/4.

By definition, the Wasserstein distance W_2(A,B) from A to B is given by

    W_2(A,B)^2=inf_π∈Π(A,B)∫_ℝ^2|x-y|^2 dπ(x,y),

where Π(A,B) denotes the set of probability measures on ℝ^2 with marginals the distributions of A and of B.
The free Fisher information and the free entropy dominate the Wasserstein distance of the distribution of F to a semicircular variable S thanks to the following free analogues of Talagrand's inequality and log-Sobolev inequality.
For all self-adjoint element F of (𝒜,τ), we have

    W_2(F,S)^2≤ 2 χ^*(F|S)≤Φ^*(F|S).


Note that the proof of the first inequality above uses the following lemma, where ^+/ t stands for the upper right derivative.
For all self-adjoint element F of (𝒜,τ), we have

    ^+/ tW_2(F,F_t)≤ (Φ^*(F_t|S))^1/2.


Finally, let us state the free version of the de Bruijn’s formula.
For all self-adjoint element F of (𝒜,τ), we have

    / tχ^*(F_t|S)= -Φ^*(F_t|S).




 §.§ Free Stein discrepancy

A free Stein kernel of F, if it exists, is an element A∈ L^2(𝒜,τ)⊗ L^2(𝒜,τ) such that, for all P∈ℂ[X],

    ⟨ F,P(F) ⟩_L^2(𝒜,τ)=⟨ A,∂ P(F) ⟩_L^2(𝒜,τ)⊗ L^2(𝒜,τ).

The free Stein discrepancy Σ^*(F|S) of F relative to S is defined as the constant

    Σ^*(F|S)=inf_AA-1_𝒜⊗ 1_𝒜_L^2(𝒜,τ)⊗ L^2(𝒜,τ),

where the infimum is taken over all the free Stein kernels A of F. We have the following free analogue of the HSI inequality, which is an improvement of the free log-Sobolev inequality 2 χ^*(F|S)≤Φ^*(F|S), because log(1+x)≤ x.
For all self-adjoint element F of (𝒜,τ), we have

    2 χ^*(F|S)≤Σ^*(F|S)^2 log(1+Φ^*(F|S)/Σ^*(F|S)^2).


Note that, if A is a free Stein kernel of F, the orthogonal projection of A into L^2(W^*(F),τ)⊗ L^2(W^*(F),τ) is also a free Stein kernel of F. Furthermore, if F is free from a self-adjoint element G∈𝒜, and A∈ L^2(W^*(F),τ)⊗ L^2(W^*(F),τ) is a free Stein kernel of F, the intertwining relation between ∂ and the conditional expectation found by Voiculescu in <cit.> (see also <cit.>) gives the following fact: for all P∈ℂ[X],

    ⟨ F,P(F+G) ⟩_L^2(𝒜,τ)=⟨ A,∂ P(F+G) ⟩_L^2(𝒜,τ)⊗ L^2(𝒜,τ).

As a consequence, we get the following decay of the Stein discrepancy.
Let F be a self-adjoint element of (𝒜,τ), S be a standard semicircular variable free from F and, for all t≥ 0, set F_t:=e^-tF+√(1-e^-2t)S.
Then, for all t≥ 0, the element 
    e^-2tA+(1-e^-2t)· 1_𝒜⊗ 1_𝒜
 is a free Stein kernel of F_t whenever A∈ L^2(W^*(F),τ)⊗ L^2(W^*(F),τ) is a free Stein kernel of F. Moreover,

    Σ^*(F_t|S)≤ e^-2tΣ^*(F|S).





 §.§ Wasserstein distance and free Stein discrepancy

We have the following inequality, which is an extension of  <cit.> in the semicircular case.
Let F be a self-adjoint element of (𝒜,τ) and let S be a standard semicircular variable free from F. For all t≥ 0, setting F_t=e^-tF+√(1-e^-2t)S, we have

    Φ^*(F_t|S)^1/2≤e^-2t/√(1-e^-2t)Σ^*(F|S) .


We can assume that Σ^*(F|S)<+∞. Let us denote by A a Stein kernel of F and, because the orthogonal projection of a Stein kernel into L^2(W^*(F),τ)⊗ L^2(W^*(F),τ) is a Stein kernel, we assume without loss of generality  that A∈ L^2(W^*(F),τ)⊗ L^2(W^*(F),τ). Because F is free from S, we can consider the map 
    ∂_S:L^2(W^*(F,S),τ)→ L^2(W^*(F,S),τ)⊗ L^2(W^*(F,S),τ)
 (respectively its adjoint ∂_S^*) defined in Section 3 of <cit.> as a closed unbounded operator whose domain contains ℂ⟨ F,S⟩ (respectively ℂ⟨ F,S⟩^⊗ 2) . The map ∂_S is given, for all monomial M in the variables F and S, by

    ∂_SM=∑_M=M_1SM_2M_1⊗ M_2,

where the sum runs over all decompositions of
M in the form M=M_1SM_2 with some monomials M_1 and M_2. By <cit.> The map ∂_S^* is given, for polynomials P_1 and P_2 in the variables F, by

    ∂_S^*(P_1⊗ P_2)=P_1SP_2.
 
As a consequence, ∂_S∂_S^* is the identity on ℂ⟨ F⟩^⊗ 2, and, for all element B∈ℂ⟨ F⟩^⊗ 2, we have

    ∂_S^*B_L^2(𝒜, τ)^2=⟨∂_S∂_S^*B,B⟩_L^2(𝒜, τ)⊗ L^2(𝒜, τ)^2=B_L^2(𝒜, τ)⊗ L^2(𝒜, τ)^2.

It implies that all elements in L^2(W^*(F),τ)⊗ L^2(W^*(F),τ) are in the domain of ∂_S^*, and that ∂_S^* maps isometrically L^2(W^*(F),τ)⊗ L^2(W^*(F),τ) into L^2(W^*(F,S),τ).

As particular cases, 1_𝒜⊗ 1_𝒜 and A are in the domain of ∂_S^*. We know that ∂_S^*(1_𝒜⊗ 1_𝒜)=S, but the computation of ∂_S^*(A) is more difficult. However, we can say that under the conditional expectation τ(·|F_t), we have

    √(1-e^-2t)τ(F|F_t)=e^-tτ(∂_S^*A|F_t).

