Multivariate Second-Order $p$-Poincaré Inequalities
Tara Trauthwein

















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    equationsectionMultivariate Second-Order p-Poincaré Inequalities
            Tara TrauthweinDepartment of Mathematics, University of Luxembourg and Department of Statistics, University of Oxford, tara.trauthwein@stats.ox.ac.uk. The author was supported by the Luxembourg National Research Fund (PRIDE17/1224660/GPS) and by the UK Engineering and Physical Sciences Research Council (EPSRC) grant (EP/T018445/1).
            
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        § ABSTRACT 
        	In this work, we discuss new bounds for the normal approximation of multivariate Poisson functionals under minimal moment assumptions. Such bounds require one to estimate moments of so-called add-one costs of the functional. Previous works required the estimation of 4^th moments, while our result only requires (2+ϵ)-moments, based on recent improvements introduced by (Trauthwein 2022). As applications, we show quantitative CLTs for two multivariate functionals of the Gilbert, or random geometric, graph. These examples were out of range for previous methods.
        Keywords: Central Limit Theorem; Gilbert Graph; Malliavin Calculus; Multivariate Central Limit Theorem; Poincaré Inequality; Poisson Process; Stein's Method; Stochastic Geometry.
        Mathematics Subject Classification (2020): 60F05, 60H07, 60G55, 60D05
        
        
        
        § INTRODUCTION
        
        The present paper establishes new distance bounds for multivariate Poisson functionals, allowing to derive quantitative Central Limit Theorems under minimal moment assumptions. It thus provides a multivariate counterpart to the improved univariate second-order Poincaré inequalities recently introduced in <cit.>. As such, the paper extends some of the results from <cit.> and <cit.>, providing comparable probabilistic inequalities but substantially reducing the moment conditions. The method used to achieve these minimal assumptions is to combine, as was done in <cit.>, Stein's and interpolation methods with Malliavin Calculus, and to make use of moment inequalities recently proposed in <cit.>. Applications include, but are not limited to, the study of random geometric objects such as spatial random graphs.
        
        The bounds in <Ref> are given in terms of the so-called add-one cost operator. Given a Poisson measure η of intensity measure λ on the σ-finite space (,λ), let F be a measurable, real-valued function of η. For any x∈, we define the add-one cost operator evaluated at x by
        
            _xF = _xF(η) := F(η + δ_x) - F(η),
        
        where δ_x is the Dirac measure at x. This operator describes the change in the functional F when a point x is added to the measure η. Under additional assumptions on F, the operator  corresponds to the Malliavin derivative of F at x.  The definition can be iterated to give the second derivative
        
            _x,yF := _y(_xF).
        
        Our main result <Ref> allows one to derive quantitative CLTs for vector-valued F=(F_1,…,F_m) by controlling the covariance matrix (F) and (2+ϵ)-moments of the terms _xF_i and _x,yF_i for i=1,…,m. Previous multivariate bounds as in <cit.> typically asked one to uniformly bound moments of the order 4+ϵ.
        
        This type of bound relying on the add-one cost operators is particularly useful for quantities exhibiting a type of `local dependence', more generally known as stabilization. See e.g. <cit.> for further details on this topic. On a heuristic level, the first add-one cost _x F quantifies the amount of local change induced when adding the point x, while the second add-one cost _x,yF controls the dependence of points x and y which are further apart.
        
        
        <Ref> provides bounds for distances of the type
        
            d(F,N) = sup_h ∈ℋ | h(F) -  h(N)|
        
        where F is a multivariate Poisson functional and N a multivariate Gaussian with covariance matrix C. The distances we treat are the d_2 and the d_3 distances, where the test functions h are chosen to be 𝒞^2, resp. 𝒞^3, with boundedness conditions (see (<ref>) and (<ref>) for precise definitions). The bound of the d_2 distance uses the multivariate Stein method, which comes at the detriment of needing the matrix C to be positive-definite. The bound for the d_3 distance uses an interpolation method to circumvent this problem, but it comes at the cost of needing a higher degree of regularity in the test functions. In <cit.>, the authors also provide a bound in the convex distance, where test functions are indicator functions of convex sets. Such a bound under minimal moment assumptions is out of reach for now, as it utilizes an involved recursive estimate which introduces new terms needing to be bounded by moments of the add-one costs. These terms cannot be treated with the currently known methods of reducing moment conditions.
        
        The passage from the bounds achieved by Stein's method (resp. the interpolation method) to a bound involving add-one cost operators is achieved using Malliavin Calculus. The first combination of the two methods dates back to <cit.> in a Gaussian context, and to <cit.> in a Poisson context, and has since seen countless applications. The first appearance of a bound relying solely on moments of add-one costs was in the seminal paper <cit.>.
        
        We study two applications in this paper in <Ref>, both relating to the random geometric graph, or Gilbert graph, the first study of which dates back to <cit.>. In this model, two points are connected if their distance is less than some threshold parameter. We study functionals of the type
        
            F = ∑_edges  e  in  G_W |e|^α,
        
        where G_W is the Gilbert graph restricted to vertices lying in some convex body W and |e| denotes the length of the edge e. We look at two types of vectors, one where the exponent α varies over different components, with fixed set W, and the other where the set W varies for different components and α is fixed. In both settings we derive quantitative CLTs when we let the intensity of the underlying Poisson process grow to infinity. The setting with varying exponents has also been studied in <cit.>, where a qualitative CLT has been derived in <cit.> and the limit of the covariance matrix was given in <cit.>. We provide a quantiative analogue to their result. In the setting with varying underlying sets, even the qualitative result is new.
        Plan of the paper. We present our main results in Section <ref> and its applications in Section <ref>. The proof of <Ref> is given in Section <ref> and the ones for <Ref> in Section <ref>. We introduce the necessary Malliavin operators and results about Malliavin Calculus and Poisson measures in Appendix <ref>.
        
        Acknowledgment. I would like to thank Giovanni Peccati and Gesine Reinert for their support and helpful comments on this project.
        
        
        
        § MAIN RESULT
        
        We now present our main results. For a function ϕ:^m → which is k times continuously differentiable, denote by ϕk (resp. ϕ'_∞ if k=1) the supremum of the absolute values of all k^th partial derivatives, i.e.
        
            ϕk := max_1 ≤ i_1,…,i_k ≤ msup_x ∈^m|∂^k ϕ/∂ x_i_1…∂ x_i_k(x)|.
        
        For a vector x ∈^m, we denote by x the Euclidean norm of x.
        
