Berry-Esseen bounds in the Breuer-Major CLT and Gebelein's inequality
aa r X i v : . [ m a t h . P R ] M a y BERRY-ESSEEN BOUNDS IN THE BREUER-MAJOR CLTAND GEBELEIN’S INEQUALITY
IVAN NOURDIN, GIOVANNI PECCATI, AND XIAOCHUAN YANG
Abstract.
We derive explicit Berry-Esseen bounds in the totalvariation distance for the Breuer-Major central limit theorem, inthe case of a subordinating function ϕ satisfying minimal regu-larity assumptions. Our approach is based on the combinationof the Malliavin-Stein approach for normal approximations withGebelein’s inequality, bounding the covariance of functionals ofGaussian fields in terms of maximal correlation coefficients. Keywords:
Breuer-Major theorem, rate of convergence, Gebelein’sinequality, Malliavin-Stein approach. Introduction
Motivation and main results.
Let X = ( X k ) k ∈ Z be a centeredstationary Gaussian sequence with covariance function E [ X k X j ] = ρ ( k − j ) satisfying ρ (0) = 1. Let ϕ ∈ L ( R , γ ), where γ is the standard Gauss-ian measure on the real line, and assume without loss of generality that E [ ϕ ( X )] = R R ϕdγ = 0. By exploiting the orthogonality and complete-ness of Hermite polynomials in L ( R , γ ) (see, e.g., [12, p. 13]), we canwrite(1) ϕ = X ℓ ≥ d a ℓ H ℓ , where H ℓ is the Hermite polynomial of order ℓ , the coefficient a d isdifferent from zero, d ≥ Hermite rank of ϕ , and the seriesconverges in L ( R , γ ). Consider the sequence of normalized sums F n = 1 √ n n X k =1 ϕ ( X k ) , n ≥ . (2)The celebrated Breuer-Major theorem [2], stated below, provides suffi-cient conditions on the covariance function ρ , in order for F n to exhibitGaussian fluctuations, as n → ∞ (see also Taqqu [22] for a relatedwork). Throughout the paper, the symbol N ( a, b ) denotes a Gaussian Date : May 9, 2019. random variable with mean a ∈ R and variance b ≥
0, and d → theconvergence in distribution. Theorem 1 (Breuer-Major Theorem) . Let the previous assumptionson X and ϕ prevail, and suppose moreover that P k ∈ Z | ρ ( k ) | d < ∞ .Then F n d → N (0 , σ ) , where σ = X ℓ ≥ d a ℓ ℓ ! X k ∈ Z ρ ( k ) ℓ < ∞ . (3) Here, and for the rest of the paper, d → denotes convergence in distribu-tion of random variables. The Breuer-Major theorem has far-reaching applications in manydifferent areas, such as mathematical statistics, signal processing orgeometry of random nodal sets, see e.g. [5, 16, 21, 23] and referencestherein. It has been generalized and refined in various aspects [3, 4,11, 13, 14].Now let σ n := V ar( F n ) and V n := F n / p V ar( F n ). The aim of thepresent paper is develop a novel method for obtaining explicit upperbounds on the sequence d TV ( V n , N (0 , A ∈B ( R ) | P ( V n ∈ A ) − P ( N (0 , ∈ A ) | , n ≥ , where B ( R ) is the Borel σ -algebra on R , under minimal regularity as-sumptions on the function ϕ . Our strategy for doing so is to com-bine the Malliavin-Stein method for probabilistic approximations (asdescribed in Section 2.2 below) and the powerful
Gebelein’s inequality for correlation of Gaussian functionals (see [6, 24], as well as Section5, for a self-contained proof), as applied to non-linear transformationsof correlated Gaussian sequences. To the best of our knowledge, ouruse of Gebelein’s inequality is new: it is reasonable to expect that thecontent of the present work might constitute the blueprint for furtherapplications of such general a bound to probabilistic approximationsin a Gaussian setting.We recall that, for every n , the quantity d TV ( V n , N (0 , total variation distance between the distributions of V n and N (0 ,
1) — see e.g. [13, Appendix C], and the references therein,for a discussion of the properties of d TV . Any statement yielding theexistence of an explicit numerical sequence { α n } such that α n → d ( V n , N (0 , ≤ α n , for some distance d , is called a quantitativeBreuer-Major Theorem . ATE FOR THE BREUER-MAJOR THEOREM 3