Indeed, for all polynomial P, we have

    √(1-e^-2t)⟨ P(F_t),F⟩_L^2(𝒜, τ)=√(1-e^-2t)e^-t⟨ (∂ P)(F_t),A⟩_L^2(𝒜, τ)
        =e^-t⟨∂_S (P(F_t)),A⟩_L^2(𝒜, τ)=e^-t⟨ P(F_t),∂_S^*A⟩_L^2(𝒜, τ).
        
        Finally, <cit.> tells us that
    ξ_t:=1/√(1-e^-2t)τ(S|F_t)
        is the conjugate variable of F_t.
        
        In the following computation, the right member of the scalar product is always in W^*(F_t). Thus, we have the freedom of replacing ξ_t by 1/√(1-e^-2t)S and √(1-e^-2t)F by e^-t∂_S^*A in the left member of the scalar product because they coincide under the conditional expectation τ(·|F_t). We can compute
        
            Φ^*(F_t|S)=   ⟨ξ_t-F_t,ξ_t-F_t⟩_L^2(𝒜, τ)
            
            =   ⟨ξ_t-e^-tF-√(1-e^-2t)S,ξ_t-F_t⟩_L^2(𝒜, τ)
            
            =   1/√(1-e^-2t)⟨ S-√(1-e^-2t)e^-tF-(1-e^-2t)S,ξ_t-F_t⟩_L^2(𝒜, τ)
            
            =   1/√(1-e^-2t)⟨ e^-2tS-e^-2t∂_S^*A,ξ_t-F_t⟩_L^2(𝒜, τ)
            
            =   e^-2t/√(1-e^-2t)⟨∂_S^*(1_𝒜⊗ 1_𝒜-A),ξ_t-F_t⟩_L^2(𝒜, τ)
            ≤   e^-2t/√(1-e^-2t)∂_S^*(1_𝒜⊗ 1_𝒜-A)_L^2(𝒜, τ)Φ^*(F_t|S)^1/2
                =e^-2t/√(1-e^-2t)1_𝒜⊗ 1_𝒜-A_L^2(𝒜, τ)⊗ L^2(𝒜, τ)Φ^*(F_t|S)^1/2,
        
        and we conclude by minimizing over A.
        
        We get the following proposition, which is the free counterpart of <cit.>.
        For all self-adjoint random variable F of (𝒜,τ), we have
        
            W_2(F,S) ≤Σ^*(F|S).
        Let S be a standard semicircular variable free from F. For all t≥ 0, set F_t=e^-tF+√(1-e^-2t)S. Lemma <ref> and Lemma <ref> tell us that
        
            d^+/dtW_2(F,F_t)≤ (Φ^*(F_t|S))^1/2≤e^-2t/√(1-e^-2t)Σ^*(F|S).
        
        Finally, we integrate on [0,+∞[ and we get W_2(F,S) ≤Σ^*(F|S).
        
         §.§ The free WSH inequality
        We have also the following free analogue of the WSH inequality (see <cit.>), which is an improvement of the free Talagrand inequality W_2(F,S)^2≤ 2 χ^*(F|S), because arccos(e^-x)≤√(2x).
        For all self-adjoint element F of (𝒜,τ), we have
        
            W_2(F,S) ≤Σ^*(F|S)arccos(e^-χ^*(F|S)/Σ^*(F|S)^2).
        
        The proof is mutatis mutandis the one of <cit.>.
        Let S be a standard semicircular variable free from F. For all t≥ 0, set F_t=e^-tF+√(1-e^-2t)S. Theorem <ref> applied to F_t gives us
        
            χ^*(F_t|S)≤1/2Σ^*(F_t|S)^2 log(1+Φ^*(F_t|S)/Σ^*(F_t|S)^2).
        
        Now, Σ^*(F_t|S)≤Σ^*(F|S) by Lemma <ref> and r↦ rlog(1+Φ^*(F_t|S)/r) is increasing, from which it follows that
        
            χ^*(F_t|S)≤1/2Σ^*(F|S)^2 log(1+Φ^*(F_t|S)/Σ^*(F|S)^2).
        
        This inequality is equivalent to
        
            √(Φ^*(F_t|S))≤Φ^*(F_t|S)/Σ^*(F|S)√(e^2χ^*(F_t|S)/Σ^*(F|S)^2-1).
        
        Using Lemma <ref> and Lemma <ref>, we get
        
            d^+/dtW_2(F,F_t)≤√(Φ^*(F_t|S))≤-/ tχ^*(F_t|S)/Σ^*(F|S)√(e^2χ^*(F_t|S)/Σ^*(F|S)^2-1)=-/ t(Σ^*(F|S)arccos(e^-χ^*(F_t|S)/Σ^*(F|S)^2)).
        
        Finally, we integrate on [0,+∞[ and we get the result because χ^*(S|S)=0.
        
        
         §.§ The multivariate case
        
        All the definitions of the current section make perfectly sense for a n-tuple F=(F_1,…,F_n) of self-adjoint elements in (𝒜, τ) and a n-tuple of free standard semicircular variables S=(S_1,…,S_n).
        We refer to <cit.> for the precise definitions of the relative free Fisher informationΦ^*(F|S), the relative free entropyχ^*(F|S) and the relative free Stein discrepancyΣ^*(F|S) in the multivariate setting. Note the following difference of notation:  in <cit.>, Φ^*(F|S) is denoted by Φ^*(F|V_1), χ^*(F|S) corresponds to χ^*(F|V_1)-χ^*(S|V_1) and Σ^*(F|S) is denoted by Σ^*(F|V_1).
        
        Beyond the definitions, the results of the current section extend to n-tuples F=(F_1,…,F_n) of self-adjoint elements. Lemma <ref> and Theorem <ref> are stated in the multivariate case in the original article <cit.>. The free Talagrand inequality and  Lemma <ref> are also true in the multivariate setting, as proved in <cit.>. Furthermore, it is straightforward to adapt the proof of Lemma <ref>, Lemma <ref>, Proposition <ref> and Proposition <ref>.
        