        
         We can now introduce the distances to be used in this context. Define the following two sets of functions:
        
          *  Let ℋ^(2)_m be the set of all 𝒞^2 functions h:^m → such that |h(x)-h(y)| ≤x-y for all x,y∈^m and sup_x∈^m h(x)_op≤ 1;
        	
          *  let ℋ^(3)_m be the set of all 𝒞^3 functions h:^m→ such that h2 and h3 are bounded by 1.
        
        Here h(x) denotes the Hessian matrix of h at the point x and ._∞ denotes the operator norm.
        
        Let X,Y be two m-dimensional random vectors. Define the d_2 and d_3 distances between X and Y as follows:
        
            d_2(X,Y) := sup_h ∈ℋ^(2)_m| h(X) -  h(Y)|,   if X,Y < ∞
               d_3(X,Y) := sup_h ∈ℋ^(3)_m| h(X) -  h(Y)|,   if X^2,Y^2 < ∞.
        
        
        Recall the definition of the add-one cost from (<ref>). We say that a measurable functional F of η is in L^2(_η) if F^2 <∞ and that it is in  if in addition
        
            ∫_ (_x F)^2 λ(dx) < ∞.
        
        
        With these definitions, we can state the main result of this paper.
        
        Formally define the terms ζ_1^(p),...,ζ_4^(q) by
        
        
            ζ_1^(p) := ∑_i,j=1^m |C_ij - (F_i,F_j)| 
               ζ_2^(p) := 2^2/p-1∑_i,j=1^m (∫_(∫_[|_x,y F_i|^2p]^1/2p[|_x,y F_j|^2p]^1/2pλ(dx))^pλ(dy))^1/p
               ζ_3^(p) := 2^2/p∑_i,j=1^m (∫_(∫_[|_x F_i|^2p]^1/2p[|_x,y F_j|^2p]^1/2pλ(dx))^pλ(dy))^1/p
               ζ_4^(q) := m^q-1∑_i,j=1^m ∫_[|_x F_i|^q+1]^1/(q+1)[|_x F_j|^q+1]^1-1/(q+1)λ(dx).
        
        
        Then the following statement holds.
        
        
        	Let (,λ) be a σ-finite measure space and η be a (,λ)-Poisson measure. Let m ≥ 1 and let F=(F_1,...,F_m) be an ^m-valued random vector such that for 1 ≤ i ≤ m, we have F_i ∈ L^2(_η) ∩ and F_i=0. Let C=(C_ij)_1 ≤ i,j ≤ m be a symmetric positive-semidefinite matrix and let X ∼𝒩(0,C). Then for all p,q ∈ [1,2],
        	
            d_3(F,X) ≤12(ζ_1^(p) + ζ_2^(p) + ζ_3^(p)) + ζ_4^(q).
        
        	If moreover the matrix C is positive-definite, then for all p,q ∈ [1,2],
        	
            d_2(F,X) ≤C^-1_opC_op^1/2 (ζ_1^(p)+ζ_2^(p)+ζ_3^(p)) + (2C^-1_opC_op^1/2∨√(2π)8C^-1_op^3/2C_op) ζ_4^(q).
        
        
        The proof of these bounds is located in <Ref>. It relies on the work <cit.>, where the bound on the d_2 distance uses the Malliavin-Stein method, whereas the one for the d_3 distance uses an interpolation technique taken from the context of spin glasses (see <cit.>). The advantage of using an interpolation technique is that the limiting covariance matrix C does not necessarily need to be positive-definite, but it comes at the cost of a higher regularity requirement for the test functions h. Building on the results of <cit.>, we adapt the univariate improvement of <cit.> to a multivariate setting and make use of the p-Poincaré inequality introduced in <cit.> (see also <Ref> in the Appendix).
        
        
        
        § APPLICATIONS
        
        
        In this section, we study two multivariate functionals of the Gilbert graph. Both of these functionals consist of sums of power-weighted edge-lengths — in the first functional, we vary the powers of the edges-lengths, and in the second functional, we restrict the graph to different domains.
        
        We start by setting the general framework. Let t≥ 1 and (ϵ_t)_t>0 a sequence of positive real numbers such that ϵ_t → 0 as t →∞. Define η^t to be a Poisson measure on ^d of intensity t  dx. For any convex body (i.e. a convex compact set with non-empty interior) W ⊂^d and any exponent α∈, define
        
            (W) := 1/2∑_x,y ∈η^t_|W0 < x-y < ϵ_tx-y^α,
        
        where η^t_|W denotes the restriction of η to the set W. The quantity (W) is the sum of all edge-lengths to the power α in the Gilbert graph with parameter ϵ_t>0 and points in W.
        
        
        
         §.§ Varying the Exponents
        
        
         We fix W ⊂^d to be a convex body and consider real numbers α_1,...,α_m such that α_i+α_j>-d for all i,j ∈{1,...,m}. For every 1 ≤ i ≤ m, define i(W)=i as in (<ref>) and set
        
            i := (tϵ_t^α_i + d/2∨ t^3/2ϵ_t^α_i+d)^-1(i-i).
        
        
        Set furthermore
        
            σ_ij^(1) := dκ_d/2|α_i+α_j+d|  and  σ_ij^(2) := d^2κ_d^2/(α_i+d)(α_j+d),
        
        where κ_d denotes the volume of a unit ball in ^d, and define the matrix C=(C_ij)_1 ≤ i,j ≤ m by C_ij = |W|c_ij, where
        
            c_ij := 
            σ_ij^(1)    if lim_t→∞ tϵ_t^d = 0
            (σ_ij^(1) + θσ_ij^(2))     if lim_t→∞ tϵ_t^d = θ≤ 1
            (1θσ_ij^(1) + σ_ij^(2))     if lim_t→∞ tϵ_t^d = θ > 1
            σ_ij^(2)    if lim_t→∞ tϵ_t^d = ∞,
        
        and |W| denotes the volume of W. Defining the vector L̃_t as 
        
            L̃_t := (1,...,m),
        
        we recall that a CLT for L̃_t has been shown in <cit.>, as well as convergence of the covariance matrix of L̃_t to C in <cit.>. The matrix C is positive-definite in the sparse and thermodynamic regime (i.e. if tϵ_t^d → 0 or tϵ_t^d → c>0), while it is singular in the dense regime (i.e. if tϵ_t^d →∞), see <cit.>.
        Define also
        
            β_ij^(t) := σ_ij^(1) + σ_ij^(2)tϵ_t^d/1 ∨ tϵ_t^d
        
        and note that β_ij^(t)→ c_ij as t →∞.
        
