Rate of Convergence in the Breuer-Major Theorem via Chaos Expansions
aa r X i v : . [ m a t h . P R ] M a y RATE OF CONVERGENCE IN THE BREUER-MAJOR THEOREM VIACHAOS EXPANSIONS
SEFIKA KUZGUN AND DAVID NUALART
Abstract.
We show new estimates for the total variation and Wasserstein distances inthe framework of the Breuer-Major theorem. The results are based on the combinationof Stein’s method for normal approximations and Malliavin calculus together with Wienerchaos expansions.
Mathematics Subject Classifications (2010) : 60H15, 60H07, 60G15, 60F05. Introduction
Suppose that X = { X n , n ≥ } is a centered stationary Gaussian sequence of randomvariables with unit variance. For all k ∈ Z , set ρ ( k ) = E ( X X k ) if k ≥ ρ ( k ) = ρ ( − k ) if k <
0. We say that a function g ∈ L ( R , γ ), where γ is the standard Gaussian measure, has Hermite rank d ≥ g ( x ) = ∞ X q = d c q H q ( x ) , where c d = 0 and H q is the q -th Hermite polynomial. We will make use of the followingcondition that relates the covariance function ρ to the Hermite rank of a function g :(1.2) X j ∈ Z | ρ ( j ) | d < ∞ . The Breuer-Major theorem (see [4]) says that, under condition (1.2), the sequence(1.3) F n := 1 √ n n X i =1 g ( X i )converges in law to the normal distribution N (0 , σ ), where(1.4) σ = ∞ X q = d q ! c q X k ∈ Z ρ ( k ) q . The aim of this paper is to estimate the rate of convergence to zero of the total variationand Wasserstein distances between the normalized sequence(1.5) Y n := F n p Var( F n )and the standard normal law N (0 , g . To show these results we will apply a combination of Stein’s method fornormal approximations and techniques of Malliavin calculus, and we will make use of the D. Nualart is supported by the NSF Grant DMS 1811181.
Wiener chaos expansion of the random variable F n . The combination of Stein’s methodwith Malliavin calculus to study normal approximations was first developed by Nourdin andPeccati (see the pioneering work [9] and the monograph [10]). For random variables on afixed Wiener chaos, these techniques provide a quantitative version of the Fourth MomentTheorem proved by Nualart and Peccati in [16].Given a function g ∈ L ( R , γ ) with expansion (1.1), we denote by A ( g ) the function in L ( R , γ ), whose Hermite coefficients are the absolute values of the coefficients of g , that is,(1.6) A ( g )( x ) = ∞ X q = d | c q | H q ( x ) . For any integer k ≥ p ≥
1, we denote by D k,p ( R , γ ) the Sobolev space offunctions which are k times weakly differentiable, such that together with their derivativesup to order k , they have finite moments of order p with respect to the measure γ . Also, wedenote by d TV and d W the total variation and Wasserstein distances, respectively. Along thepaper, Z will denote a N (0 ,
1) random variable. Our first result is the following.
Theorem 1.1.
Assume that g ∈ L ( R , γ ) has Hermite rank d ≥ and satisfies A ( g ) ∈ D , ( R , γ ) . Suppose that (1.2) holds true and let Y n be the random variable defined in (1.5).Then we have the following estimates: (i) If d = 2 , then d TV ( Y n , Z ) ≤ Cn − X | k |≤ n | ρ ( k ) | + Cn − X | k |≤ n | ρ ( k ) | . (1.7)(ii) If d ≥ , we have d TV ( Y n , Z ) ≤ Cn − X | k |≤ n | ρ ( k ) | d − X | k |≤ n | ρ ( k ) | + Cn − X | k |≤ n | ρ ( k ) | X | k |≤ n | ρ ( k ) | . (1.8)The proof of these results is based on Proposition 2.1, that requires the estimation ofVar( h DF n , u n i H ), where u n is such that F n = δ ( u n ). Here D and δ are the derivative anddivergence operators associated with the Malliavin calculus for the Gaussian sequence X .Following the ideas developed in [8] and [17], we construct the sequence u n using the operator T ( g ) that shifts in one unit the Hermite expansion of g . A basic ingredient of the proof is anexplicit computation of the variance Var( h DF n , u n i H ), using Wiener chaos expansions. Forthis we need a result on the convergence in L of powers of truncated Wiener chaos expansionsestablished in Proposition 3.1, which has its own interest. A sufficient condition for a function g to satisfy A ( g ) ∈ D k,M ( R , γ ) for any integer k ≥ M ≥ d = 2, theestimate (1.7) coincides with the estimate obtained in [17] (see Theorem 4.3 (iii)), assuming g ∈ D , ( R , γ ). This is the best estimate that one can obtain using Proposition 2.1 (itcoincides with the bound for g ( x ) = x − ATE OF CONVERGENCE IN THE BM THEOREM 3 for this reason one requires the function g to be four times differentiable. Here, we onlyneed one derivative, but for the function A ( g ). In a recent note (see [12]), the authors haveobtained the weaker bound(1.9) d TV ( Y n , Z ) ≤ Cn − X | k |≤ n | ρ ( k ) | assuming only g ∈ D , ( R , γ ) and applying Gebelein’s inequality, instead of Poincar´e’s inequal-ity, to estimate the variance of h DF n , u n i H . Notice that the bound (1.9) holds, for example,for the function g ( x ) = | x | − E ( | Z | ), which belongs to D , ( R , γ ).In the case d ≥
3, the estimate (1.8) coincides with the estimate obtained in [17, Theorem4.5], assuming g ∈ D d − , ( R , γ ), and applying the integration-by-parts argument severaltimes. Again our estimate requires only one derivative (for A ( g )) instead of 3 d − g = H d , leads to the optimalbound (see [2]) d TV ( Y n , Z ) ≤ Cn X | k |≤ n | ρ ( k ) | d − X | k |≤ n | ρ ( k ) | + C √ n X | k |≤ n | ρ ( k ) | d { d even } . The second part of the paper is devoted to showing two improvements of the above boundfor d = 2. First we establish the following upper bound for the Wasserstein distance, usinga new estimate (see Proposition 2.3) and the representation of F n as an iterated divergence F n = δ ( v n ). Theorem 1.2.
Assume that g ∈ L ( R , γ ) has Hermite rank d = 2 and satisfies A ( g ) ∈ D , ( R , γ ) . Suppose that (1.2) holds true and let Y n be the random variable defined in (1.5).Then we have the following estimate (1.10) d W ( Y n , Z ) ≤ Cn − X | k |≤ n | ρ ( k ) | + Cn − X | k |≤ n | ρ ( k ) | . Going back to the total variation distance, we recall first that the optimal bound for d = 2is(1.11) d TV ( Y n , Z ) ≤ Cn − X | k |≤ n | ρ ( k ) | . This estimate was obtained for g = H in [11], with a matching lower bound, and it wasextended to g ∈ D , ( R , γ ) in [17]. This upper bound, however, cannot be obtained as aconsequence of Proposition 2.1 and requires a more intensive application of Stein’s method(see [11, 17]). Using Proposition 2.2, we have obtained the following result. Theorem 1.3.
Assume that g ∈ L ( R , γ ) has Hermite rank d = 2 and satisfies A ( g ) ∈ D , ( R , γ ) . Suppose that (1.2) holds true and let Y n be the random variable defined in (1.5).Then the estimate (1.11) holds true. Notice that the first term in (1.10) coincides with the first term in (1.7), while the secondterm is precisely the optimal rate for the total variation distance (1.11).
S. KUZGUN AND D. NUALART
The paper is organized as follows. Section 2 reviews some preliminaries on the Malliavincalculus for an isonormal Gaussian process and Stein’s method. Section 3 presents a newresult on the convergence in L (Ω) of powers of Wiener chaos expansions, which has itsown interest. Finally, Sections 4, 5 and 6 contain the proofs of Theorems 1.1, 1.2 and 1.3,respectively.Along the paper we will denote by C a generic constant that may vary from line to line.2. Preliminaries
In this section, we briefly recall some elements of the Malliavin calculus associated with aGaussian family of random variables. We refer the reader to [10, 13, 14] for a detailed accounton this topic. We will also recall two basic inequalities for the total variation distance provedusing Stein’s method and we present a new inequality for the Wasserstein distance.2.1.