        Finally, the free WS inequality and the free WSH inequality are true: for all n-tuples F=(F_1,…,F_n) of self-adjoint elements of (𝒜,τ), we have
     W_2(F,S) ≤Σ^*(F|S)   and   W_2(F,S) ≤Σ^*(F|S)arccos(e^-χ^*(F|S)/Σ^*(F|S)^2),
        where S is a n-tuple of free standard semicircular variables and W_2 is the free 2-Wasserstein distance of <cit.>.
        
        
        § STEIN DISCREPANCY IN THE WIGNER CHAOS
        
        Let 𝒫_2(n) be the set of pairings of {1,…,n}. Let π be a pairing of {1,…,n}. A quadruplet 1≤ i< j < k < l ≤ n is called a crossing of π if {i,k}∈π and {j,l}∈π. We denote by 𝒩𝒞_2(n) the set of pairings of {1,…,n} which have no crossings. A subset 𝒮 of self-adjoint elements of 𝒜 is said to be centred jointly semicircular if, for all S_1,…,S_n∈𝒮, we have
        
            τ(S_1⋯ S_n)=∑_π∈𝒩𝒞_2(n)∏_{i,j}∈πτ(S_i S_j).
        
        Note that this definition is the analogue of the Wick formula for a family 𝒩 of centred jointly Gaussian random variables: for all N_1,…,N_n∈𝒩, we have
        
            𝔼(N_1⋯ N_n)=∑_π∈𝒫_2(n)∏_{i,j}∈π𝔼(N_i N_j).
        
        Furthermore, every element of a family of centred jointly semicircular variables is a standard semicircular variable up to a multiplicative constant.
        
        
        In order to study families of centred jointly semicircular variables, we will use free stochastic analysis, and for this reason, in the three next sections, we introduce the general framework of Wigner-Itô integrals and the very useful tool of free Malliavin derivative.
        
        
        
         §.§ The semicircular field over L^2_ℝ(ℝ_+)
        
        Let 𝒮 be a family of centred jointly semicircular variables in some tracial W^*-probability space (𝒜,τ). For any integer n≥ 0, we denote by 𝒫_n the Wigner chaos of order n, that is to say the Hilbert space in L^2(𝒜,τ) generated by the set
        {1_𝒜}∪{S_1⋯ S_k:1≤ k≤ n,S_1,…, S_k ∈𝒮}.
        We denote by ℋ_n the homogeneous Wigner chaos of order n, defined by
    ℋ_n:=𝒫_n⊖𝒫_n-1  (=𝒫_n∩𝒫_n-1^⊥),
        and by π_n the orthogonal projection of L^2(𝒜,τ) onto ℋ_n. It follows that
        𝒫_n=⊕_k=0^nℋ_k.
        As examples, ℋ_0 is the Hilbert space ℂ· 1_𝒜, and ℋ_1 is the Hilbert space in L^2(𝒜,τ) generated by 𝒮.
        
        From now, we will assume that ℋ_1 is separable.  Moreover, enlarging 𝒮 (and 𝒜) if necessary, we will assume that ℋ_1 is an infinite separable Hilbert space and that 𝒮 is the set of self-adjoint elements of ℋ_1. In this case, 𝒮 is an infinite separable real Hilbert space. Every infinite separable real Hilbert space is isometrically isomorphic to L^2_ℝ(ℝ_+), the space of real-valued measurable functions on ℝ_+ which are square-integrable. As a consequence, 𝒮 is a semicircular field over L^2_ℝ(ℝ_+), in the sense that 𝒮 is a centred jointly semicircular family which can be encoded as 𝒮={S(h):h∈ L^2_ℝ(ℝ_+)}, where h↦ S(h) is an isomorphism of Hilbert space  from L^2_ℝ(ℝ_+) to 𝒮. This convention will make life easier in the future, and has the advantage to stay in the framework of <cit.>.
        
        
        
        
        
         §.§ The Wigner chaos
        Our references for this section and the next one are <cit.>. The fact that 𝒮 is a semicircular field over L^2_ℝ(ℝ_+) allows us to encode the homogeneous Wigner chaoses ℋ_n by the spaces L^2(ℝ_+^n), the spaces of complex-valued measurable functions on ℝ_+^n which are square-integrable.
        
        More precisely, we denote by H=L^2(ℝ_+) the complexified of L^2_ℝ(ℝ_+). We set H^⊗ 0=ℂ.
        
        The mapping
    I_n:h_1⊗⋯⊗ h_n↦π_n(S(h_1)⋯ S(h_n))
        for h_1,…,h_n∈ L^2_ℝ(ℝ_+) can be uniquely extended to a linear isometry between H^⊗ n and ℋ_n, or between L^2(ℝ_+^n)  (which is a concrete realization of H^⊗ n) and ℋ_n. This map I_n:L^2(ℝ_+^n)→ℋ_n is called the Wick map, or the multiple Wigner-Itô integral (with the convention that I_0(1)=1_𝒜). A priori, a multiple Wigner-Itô integral is not supposed to belongs to 𝒜, but, thanks to the following version of Haagerup's inequality, it does.
        For all A∈ℋ_n, A is in 𝒜 and we have
         
            A_𝒜≤ (n+1)A_L^2(𝒜,τ).
        Let n≥ 0. Then, 𝒫_n⊂𝒜 holds. Furthermore, if A,B∈𝒫_n, then AB∈𝒫_2n and 
        
            AB_L^2(𝒜,τ)≤ (n+1)^2A_L^2(𝒜,τ)B_L^2(𝒜,τ).
        Let us write A=∑_k=0^nA_k, where each A_k∈ℋ_k. By Theorem <ref>, we have
        
            A_𝒜≤∑_k=0^nA_k_𝒜≤ (n+1)∑_k=0^nA_k_L^2(𝒜,τ)=(n+1)A_L^2(𝒜,τ).
        
        Finally,
        AB_L^2(𝒜,τ)≤AB_𝒜≤A_𝒜B_𝒜≤ (n+1)^2A_L^2(𝒜,τ)B_L^2(𝒜,τ). Moreover, if A,B∈{1_𝒜}∪{S_1⋯ S_k:1≤ k≤ n,S_1,…, S_k ∈𝒮}, then AB∈𝒫_2n, and we conclude by linearity and continuity.
        
        There is no reason that the product should behave well with the Wick map I_n. It is possible to get rid of this difficulty thanks to the following notion of contraction. Given f∈ L^2(ℝ_+^n) and g∈ L^2(ℝ_+^m), for every 0≤ p ≤ n ∧ m, the contraction of f and g of order p is the element of L^2(ℝ_+^n+m-2p) defined almost everywhere by
        
            fp⌢ g(t_1,…,t_n+m-2p) 
            =∫_ℝ_+^pf(t_1,…,t_n-p,s_p,…,s_1)g(s_1,…,s_p,t_n-p+1,…,t_n+m-2p) s_1 ⋯ s_p.
        For all f∈ L^2(ℝ_+^n) and g∈ L^2(ℝ_+^m),
         
            I_n(f)I_m(g)=∑_p=0^n ∧ mI_n+m-2p(fp⌢ g).
        
        In particular,
         τ(I_n(f)I_m(g))=δ_n=m· (fn⌢ g). Given f∈ L^2(ℝ_+^n), the adjoint of f is the function f^*(t_1,…,t_n)=f(t_n,…,t_1) in L^2(ℝ_+^n), in such a way that I_n(f)^*=I_n(f^*). In particular, I_n(f) is self-adjoint if and only if f=f^*. The notion of contraction is useful to compute the norm of Wigner-Itô integrals.
          Let f∈ L^2(ℝ_+^n) such that f_L^2(ℝ_+^n)=1. For all 1≤ p ≤ n-1, we have
          
            τ(|I_n(f)|^4)=2+∑_p=1^n-1f p⌢ f^*_L^2(ℝ_+^2n-2p)^2.
        
         It is also possible to define more sophisticated contractions. Let n_1,…,n_r be positive integers. We denote by 𝒫_2(n_1⊗⋯⊗ n_r) the set of pairings π of 𝒫_2(n_1+⋯+n_r) such that no block of π contains more than one element from each interval set
    {1,…,n_1},{n_1+1,…,n_1+n_2},…,{n_1+⋯+n_r-1+1,…,n_1+⋯+n_r}.
        For all f_1∈ L^2(ℝ_+^n_1),…,f_r∈ L^2(ℝ_+^n_r), we define the pairing integral of f_1⊗⋯⊗ f_r with respect to π∈𝒫_2(n_1⊗⋯⊗ n_r) to be the constant
    ∫_π f_1⊗⋯⊗ f_r:=_ℝ_+^(n_1+⋯+n_r)/2(f_1⊗⋯⊗ f_r)(t_1,t_2,…,t_n_1+⋯+n_r)∏_{i,j}∈π t_i,
        where t_i and t_j are identified whenever {i,j}∈π.
         Let n_1,…,n_r>0, f_1∈ L^2(ℝ_+^n_1),…,f_r∈ L^2(ℝ_+^n_r) and π∈𝒫_2(n_1⊗⋯⊗ n_r). Then,
         
         
            |∫_π f_1⊗⋯⊗ f_r|≤f_1_L^2(ℝ_+^n_1)⋯f_r_L^2(ℝ_+^n_r).
        
        
         §.§ The free Malliavin derivative
        
        The (free) Malliavin derivative is the unique unbounded operator
    [                          ∇:                    L^2(𝒮,τ)                           → L^2(ℝ_+;L^2(𝒮,τ)⊗ L^2(𝒮,τ));                                                       A                           ↦            ∇ A=(∇_t A)_t≥ 0;                             ]
        such that, for all h∈ L^2_ℝ(ℝ_+), we have
        ∇(S(h))=h· 1 ⊗ 1 and such that ∇ satisfies the following derivation rule: for all A and B in the algebra generated by 𝒮, we have
        ∇(AB)=∇(A)· B+A·∇(B), where the left and right actions of 𝒮 are given by multiplication on the left leg and by opposite multiplication on the right leg.
        Let n≥ 0. The domain of ∇ contains 𝒫_n, and the restriction of ∇ to 𝒫_n is a bounded linear operator.
        
        It is possible to describe explicitely the action of ∇ on 𝒫_n. For all n,m≥ 0, I_n and I_m induce the linear map I_n⊗ I_m from the Hilbert space tensor product L^2(ℝ_+^n)⊗ L^2(ℝ_+^m) to ℋ_n⊗ℋ_m. The concrete realization L^2(ℝ_+^n+m) of the tensor product L^2(ℝ_+^n)⊗ L^2(ℝ_+^m) allows us to consider I_n⊗ I_m as a map
    I_n⊗ I_m:L^2(ℝ_+^n+m)→ℋ_n⊗ℋ_m
        defined on L^2(ℝ_+^n+m) such that
    I_n⊗ I_m:h_1⊗⋯⊗ h_n+m↦π_n(S(h_1)⋯ S(h_n))⊗π_n(S(h_n+1)⋯ S(h_n+m))
        for h_1,…,h_n+m∈ L^2_ℝ(ℝ_+).
        
        Finally, for all f∈ L^2(ℝ_+^n), and for almost all t∈ℝ_+, we define  f^k_t∈ L^2(ℝ_+^n-1) by the following equality, true for almost all t≥ 0:
    f(t_1,…,t_k-1,t,t_k+1,…,t_n)=f^k_t(t_1,…,t_k-1,t_k+1,…,t_n).
        Let n≥ 0. The Malliavin derivative ∇ maps 𝒫_n into L^2(ℝ_+;𝒫_n⊗𝒫_n). More precisely, if f∈ L^2(ℝ_+^n), then, for almost all t≥ 0,
        
            ∇_t(I_n(f))=∑_k=1^n (I_k-1⊗ I_n-k)(f^k_t).
        