One of the first quantitative Breuer-Major theorems is contained inthe work by Nourdin, Peccati and Podolskij [14] — see, in particular,[14, Cor. 2.4], where the focus is on the Kolmogorov and 1-Wassersteindistances and on the case where ϕ is a Hermite polynomial of order q . The rates obtained in [14] are, in general, not optimal. We stressthat, according to [14, Corollary 2.4], the convergence in distributionin Theorem 1 always takes place in the sense of the Kolmogorov and1-Wasserstein distances.Determining whether the Breuer-Major CLT holds in the topologyof the distance d TV is a much more delicate matter, since – unlike con-vergence in the Kolmogorov or 1-Wasserstein distances – convergencein total variation cannot take place in full generality, and requires extraregularity assumptions on ϕ . Our specific aim is therefore to tackle thefollowing problem: Problem P : Letting the notation and assumptions of Theorem 1 pre-vail, find conditions on ϕ and ρ in order to have that d TV ( F n / p V ar( F n ) , N (0 , → n → ∞ . To appreciate the subtlety of Problem P, one should recall the fol-lowing two facts:(i) according to the main findings in [17], if ϕ is a polynomial, thenthe convergence in Theorem 1 always takes place in the senseof total variation;(ii) on the other hand, if one considers independent X k ∼ N (0 , ϕ ( x ) = sign( x ) — in which case the assumptions ofTheorem 1 are satisfied, but d TV ( F n / p V ar( F n ) , N (0 , n .As anticipated, the content of Points (i) and (ii) suggests that thereexists a minimal amount of regularity for the function ϕ , below whichconvergence in total variation in the Breuer-Major Theorem ceases totake place. Exactly locating such a threshold is the ultimate goal ofthe line of research inaugurated by the present paper.As already discussed, in what follows we will be concerned withupper bounds on the rate of convergence in the Breuer-Major theoremwhen the function ϕ possibly displays an infinite Hermite expansion(1), and belongs to the Sobolev space D , — where we adopted theusual notation D p,q in order to indicate the Sobolev space of thoserandom variables on a Gaussian space that are p times differentiable inthe sense of Malliavin, and whose Malliavin derivative is q -integrable IVAN NOURDIN, GIOVANNI PECCATI, AND XIAOCHUAN YANG (see Section 2 for a precise definition). We consider that the propertyof belonging to some space ϕ ∈ D ,q , q ≥
1, is somehow unavoidable,in the sense that it is the least requirement on ϕ that allows one todirectly apply the Malliavin-Stein method outlined in Section 2.The following statement is the main result of the paper: Theorem 2.
Let X = ( X k ) k ∈ Z be a centered stationary Gaussian se-quence with covariance function E [ X k X j ] = ρ ( k − j ) satisfying ρ (0) = 1 ,and let ϕ ∈ D , ⊂ L ( R , γ ) be such that E [ ϕ ( X )] = R R ϕdγ = 0 . Let F n be given by (2) and set σ n = V ar( F n ) and V n = F n /σ n . Then, fora finite constant C ( ϕ ) , whose explicit value is given in (11) below: (i) For every n , d TV ( V n , N (0 , ≤ C ( ϕ ) σ n n − X | k |