        	Assume that t^2ϵ_t^d →∞ as t →∞. Let X∼𝒩(0,C) be a centred Gaussian with covariance matrix C. Then
        	
          *  if α_1,...,α_m>-d/4, there is a constant c_1>0 such that for all t≥ 1 large enough
        		
            d_3(L̃_t, X) ≤ c_1 (ϵ_t + max_1 ≤ i,j ≤ m|β_ij^(t)-c_ij| + (t^-1/2∨ (t^2ϵ_t^d)^-1/2)).
        
          *  if -d/2<min{α_1,...,α_m}≤ -d/4, then for any 1<p<-d/2min{α_1,...,α_m}^-1, there is a constant c_2>0 such that for all t≥ 1 large enough
        		
            d_3(L̃_t, X) ≤ c_2 (ϵ_t + max_1 ≤ i,j ≤ m|β_ij^(t)-c_ij| + (t^-1+1/p∨ (t^2ϵ_t^d)^-1+1/p)).
        
        		If lim_t →∞ tϵ_t^d < ∞, then the bounds (<ref>) and (<ref>) apply to d_2(L̃_t,X) as  well for different constants c_1,c_2>0.
        	
        The proof of this theorem can be found in <Ref>. The bounds given in <Ref> vary according to the limit of tϵ^d. We give a precise discussion of the bounds (<ref>) in Table <ref>. The bounds for (<ref>) follow when setting p=2.
        
        
        
         §.§ Varying the Domains
        
         Fix α>-d/2 to be a real number and let W_1,...,W_m ⊂^d be convex bodies. For every 1 ≤ i ≤ m, define F^(i)_t := (W_i) using the definition (<ref>) and set
        
            F̃_t^(i) := t^-1ϵ_t^-α-d/2(12σ_1  + σ_2 tϵ_t^d)^-1/2(F^(i)_t -  F^(i)_t)
        
        and
        
            F̃_t := (F̃_t^(1),...,F̃_t^(m)),
        
        where
        
            σ_1 := dκ_d/d+2α  and  σ_2 := (dκ_d/α+d)^2.
        
        Define the matrix C=(C_ij)_1 ≤ i,j≤ m by 
        
            C_ij := |W_i ∩ W_j|.
        
        	Under the above conditions, the matrix C is the asymptotic covariance matrix of the m-dimensional random vector F̃_t.
        	
        	Moreover, assume that t^2ϵ_t^d →∞ as t →∞. Let X∼𝒩(0,C) be a centred Gaussian with covariance matrix C. Then 
        	
          *  if α>-d/4, there is a constant c_1>0 such that for all t ≥ 1 large enough,
        		
            d_3(F̃_t, X) ≤ c_1 (ϵ_t + (t^-1/2∨ (t^2ϵ_t^d)^-1/2)).
        
          *  if -d/2<α≤ -d/4, then for any 1<p<-d/2α, there is a constant c_2>0 such that for all t ≥ 1 large enough,
        		
            d_3(F̃_t, X) ≤ c_2 (ϵ_t + (t^-1+1/p∨ (t^2ϵ_t^d)^-1+1/p)).
        
        		If the matrix C is positive definite, then the bounds (<ref>) and (<ref>) apply to d_2(F̃_t,X) as  well for different constants c_1,c_2>0.
        	
        The speed of convergence varies according to the asymptotic behaviour of tϵ_t^d. In particular, one has
        
            t^-1+1/p∨ (t^2ϵ_t^d)^-1+1/p =
            	
            		(t^2ϵ_t^d)^-1+1/p    if tϵ_t^d → 0 (sparse regime) 
            
            		t^-1+1/p    if tϵ_t^d →θ>0 (thermodynamic regime) 
            
            		t^-1+1/p    if tϵ_t^d →∞ (dense regime).
        
        The proof of <Ref> can be found in <Ref>.
        
        
        	The question whether C is positive-definite is not entirely straightforward, but some things can be said. For a vector x∈^m, we have
        	
            x^TCx = ∑_i,j=1^m x_ix_j ∫_^d_W_i(z) _W_j(z) dz = ∫_^d(∑_i=1^m x_i _W_i(z) )^2 dz.
        
        	Since W_1,…,W_m form a collection of convex bodies, it is clearly a necessary and sufficient condition that the family of indicators _W_1(z),...,_W_m(z) is linearly independent in L^2(^d), which translates (to some extent) to none of the sets being obtainable from the other sets via certain combinations of unions, intersections and complements, disregarding sets of measure zero.
        	
        	
        	
        		
        		
        	
        	
        	A simple sufficient condition for positive definiteness is that for each 1 ≤ i ≤ m-1, we have
        	
            W_i ∖⋃_j>i W_j ≠∅,
        
        	i.e. the sets W_1,...,W_m can be ordered in a sequence such that each set has a point not included in subsequent sets. This implies that the family of indicators _W_1(z),...,_W_m(z) is linearly independent, and it also entails positive-definiteness of C. Indeed, let z_1 ∈ W_1 ∖⋃_j>1 W_j. Since ⋃_j>1 W_j is closed, there is an open set U ⊂(⋃_j>1 W_j)^c such that z_1 ∈ U. As W_1 is a convex body, the intersection V of the interior of W_1 with U is open and non-empty. Assume that x^TCx=0, then the function f=x_1_W_1 + … + x_m_W_m is zero almost everywhere. In particular, it is constant f ≡ x_1 on V ⊂ W_1 ∖⋃_j>1 W_j, hence we must have x_1=0. One can now iterate this argument to show that x_2=…=x_m=0.
        	
        	
        	
        	If the collection W_1,…,W_m consists of distinct balls, we can prove positive-definiteness of C using condition (<ref>). Indeed, let B_1,…,B_m be a collection of closed distinct balls in ^d. The union ⋃_i≥ 1 B_i is closed and bounded, and admits thus at least one point z whose first coordinate achieves the maximum over all first coordinates of points in ⋃_i≥ 1 B_i. Let B_i_1,…,B_i_n be the balls tangent to the hyperplane x_1=z_1, sorted by decreasing radius (i.e. B_i_1 is the largest ball among B_i_1,…,B_i_n). There is an open ball B(z,ϵ) ⊂(⋃_j ≠ i_1,…,i_nB_j)^c, and the intersection B(z,ϵ) ∩ B_i_1 contains a point not included in B_i_2,…,B_i_n. The ball B_i_1 can thus be made the first ball in the sequence. This construction can be iterated to satisfy condition (<ref>). See Figure <ref> for an illustration.
        	