        
        Let n≥ 0. If A_1⊗  B_1∈𝒫_n⊗𝒫_n and A_2⊗ B_2∈𝒫_n ⊗𝒫_n, we set
    (A_1⊗ B_1)^*:=A_1^*⊗ B_1^* and (A_1⊗ B_1)♯ (A_2⊗  B_2):=A_1A_2⊗ B_2B_1.
        By anti-linearity and continuity, 
        the mapping A↦ A^* extends to an antilinear map
        from 𝒫_n ⊗𝒫_n to itself. By bilinearity and continuity, we extend 
        the mapping (A, B)↦ A♯ B to a continuous bilinear map
        from (𝒫_n ⊗𝒫_n)^2 to 𝒫_2n⊗𝒫_2n. Indeed, if A and B are finite sums A=∑_k A_k⊗ C_k and B=∑_k B_k⊗ C_k, with {C_k} an orthonormal family, we have
        
            A♯ B_L^2(𝒜,τ)⊗ L^2(𝒜,τ)=∑_i,jA_iB_j⊗ C_jC_i_L^2(𝒜,τ)⊗ L^2(𝒜,τ)≤∑_i,jA_iB_j_L^2(𝒜,τ) C_jC_i_L^2(𝒜,τ)
            ≤∑_i,j(n+1)^4A_i_L^2(𝒜,τ)B_j_L^2(𝒜,τ) C_j_L^2(𝒜,τ)C_i_L^2(𝒜,τ)=(n+1)^4A_L^2(𝒜,τ) B_L^2(𝒜,τ)
            .
        
        By construction, we have (A♯ B)^*=B^*♯ A^*, and the map
        (A,B)↦ A♯ B is associative on ∪_n (𝒫_n ⊗𝒫_n)^2. Moreover, the relation
    ⟨ A,B ⟩_L^2(𝒜,τ)⊗ L^2(𝒜,τ)=τ⊗τ (B♯ A^*),
        which is true for A=A_1⊗  B_1∈𝒫_n⊗𝒫_n and B=A_2⊗ B_2∈𝒫_n ⊗𝒫_n, extends by (anti)linearity and continuity to all A, B ∈𝒫_n ⊗𝒫_n.
        
        From the derivation rule of ∇, we deduce the following chain rule for A in the algebra generated by 𝒮, and by density, for A∈𝒫_n.
        Let A∈𝒫_n and P∈ℂ[X]. For almost all t≥ 0,
        ∇_t(P(A))=(∂ P(A))♯ (∇_t A).
        
         §.§ A free Stein kernel for elements of the Wigner chaos
        
        Let n≥ 0. The bounded linear maps τ⊗𝕀:𝒫_n⊗𝒫_n→𝒫_n and 𝕀⊗τ:𝒫_n⊗𝒫_n→𝒫_n are defined by τ⊗𝕀(A⊗ B)=τ(A)· B and 𝕀⊗τ(A⊗ B)=τ(B)· A for all A,B∈𝒫_n.
        
        For all A,B∈𝒫_n such that τ(A)=0 or τ(B)=0, we have
        
            τ(AB)=τ(∫_ℝ_+(𝕀⊗τ(∇_t A))· (τ⊗𝕀(∇_t B)) t).
        If τ(A)=0, we can assume that τ(A)=τ(B)=0 by replacing B by B-τ(B)1_𝒜 if necessary: indeed, τ(Aτ(B))=0 and ∇_t(τ(B)1_𝒜)=0. For the same reason, if τ(B)=0, we can assume that τ(A)=τ(B)=0.
        
        By linearity, it suffices to prove the result when A=I_n(f)∈ℋ_n and B=I_m(g)∈ℋ_m with n,m>0. By linearity and continuity of both side, we can assume that f=f_1⊗⋯⊗ f_n and g=g_1⊗⋯⊗ f_m. We have
        
            ∇_t A=∑_i=1^nf_i(t)·  I_i-1(f_1⊗⋯⊗ f_i-1)⊗  I_n-i(f_i+1⊗⋯⊗ f_n))
        
        and applying 𝕀⊗τ gives us 𝕀⊗τ(∇_t A)=f_n(t)I_n-1(f_1⊗⋯⊗ f_n-1). Similarly, τ⊗𝕀(∇_t B)=g_1(t)I_m-1(g_2⊗⋯⊗ g_m-1).
        Finally,
        
            τ(∫_ℝ_+(𝕀⊗τ(∇_t A))· (τ⊗𝕀(∇_t B)) t)
               =∫_ℝ_+f_n(t)g_m(t) t ·τ(I_n-1(f_1⊗⋯⊗ f_n-1)I_m-1(g_2⊗⋯⊗ g_m-1))
               =∫_ℝ_+f_n(t)g_m(t) t ·δ_n-1=m-1(f_1⊗⋯⊗ f_n-1)n⌢ (g_2⊗⋯⊗ g_n-1)
               =δ_n=mfn⌢ g
                =τ(AB).
        
        Recall that, if A ∈𝒫_n and B ∈𝒫_n⊗𝒫_n, the left action A· B is the multiplication on the left leg:
    A· B=(A⊗ 1_𝒜) ♯ B.
        For all F∈𝒫_n such that τ(F)=0, we have, for all polynomial P∈ℂ[X],
        
            τ(P(F)F)=τ⊗τ(∂ P(F)♯∫_ℝ_+∇_tF♯((τ⊗𝕀(∇_t F))⊗ 1_𝒜) t).
        
        In other words, if F is self-adjoint,
        
            ∫_ℝ_+(𝕀⊗τ(∇_tF))· (∇_tF)^* t
        
        is a Stein kernel of F which belongs to 𝒫_2n.
        
        Before the demonstration, let us remark that the use of the divergence operator δ (the adjoint of the Malliavin derivative, which is not used in this article) gives an alternative proof of the above corollary. Indeed, one can compute δ ((𝕀⊗τ(∇ F))⊗ 1_𝒜)=F, from which we deduce the following computation: 
        
            τ(P(F)F^*)= τ( P(F)(δ ((𝕀⊗τ(∇ F))))^*)=∫_ℝτ⊗τ( ∇_t(P(F))♯ (𝕀⊗τ(∇_tF))⊗ 1_𝒜)^*) s
            =τ⊗τ(∂ P(F)♯∫_ℝ_+∇_tF♯((τ⊗𝕀(∇_t F))⊗ 1_𝒜) t).
        