Malliavin calculus.
Let H be a real separable Hilbert space. For any integer m ≥ H ⊗ m and H ⊙ m to denote the m -th tensor product and the m -th symmetric tensorproduct of H , respectively. Let W = { W ( φ ) , φ ∈ H } denote an isonormal Gaussian processover the Hilbert space H . That means, W is a centered Gaussian family of random variables,defined on some probability space (Ω , F , P ), with covariance E ( W ( φ ) W ( ψ )) = h φ, ψ i H , φ, ψ ∈ H . We assume that F is generated by W .We denote by H m the closed linear subspace of L (Ω) generated by the random variables { H m ( W ( ϕ )) : ϕ ∈ H , k ϕ k H = 1 } , where H m is the m -th Hermite polynomial defined by H m ( x ) = ( − m e x d m dx m e − x , m ≥ , and H ( x ) = 1. The space H m is called the Wiener chaos of order m . The m -th multipleintegral of φ ⊗ m ∈ H ⊙ m is defined by the identity I m ( φ ⊗ m ) = H m ( W ( φ )) for any φ ∈ H with k φ k H = 1. The map I m provides a linear isometry between H ⊙ m (equipped with the norm √ m ! k · k H ⊗ m ) and H m (equipped with L (Ω) norm). By convention, H = R and I ( x ) = x .The space L (Ω) can be decomposed into the infinite orthogonal sum of the spaces H m .Namely, for any square integrable random variable F ∈ L (Ω), we have the following expan-sion,(2.1) F = ∞ X m =0 I m ( f m ) , where f = E ( F ), and f m ∈ H ⊙ m are uniquely determined by F . This is known as the Wienerchaos expansion.For a smooth and cylindrical random variable F = f ( W ( ϕ ) , . . . , W ( ϕ n )), with ϕ i ∈ H and f ∈ C ∞ b ( R n ) ( f and its partial derivatives are bounded), we define its Malliavin derivative asthe H -valued random variable given by DF = n X i =1 ∂f∂x i ( W ( ϕ ) , . . . , W ( ϕ n )) ϕ i . By iteration, we can also define the k -th derivative D k F , which is an element in the space L (Ω; H ⊗ k ). For any real p ≥ k ≥
1, the Sobolev space D k,p is defined as ATE OF CONVERGENCE IN THE BM THEOREM 5 the closure of the space of smooth and cylindrical random variables with respect to the norm k · k k,p defined by k F k pk,p = E ( | F | p ) + k X i =1 E ( k D i F k p H ⊗ i ) . We define the divergence operator δ as the adjoint of the derivative operator D . Namely, anelement u ∈ L (Ω; H ) belongs to the domain of δ , denoted by Dom δ , if there is a constant c u > u and satisfying | E ( h DF, u i H ) | ≤ c u k F k L (Ω) for any F ∈ D , . If u ∈ Dom δ , the random variable δ ( u ) is defined by the duality relationship(2.2) E ( F δ ( u )) = E ( h DF, u i H ) , which is valid for all F ∈ D , . In a similar way, for each integer k ≥
2, we define the iterateddivergence operator δ k through the duality relationship(2.3) E ( F δ k ( u )) = E (cid:16) h D k F, u i H ⊗ k (cid:17) , valid for any F ∈ D k, , where u ∈ Dom δ k ⊂ L (Ω; H ⊗ k ).Let γ be the standard Gaussian measure on R . The Hermite polynomials { H m ( x ) , m ≥ } form a complete orthonormal system in L ( R , γ ) and any function g ∈ L ( R , γ ) admits anorthogonal expansion of the form (1.1). If g has Hermite rank d , for any integer 1 ≤ k ≤ d ,we define the operator T k by(2.4) T k ( g )( x ) = ∞ X m = d c m H m − k ( x ) . To simplify the notation we will write T k ( g ) = g k .Suppose that F is a random variable in the first Wiener chaos of W of the form F = I ( ϕ ),where ϕ ∈ H has norm one. Then g k ( F ) has the representation(2.5) g ( F ) = δ k ( g k ( F ) ϕ ⊗ k ) . Moreover, if g ( F ) ∈ D j,p for some j ≥ p >
1, then g k ( F ) ∈ D j + k,p . We refer to [17] forthe proof of these results.Consider H = R , the probability space (Ω , F , P ) = ( R , B ( R ) , γ ) and the isonormal Gaussianprocess W ( h ) = h . For any k ≥ p ≥
1, denote by D k,p ( R , γ ) the corresponding Sobolevspaces of functions. Notice that if F = I ( ϕ ) is an element in the first Wiener chaos with k ϕ k H = 1, then g ∈ D k,p ( R , γ ) if and only if g ( F ) ∈ D k,p .2.2. Stein’s method.
We refer to [6] for a complete presentation of this topic. Let h : R → R be a Borel function such that h ∈ L ( R , γ ). The ordinary differential equation(2.6) f ′ ( x ) − xf ( x ) = h ( x ) − E ( h ( Z ))is called the Stein’s equation associated with h . The function f h ( x ) := e x / Z x −∞ ( h ( y ) − E ( h ( Z ))) e − y / dy is the unique solution to the Stein’s equation satisfying lim | x |→∞ e − x / f h ( x ) = 0. Moreover,if h is bounded by 1, f h satisfies k f h k ∞ ≤ p π/ k f ′ h k ∞ ≤
2. On the other hand, if h ∈ Lip(1) ( h is Lipschitz with a Lipschitz constant bounded by 1), then f h is continuously S. KUZGUN AND D. NUALART differentiable, k f ′ h k ∞ ≤ p /π and (see [19, Lemma 3]) k f ′′ h k ∞ ≤
2. We refer to [10] and thereferences therein for a complete proof of these results.We recall that the total variation distance between the laws of two random variables
F, G is defined by d TV ( F, G ) = sup B ∈B ( R ) | P ( F ∈ B ) − P ( G ∈ B ) | , where the supremum runs over all Borel sets B ⊂ R . Substituting x by F in Stein’s equation(2.6) and using the estimate for k f ′ h k ∞ lead to the fundamental estimate(2.7) d TV ( F, Z ) ≤ sup f ∈C ( R ) , k f ′ k ∞ ≤ | E ( f ′ ( F ) − F f ( F )) | . Furthermore, the Wasserstein distance between the laws of two random variables
F, G isdefined by d W ( F, G ) = sup f ∈ Lip(1) | E ( f ( F )) − E ( f (( G )) | and using Stein’s equation leads to(2.8) d W ( F, G ) ≤ sup f ∈F W | E ( f ′ ( F ) − F f ( F )) | , where F W is the set of functions f ∈ C ( R ) such that k f ′ h k ∞ ≤ p /π and k f ′′ h k ∞ ≤ W , we can use Stein’s equation toestimate the total variation distance between a random variable F = δ ( u ) and Z . A basicresult is given in the next proposition (see [15, 10]), which is an easy consequence of (2.7) andthe duality relationship (2.2). Proposition 2.1.
Assume that u ∈ Dom δ , F = δ ( u ) ∈ D , and E ( F ) = 1 . Then, d TV ( F, Z ) ≤ p Var( h DF, u i H ) . An iterative application of the Stein-Malliavin approach leads to the following result, whichrequires the random variable F to be three times differentiable (see [17, Proposition 3.2.]). Proposition 2.2.
Assume that u ∈ Dom δ , F = δ ( u ) ∈ D , and E ( F ) = 1 . Then, d T V ( F, Z ) ≤ (8 + √ π )Var( h DF, u i H ) + √ π | E ( F ) | + √ π E ( | D u F | ) + 4 π E ( | D u F | ) , where we have used the notation D u F = h u, DF i H and D i +1 u F = h u, D ( D iu F ) i H for i ≥ . In the next proposition we present a new estimate for the Wasserstein’s distance betweena random variable F = δ ( v ) and a N (0 ,
1) random variable obtained using Stein’s methodand Malliavin calculus.