        	The condition (<ref>) is however not necessary for the matrix C to be positive definite. Indeed, consider a superposition of the sets W_1,…,W_4 depicted in Figure <ref>. One can show easily that if x_1_W_1 + … + x_4_W_4 =0 almost everywhere, then x_1=…=x_4=0, hence the indicators are linearly independent in L^2(^d) and the resulting matrix is positive-definite. However, these sets do not fulfill condition (<ref>).
        
        
        § PROOF OF <REF>
        
        
        The proof of <Ref> uses both an interpolation technique and the multivariate Stein method. For a positive-definite symmetric matrix C=(C_ij)_1 ≤ i,j ≤ m and a function g:^m →, the multivariate Stein equation is given by
        
            g(x) -  g(X) = ⟨ x , ∇ f(x) ⟩_^m - ⟨ C ,  f(x) ⟩_H.S.,
        
        for x ∈^m and X ∼𝒩(0,C). The inner product ⟨ .,. ⟩_H.S. is the Hilbert-Schmidt inner product defined as ⟨ A,B ⟩_H.S.:= Tr(AB^T) for real m × m matrices A and B and where Tr(.) denotes the trace function. If g∈𝒞^2(^m) has bounded first- and second-order partial derivatives, then a solution to (<ref>) is given by
        
            f_g(x) := ∫_0^1 1/2t[g(√(t)x + √(1-t)X) - g(X)]  dt.
        
        The solution f_g satisfies the following bounds:
        
            f_h2≤C^-1_opC_op^1/2g'_∞
            and   f_g3≤√(2π)/4C^-1_op^3/2C_opsup_x ∈^m g(x)_op,
        
        where g(x) denotes the Hessian matrix of the function g evaluated at x and ._op denotes the operator norm.
        These results can be found in <cit.>.
        
        Before we can give the proof of <Ref>, we also need some technical estimates which improve the corresponding estimates given in <cit.>. The first lemma is an extension of Lemma 3.1 in <cit.>: contrary to what was done in <cit.>, we bound the rest term R_x in the development below by a q^th power of the add-one costs, with q ∈ [1,2]. Lemma 3.1 in <cit.> corresponds to the choice of q=2.
        
        	Let F=(F_1,...,F_m) for some m ≥ 1, where F_i ∈ L^2(_η) ∩ and F_i=0 for 1 ≤ i ≤ m. Then for all ϕ∈𝒞^2(^m) with ϕ'_∞,ϕ2<∞, it holds that for a.e. x∈ and all q∈[1,2],
        	
            _x ϕ(F) = ∑_i=1^m∂ϕ/∂ x_i(F) _x F_i + R_x,
        
        	where
        	
            |R_x| ≤(2ϕ'_∞∨12ϕ2) (∑_i=1^m |_x F_i|)^q.
        
        	One has that
        	
            _x ϕ(F)    = ϕ(F + _xF) - ϕ(F) 
               = ∫_0^1 ∑_i=1^m∂ϕ/∂ x_i (F+t_x F_i) ·_x F_i  dt 
               = ∑_i=1^m∂ϕ/∂ x_i (F) ·_x F_i + R_x,
        
        	where
        	
            R_x = ∑_i=1^m∫_0^1 (∂ϕ/∂ x_i(F+t_xF) - ∂ϕ/∂ x_i(F)) ·_x F_i  dt.
        
        	Note that
        	
            |∂ϕ/∂ x_i(F+t_xF) - ∂ϕ/∂ x_i(F)| ≤(2 ϕ'_∞) ∧(tϕ2_xF)
        
        	by the mean value theorem. Hence
        	
            |R_x|    ≤(2 ϕ'_∞∑_i=1^m |_x F_i|) ∧(12ϕ2_xF∑_i=1^m |_xF_i|) 
               ≤(2ϕ'_∞∨1/2ϕ2) [∑_i=1^m |_xF_i| ∧(∑_i=1^m |_xF_i|)^2]
               ≤(2ϕ'_∞∨1/2ϕ2) (∑_i=1^m |_xF_i|)^q.
        
        The next lemma is an improvement of Lemma 4.1 in <cit.>, where the improvement comes from the fact that we use <Ref> in the final step. For the definition of the operator , see (<ref>).
        
        	Let m ≥ 1 and for 0 ≤ i ≤ m, let F_i ∈ L^2(_η) ∩ and assume F_i=0. Then for all g ∈𝒞^2(^m) such that g'_∞,g2<∞, we have
        	
            g(F_1,...,F_m)F_0 = ∑_i=1^m∂ g/∂ x_i(F_1,...,F_m) ⟨ F_i, - F_0 ⟩_L^2(λ) + R,
        
        	where for all q∈[1,2],
        	
            |R| ≤(2g'_∞∨12g2) ∫_(∑_k=1^m |_xF_k|)^q |_x F_0| λ(dx).
        
        	Using integration by parts (see <Ref>) as detailed in the proof of <cit.>, we derive that
        	
            g(F_1,...,F_m)F_0    = ⟨ g(F_1,...,F_m),- F_0⟩_L^2(λ)
               = ∑_i=1^m∂ g/∂ x_i (F_1,...,F_m) ⟨ F_i, - F_0 ⟩_L^2(λ) + ⟨ R_*,- F_0 ⟩_L^2(λ),
        
        	where R_* is a rest term satisfying the bound in <Ref>. The claimed result follows with R:= ⟨ R_*,- F_0 ⟩_L^2(λ) .
        
        The next proposition is an extension of Theorems 3.3 and 4.2 in <cit.> and exploits much of the arguments rehearsed in the proofs of these theorems, which are combined with the content of <Ref>.
        
        	Let m ≥ 1 and let F=(F_1,...,F_m) be an ^m-valued random vector such that for 1≤ i ≤ m, we have F_i ∈ L^2(_η) ∩ and F_i=0. Let C=(C_ij)_1≤ i,j ≤ m be a symmetric positive-semidefinite matrix and let X∼𝒩(0,C). Then for all q ∈ [1,2],
        	
            d_3(F,X) ≤12∑_i,j=1^m[|C_ij - ⟨ F_i, - F_j ⟩_L^2(λ)|] 
            + ∑_i=1^m∫_[(∑_j=1^m |_x F_j|)^q |_x  F_i|] λ(dx).
        