        The same kind of computation allows Kemp, Nourdin, Peccati and Speicher to find the following Stein kernel (see <cit.>): for all F=F^*∈ℋ_n such that τ(F)=0,
    1/n∫_ℝ_+(∇_tF)· (∇_tF)^* t
        is a Stein kernel of F. However, as explained in the next section, it is more convenient to replace it by
    ∫_ℝ_+(𝕀⊗τ(∇_tF))· (∇_tF)^* t
        in order to control the Stein discrepancy by the fourth moment.
        If A⊗ B∈𝒫_n⊗𝒫_n and C∈𝒫_n, we have τ((𝕀⊗τ(A⊗ B))· C)=τ(AC)τ(B)=τ⊗τ((A ⊗ B)♯ (C⊗ 1_𝒜)).
        As a consequence, τ((𝕀⊗τ(D))· C)=τ⊗τ(D♯ (C⊗ 1_𝒜)) for all D∈𝒫_n⊗𝒫_n and B∈𝒫_n. Applied to Lemma <ref>, it gives us
        
            τ(P(F)F)= τ⊗τ(∫_ℝ_+∇_t(P(F))♯((τ⊗𝕀(∇_t F))⊗ 1_𝒜) t).
        
        The first equality of the corollary is then a consequence of the chain rule of Proposition <ref>:  ∇_t(P(F))=∂ P(F)♯∇_t(F).
        
        Because ⟨ A^*,B ⟩_L^2(𝒜,τ)⊗ L^2(𝒜,τ)=τ⊗τ (B♯ A) for all A,B∈∪_n 𝒫_n⊗𝒫_n, we know that the adjoint
        ∫_ℝ_+((τ⊗𝕀(∇_t F))⊗ 1_𝒜)^*♯ (∇_tF)^* t
        of ∫_ℝ_+∇_tF♯((τ⊗𝕀(∇_t F))⊗ 1_𝒜) t
        is a Stein kernel of F. We conclude by remarking that
        
            ((τ⊗𝕀(∇_t F))⊗ 1_𝒜)^*=(𝕀⊗τ)(∇_tF)⊗ 1_𝒜,
        
        where we used that (τ⊗𝕀(∇_t F))^*=𝕀⊗τ(∇_t (F^*))=𝕀⊗τ(∇_t F) as shown in <cit.>.
        
        
         §.§ The Stein discrepancy and the fourth moment
        Let n≥ 2, and let F be a self-adjoint element in the  homogeneous Wigner chaos ℋ_n such that τ(F^2)=1. Then,
        
            ∫_ℝ_+(𝕀⊗τ(∇_tF))· (∇_tF)^* t-1_𝒜⊗ 1_𝒜^2_L^2(𝒜,τ)⊗ L^2(𝒜,τ)≤ n^3/2(τ(F^4)-2)^1/2,
        
        and in particular, we have the following bound on the Stein discrepancy:
        
            Σ^*(F|S) ≤ n^3/4(τ(F^4)-2)^1/4.
        
        This proposition has to be compared with <cit.>, which states that there exists a certain constant C_n such that
    1/n∫_ℝ_+(∇_tF)· (∇_tF)^* t-1_𝒜⊗ 1_𝒜^2_L^2(𝒜,τ)⊗ L^2(𝒜,τ)≤ C_n (τ(F^4)-2)
        for F in a particular subset of the self-adjoint elements in ℋ_n (the fully symmetric elements). Unfortunately, as explained in <cit.>, this inequality fails to be true for all self-adjoint element in ℋ_n. This explains the necessity of replacing the Stein kernel 1/n∫_ℝ_+(∇_tF)· (∇_tF)^* t by ∫_ℝ_+(𝕀⊗τ(∇_tF))· (∇_tF)^* t.
        Thanks to Proposition <ref>,
        
            ∫_ℝ_+(𝕀⊗τ(∇_tF))· (∇_tF)^* t
         
        is a Stein kernel, and it suffices to prove the first inequality.
        
        Let us write F=I_n(f) for some f=f^*∈ L^2(ℝ_+^n). We define 
        
        
        f̃^k_t∈ L^2(ℝ_+^n-1)  (for almost all t∈ℝ_+) by
        
            f(t_1,…,t_k-1,t,t_k+1,…,t_n)=f̃^k_t(t_k-1,t_k-2,…,t_1,t_n,t_n-1,…,t_k+1).
        
        Remark that ([I_k-1⊗ I_n-k](f^k_t))^*=[I_k-1⊗ I_n-k](f̃^k_t) holds for almost all t≥ 0 if f=f_1⊗⋯⊗ f_n, and by linearity and density, it holds for all f∈ L^2(ℝ_+^n).
        
        Let us compute
        
            (𝕀⊗τ)(∇_tF)· (∇_tF)^*   =((𝕀⊗τ)(∑_k=1^n[I_k-1⊗ I_n-k](f^k_t)))·(∑_k=1^n[I_k-1⊗ I_n-k](f^k_t))^*
            
               =I_n-1(f^n_t)·(∑_k=1^n[I_k-1⊗ I_n-k](f̃^k_t))
               =∑_k=1^n∑_p=0^k-1[I_n-1+k-1-2p⊗ I_n-k](f^n_t p⌢f̃^k_t)
        
        for almost all t≥ 0,
        where the equality 
            I_n-1(f^n_t)·[I_k-1⊗ I_n-k](f̃^k_t)=∑_p=0^k-1[I_n-1+k-1-2p⊗ I_n-k](f^n_t p⌢f̃^k_t)
         is proved by density, starting from functions of type f^n_t=h_1⊗⋯⊗ h_n-1 and f̃^k_t=g_1⊗⋯⊗ g_n-1 thanks to Proposition <ref>.
        