In the case where ϕ has a possibly infinite Hermiteexpansion (1), and under some extra smoothness assumptions, Nour-din, Peccati and Reinert [15], Nualart and Zhou [20] and Vidotto [25]obtained total variation error bounds that are better than those derivedin [14]. The rates of convergence deduced in [14] and [15, 20, 25] (thatare sometimes optimal, and sometimes not) are all obtained via somevariation of the Malliavin-Stein approach described in Section 2.2.In [20] (the closest reference to the present note), the following gen-eral quantitative result is proved (see [20, Th. 4.2 and Th. 4.3(v)]): as n → ∞ , one has that d TV ( V n , N (0 , O ( n − / ) , provided that(a) either ϕ has Hermite rank 1 ( d = 1 in (1)), ϕ ∈ D , and ρ ∈ ℓ ( Z ), or(b) ϕ has Hermite rank 2 ( d = 2 in (1)), ϕ ∈ D , and ρ ∈ ℓ ( Z ) ⊂ ℓ ( Z ).The regularity assumptions on ϕ required at Points (a) and (b) aboveare clearly more restrictive than ours. On the other hand, disregardingthe regularity of ϕ , the upper bound of the order n − / obtained in[20] is optimal for the set of assumptions at Point (a) and (b) above.The optimality for Point (a) follows from the same argument used inRemark 1-(1). Similarly, the order n − / under the set of assumptionsat Point (b) cannot be improved in general, since it coincides with the third/fourth cumulant barrier for the total variation distance, betweenthe laws of a sequence of random variables in a fixed chaos and thestandard normal distribution. Such a result was established in fullgenerality in [13, Theorem 11.2], and is presented in the next propo-sition in the simple case of polynomials of order 2. Here and after, a ( n ) ≍ b ( n ) means that the ratio a ( n ) /b ( n ) is bounded from above andbelow by positive finite constants. IVAN NOURDIN, GIOVANNI PECCATI, AND XIAOCHUAN YANG
Proposition 3. [13, Proposition 4.2]
Let F n be given by (2) with ϕ = H . Set V n = F n / p V ar( F n ) . Then, d TV ( V n , N (0 , ≍ √ n X | k | 1, but ϕ / ∈ D , . Also, ϕ hasan infinite expansion (1) with Hermite rank d = 2. Such a case is notcovered by the findings of [20] or [14, 15, 24] (due to the lack of sufficientregularity for the function ϕ ), and enters indeed the framework of ourmain result, stated in Theorem 2. The case of such a mapping isalso covered by the recent reference [9], where convergence in totalvariation is deduced for a class much smaller than D , , containinghowever ϕ ( x ) = | x | − p /π .The higher regularity requirement for ϕ which is necessary in [20]stems from the method, used therein, of applying integration by partsseveral times. On the other hand, our approach requires that we onlyperform one integration by parts in the Malliavin-Stein approach, sinceour final estimate makes use of the intrinsic correlation bound givenby Gebelein’s inequality. The use of Gebelein’s inequality, which is themain technological breakthrough of the present paper, requires muchless regularity on ϕ .Although the focus of our paper is on finding minimal regularity as-sumptions on ϕ for having convergence in total variation in the Breuer-Major Theorem, a natural question one might ask is whether the ratesof convergence implied by our bounds are optimal. In view of Proposi-tion 3, applying the upper bound in Theorem 2-(ii) to the case ϕ = H (and ρ ∈ ℓ b ( Z ), for some 1 ≤ b < ϕ ( x ) = | x | − p /π . Furtherdiscussions around this problem are gathered at the end of the paper— see Section 4.The present paper is organised as follows. We start by reviewingsome basic elements of stochastic analysis on the Wiener space and of ATE FOR THE BREUER-MAJOR THEOREM 7 the Malliavin-Stein approach. Then we introduce the new ingredient,Gebelein’s inequality for correlated isonormal Gaussian processes, inSection 2. We apply a Gebelein-Malliavin-Stein bound to prove ourmain theorem in Section 3. A discussion on optimality is provided inSection 4, thus concluding the paper.Every random object considered below is defined on a common prob-ability space (Ω , F , P ), with E denoting mathematical expectation withrespect to P . 