        	If moreover the matrix C is positive-definite, then for all q ∈ [1,2],
        	
            d_2(F,X) ≤   C^-1_opC_op^1/2∑_i,j=1^m[|C_ij - ⟨ F_i, - F_j ⟩_L^2(λ)|] 
               + (2C^-1_opC_op^1/2∨√(2π)/8C^-1_op^3/2C_op) 
               ∑_i=1^m∫_[(∑_j=1^m |_x F_j|)^q |_x  F_i|] λ(dx).
        
        	To show the bound on the d_3 distance, we proceed as in the proof of Theorem 4.2 in <cit.>, but we replace the use of Lemma 4.1 therein with our <Ref>. Indeed, we only need to show that
        	
            |[ϕ(F)] - [ϕ(X)]| ≤12ϕ2∑_i,j=1^m[|C_ij - ⟨ F_i, - F_j ⟩_L^2(λ)] 
            + 12(2ϕ2∨12ϕ3) ∑_i=1^m∫_[(∑_j=1^m |_x F_j|)^p |_x  F_i|] λ(dx)
        
        	for any ϕ∈𝒞^3(^m) with bounded second and third partial derivatives. Defining
        	
            ψ(t):= ϕ(√(1-t)F+√(t)X),
        
        	it is clear that 
        	
            |ϕ(F) - ϕ(X)| ≤sup_t ∈ (0,1) |ψ'(t)|.
        
        	Defining moreover
        	
            ϕ_i^t,b(x) := ∂ϕ/∂ x_i(√(1-t)x+√(t)b)
        
        	for any vector b ∈^m, it is shown in the proof of <cit.> that ψ'(t) can be written as
        	
            ψ'(t)=1/2√(t)𝒜 - 1/2√(1-t)ℬ,
        
        	where
        	
            𝒜 = √(t)∑_i,j=1^m C_ij∂^2ϕ/∂ x_i ∂ x_j (√(1-t)F + √(t)X)
        
        	and
        	
            ℬ = ∑_i=1^m∂ϕ/∂ x_i (√(1-t)F + √(t)X) F_i.
        
        	Conditioning on X in ℬ, one can apply <Ref> and deduce that
        	
            ℬ = √(1-t)∑_i,j=1^m∂^2 ϕ/∂ x_i ∂ x_j (√(1-t)F + √(t)X) ⟨ F_i, - F_j ⟩ + ∑_i=1^m[R_X^i],
        
        	where R_X^i satisfies
        	
            |R_X^i| ≤(12(ϕ_i^t,X)^(2)_∞∨ 2 (ϕ_i^t,X)'_∞) ∫_(∑_j=1^m |_x F_j|)^q |_x  F_i| λ(dx).
        
        	It suffices now to observe that
        	
            |∂^2 ϕ/∂ x_i ∂ x_j (√(1-t)F + √(t)X)| ≤ϕ2
            and   |∂ϕ_i^t,b/∂ x_j (x)| = √(1-t)|∂^2 ϕ/∂ x_i ∂ x_j (√(1-t)x + √(t)b)| ≤√(1-t)ϕ2
            and   |∂^2 ϕ_i^t,b/∂ x_j∂ x_k (x)| = (1-t) |∂^3 ϕ/∂ x_i ∂ x_j ∂ x_k (√(1-t)x + √(t)b)| ≤ (1-t) ϕ3
        
        	to deduce that
        	
            sup_t ∈ (0,1) |ψ'(t)| ≤   12ϕ2∑_i,j=1^m[|C_ij - ⟨ F_i, - F_j ⟩_L^2(λ)|] 
               + sup_t ∈ (0,1)1/2 √(1-t)(2√(1-t)ϕ2∨12 (1-t) ϕ3)
               ∑_i=1^m∫_[(∑_j=1^m |_x F_j|)^q |_x  F_i|] λ(dx).
        
        	The bound for the d_3 distance now follows.
        	
        	For the d_2 distance, as argued in the proof of Theorem 3.3 in <cit.>, it is enough to show that
        	
            |[g(F) - g(X)]| ≤    A C^-1_opC_op^1/2∑_i,j=1^m |C_ij - ⟨ F_i, - F_j ⟩_L^2(λ)| 
               + (2A C^-1_opC_op^1/2∨√(2π)/8B C^-1_op^3/2C_op)
               ∑_i=1^m∫_[(∑_j=1^m |_x F_j|)^q |_x  F_i|] λ(dx),
        
        	for smooth functions g ∈𝒞^∞(^m) whose first- and second-order derivatives are bounded in such a way that g'_∞≤ A and sup_x ∈^m g(x)_op≤ B. We now proceed as in the proof of Theorem 3.3 in <cit.> to deduce that
        	
            g(F) -  g(X) = ∑_i,j=1^m[C_ij∂^2f_g/∂ x_i ∂ x_j (F)] - ∑_k=1^m[⟨(∂ f_g/∂ x_k(F)),- F_k ⟩_L^2(λ)],
        
        	where f_g is the canonical solution (<ref>) to the multivariate Stein equation (<ref>). Define ϕ_k(x) := ∂ f_g/∂ x_k(x). By <Ref>, we have that
        	
            _x ϕ_k(F) = ∑_i=1^m∂ϕ_k/∂ x_i(F) ·_x F_i + R_x,k,
        
        	where
        	
            |R_x,k| ≤(2 ϕ_k'_∞∨12ϕ_k^(2)_∞) (∑_i=1^m |_x F_i|)^q.
        
        	It follows that
        	
            | g(F) -  g(X)| ≤f_g^(2)_∞∑_i,j=1^m |C_ij - ⟨ F_i, - F_j ⟩_L^2(λ)| 
            
            		+ sup_1≤ k ≤ m(2 ϕ_k'_∞∨12ϕ_k^(2)_∞) ∑_i=1^m∫_[(∑_j=1^m |_x F_j|)^q |_x  F_i|] λ(dx).
        
        	To see that (<ref>) holds, it suffices now to see that by (<ref>) and (<ref>), we have
        	
            f_g2≤C^-1_opC_op^1/2g'_∞≤ AC^-1_opC_op^1/2
               ϕ_k'_∞≤f_g2≤ AC^-1_opC_op^1/2
               ϕ2≤f_g3≤√(2π)/4C^-1_op^3/2C_opsup_x ∈^m g(x)_op≤√(2π)/4B C^-1_op^3/2C_op.
        
        	This concludes the proof.
        