        Remark that, when k=n and p=n-1, f^n_t p⌢f̃^k_t is the constant
        
            ∫_ℝ_+^n-1|f(t_1,…,t_n-1,t)|^2 t_1 ⋯ t_n-1
        
        and, when integrating the corresponding term [I_n-1+k-1-2p⊗ I_n-k](f^n_t p⌢f̃^k_t)=(f^n_t p⌢f̃^k_t)· 1_𝒜⊗ 1_𝒜 over t, we get f_L^2(ℝ_+^n)^2· 1_𝒜⊗ 1_𝒜=τ(F^2)· 1_𝒜⊗ 1_𝒜=1_𝒜⊗ 1_𝒜. Consequently, it will disappear in the following computation:
        
        
            ∫_ℝ_+(𝕀⊗τ)(∇_tF)· (∇_tF)^* t-1_𝒜⊗ 1_𝒜^2_L^2(𝒜,τ)⊗ L^2(𝒜,τ)
               =∑_k=1^n-1∑_p=0^k-1∫_ℝ_+[I_n-1+k-1-2p⊗ I_n-k](f^n_t p⌢f̃^k_t) t.
               .+∑_p=0^n-1∫_ℝ_+[I_2n-2p-2⊗ I_n-k](f^n_t p⌢f̃^n_t) t-1_𝒜⊗ 1_𝒜^2_L^2(𝒜,τ)⊗ L^2(𝒜,τ)
               =∑_k=1^n-1∑_p=0^k-1∫_ℝ_+(f^n_t p⌢f̃^k_t) t^2_L^2(ℝ_+^2n-2p-2)+∑_p=0^n-2∫_ℝ_+(f^n_t p⌢f̃^n_t) t^2_L^2(ℝ_+^2n-2p-2).
        
        Now, fixing k and p, let us compute
        
            f^n_t p⌢f̃^k_t(t_1,…,t_n-p-1,r_1,…,r_n-p-1)
               =∫_ℝ_+^pf^n_t(t_1,…,t_n-p-1,s_p,…,s_1)f̃^k_t(s_1,…,s_p,r_1,…,r_n-p-1) s_1⋯ s_p
               =∫_ℝ_+^pf(t_1,…,t_n-p-1,s_p,…,s_1,t)f(r_k-p-1,⋯,r_1,s_p,…,s_1,t,r_n-p-1,…,r_k-p) s_1⋯ s_p
        
        from which we deduce that
        
            ∫_ℝ_+(f^n_t p⌢f̃^k_t) t^2_L^2(ℝ_+^2n-2p-2)
            
            =∫_ℝ_+^2n    f(t_1,…,t_n-p-1,s_p,…,s_1,t)f(r_k-p-1,⋯,r_1,s_p,…,s_1,t,r_n-p-1,…,r_k-p)
               ·f(t_1,…,t_n-p-1,s'_p,…,s'_1,t')f(r_k-p-1,⋯,r_1,s'_p,…,s'_1,t',r_n-p-1,…,r_k-p)
                s_1⋯ s_p  t  s'_1⋯ s'_p  t'  t_1 … t_n-p-1  r_1 … r_n-p-1.
        
        Because f=f^*, we have
        
            ∫_ℝ_+(f^n_t p⌢f̃^k_t) t^2_L^2(ℝ_+^2n-2p-2)
            
            =∫_ℝ_+^2n    f(t_1,…,t_n-p-1,s_p,…,s_1,t)f(r_k-p,…,r_n-p-1,t,s_1,…,s_p,r_1,…,r_k-p-1)
               · f(t',s'_1,…,s'_p, t_n-p-1,…,t_1)f(r_k-p-1,⋯,r_1,s'_p,…,s'_1,t',r_n-p-1,…,r_k-p)
                s_1⋯ s_p  t  s'_1⋯ s'_p  t'  t_1 … t_n-p-1  r_1 … r_n-p-1.
        
        If we integrate over t_1 … t_n-p-1, we get f n-p-1⌢ f.
        
        Thus, we are left to
        ∫_π  (f n-p-1⌢ f)⊗ f ⊗ f
        for some pairing π∈𝒫_2((2p+2)⊗ n ⊗ n). Using Proposition <ref> leads to the following bounds
        
            ∫_ℝ_+(f^n_t p⌢f̃^k_t) t^2_L^2(ℝ_+^2n-2p-2)≤|∫_π  (f n-p-1⌢ f)⊗ f⊗ f|
               ≤f n-p-1⌢ f_L^2(ℝ_+^2p+2)·f_L^2(ℝ_+^n)·f_L^2(ℝ_+^n)= f n-p-1⌢ f_L^2(ℝ_+^2p+2).
        
        
        The norm of the contraction which appears can be controlled by τ(F^4)-2 in the following way: Proposition <ref> ensures that ∑_i=1^n-1f i⌢ f__L^2(ℝ_+^2n-2i)^2 is bounded by τ(I_n(f)^4)-2=τ(F^4)-2, and consequently,
        
            ∑_i=1^n-1f i⌢ f_L^2(ℝ_+^2n-i)≤√(n-1)(∑_i=1^n-1f i⌢ f_L^2(ℝ_+^2n-i)^2)^1/2≤√(n)(τ(F^4)-2)^1/2
            .
        
        
        Finally, 
            ∫_ℝ_+(𝕀⊗τ)(∇_tF)· (∇_tF)^* t-1⊗ 1^2_L^2(𝒜,τ)⊗ L^2(𝒜,τ)
               =
            ∑_k=1^n-1∑_p=0^k-1∫_ℝ_+(f^n_t p⌢f̃^k_t) t^2_L^2(ℝ_+^2n-2p-2)+∑_p=0^n-2∫_ℝ_+(f^n_t p⌢f̃^n_t) t^2_L^2(ℝ_+^2n-2p-2)
               ≤∑_k=1^n-1∑_p=0^k-1f n-p-1⌢ f_L^2(ℝ_+^2p+2)+∑_p=0^n-2f n-p-1⌢ f_L^2(ℝ_+^2p+2)
               ≤∑_k=1^n-1√(n)(τ(F^4)-2)^1/2+√(n)(τ(F^4)-2)^1/2
               ≤ n^3/2(τ(F^4)-2)^1/2.
        
        
         §.§ Fourth moment theorems in the Wigner chaos
        Given n≥ 1, f∈ L^2(ℝ_+^n) is called mirror symmetric if f=f^*. An element F of the homogeneous Wigner chaos ℋ_n is self-adjoint if and only if F=I_n(f) for a certain mirror symmetric f∈ L^2(ℝ_+^n), which allows us to state the free fourth moment theorem, due to Kemp, Nourdin, Peccati and Speicher, in the following form.
        