2. Preliminaries Stochastic analysis on the Wiener space. The content of thissubsection can be found in [12] or [11]. An isonormal Gaussian process { W ( h ) : h ∈ H } is a family of centered Gaussian random variablesindexed by a real separable Hilbert space H such that the covariancesatisfies E [ W ( g ) W ( h )] = h g, h i H . Let F be a square-integrable functional of an isonormal Gaussian pro-cess W . Then, F has a unique Wiener-Itˆo chaos expansion F = E [ F ] + X k ≥ I k ( f k ) in L (Ω) , (6)where f k ∈ H ⊗ k is a symmetric kernel, and I k ( f k ) is the k -th multipleWiener-Itˆo integral, k ≥ 1. By convention we write I ( f ) = f = E [ F ]. By orthogonality between multiple integrals of different orders,we have E [ F ] = P k ≥ k ! k f k k H ⊗ k . Let f : R n → R be of class C ∞ , andsuch that all its partial derivatives have at most polynomial growth.Consider a smooth functional of the form F = f ( W ( h ) , ..., W ( h n ))with h , .., h n ∈ H . We define the Malliavin derivative of F as DF = n X i =1 ∂ i f ( W ( h ) , ..., W ( h n )) h i . The set of smooth functionals F introduced above is dense in L q (Ω), q ≥ 1, and the operator D is closable. Therefore, D can be extended to D ,q , the set of F such that there exists a sequence of smooth functionals( F n ) n ≥ satisfying E [ | F n − F | q ] → E [ k DF n − η k q H ] → 0, for some η ∈ L q (Ω , H ), that we rewrite as η := DF . One defines similarly D p and D p,q . When q = 2, these spaces are Hilbert spaces and we have thefollowing characterization in terms of the chaos expansion (6): D p, = { F ∈ L (Ω) : X k ≥ p k p k ! k f k k H ⊗ k < ∞} . IVAN NOURDIN, GIOVANNI PECCATI, AND XIAOCHUAN YANG The adjoint of D , customarily called the divergence operator or the Skorohod integral , is denoted by δ and satisfies the duality formula, E [ δ ( u ) F ] = E [ h u, DF i H ](7)for all F ∈ D , , whenever u : Ω → H is in the domain of δ . The Ornstein-Uhlenbeck semigroup ( P t ) t ≥ is defined by Mehler’s formulafor all F ∈ L (Ω) by P t F = E ′ [ F ( e − t W + √ − e − t W ′ )] , where W ′ is an independent copy of W and E ′ denotes the expectationwith respect to W ′ . For F ∈ L (Ω) given by the chaos expansion (6),the Ornstein-Uhlenbeck semigroup takes the form P t F = X k ≥ e − kt I k ( f k ) . The generator of ( P t ) t ≥ is denoted by L and acts on the chaos expan-sion in a simple way, − LF = X k ≥ kI k ( f k ) , with dom L = { F : P k ≥ k k ! k f k k H ⊗ k < ∞} . The pseudo-inverse of L is defined by − L − F = X k ≥ k I k ( f k )for all F ∈ L (Ω). We have LL − F = F − E [ F ] for all F ∈ L (Ω).The key identity that links the objects defined above is L = − δD ; inparticular, we have − DL − F ∈ dom( δ ) for all F ∈ L (Ω).We end this subsection with a fundamental product formula for mul-tiple integrals. Proposition 4 (Product formula) . Let p, q be non-negative integers.Let f ∈ H ⊗ p and g ∈ H ⊗ q be symmetric kernels. We have I p ( f ) I q ( g ) = p ∧ q X r =0 r ! (cid:18) pr (cid:19)(cid:18) qr (cid:19) I p + q − r ( f e ⊗ r g ) where f e ⊗ r g is the symmetrized r -th contraction of f and g , see [12,p. 208] for a definition. ATE FOR THE BREUER-MAJOR THEOREM 9 Malliavin-Stein approach. We make use of an identity (labeledbelow as (8)) first noted by Jaramillo and Nualart in [8].First of all, we observe that any stationary centered Gaussian se-quence X = { X k : k ∈ Z } is embedded in an isonormal Gaussianprocess W = { W ( h ) : h ∈ H } . This means that that there always ex-ists a Hilbert space H and an isonormal Gaussian process W (definedon the same probability space) such that, for some { e k : k ≥ } ⊂ H , W ( e k ) = X k for all k , and consequently E [ W ( e k ) W ( e l )] = h e k , e l i H = ρ ( k − l ), for all k, l (see, e.g., [10, Section 1] for a justification of thisfact).For ϕ = P ℓ ≥ a ℓ H ℓ ∈ L ( R , γ ), we define the shift mapping ϕ := P ℓ ≥ a ℓ H ℓ − and set u n := 1 σ n √ n n X m =1 ϕ ( X m ) e m Then, δu n = V n . (8)To