        	Using <Ref>, it suffices to show that
        	
            ∑_i,j=1^m[|C_ij - ⟨ F_i, - F_j ⟩_L^2(λ)|] ≤ζ_1^(p) + ζ_2^(p) + ζ_3^(p)
        
        	and
        	
            ∑_i=1^m∫_[(∑_j=1^m |_x F_j|)^q |_x  F_i|] λ(dx) ≤ζ_4^(q).
        
        	
        	Fix i,j∈{1,...,m}. By the triangle inequality, we have that
        	
            [|C_ij - ⟨ F_i, - F_j ⟩_L^2(λ)|] 
            ≤[|C_ij - (F_i,F_j)|] + [|(F_i,F_j) - ⟨ F_i, - F_j ⟩_L^2(λ)|].
        
        	
        	Define G_ij:= (F_i,F_j) - ⟨ F_i, - F_j ⟩_L^2(λ). Then since F_i,F_j ∈, we have G_ij∈ L^1(_η) and by <Ref>, one has G_ij=0.
        	
        	Using the p-Poincaré inequality (<ref>) given in <Ref>, we deduce that for any p ∈ [1,2],
        	
            |G_ij| ≤[|G_ij|^p]^1/p≤ 2^2/p-1[∫_ |_xG_ij|^pλ(dx)]^1/p.
        
        	Now note that
        	
            _x G_ij = _x ∫_(_y F_i) ·(-_y  F_j) λ(dy),
        
        	and by the argument in the proof of <cit.>, we have that
        	
            |_x ∫_(_y F_i) ·(-_y  F_j) λ(dy)| ≤∫_|_x((_y F_i) ·(-_y  F_j))| λ(dy).
        
        	Using Minkowski's integral inequality, it follows that
        	
            [|(F_i,F_j) - ⟨ F_i, - F_j ⟩_L^2(λ)|] 
            ≤(∫_(∫_[|_x((_y F_i) ·(-_y  F_j))|^p]^1/pλ(dy))^pλ(dx))^1/p.
        
        	By (<ref>), we have that
        	
            _x((_y F_i) ·(-_y  F_j))
             = _x,yF_i·(-_y  F_j) + _y F_i·(-_x,y F_j) + _x,yF_i·(-_x,y F_j).
        
        	Using Minkowski's norm inequality, the Cauchy-Schwarz inequality and <Ref>, one sees that
        	
            [|_x((_y F_i) ·(-_y  F_j))|^p]^1/p≤   [|_x,yF_i|^2p]^1/2p [|_yF_j|^2p]^1/2p
               + [|_yF_i|^2p]^1/2p [|_x,yF_j|^2p]^1/2p
               + [|_x,yF_i|^2p]^1/2p [|_x,yF_j|^2p]^1/2p.
        
        	Combining (<ref>), (<ref>) and (<ref>) yields (<ref>).
        	
        	To show (<ref>), note that
        	
            (∑_j=1^m |_x F_j|)^q ≤ m^q-1∑_j=1^m |_x F_j|^q.
        
        	The bound (<ref>) now follows by Hölder inequality and <Ref>.
        
        
        § PROOFS OF <REF>
        
        Throughout this section, we denote by c a positive absolute constant whose value can change from line to line. We will need some technical bounds derived in <cit.> and presented in the next lemma.
        
        	Let α>-d/2 and r≥ 1 such that d+rα>0. Then (W) ∈ and there is a constant c_0>0 such that for all x,y ∈ W and t>0,
        	
            [(_x )^r]^1/r≤  c_0 ϵ_t^α (tϵ_t^d)^1/r (1 ∨ t ϵ_t^d)^1-1/r
            and   _x,y = 0<x-y<ϵ_tx-y^α.
        
        
        Moreover, we will need some properties of convex bodies. Let W ⊂^d be a convex body and define its inner parallel set W_ϵ by 
        
            W_ϵ := {x ∈ W : dist(x,∂ W)>ϵ},
        
        for ϵ>0 and where dist denotes the Euclidean distance and ∂ W the boundary of W. Since W is a convex set with non-empty interior, the set W_ϵ is non-empty for ϵ>0 small enough. Combining <cit.> with Steiner's formula (cf. <cit.>), one sees that there is a constant γ_W>0 such that
        
            |W ∖ W_ϵ| ≤γ_W ϵ.
        
        We can now start with the proof of <Ref>.
        
        
        	It suffices to bound the terms ζ_1^(p),...,ζ_4^(q) of <Ref>. We start by giving a bound for ζ_1^(p). Note that is has been shown in <cit.> that
        	
            |W| - γ_W ϵ_t < (i,j)/σ_ij^(1)t^2ϵ_t^α_i+α_j+d + σ_ij^(2)t^3 ϵ_t^α_i+α_j+2d≤ |W|,
        
        	where γ_W is a constant depending on W such that (<ref>) holds.
        	This implies that
        	
            β_ij^(t)(|W|-γ_Wϵ_t) < (i,j) ≤β_ij^(t) |W|,
        
        	where
        	
            β_ij^(t) = σ_ij^(1)t^2ϵ_t^α_i+α_j+d + σ_ij^(2)t^3 ϵ_t^α_i+α_j+2d/(tϵ_t^α_i+d/2∨ t^3/2ϵ_t^α_i+d)(tϵ_t^α_j+d/2∨ t^3/2ϵ_t^α_j+d) = σ_ij^(1) + σ_ij^(1) tϵ_t^d/1 ∨ tϵ_t^d,
        
        	as defined in (<ref>). Hence we have
        	
            |(i,j) - C_ij| ≤β_ij^(t)γ_W ϵ_t + ||W|β_ij^(t)-C_ij|
        
        	and thus
        	
            ζ_1^(p)≤∑_i,j=1^m(β_ij^(t)γ_W ϵ_t + ||W|β_ij^(t)-C_ij|) ≤ c (ϵ_t + max_1 ≤ i,j ≤ m|β_ij^(t) - c_ij|)
        
        	for some constant c>0.
        	
        	Next, we bound ζ_2^(p). For this we use the expressions given in <Ref>. We deduce from (<ref>) that
        	
            ζ_2^(p) = 2^2/p-1∑_i,j=1^m (t ϵ_t^α_i+d/2∨ t^3/2ϵ_t^α_i+d)^-1(t ϵ_t^α_j+d/2∨ t^3/2ϵ_t^α_j+d)^-1
            (∫_W (∫_W |x-y|<ϵ_t |x-y|^α_i + α_jtdx)^ptdy)^1/p.
        
        	Note that the inner integral is upper bounded by
        	
            dκ_d/d+α_i+α_j tϵ_t^d+α_i+α_j,
        
        	and hence we deduce, after simplification,
        	
            ζ_2^(p)≤ 2^2/p-1 |W|^1/p(∑_i,j=1^m dκ_d/d+α_i+α_j) t^-1+1/p(1 ∨ tϵ_t^d)^-1.
        
        	For ζ_3^(p), after plugging in the bounds from <Ref>, we get
        	
            ζ_3^(p)≤ 2^2/pc ∑_i,j=1^m (t ϵ_t^α_i+d/2∨ t^3/2ϵ_t^α_i+d)^-1(t ϵ_t^α_j+d/2∨ t^3/2ϵ_t^α_j+d)^-1
            (∫_W (∫_W ϵ_t^α_i (tϵ_t^d)^1/(2p)(1 ∨ tϵ_t^d)^1-1/(2p)|x-y|<ϵ_t |x-y|^α_jtdx)^ptdy)^1/p.
        
        	After simplification, this bound yields
        	
            ζ_3^(p)≤ 2^2/pc|W|^1/p m(∑_i=1^m dκ_d/d+α_i) t^-1+1/p(1 ∧ tϵ_t^d)^1/(2p).
        
        	For ζ_4^(q), we plug in the first bound from <Ref> and deduce
        	
            ζ_4^(q)≤ m^q-1c^2 ∑_i,j=1^m (t ϵ_t^α_i+d/2∨ t^3/2ϵ_t^α_i+d)^-1(t ϵ_t^α_j+d/2∨ t^3/2ϵ_t^α_j+d)^-q
            ∫_W ϵ_t^α_i (tϵ_t^d)^1/(q+1) (1 ∨ tϵ_t^d)^1-1/(q+1)ϵ_t^qα_j (tϵ_t^d)^q/(q+1) (1 ∨ tϵ_t^d)^q-q/(q+1) tdx,
        
        	which, after simplification, yields
        	
            ζ_4^(q)≤ m^q c^2 |W| t^(1-q)/2 (1 ∧ tϵ_t^d)^(1-q)/2.
        
        	If we take q=3-2/p, then q ∈ (1,2] and (q+1)α_i + d>2pα_i+d>0 and we get
        	
            ζ_2^(p) + ζ_3^(p) + ζ_4^(q)≤ c (t^-1+1/p∨ (t^2ϵ_t^d)^-1+1/p),
        
        	which concludes the proof.
        
        	As a first step, we compute the asymptotic covariance matrix of the vector F̃_t. Define the functions
        	
            h_i(x,y) := |x-y|<ϵ_tx ∈ W_iy ∈ W_i |x-y|^α,    i=1,…,m.
        
        	Then it holds that
        	
            F_t^(i) = 1/2∬_(^d)^2 h_i(x,y) (η^t)^(2)(dx,dy),
        
        	where (η^t)^(2) denotes the product measure η^t ⊗η^t. We deduce that for i,j ∈{1,…,m},
        	
            (F_t^(i),F_t^(j)) = 1/4[∬_(^d)^2 h_i(x,y) (η^t)^(2)(dx,dy) ∬_(^d)^2 h_j(z,w) (η^t)^(2)(dz,dw)] 
             - 1/4[∬_(^d)^2 h_i(x,y) (η^t)^(2)(dx,dy)][∬_(^d)^2 h_j(z,w) (η^t)^(2)(dz,dw)].
        
        	Since η^t is a point measure, in the first term on the RHS of (<ref>), it is possible to have x=z or x=w or similar equalities, which constitute the diagonals of the sets we are summing over. Using the Mecke formula (<ref>) and isolating these diagonals, one sees that 
        	
            (F_t^(i),F_t^(j))    = 1/4∬_(^d)^2 h_i(x,y) t^2  dxdy ·∬_(^d)^2 h_j(z,w) t^2  dzdw 
               =+ ∭_(^d)^3 h_i(x,y)h_j(y,w) t^3  dxdydw 
               =+ 1/2∬_(^d)^2 h_i(x,y)h_j(x,y) t^2  dxdy
               =-1/4∬_(^d)^2 h_i(x,y) t^2  dxdy ·∬_(^d)^2 h_j(z,w) t^2  dzdw.
        
        	The first and the last term cancel, thus we are left with
        	
            (F_t^(i),F_t^(j)) = ∭_(^d)^3 h_i(x,y)h_j(y,w) t^3  dxdydw + 1/2∬_(^d)^2 h_i(x,y)h_j(x,y) t^2  dxdy.
        
        	We start by computing the first term on the RHS of (<ref>). We have
        	
            ∭_(^d)^3 h_i(x,y)h_j(y,w) t^3  dxdydw 
            = t^3 ∫_W_i ∩ W_jdy∫_W_idx∫_W_jdw x ∈ B(y,ϵ_t)w ∈ B(y,ϵ_t) |x-y|^α |w-y|^α.
        
        	Recall from (<ref>) the definition of the (possibly empty) inner parallel set
        	
            W_ϵ := {z ∈ W : (x,∂ W)>ϵ},
        
        	for ϵ>0. Recall also from (<ref>) that there is a constant γ_W>0 such that
        	
            |W ∖ W_ϵ| ≤γ_W ϵ.
        
        	We can now rewrite (<ref>) as 
        	
            t^3 ∭_(^d)^3 h_i(x,y)h_j(y,w)  dxdydw = t^3 ∫_(W_i ∩ W_j)_ϵ_tdy (∫_B(y,ϵ_t)dx  |x-y|^α)^2 + R_t,
        
        	where R_t is given by
        	
            R_t := t^3 ∫_(W_i ∩ W_j)∖ (W_i ∩ W_j)_ϵ_tdy∫_W_idx∫_W_jdw x ∈ B(y,ϵ_t)w ∈ B(y,ϵ_t) |x-y|^α |w-y|^α.
        
        	The first term on the RHS of (<ref>) is given by
        	
            t^3∫_(W_i ∩ W_j)_ϵ_tdy (∫_B(y,ϵ_t)dx  |x-y|^α)^2 = |(W_i ∩ W_j)_ϵ_t| (dκ_d/d+α)^2 t^3ϵ_t^2d+2α.
        
        	Using (<ref>), one sees that
        	
            0 ≤ R_t ≤ t^3|(W_i ∩ W_j)∖ (W_i ∩ W_j)_ϵ_t|(∫_B(y,ϵ_t)dx  |x-y|^α)^2 
            ≤γ_W_i ∩ W_jt^3ϵ_t^1+2d+2α(dκ_d/d+α)^2.
        