Proposition 2.3.
Assume that v ∈ Dom δ , F = δ ( v ) ∈ D , and E ( F ) = 1 . Then, d W ( F, Z ) ≤ p /π q Var ( h D F, v i H ⊗ ) + 2 E ( |h DF ⊗ DF, v i H ⊗ | ) . ATE OF CONVERGENCE IN THE BM THEOREM 7
Proof.
By the duality relation (2.3), E (cid:0) F δ ( v ) (cid:1) = E (cid:0) h D F, v i H ⊗ (cid:1) . As a consequence, using(2.8) we can write d W ( F, Z ) ≤ sup f ∈F W | E (cid:0) f ′ ( F ) (cid:1) − E ( F f ( F )) | = sup f ∈F W | E (cid:0) f ′ ( F ) (cid:1) − E (cid:0) δ ( v ) f ( F ) (cid:1) | = sup f ∈F W | E (cid:0) f ′ ( F ) (cid:1) − E (cid:0) h D ( f ( F )) , v i H ⊗ (cid:1) | = sup f ∈F W | E (cid:0) f ′ ( F ) (cid:1) − E (cid:0) f ′ ( F ) h D F, v i H ⊗ (cid:1) − E (cid:0) f ′′ ( F ) h DF ⊗ DF, v i H ⊗ (cid:1) |≤ p /π E (cid:0) | − h D F, v i H ⊗ | (cid:1) + 2 E ( |h DF ⊗ DF, v i H ⊗ | ) . Now, since 1 = E (cid:0) F (cid:1) = E (cid:0) F δ ( v ) (cid:1) = E (cid:0) h D F, v i H ⊗ (cid:1) , using Cauchy-Schwarz inequality, weget E (cid:0) | − h D F, v i H ⊗ | (cid:1) ≤ r E (cid:16) | E ( h D F, v i H ⊗ ) − h D F, v i H ⊗ | (cid:17) = q Var( h D F, v i H ⊗ ) , which concludes our proof. (cid:3) Some basic inequalities.
In this subsection we recall several inequalities proved in [17](see Lemmas 6.6, 6.7 and 6.8), which can be deduced from the Brascamp-Lieb inequality (see[3]) or just using H¨older’s and Young’s convolution inequalities.
Lemma 2.1.
Fix an integer M ≥ . Let f be a non-negative function on the integers andset k = ( k , . . . , k M ) . Then, we have: (i) For any vector v ∈ R M whose components are or − X k ∈ Z M f ( k · v ) M Y j =1 f ( k j ) ≤ C X k ∈ Z f ( k ) M ! M . (ii) For any vector v ∈ R M whose components are , or − , assuming P k ∈ Z f ( k ) < ∞ , (2.10) X k ∈ Z M f ( k · v ) M Y j =1 f ( k j ) ≤ C X k ∈ Z f ( k ) ! M − . (iii) Suppose M ≥ . Let v , w ∈ R M be linearly independent vectors, whose componentsare , or − . Suppose P k ∈ Z f ( k ) < ∞ . Then, (2.11) X k ∈ Z M f ( k · v ) f ( k · w ) M Y j =1 f ( k j ) ≤ C X k ∈ Z f ( k ) ! M − . Some remarks on Wiener chaos expansions
In this section we present some useful results on Wiener chaos expansions. We first recalla formula for the expectation of the product of multiple stochastic integrals.
Lemma 3.1.
Let q i ≥ be integers, and consider functions f i ∈ H ⊙ q i , i = 1 , . . . , M . Then, E M Y i =1 I q i ( f i ) ! = X β ∈D q C q,β (cid:0) ⊗ Mi =1 f i (cid:1) β , S. KUZGUN AND D. NUALART where C q,β = Q Mi =1 q i ! Q ≤ j In general, given a random variable F ∈ L (Ω) with chaos expansion (2.1), the fact that E ( | F | p ) < ∞ for some p > L p (Ω). The next proposition provides a partial result inthis direction for p = 2 M and in the one-dimensional case, assuming that all the coefficientsare nonnegative. Proposition 3.1. Consider a function g ∈ L ( R , γ ) , with an expansion of the form g ( x ) = P ∞ q =0 c q H q ( x ) . Suppose that c q ≥ for each q ≥ and g ∈ L M ( R , γ ) for some M ≥ .Consider the truncated sequence (3.2) g ( N ) := N X q =0 c q H q . Then ( g ( N ) ) M converges in L ( R , γ ) to g M .Proof. The proof will be done by induction on M . The result is clearly true for M = 1.Suppose that M ≥ M − 1. Using the product formula for Hermitepolynomials, which is a particular case of (3.1), we can write( g ( N ) ) M = N X q ,...,q M =0 M Y i =1 c q i H q i = N X q ,...,q M =0 M Y i =1 c q i ! X ( β,γ ) ∈ b D q C q,β,γ H γ + ··· + γ M , ATE OF CONVERGENCE IN THE BM THEOREM 9 where C q,β,γ = Q Mi =1 q i ! Q Mi =1 γ i ! Q ≤ j 1. It follows that lim N →∞ d m,N = d m . This impliesthat ( g ( N ) ) M converges in L ( R , γ ) to g M and allows us to complete the proof. (cid:3) The absolute value operator. Recall that A , defined in (1.6) is the operator actingon L ( R , γ ) which replace the Hermite coefficients by its absolute values. Clearly, for anyinteger k ≥ 0, and for any g ∈ D k, ( R , γ ), we have k A ( g ) k k, = k g k k, . Therefore, g belongs to D k, ( R , γ ) if and only if A ( g ) ∈ D k, ( R , γ ). If we consider functionsin L p ( R , γ ) for some real number p > 2, we do not know whether g ∈ L p ( R , γ ) implies A ( g ) ∈ L p ( R , γ ). However, the following result holds. Lemma 3.2. Suppose that A ( g ) ∈ D k, M ( R , γ ) for some integers M ≥ and k ≥ . Then g ∈ D k, M ( R , γ ) .Proof. We will show the result only for k = 0, the case k ≥ g = P ∞ q = d c q H q and define g + = P ∞ q = d c q { q : c q > } H q and g − = P ∞ q = d c q { q : c q < } H q . Then g = g + + g − . Wewill show that g + ∈ L M ( R , γ ), and in the same way one can prove that g − ∈ L M ( R , γ ).Using Proposition 3.1, we can write E (cid:0) g M + (cid:1) = lim N →∞ E (cid:16) ( g ( N )+ ) M (cid:17) = ∞ X q ,...,q M =0 M Y i =1 c q i { q : c q > } ! X β ∈D q Q Mi =1 q i ! Q ≤ j Fix integers ℓ ≥ and M ≥ . Let g be a function in g ∈ D ℓ, ( R , γ ) , withHermite expansion g = P ∞ k =0 c q H q . Then, A ( g ) ∈ D ℓ,M ( R , γ ) if (3.4) ∞ X q =0 | c q | q ℓ − p q !( M − q < ∞ . Proof. We have D ℓ A ( N ) ( g ) = N X q = ℓ | c q | q ( q − · · · ( q − ℓ + 1) H q − ℓ . Applying the estimate (see, for instance, [7]) k H q k L M ( R ,γ ) = c ( M ) q − p q !( M − q (1 + O ( q − )) , ATE OF CONVERGENCE IN THE BM THEOREM 11 we obtain k D ℓ A ( N ) ( g ) k L M ( R ,γ ) ≤ c ( M ) | c ℓ | N X q = ℓ | c q | q ( q − · · · ( q − ℓ + 1)( q − ℓ ) − × p ( q − ℓ )!( M − q − ℓ (1 + O ( q − )) ! ≤ c ( M, ℓ ) | c ℓ | + N X q = ℓ | c q | q ℓ − p q !