prove this, just observe that u n = − DL − V n , and then apply therelations L = − δD and LL − F = F , valid for any centered randomvariable F ∈ L (Ω). By Stein’s lemma (see [12, Th. 3.3.1]) for d TV andthen by integration by parts via (7), we have that d TV ( V n , N (0 , ≤ sup g ∈G | E [ V n g ( V n )] − E g ′ ( V n ) | = sup g ∈G | E [ δ ( u n ) g ( V n )] − E g ′ ( V n ) | = sup g ∈G | E g ′ ( V n )(1 − h DV n , u n i H ) |≤ p V ar( h DV n , u n i H ) . (9)where we used the fact that E h DV n , u n i H = E V n = 1, and the class G is composed of those g : R → R such that k g k ∞ < √ π and k g ′ k ∞ ≤ DX k = e k , h DV n , u n i H = 1 σ n n n X k,ℓ =1 ϕ ′ ( X k ) ϕ ( X ℓ ) ρ ( k − ℓ ) . Hence, V ar( h DV n , u n i H )(10)= 1 σ n n n X k,ℓ,k ′ ,ℓ ′ =1 C ov( ϕ ′ ( X k ) ϕ ( X ℓ ) , ϕ ′ ( X k ′ ) ϕ ( X ℓ ′ )) ρ ( k − ℓ ) ρ ( k ′ − ℓ ′ ) . The following relation is a consequence of Meyer’s inequality and ofthe equivalence of Sobolev norms [18, p.72], justifying our integrabilityassumption on ϕ . Its proof is given in [20, Lem. 2.2]. Lemma 5. Let q > . The shift ϕ ϕ is a bounded operator from L q ( R , γ ) to L q ( R , γ ) . Note that p V ar( ϕ ′ ( X k ) ϕ ( X ℓ )) ≤ p E ϕ ′ ( X k ) ϕ ( X ℓ ) ≤ E [ ϕ ′ ( X ) ] / E [ ϕ ( X ) ] / =: C ( ϕ ) < ∞ , (11)so that the covariance in (10) is finite.2.3. Gebelein’s inequality. Up to some slight adaptation, Theorem6 can be deduced from Veraar’s paper [24]. For the sake of complete-ness, in the Appendix contained in Section 5 we will however presentan independent proof of such a result (inspired by the approach of [24]),using tools and concepts that are directly connected to the frameworkof isonormal Gaussian processes.Recall that an L functional of an isonormal Gaussian process is saidto have Hermite rank d if its projection to the first d − d -th chaos is non trivial. Theorem 6 (Gebelein’s inequality for isonormal processes) . Let W = { W ( h ) : h ∈ H } be an isonormal Gaussian process over some realseparable Hilbert space H , and let H , H be two Hilbert subspaces of H .Define W and W , respectively, to be the restriction of W to H and H . Now consider two measurable mappings F i : R H i → R , i = 1 , ,and assume that each F i ( W i ) is centred and square-integrable. If F has Hermite rank equal to p ≥ , one has that | E [ F ( W ) F ( W )] | ≤ θ p V ar( F ( W )) / V ar( F ( W )) / , (12) where θ := sup h ∈ H ,g ∈ H : k g k , k h k =1 |h h, g i| ∈ [0 , . ATE FOR THE BREUER-MAJOR THEOREM 11 Proof of the main result k -sparsity. As we will see in the next subsection, combiningGebelein’s inequality with the Malliavin-Stein approach will lead toeffective upper bounds for the total variation distance in the Breuer-Major CLT. To this end, we need information on the Hermite rank offunctionals of the type F := ϕ ′ ( W ( h )) ϕ ( W ( g )) for h, g ∈ H with unitnorm, and ϕ ∈ D , . We introduce the notion of k - sparsity . Definition 1. Let ϕ ∈ L ( R , γ ) be given by the series expansion ϕ = P q ≥ d a q H q . We say the ϕ is k - sparse if min { j − i : j > i ≥ d, a i =0 , a j = 0 } ≥ k . Remark 2. Symmetric functions are 2-sparse. Indeed, since H q ( − x ) =( − q H ( x ) for all q ≥ 1, the expansion of a symmetric function satisfies a k − = 0 for k ∈ N . Lemma 7. Assume that ϕ ∈ D , is -sparse and set F := ϕ ′ ( W ( h )) ϕ ( W ( g )) ,for h, g ∈ H with unit norm. Then F − E [ F ] has Hermite rank at least .Proof. By [12, Th. 2.7.7], we have H p ( W ( e )) = I p ( e ⊗ p ) for e ∈ H with k e k H = 1. Thus, ϕ ′ ( W ( h )) ϕ ( W ( g )) = X q ≥ d X p ≥ d qa q a p I q − ( h ⊗ q − ) I p − ( g ⊗ p − ) , where the series convergence in L (Ω). By 2-sparsity, only those prod-ucts of multiple integrals with indices ( p, q ) satisfying p = q or | p − q | ≥ | p − q | ≥ 