        Let n≥ 2, and let (F_k)_k≥ 1 be a sequence of self-adjoint element in the  homogeneous Wigner chaos ℋ_n, each with τ(F_k^2)=1. The distribution of F_k converges weakly to the distribution of a semi-circular variable S if and only if τ(F_k^4) converges to τ(S^4)=2 as k tends to ∞.
        
        A natural question is whether it is possible to quantify the convergence in the fourth moment theorem, as in the classical case. Kemp, Nourdin, Peccati and Speicher give an answer in the 2-th chaos, using the following distance. For all self-adjoint random variables A and B in (𝒜,τ), the distance d_𝒞_2, introduced in <cit.>, is defined by
    d_𝒞_2(A,B):=sup_h∈𝒞_2
    ℐ_2(h)≤ 1|τ(h(A))-τ(h(B))|,
        where 𝒞_2 is the set of Fourier transforms h:ℝ→ℂ of complex measures μ such that the quantity ℐ_2(h):=∫_ℝx^2|μ|( x) is finite.
        
        Let F be a self-adjoint element in the  homogeneous Wigner chaos ℋ_2, such that τ(F^2)=1. Then,
        
            d_𝒞_2(F,S)  ≤1/2√(3/4)(τ(F^4)-2)^1/2.
        
        Given n≥ 1, f∈ L^2(ℝ_+^n) is called (fully) symmetric if f is real-valued and, for any permutation σ of {1,…,n}, f=f_σ in L^2(ℝ_+^n), where f_σ(t_1,…,t_n)=f(t_σ(1),…,t_σ(n)).  A fully symmetric function is mirror symmetric, however, the set of element I_n(f) where f is fully symmetric is a strict subset of the self-adjoint elements of ℋ_n. By restricting themselves to the case of fully symmetric kernels,  Bourguin and Campese prove the following quantitative bound.
        Let n≥ 2, and let F be a self-adjoint element in the  homogeneous Wigner chaos ℋ_n, which can be written F=I_n(f) for a fully symmetric f∈ L^2(ℝ_+^n), and such that τ(F^2)=1. Then,
        
            d_𝒞_2(F,S)  ≤√(C_n)(τ(F^4)-2)^1/2
        
        for a constant C_n which grows asymptotically linearly with n.
        
        
        In fact, the following theorem shows that it is possible to control the distance d_𝒞_2(F,S), and even the Wasserstein distance W_2(F,S) of every self-adjoint element of a homogeneous Wigner chaos.
        Let n≥ 2, and let F be a self-adjoint element in the  homogeneous Wigner chaos ℋ_n such that τ(F^2)=1. Then,
        
            d_𝒞_2(F,S)≤ W_2(F,S)  ≤Σ^*(F|S) ≤ n^3/4(τ(F^4)-2)^1/4.
        Thanks to Proposition <ref> and Proposition <ref>, it only remains to prove that d_𝒞_2(F,S)≤ W_2(F,S). It is a consequence of Lemma <ref>.
        For all self-adjoint random variables A and B∈𝒜 and all function f twice differentiable, we have
            |τ(f(A))-τ(f(B))|≤ |τ(A)-τ(B)|· |f'(0)|+ 1/2(τ(A^2)^1/2+τ(B^2)^1/2)W_2(A,B)·f”_∞.
        
        In particular, if τ(A)=τ(B)=0 and τ(A^2)=τ(B^2)=1, we have
        
            d_𝒞_2(A,B)≤ W_2(A,B).
        Because the functions h of 𝒞_2 such that  ℐ_2(h)≤ 1 are twice differentiable with h”_∞≤ 1, the second inequality is a consequence of the first inequality. Let us assume that f”_∞<+∞. We rewrite
        
            f(x)-f(y)   =∫_y^xf'(t)  t
               =∫_y^x(f'(0)+∫_0^tf”(s) s)  t
               =(x-y)f'(0)+∫_y^x∫_0^tf”(s)  s  t.
        
        For all π∈Π(A,B), we can integrate the above equality and get
        
            τ(f(A))-τ(f(B))=(τ(A)-τ(B))f'(0)+∫_ℝ^2∫_y^x∫_0^tf”(s)  s  t π(x,y).
        
        Taking the absolute value yields
        
            |τ(f(A))-τ(f(B))|≤ |τ(A)-τ(B)|· |f'(0)|+f”_∞∫_ℝ^2∫_x∧ y^x ∨ y |t|  t π(x,y).
        
        Let us remark that |x∧ y|· x ∨ y-|x ∨ y|· x∧ y≥ 0, which allows to compute
        
            ∫_x∧ y^x ∨ y |t| t=   1/2(|x∨  y|· x∨  y-|x∧ y|· x∧ y)
            ≤   1/2(|x∨  y|· x ∨ y-|x∧ y|· x∧ y+|x∧ y|· x ∨ y-|x ∨ y|· x∧ y)
               =1/2(|x∨  y|+|x∧  y|)(x∨  y-x∧  y)
               =1/2(|x|+|y|)|x-  y|.
        
        Therefore, by using Hölder's inequality,
        
            ∫_ℝ^2∫_x∧ y^x ∨ y |t|  t π(x,y)≤1/2∫_ℝ^2(|x|+|y|)·|x-  y|π(x,y)
               ≤1/2(∫_ℝ^2|x|^2π(x,y)∫_ℝ^2|x-y|^2π(x,y))^1/2+1/2(∫_ℝ^2|y|^2π(x,y)∫_ℝ^2|x-y|^2π(x,y))^1/2
               = 1/2(τ(A^2)^1/2+τ(B^2)^1/2)(∫_ℝ^2|x-y|^2π(x,y))^1/2.
        
        Finally, |τ(f(X))-τ(f(Y))| is bounded by
        
            |τ(A)-τ(B)|· |f'(0)|+1/2(τ(A^2)^1/2+τ(B^2)^1/2)(∫_ℝ^2|x-y|^2π(x,y))^1/2f”_∞.
        
        and we conclude by minimizing over π∈Π(A,B).
        
        
         §.§ Acknowledgements
         The author wishes to thank Tobias Mai and Roland Speicher for plenty of useful discussions, and in particular Tobias Mai for introducing him to this problem and motivating him to establish Proposition <ref>.
        
        amsalpha