( M − q − ℓ (1 + O ( q − )) ! . Therefore, taking into account that A ( N ) ( g ) converges in L (Ω) to A ( g ) as N tends to infinity,we conclude that E ( | D ℓ A ( g ) | M ) < ∞ if (3.4) holds. (cid:3) Proof of Theorem 1.1 Proof. Consider a centered stationary Gaussian family of random variables X = { X n , n ≥ } with unit variance and covariance ρ ( k ) = E ( X X k ) for k ≥ 0. We put ρ ( − k ) = ρ ( k ) for k < H is a Hilbert space and e i ∈ H , i ≥ 0, are elements such that, for each i, j ≥ h e i , e j i H = ρ ( i − j ). In this situation, if { W ( φ ) : φ ∈ H } is an isonormal Gaussianprocess, then the sequence X = { X n , n ≥ } has the same law as { W ( e n ) , n ≥ } and we canassume, without any loss of generality, that X n = W ( e n ).Consider the sequence F n := √ n P nj =1 g ( X j ) introduced in (1.5), where g ∈ L ( R , γ ) hasHermite rank d ≥ σ n = E ( F n ). Under condition (1.2), it is well known that as n → ∞ , σ n → σ , where σ has been defined in (1.4). Set Y n = F n σ n . Notice that σ > σ n is bounded below for n large enough. Taking into account (2.5), we have therepresentation Y n = δ ( σ n u n ), where(4.1) u n = 1 √ n n X j =1 g ( X j ) e j , and g is the shifted function introduced in (2.4).As a consequence of Proposition 2.1, we have the estimate d T V ( Y n , Z ) ≤ r Var( h DY n , σ n u n i H ) ≤ C p Var( h DF n , u n i H ) . (4.2)Then, we can write h DF n , u n i H = 1 n n X i,j =1 g ′ ( X i ) g ( X j ) ρ ( i − j ) . The random variable g ′ ( X i ) g ( X j ) belongs to L (Ω), but we do not know its chaos expansion.For this reason, we need to use a limit argument. We have h DF n , u n i H = lim N →∞ Φ n,N , where the convergence holds in L (Ω) andΦ n,N = 1 n n X i,j =1 N X q ,q = d c q c q q H q − ( X i ) H q − ( X j ) ρ ( i − j ) . Therefore, by Fatou’s lemmaVar ( h DF n , u n i H ) = E (cid:0) h DF n , u n i H (cid:1) − ( E ( h DF n , u n i H )) ≤ lim inf N →∞ (cid:0) E (Φ n,N ) − ( E (Φ n,N )) (cid:1) = lim inf N →∞ Var(Φ n,N ) . We can writeVar(Φ n,N ) = 1 n n X i ,i ,i ,i =1 N X q ,q ,q ,q = d q q c q c q c q c q ρ ( i − i ) ρ ( i − i ) × Cov( H q − ( X i ) H q − ( X i ) , H q − ( X i ) H q − ( X i )) . (4.3)The next step is to compute the covariance appearing in the previous formula. To do thiswe will write the Hermite polynomials in terms of stochastic integrals and apply Lemma 3.1.That is, Cov( H q − ( X i ) H q − ( X i ) , H q − ( X i ) H q − ( X i ))= Cov( I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) , I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ))= E (cid:16) I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) (cid:17) − E (cid:16) I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) (cid:17) E (cid:16) I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) (cid:17) and using Lemma 3.1, E (cid:16) I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) I q − ( e ⊗ ( q − i ) (cid:17) = X β ∈D q C q,β Y ≤ j 4, satisfying β + β + β + β ≥ β ≥ β ≥ q ℓ = X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . This leads to the estimateVar(Φ n,N ) ≤ sup β A n,β N X q ,q ,q ,q = d X β ∈E q K q,β | c q c q c q c q | , where A n,β = 1 n n X i ,i ,i ,i =1 Y ≤ j 4, satisfying β + β + β + β ≥ β ≥ β ≥ β jk ≤ d for 1 ≤ j < k ≤ d ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . To complete the proof we need to show the following claims:(a) We have(4.8) ∞ X q ,q ,q ,q = d X β ∈E q K q,β | c q c q c q c q | < ∞ . (b) If d = 2, then sup β A n,β is bounded by a constant times the right-hand side of (1.7).(c) If d ≥ 3, then sup β A n,β is bounded by a constant times the right-hand side of (1.8). Proof of (4.8): The main idea here is to identify the sum in (4.8) as the variance of a truncatedfunction composed with a fixed random variable X . From our previous computations itfollows that N X q ,q ,q ,q = d X β ∈E q K q,β | c q c q c q c q | = N X q ,q ,q ,q = d q q | c q c q c q c q |× Cov( H q − ( X ) H q − ( X ) , H q − ( X ) H q − ( X ))= Var( A ( g ′ ) ( N ) ( X ) A ( g ) ( N ) ( X )) , where for each integer N ≥ d , we denote by A ( g ′ ) ( N ) and A ( g ) ( N ) the truncated expansionsof A ( g ′ ) and A ( g ), respectively, introduced in (3.2). By Proposition 3.1, ( A ( g ′ ) ( N ) ) and( A ( g ) ( N ) ) are convergent in L ( R , γ ) to A ( g ′ ) and A ( g ) , respectively. Therefore, ∞ X q ,q ,q ,q = d X β ∈E q K q,β | c q c q c q c q | = Var( A ( g ′ )( X ) A ( g )( X )) < ∞ . Proof of (b): We will use ideas from graph theory to show the bound in the first part ofTheorem 1. Recall the supremum is taken over all sets of nonnegative integers β jk , 1 ≤ j 4, satisfying β + β + β + β ≥ β ≥ β ≥ β jk ≤ ≤ j < k ≤ ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . The exponents β jk induce an unordered simple graph on the set of vertices V = { , , , } byputting an edge between j and k if β jk = 0. There are edges connecting the pairs of vertices(1 , 2) and (3 , 4) and condition β + β + β + β ≥ β jk are equal to one. So far we have edges in (1 , , 4) and (2 , β = 1. In this case we have A n,β ≤ n n X i ,i ,i ,i =1 | ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) | . After making the change of variables i = i , k = i − i , k = i − i and k = i − i and using the inequality (2.9) with M = 3 and v = (1 , , A n,β ≤ n X | k i |≤ n,i =1 , , | ρ ( k ) ρ ( k ) ρ ( k ) ρ ( k + k + k ) | ≤ Cn X | k |≤ n | ρ ( k ) | . (ii) Suppose that we add two more edges to the graph formed by the edges (1 , , , A n,β ≤ n n X i ,i ,i ,i =1 | ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i α − i β ) ρ ( i α − i β ) | . ATE OF CONVERGENCE IN THE BM THEOREM 15 Making the change of variables i = i , k = i − i , k = i − i and k = i − i , weobtain A n,β ≤ n X | k i |≤ n,i =1 , , | ρ ( k ) ρ ( k ) ρ ( k ) ρ ( k · v ) ρ ( k · w ) | , where v and w are two linearly independent vectors in Z and k = ( k , k , k ). Using(2.11), we obtain A n,β ≤ Cn X | k |≤ n | ρ ( k ) | , which completes the proof of (b). Proof of (c): This estimate can be obtained by exactly the same arguments as in the proofof Theorem 4.5 in [17]. We omit the details. (cid:3) Remark 4.1. We can show that both bounds in (1.7) are not comparable. In the particularcase | ρ ( k ) | ∼ | k | − α as | k | → ∞ , with α > , we obtain: d TV ( Y n , Z ) ≤ Cn − α if < α < ,Cn − α if ≤ α < ,Cn − (log n ) if α = 1 ,Cn − if α > . Proof of Theorem 1.2 Proof. As in the proof of Theorem 1.1, we can assume that X n = W ( e n ), where e i ∈ H , i ≥ H such that, for each i, j ≥ 0, we have h e i , e j i H = ρ ( i − j ) and W = { W ( φ ) : φ ∈ H } is an isonormal Gaussian process.Consider the sequence F n := √ n P nj =1 g ( X j ) introduced in (1.5), where g ∈ L ( R , γ ) hasHermite rank d = 2 and let σ n = E ( F n ). Set Y n = F n σ n . Taking into account (2.5), we havethe representation Y n = δ ( σ n v n ), where(5.1) v n = 1 √ n n X j =1 g ( X j ) e j ⊗ e j . Under condition (1.2), it is well known that as n → ∞ , σ n → σ , where σ has been definedin (1.4). As a consequence of Proposition 2.3, we have the estimate d W ( Y n , Z ) ≤ C q Var ( h D F n , v n i H ⊗ ) + C E ( |h DF n ⊗ DF n , v n i H ⊗ | ) . (5.2)Therefore, we need to estimate the quantities Var( h D F n , v n i H ⊗ ) and E ( |h DF n ⊗ DF n , v n i H ⊗ | ). (i) Estimation of Var( h D F n , v n i H ⊗ ). We will follow similar arguments as in the proof ofTheorem 1.2. First, we write h D F n , v n i H ⊗ = 1 n n X i,j =1 g ′′ ( X i ) g ( X j ) ρ ( i − j ) . Using a limit argument, we obtain h D F n , V n i H ⊗ = lim N →∞ Φ n,N , where the convergence holds in L (Ω) andΦ n,N = 1 n n X i,j =1 N X q ,q =2 c q c q q ( q − H q − ( X i ) H q − ( X j ) ρ ( i − j ) . Therefore, by Fatou’s lemmaVar( h D F n , v n i H ⊗ ) ≤ lim inf N →∞ Var(Φ n,N ) . We can writeVar(Φ n,N ) = 1 n n X i ,i ,i ,i =1 N X q ,q ,q ,q =2 q ( q − q ( q − c q c q c q c q ρ ( i − i ) ρ ( i − i ) × Cov( H q − ( X i ) H q − ( X i ) , H q − ( X i ) H q − ( X i )) . (5.3)With a very similar calculation as in the proof of Theorem 1.1, we have(5.4) Cov( H q − ( X i ) H q − ( X i ) , H q − ( X i ) H q − ( X i )) = X β ∈D ′ q C q,β Y ≤ j 4, satisfying(5.5) q ℓ − X j or k = ℓ β jk , for 1 ≤ ℓ ≤ β + β + β + β ≥ . Substituting (5.4) into (5.3) yieldsVar(Φ n,N ) = 1 n n X i ,i ,i ,i =1 N X q ,q ,q ,q =2 X β ∈D ′ q C q,β q ( q − q ( q − c q c q c q c q × ρ β +2 ( i − i ) ρ β ( i − i ) ρ β ( i − i ) ρ β ( i − i ) ρ β ( i − i ) ρ β +2 ( i − i ) . Replacing β + 2 and β + 2 by β and β , the above equality can be rewritten asVar(Φ n,N ) = 1 n n X i ,i ,i ,i =1 N X q ,q ,q ,q =2 X β ∈E q K q,β c q c q c q c q Y ≤ j 4, satisfying β + β + β + β ≥ β ≥ β ≥ q ℓ = X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . We can write Var(Φ n,N ) ≤ sup β A n,β N X q ,q ,q ,q =2 X β ∈E q K q,β | c q c q c q c q | , ATE OF CONVERGENCE IN THE BM THEOREM 17 where A n,β = 1 n n X i ,i ,i ,i =1 Y ≤ j 4, satisfying β + β + β + β ≥ β ≥ β ≥ 2, for 1 ≤ j < k ≤ ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . Then, in this case we have A n,β ≤ n n X i ,i ,i ,i =1 (cid:12)(cid:12) ρ ( i − i ) ρ ( i α − i α ) ρ ( i − i ) (cid:12)(cid:12) where α ∈ { , } and α ∈ { , } . After making the change i = i , k = i − i , k = i α − i α and k = i − i , we obtain A n,β ≤ n X | k i |≤ n,i =1 , , (cid:12)(cid:12) ρ ( k ) ρ ( k ) ρ ( k ) (cid:12)(cid:12) ≤ Cn X | k |≤ n | ρ ( k ) | . Now, it is left to show that N X q ,q ,q ,q =2 X β ∈E q K q,β | c q c q c q c q | < ∞ . (5.6)We have N X q ,q ,q ,q =2 X β ∈E q K q,β | c q c q c q c q | = N X q ,q ,q ,q =2 q ( q − q ( q − | c q c q c q c q |× E ( H q − ( X ) H q − ( X ) H q − ( X ) H q − ( X ))= E (cid:16) ( A ( g ′′ ) ( N ) ) ( A ( g ) ( N ) ) (cid:17) . By H¨older’s inequality, we obtain N X q ,q ,q ,q =2 X β ∈E q K q,β | c q c q c q c q | ≤ k A ( g ′′ ) ( N ) k / L ( R ,γ ) k A ( g ) ( N ) k / L ( R ,γ ) . From the hypothesis and the Proposition 3.1, ( A ( g ′′ ) ( N ) ) and ( A ( g ) ( N ) ) converge to A ( g ′′ ) and A ( g ) in L ( R , γ ) respectively. Hence, (5.6) holds. (ii) Estimation of E ( |h DF n ⊗ DF n , v n i H ⊗ | ) . We can write h DF n ⊗ DF n , v n i H ⊗ = n − n X i,j,k =1 g ′ ( X i ) g ′ ( X j ) g ( X k ) ρ ( i − k ) ρ ( j − k ) . We have, in the L (Ω) sense, h DF n , u n i H = lim N →∞ Ψ n,N , whereΨ n,N = n − n X i,j,k =1 N X q ,q ,q =2 c q c q c q q q H q − ( X i ) H q − ( X j ) H q − ( X k ) ρ ( i − k ) ρ ( j − k ) . Therefore, by Fatou’s lemma E (cid:0) h DF ⊗ DF, v i H ⊗ (cid:1) ≤ lim inf N →∞ E (cid:0) Ψ n,N (cid:1) . We can write E (cid:0) Ψ n,N (cid:1) = n − n X i ,...,i =1 N X q ,...,q =2 Y i =1 c q i ! q q q q × E ( H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i )) × ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) . (5.7)Using Lemma 3.1, we obtain E ( H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ))= X β ∈D q C q,β Y ≤ j 6, satisfying q ℓ − X j or k = ℓ β jk , for ℓ = 1 , , , ,q − X j or k =3 β jk ,q − X j or k =6 β jk . (5.9)Replacing (5.8) into (5.7) yields E (Ψ n,N ) = n − n X i ,...,i =1 N X q ,...,q =2 X β ∈D q C q,β Y i =1 c q i ! q q q q × ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) Y j,k =1 ,j 6, satisfying q ℓ = X j or k = ℓ β jk , for ℓ = 1 , . . . , . Hence E (Ψ n,N ) ≤ sup β A n,β N X q ,...,q =2 X β ∈E q K q,β Y i =1 | c q i | ! q q q q , where A n,β = n − n X i ,...,i =1 Y ≤ j 1, there are edges connecting the pairs of vertices (1 , , , 6) and (5 , Case 1: Suppose that the graph is not connected. This implies that β ≥ β ≥ V = { , , } and V = { , , } . The worse case is when β = β = β = β = β = β = 1 and all the other exponents are zero. In this case wehave the estimate A n,β ≤ n − X | k | , | k |≤ n | ρ ( k ) ρ ( k ) ρ ( k − k ) | . Using (2.9), we obtain A n,β ≤ Cn − X | k |≤ n | ρ ( k ) | . Case 2: Suppose that the graph is connected. This means that there is an edge connectingthe sets V and V . Suppose that β α δ ≥ 1, where α ∈ { , , } and δ ∈ { , , } . We havethen 5 nonzero coefficients β : β , β , β , β and β α δ . Because all the edges have at leastdegree 2, there must be at least two more nonzero coefficients β . Let us denote them by β α δ and β α δ .Then, the worse case will be when β = β = β = β = β α δ = β α δ = β α δ = 1 andall the other coefficients are zero. Consider the change of variables i − i = k , i − i = k , i − i = k , i − i = k , i α − i δ = k . Then, i α − i δ = k · v and i α − i δ = k · w , where k = ( k , . . . , k ) and v , w are 5-dimensional linearly independent vectors whose componentsare 0, 1 or − 1. Then, we can write, using (2.11) and H¨older’s inequality, A n,β ≤ n − X | k i |≤ n, ≤ i ≤ Y i =2 | ρ ( k i ) || ρ ( k · v ) ρ ( k · w ) | ≤ Cn − X | k |≤ n | ρ ( k ) | ≤ Cn − X | k |≤ n | ρ ( k ) | . (cid:3) Remark 5.1. In the case g ( x ) = x − , the term