        	Combining (<ref>), (<ref>) and (<ref>) with (<ref>), one sees that
        	
            |t^3∭_(^d)^3 h_i(x,y)h_j(y,w)   dxdydw - |W_i ∩ W_j| (dκ_d/d+α)^2 t^3ϵ_t^2d+2α| 
            ≤γ_W_i ∩ W_jt^3ϵ_t^1+2d+2α(dκ_d/d+α)^2.
        
        	For the second term in (<ref>), we proceed similarly. We have
        	
            12t^2∬_(^d)^2 h_i(x,y)h_j(x,y)  dxdy    = 12t^2∬_W_i∩ W_j|x-y|<ϵ_t |x-y|^2α dxdy
               =12t^2∫_(W_i ∩ W_j)_ϵ_tdy∫_B(y,ϵ_t)dx |x-y|^2α + R_t',
        
        	where
        	
            0 ≤ R_t' ≤12t^2γ_W_i ∩ W_jdκ_d/d+2αϵ_t^d+2α+1.
        
        	Hence we get
        	
            |12t^2∬_(^d)^2 h_i(x,y)h_j(x,y)  dxdy - 12|W_i ∩ W_j| dκ_d/d+2αt^2ϵ^d+2α| 
            ≤12γ_W_i ∩ W_jdκ_d/d+2αt^2ϵ^d+2α+1.
        
        	From (<ref>), (<ref>) and (<ref>), we deduce that
        	
            |(F_t^(i),F_t^(j))/12dκ_d/d+2α t^2ϵ_t^d+2α + (dκ_d/d+α)^2 t^3 ϵ_t^2d+2α - |W_i ∩ W_j|| ≤γ_W_i ∩ W_jϵ_t.
        
        	Now we use <Ref> and provide bounds for the terms ζ_1^(p),…,ζ_4^(q). 
        	For the term ζ_1^(p), we have by (<ref>) that
        	
            ζ_1^(p)≤β_W_i ∩ W_jϵ_t.
        
        	Plugging in the bounds from <Ref>, we get for ζ_2^(p) that
        	
            ζ_2^(p)≤ 2^2/p-1c t^-2ϵ_t^-2α-d(1 ∨ tϵ_t^d)^-1
            ∑_i,j=1^m(∫_W_i∩ W_j(∫_W_i ∩ W_j|x-y|<ϵ_t |x-y|^2α tdx )^p tdy)^1/p.
        
        	Simplifying, we deduce
        	
            ζ_2^(p)≤ c t^-1+1/p (1 ∨ tϵ_t^d)^-1σ_1 ∑_i,j=1^m |W_i ∩ W_j|^1/p.
        
        	We proceed in the same way for ζ_3^(p) and ζ_4^(q), deducing
        	
            ζ_3^(p)≤ 2^2/pc m t^-1+1/p (1 ∧ tϵ_t^d)^1/(2p)dκ_d/d+α∑_i=1^m |W_i|^1/p
        
        	and
        	
            ζ_4^(q)≤ m^q-1c t^(1-q)/2 (1 ∧ tϵ_t^d)^(1-q)/2∑_i,j=1^m |W_i ∩ W_j|.
        
        	If we take, as in the proof of <Ref>, q=3-2/p, we get
        	
            ζ_2^(p) + ζ_3^(p) + ζ_4^(q)≤ c (t^-1+1/p∨ (t^2ϵ_t^d)^-1+1/p),
        
        	which concludes the proof.
        
        
        § SOME RESULTS ON POISSON FUNCTIONALS
        
        In this section, we collect some necessary results on Poisson functionals and Malliavin calculus. Let η be a Poisson measure of intensity λ on a σ-finite measure space (,λ). We denote by L^p(_η) the set of measurable functionals F of η such that |F|^p < ∞.
        
        Add-one cost. Let F be a measurable Poisson functional. For x ∈, we define the add-one cost operator  by
        
            _x F := F(η + δ_x) - F(η),
        
        fo x ∈. For n ≥ 1, we set inductively ^(n) := ^(n-1), where ^(0) is the identity operator and ^(1)=. We say that F ∈ if F ∈ L^2(_η) and
        
            ∫_ (_x F)^2 λ(dx) < ∞.
        
        Note that the following product formula holds for F,G measurable functionals of η:
        
            _x (FG) = _xF · G + F ·_x G + _xF ·_xG.
        Chaotic decomposition. Let n ∈ and h∈ L^2(^n,λ^(n)). Denote by I_n(h) the n^th Wiener-Itô integral (see <cit.>). Then for any F ∈ L^2(_η), it holds that
        
            F = ∑_n=0^∞ I(f_n),
        
        where f_n(x_1,…,x_n) = 1/n!_x_1,…,x_n^(n) F and f_0 = I(f_0) =  F and the series converges in L^2(_η) (see <cit.>).
        
        Mecke formula. Let h=h(x,η) be a measurable, non-negative function of η and x ∈ such that 
        
            ∫_ |h(x,η)| λ(dx) <∞.
        
        
        Then it holds that
        
            ∫_ h(x,η) η(dx) = ∫_ h(x,η + δ_x) λ(dx).
        The operator . For functionals F ∈ L^2(_η) having expansion (<ref>), we define the (pseudo) inverse of the Ornstein-Uhlenbeck generator  by
        
            F := - ∑_n=1^∞1/n I_n(f_n).
        
        See also <cit.> for further details. We also use the following properties of the operator , which can be found in <cit.>.
        
        	For any F ∈ L^2(_η) and r ≥ 1, it holds that
        	
            |_x F |^r ≤ |_x F|^r,   for λ-a.e.  x ∈,
            and    |_x,y F |^r ≤ |_x,y F|^r,   for λ^(2)-a.e.  (x,y) ∈^2.
        
        	Moreover, for F,G ∈ with F =  G =0, we have
        	
            (F,G) = ∫_ (_x F) · (-_x  F) λ(dx).
        
        
        The improvement to the moment conditions given by the bounds of Theorem <ref> comes from the following inequality, the so-called p-Poincaré inequality.
        
        
        	Let F ∈ L^1(_η) and p ∈ [1,2]. Then
        	
            |F|^p - | F|^p ≤ 2^2-p∫_ |_x F|^p λ(dx).
        
        When p=2, this inequality reduces to the classical Poincaré inequality, see e.g. <cit.>.
        
        alpha