Var( h D F n , v n i H ⊗ ) is zero because h D F n , v n i H ⊗ is deterministic, and for the second term we get the estimate (1.11). Remark 5.2. We can show that both bounds in (1.10) are not comparable. In the particularcase | ρ ( k ) | ∼ | k | − α as | k | → ∞ , with α > , we obtain: d W ( Y n , Z ) ≤ Cn − α if < α ≤ ,Cn − α if < α ≤ ,Cn − (log n ) if α = 1 ,Cn − if α > . ATE OF CONVERGENCE IN THE BM THEOREM 21 Proof of Theorem 1.3 Proof. With the notation used in the proof of Theorem 1.1 and using Proposition 2.2, we canwrite d T V ( Y n , Z ) ≤ (8 + √ π )Var( h DY n , u n /σ n i H ) + √ π | E ( Y n ) | + √ π E ( | D u n /σ n Y n | )(6.1) + 4 π E ( | D u n /σ n Y n | ) ≤ C (cid:16) Var( h DF n , u n i H + | E ( F n ) | + E ( | D u n F n | ) + q E ( | D u n F n | ) (cid:17) . Now, we want to estimate each of these terms separately. Step 1. From Theorem 1.1 we know that(6.2) Var( h DF n , u n i H ≤ Cn − X | k |≤ n | ρ ( k ) | + Cn − X | k |≤ n | ρ ( k ) | . Step 2. We claim that(6.3) | E ( F n ) | ≤ C √ n X | k |≤ n | ρ ( k ) | . We can write F n = 1 n / n X i,j,k =1 g ( X i ) g ( X j ) g ( X k ) . Truncating the Wiener chaos expansion of the random variables g ( X i ), as in the proof ofTheorem 1.1, we obtain F n = lim N →∞ Ψ n,N := lim N →∞ √ n n X i =1 N X q =2 c q H q ( X i ) , where the convergence holds in L (Ω) due to Proposition 3.1 because g ∈ L ( R , γ ). Therefore, E ( F n ) = lim N →∞ E (Ψ n,N ) . We can write E (Ψ n,N ) = 1 n / n X i ,i ,i =1 N X q ,q ,q =2 c q c q c q E ( H q ( X i ) H q ( X i ) H q ( X i ))= 1 n / n X i ,i ,i =1 N X q ,q ,q =2 c q c q c q E (cid:16) I q ( e ⊗ q i ) I q ( e ⊗ q i ) I q ( e ⊗ q i ) (cid:17) . (6.4)Using Lemma 3.1, we obtain(6.5) E (cid:16) I q ( e ⊗ q i ) I q ( e ⊗ q i ) I q ( e ⊗ q i ) (cid:17) = X β ∈D q C q,β Y ≤ j 3, satisfying(6.6) q ℓ = X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . Then, | E (Ψ n,N ) | ≤ sup β A n,β N X q ,q ,q =2 X β ∈E q C q,β | c q c q c q | , where A n,β = 1 n / n X i ,i ,i =1 Y ≤ j 3, satisfying β jk ≤ ≤ j < k ≤ ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . It is easy to see that to satisfy the above conditions, β jk ≥ ≤ j < k ≤ 3. Hence,we have A n,β ≤ n / n X i ,i ,i =1 | ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) | . After making the change of variables i = i , k = i − i , k = i − i and using the inequality(2.9) with M = 2 and v = ( − , A n,β ≤ n / X | k | , | k |≤ n | ρ ( k ) ρ ( k ) ρ ( k − k ) | ≤ C √ n X | k |≤ n | ρ ( k ) | . To complete the proof of (6.3), we need to show that: ∞ X q ,q ,q =2 X β ∈D q C q,β | c q c q c q | < ∞ . In fact, lim N →∞ N X q ,q ,q =2 X β ∈D q C q,β | c q c q c q | = lim N →∞ E (cid:0) A ( g ) N ) (cid:1) = E (cid:0) ( A ( g )) (cid:1) < ∞ , taking into account Proposition 3.1 and the fact that A ( g ) ∈ L ( R , γ ). Step 3. We proceed now with the estimation of E ( | D u n F n | ). We can write D u n F n = h DF n , u n i H = 1 n n X i,j =1 g ′ ( X i ) g ( X j ) ρ ( i − j )and D ( h DF n , u n i H ) = 1 n n X i,j =1 ( g ′′ ( X i ) g ( X j ) e i + g ′ ( X i ) g ′ ( X j ) e j ) ρ ( i − j ) . ATE OF CONVERGENCE IN THE BM THEOREM 23 Therefore, D u n F n = h u n , D ( h DF n , u n i H ) i H = 1 n / n X i,j,k =1 ( g ′′ ( X i ) g ( X j ) g ( X k ) ρ ( i − k ) + g ′ ( X i ) g ′ ( X j ) g ( X k ) ρ ( j − k )) ρ ( i − j ) . (6.7)Because the random variables g ′′ ( X i ), g ( X j ), g ( X k ), g ′ ( X i ) and g ′ ( X j ) appearing in theabove expression belong to L (Ω), their truncated Wiener chaos expansions convergence in L (Ω), and, as a consequence, D u n F n = lim N →∞ Φ n,N in probability, whereΦ n,N = 1 n / n X i ,i ,i =1 N X q ,q ,q =2 c q c q c q q ( q − H q − ( X i ) H q − ( X i ) H q − ( X i ) × ρ ( i − i ) ρ ( i − i )+ c q c q c q q ( q − H q − ( X i ) H q − ( X i ) H q − ( X i ) ρ ( i − i ) ρ ( i − i ) . Making the change of variables ( q , q ) → ( q , q ) and ( i , i ) → ( i , i ) in the second sumallows us to put the two terms together, and we obtainΦ n,N = 1 n / n X i ,i ,i =1 N X q ,q ,q =2 c q c q c q ( q + q )( q − H q − ( X i ) H q − ( X i ) H q − ( X i ) × ρ ( i − i ) ρ ( i − i ) . Therefore, by Fatou’s lemma, E (cid:0) | D u n F n | (cid:1) ≤ lim inf N →∞ E (cid:0) | Φ n,N | (cid:1) . Then, | Φ n,N | = 1 n n X i ,...,i =1 N X q ,...,q =2 C q H q − ( X i ) H q − ( X i ) H q − ( X i ) × H q − ( X i ) H q − ( X i ) H q − ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) , where C q = c q c q c q c q c q c q ( q + q )( q − q + q )( q − . Using the product formula for multiple integrals (see Lemma 3.1), we get E (cid:0) | Φ n,N | (cid:1) = 1 n n X i ,...,i =1 N X q ,...,q =2 X β ∈D q K q,β Y ≤ k 6, satisfying β , β , β , β ≥ β jk ≤ ≤ j < k ≤ ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . Then, the estimation follows as in the proof of the last part of Theorem 1.2.Now, we need to show that ∞ X q ,...,q =2 X β ∈C q | L q,β | < ∞ . (6.8)In fact, N X q ,...,q =2 X β ∈C q | L q,β | = N X q ,...,q =2 Y i =1 | c q i | ! ( q + q )( q − q + q )( q − × E ( H q − ( X ) H q − ( X ) H q − ( X ) H q − ( X ) H q − ( X ) H q − ( X ))= E (cid:16) A ( g ′′ ) ( N ) ) ( A ( g ) ( N ) ) (cid:17) ≤ k A ( g ′′ ) ( N ) k L ( R ,γ ) k A ( g ) ( N ) k L ( R ,γ ) . Since A ( g ) ∈ D , , ( A ( g ′′ ) ( N ) ) and ( A ( g ) ( N ) ) converge to A ( g ′′ ) and A ( g ), respectively, in L ( R , γ ) by (3.1). Then, (6.8) is true. ATE OF CONVERGENCE IN THE BM THEOREM 25 Step 4. We proceed to the estimation of p E ( | D u n F n | ). Taking the derivative in (6.7),yields D ( D u n F n ) = 1 n / n X i,j,k =1 g ′′′ ( X i ) g ( X j ) g ( X k ) ρ ( i − j ) ρ ( i − k ) e i + g ′′ ( X i ) g ′ ( X j ) g ( X k ) ρ ( i − j ) ρ ( i − k ) e j + g ′′ ( X i ) g ( X j ) g ′ ( X k ) ρ ( i − j ) ρ ( i − k ) e k + g ′′ ( X i ) g ′ ( X j ) g ( X k ) ρ ( i − j ) ρ ( j − k ) e i + g ′ ( X i ) g ′′ ( X j ) g ( X k ) ρ ( i − j ) ρ ( j − k ) e j + g ′ ( X i ) g ′ ( X j ) g ′ ( X k ) ρ ( i − j ) ρ ( j − k ) e k . This implies h u n , D ( D u n F n i H = 1 n n X i ,i ,i ,i =1 g ′′′ ( X i ) g ( X i ) g ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ g ′′ ( X i ) g ′ ( X i ) g ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ g ′′ ( X i ) g ( X i ) g ′ ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ g ′′ ( X i ) g ′ ( X i ) g ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ g ′ ( X i ) g ′′ ( X i ) g ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ g ′ ( X i ) g ′ ( X i ) g ′ ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) . Notice that the second, third and fourth terms are identical. This allows us to write D u n F n = 1 n n X i ,i ,i ,i =1 g ′′′ ( X i ) g ( X i ) g ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ 3 g ′′ ( X i ) g ′ ( X i ) g ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ g ′ ( X i ) g ′′ ( X i ) g ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ g ′ ( X i ) g ′ ( X i ) g ′ ( X i ) g ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) . Then, we have D u n F n = lim N →∞ Φ n,N , where the convergence holds in probability andΦ n,N = 1 n n X i ,i ,i ,i =1 N X q ,q ,q ,q =2 C (1) q H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) × ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ C (2) q H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ C (3) q H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ C (4) q H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) with C (1) q = c q c q c q c q q ( q − q − ,C (2) q = 3 c q c q c q c q q ( q − q − ,C (3) q = c q c q c q c q q ( q − q − ,C (4) q = c q c q c q c q q ( q − q − . We can combine the first and third terms with the change of variables ( q , q ) → ( q , q ) and( i , i ) → ( i , i ). In this way we obtainΦ n,N = 1 n n X i ,i ,i ,i =1 N X q ,q ,q ,q =2 e C (1) q H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) × ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ e C (2) q H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )+ e C (3) q H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i )=: Φ (1) n,N + Φ (2) n.N + Φ (3) n.N with e C (1) q = c q c q c q c q ( q + q )( q − q − , e C (2) q = c q c q c q c q q ( q − q − , e C (3) q = c q c q c q c q q ( q − q − . Then, by Fatou’s lemma, E (cid:0) | D u n F n | (cid:1) ≤ lim inf N →∞ E (cid:0) | Φ n,N | (cid:1) . We are going to treat each term Φ ( i ) n,N , i = 1 , , 3, separately. Case i = 1. Let us first estimate E (cid:16) | Φ (1) n,N | (cid:17) . We have E (cid:16) (Φ (1) n,N ) (cid:17) = 1 n n X i ,...,i =1 N X q ,...,q =2 M (1) q E ( H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) × H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i )) × ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) , where M (1) q = Y j =1 c q j ( q + q )( q − q − q + q )( q − q − . This yields E (cid:16) Φ (1) n,N ) (cid:17) ≤ n n X i ,...,i =1 N X q ,...q =2 X β ∈D (1) q K (1) q,β Y ≤ k 8, satisfying β , β , β , β , β , β ≥ β jk ≤ ≤ j < k ≤ ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . We need to estimate A (1) n,β and to show that(6.9) ∞ X q ,...,q =2 X β ∈C (1) q L (1) q,β < ∞ . Estimation of A (1) n,β : We claim that(6.10) sup β A (1) n,β ≤ Cn − X | k |≤ n | ρ ( k ) | . As in the proof of Theorem 1.2, we will make use of ideas from graph theory. The exponents β jk induce an unordered simple graph on the set of vertices V = { , , , , , } by putting anedge between j and k whenever β jk = 0. Because β , β ≥ β ≥ β ≥ , β ≥ β ≥ 1, there are edges connecting the pairs of vertices (1 , , , , , 7) and(5 , Case 1: Suppose that the graph is not connected. This means that β jk = 0 if j ∈ { , , , } and k ∈ { , , , } and there is no edge between the sets V = { , , , } and V = { , , , } .Therefore, A (1) n,β ≤ ( A (0) n,β ) , where A (0) n,β = 1 n n X i ,...,i =1 Y ≤ j ≤ k ≤ | ρ ( i i − i k ) | β jk and the nonnegative integers β jk , 1 ≤ j < k ≤ 4, satisfy β , β , β ≥ β jk ≤ ≤ j < k ≤ ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . As a consequence, β + β ≥ β + β ≥ β + β ≥ 1. This means that at leasttwo of the indices β , β and β is larger or equal to 1. Considering the worst case, we canassume that β = 1 and β = 1. This leads to(6.11) A (0) n,β ≤ n − X | k | , | k | , | k |≤ n | ρ ( k ) ρ ( k ) ρ ( k ) ρ ( k − k ) ρ ( k − k ) | . Using (2.11) and H¨older’s inequality we obtain A (0) n,β ≤ Cn − X | k |≤ n | ρ ( k ) | ≤ Cn − X | k |≤ n | ρ ( k ) | . Case 2: Suppose that the graph is connected. This means that there is an edge connectingthe sets V and V . Suppose that β α δ ≥ 1, where α ∈ { , , , } and δ ∈ { , , , } . Wehave then 7 nonzero coefficients β : β , β , β , β , β , β and β α δ . Because all the edgeshave at least degree 2, there must be another nonzero coefficient β . Assume it is β α δ . Then,the worse case will be when β = β = β = β = β = β = β α δ = β α δ = 1 and allthe other coefficients are zero. Consider the change of variables i − i = k , i − i = k , i − i = k , i − i = k , i − i = k , i − i = k , i α − i δ = k . Then, it is easy to show that i α − i δ = k · v , where k = ( k , . . . , k ) and v is a 7-dimensional vector whose componentsare 0, 1 or − 1. Applying (2.10) and H¨older’s inequality yields A (1) n,β ≤ Cn − X | k |≤ n | ρ ( k ) | ≤ Cn − X | k |≤ n | ρ ( k ) | . This completes the proof of (6.10). ATE OF CONVERGENCE IN THE BM THEOREM 29 Proof of (6.9): We have ∞ X q ,...,q =2 X β ∈C (1) q L (1) q,β = E (cid:16) (cid:12)(cid:12)(cid:12) ( A ( g ′′′ ) ( N ) )( X )( A ( g ) ( N ) ( X )) +( A ( g ′ ) ( N ) )( X )( A ( g ′′ ) ( N ) )( X )( A ( g ) ( N ) ( X )) (cid:12)(cid:12)(cid:12) (cid:19) . Applying H¨older’s inequality, yields ∞ X q ,...,q =2 X β ∈C (1) q L (1) q,β ≤ k A ( g ′′′ ) ( N ) k L ( R ,γ ) k A ( g ) ( N ) k L ( R ,γ ) + 2 k A ( g ′ ) ( N ) k L ( R ,γ ) k A ( g ′′ ) ( N ) k L ( R ,γ ) k A ( g ) ( N ) k L ( R ,γ ) . By Proposition 3.2 and our hypothesis, taking the limit as N tends to infinity, it follows that ∞ X q ,...,q =2 X β ∈C (1) q L (1) q,β ≤ k A ( g ′′′ ) k L ( R ,γ ) k A ( g ) k L ( R ,γ ) + 2 k A ( g ′ ) k L ( R ,γ ) k A ( g ′′ ) k L ( R ,γ ) k A ( g ) k L ( R ,γ ) < ∞ . Case i = 2. For E [ | Φ (2) n,N | ] we have E (cid:16) (Φ (2) n,N ) (cid:17) = 1 n n X i ,...,i =1 N X q ,...,q =2 M (2) q E ( H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i ) × H q − ( X i ) H q − ( X i ) H q − ( X i ) H q − ( X i )) × ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) ρ ( i − i ) , where M (2) q = Y j =1 c q j q ( q − q − q ( q − q − . This yields E (cid:16) (Φ (2) n,N ) (cid:17) ≤ n n X i ,...,i =1 N X q ,...q =2 X β ∈D (2) q K (2) q,β Y ≤ k 8, satisfying β , β , β , β , β , β ≥ β jk ≤ ≤ j < k ≤ ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . We need to estimate A (2) n,β and to show that(6.12) ∞ X q ,...,q =2 X β ∈C (2) q L (2) q,β < ∞ . Estimation of A (2) n,β : We claim thatsup β A (2) n,β ≤ Cn − X | k |≤ n | ρ ( k ) | . As in the proof of Theorem 1.2, we will make use of ideas from graph theory. The exponents β jk induce an unordered simple graph on the set of vertices V = { , , , , , } by putting anedge between j and k whenever β jk = 0. Because β ≥ β ≥ β ≥ β ≥ , β ≥ β ≥ 1, there are edges connecting the pairs of vertices (1 , , , , 6) (5 , , ATE OF CONVERGENCE IN THE BM THEOREM 31 Case 1: Suppose that the graph is not connected. This means that β jk = 0 if j ∈ { , , , } and k ∈ { , , , } and there is no edge between the sets V = { , , , } and V = { , , , } .Therefore, A (2) n,β ≤ ( A (0) n,β ) , where A (0) n,β = 1 n n X i ,...,i =1 Y ≤ j ≤ k ≤ | ρ ( i i − i k ) | β jk and the nonnegative integers β jk , 1 ≤ j < k ≤ 4, satisfy β , β , β ≥ β jk ≤ ≤ j < k ≤ ≤ X j or k = ℓ β jk , for 1 ≤ ℓ ≤ . As a consequence, β + β ≥ β + β ≥ 1. This means β ≥ β and β are larger or equal than one. There are two possible cases:(i) Suppose β ≥ 1, Considering the worst case, we can assume that β = 1. Then,applying (2.9) and H¨older’s inequality, we obtain A (0) n,β ≤ n − X | k | , | k | , | k |≤ n | ρ ( k ) ρ ( k ) ρ ( k ) ρ ( k + k − k ) | ≤ n − X | k |≤ n | ρ ( k ) | . By H¨older’s inequality, we can show that( A (0) n,β ) ≤ Cn − X | k |≤ n | ρ ( k ) | . (ii) Suppose β ≥ β ≥ 1. Then, A (0) n,β ≤ n − X | k | , | k | , | k |≤ n | ρ ( k ) ρ ( k ) ρ ( k ) ρ ( k + k ) ρ ( k − k ) | , and this case can be treated as (6.11). Case 2: Suppose that the graph is connected. This means that there is an edge connectingthe sets V and V . Suppose that β α δ ≥ 1, where α ∈ { , , , } and δ ∈ { , , , } . Wehave then 7 nonzero coefficients β : β , β , β , β , β , β and β α δ . Because all the edgeshave at least degree 2, there must be another nonzero coefficient β . Assume it is β α δ . Then,the worse case will be when β = β = β = β = β = β = β α δ = β α δ = 1 and allthe other coefficients are zero. Consider the change of variables i − i = k , i − i = k , i − i = k , i − i = k , i − i = k , i − i = k , i α − i δ = k . Then, it is easy to show that i α − i δ = k · v , where k = ( k , . . . , k ) and v is a 7-dimensional vector whose componentsare 0, 1 or − 1. Then, using (2.10) and H¨older’s inequality, we obtain A (1) n,β ≤ Cn − X | k |≤ n | ρ ( k ) | ≤ Cn − X | k |≤ n | ρ ( k ) | . Proof of (6.12): We have ∞ X q ,...,q =2 X β ∈C (2) q L (2) q,β = 9 E (cid:18)(cid:12)(cid:12)(cid:12) A ( g ′′ ) ( N ) ( X ) A ( g ′ )( X ) A ( g )( X ) (cid:12)(cid:12)(cid:12) (cid:19) ≤ k A ( g ′′ ) ( N ) k L ( R ,γ ) k A ( g ′ ) ( N ) k L ( R ,γ ) k A ( g ) ( N ) k L ( R ,γ ) , which converges as N → ∞ to9 k A ( g ′′ ) k L ( R ,γ ) k A ( g ′ ) k L ( R ,γ ) k A ( g ) k L ( R ,γ ) < ∞ . Case i = 3 . The term E [ | Φ (3) n,N | ] can be handled in a similar way and we omit the details. (cid:3) References [1] D. Bell and D. Nualart (2017). Noncentral limit theorem for the generalized Hermite process. Electron.J. Probab. ,no. 2, 1-13.[2] H. Bierm´e, A. Bonami, I. Nourdin and G. Peccati (2012). Optimal Berry-Esseen rates on the Wienerspace: the barrier of third and fourth cumulants. ALEA , , no. 2, 473-500.[3] H. J. Brascamp and E. H. Lieb (1976). Best constants in Youngs inequality, its converse, and its general-ization to more than three functions. Adv. Math. , 151-173.[4] P. Breuer and P. Major (1983). Central limit theorems for non-linear functionals of Gaussian fields. J.Mult. Anal. , 425-441.[5] S. Campese, I. Nourdin and D. Nualart: Continuous Breuer-Major theorem: tightness and non-stationarity. Preprint.[6] L. H. Y. Chen, L. Goldstein and Q.-M. Shao (2011). Normal approximation by Stein’s method . Springer-Verlag, Berlin.[7] L. Larsson-Cohn (2002). L p -norms of Hermite polynomials and an extremal problem on Wiener chaos. Arkiv f¨or Matematik , no. 1, 133–144.[8] I. Nourdin and D. Nualart (2018). The functional Breuer-Major theorem. Preprint[9] I. Nourdin and G. Peccati (2009). Stein’s method on Wiener chaos. Probab. Theory Relat. Fields , no.1, 75-118.[10] I. Nourdin and G. Peccati (2012). Normal approximations with Malliavin calculus. From Stein’s methodto universality . Cambridge University Press.[11] I. Nourdin and G. Peccati (2015). The optimal fourth moment theorem. Proc. Amer. Math.Soc. ,3123-3133.[12] I. Nourdin, G. Peccati, X. Yang (2019). On rate of convergence for the Breuer-Major theorem. Preprint.[13] D. Nualart (2006). The Malliavin calculus and related topics. Springer-Verlag, Berlin, second edition.[14] D. Nualart (2009). Malliavin calculus and its applications. American Mathematical Society, CBMS re-gional conference series in mathematics.[15] D. Nualart and E. Nualart (2018). Introduction fo Malliavin calculus. Cambridge University Press.[16] D. Nualart and G. Peccati (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. , no. 1, 177-193.[17] D. Nualart and H. Zhou (2018). Total variation estimates in the Breuer-Major theorem. Preprint[18] G. Peccati and M. S. Taqqu (2011). Wiener chaos: moments, cumulants and diagrams - a survey withcomputer implementation. Springer-Verlag, Italia.[19] C. Stein (1986). Approximate computation of expectations. Institute of Mathematical Statistics. University of Kansas, Department of Mathematics, USA E-mail address : [email protected] University of Kansas, Department of Mathematics, USA E-mail address ::