2. By Proposition 4, the multiple integralof lowest order in the chaos expansion for the product is I | p − q | ( · ), hencethe projection of I q − ( h ⊗ q − ) I p − ( g ⊗ p − ) to the first chaos is zero. If p = q , Proposition 4 shows that the chaos expansion for the productcontains only multiple integrals of even order, ending the proof. (cid:3) Gebelein-Malliavin-Stein upper bound. Putting things to-gether, we have the following Gebelein-Malliavin-Stein upper boundfor the total variation distance. Proposition 8. Let ϕ ( X ) ∈ D , have Hermite rank d ≥ , and define V n = F n /σ n according to (2) and σ n := V ar( F n ) . We have d TV ( V n , N (0 , ≤ C ( ϕ ) σ n vuut n n − X i,j,k,ℓ =0 (cid:12)(cid:12)(cid:12)(cid:12) ρ ( j − k ) ρ ( i − j ) ρ ( k − ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) . If, in addition, ϕ is -sparse, then d TV ( V n , N (0 , ≤ C ( ϕ ) σ n vuut n n − X i,j,k,ℓ =0 (cid:12)(cid:12)(cid:12)(cid:12) ρ ( j − k ) ρ ( i − j ) ρ ( k − ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) . Proof. We evaluate the right-hand side of (10), by applying Theorem6 in the specific situation where H is the linear span of { e i , e j , e k , e ℓ } , H the linear span of { e i , e j } , H the linear span of { e k , e ℓ } . It isstraightforward that | θ | = max( | ρ ( i − k ) | , | ρ ( i − ℓ ) | , | ρ ( j − ℓ ) | , | ρ ( j − k ) | ) ≤ | ρ ( i − k ) | + | ρ ( i − ℓ ) | + | ρ ( j − ℓ ) | + | ρ ( j − k ) | . The conclusion follows from symmetry, and by using the estimate (11). (cid:3) End of the proof. We are now ready to finish the proof of The-orem 2. We set ρ n ( k ) = | ρ ( k ) | | k | Let b ∈ [1 , n − X i,j,k,ℓ =0 (cid:12)(cid:12) ρ ( j − k ) ρ ( i − j ) ρ ( k − ℓ ) (cid:12)(cid:12) ≤ n k ρ n ∗ n k ℓ bb − ( Z ) (cid:13)(cid:13) ρ n ∗ ρ n (cid:13)(cid:13) ℓ b ( Z ) ≤ n k ρ n k ℓ b k n k ℓ b b − ( Z ) k ρ n k ℓ b ( Z ) (cid:13)(cid:13) ρ n (cid:13)(cid:13) ℓ ( Z ) = n b − b (cid:13)(cid:13) ρ n (cid:13)(cid:13) ℓ ( Z ) k ρ n k ℓ b ( Z ) . The result follows from Proposition 8. (cid:3) A remark on optimality Our Gebelein-Malliavin-Stein upper bound (Proposition 8) could notprovide the rate n − / in the case where ρ is square integrable butnot summable. Indeed, restricting ourselves to the subset of indices { i = j = k } , we obtain that1 n n − X i,j,k,ℓ =0 (cid:12)(cid:12) ρ ( j − k ) ρ ( i − j ) ρ ( k − ℓ ) (cid:12)(cid:12) ≥ n n X k,ℓ =1 (cid:12)(cid:12) ρ ( k − ℓ ) (cid:12)(cid:12) = 1 + 2 n − X ℓ =1 (1 − ℓn ) | ρ ( ℓ ) | ≥ n/ X ℓ =1 | ρ ( ℓ ) | goes to infinity as n → ∞ .5. Appendix: Proof of Theorem 6 We start by proving a similar result in a simpler setting, to whichwe can reduce the general case. Proposition 9. Let ( X, Y ) be a pair of jointly isonormal Gaussianprocesses over H , such that X, Y are rigidly correlated, in the followingsense: there exists θ ∈ [ − , such that, for every h, g ∈ H , one has E [ X ( h ) Y ( g )] = θ h h, g i . Consider measurable mappings F : R H → R and G : R H → R such that F ( X ) and G ( Y ) are square-integrable andcentred, and assume that F has Hermite rank p ≥ . Then, (13) | E [ F ( X ) G ( Y )] | ≤ | θ | p V ar( F ( X )) / V ar( G ( Y )) / Proof. Let { e i : i ≥ } be an orthonormal basis of H . We write α, β, ... to indicate multi-indices; for a multi-index α , the symbol H α indicatesthe corresponding multivariate polynomial. We also write H α ( X ) = H α ( X ( e i ) : i = 1 , , ... ) = ∞ Y i =1 H a i ( X ( e i )) , where H k stands for the k th Hermite polynomial in one variable; H α ( Y )is defined analogously. From the properties of Hermite polynomials andfrom the rigid correlation assumption, we infer that, for any choice ofmulti-indices α, β , one has that E [ H α ( X ) H β ( Y )] = θ | α | α ! α = β . Now,by the chaotic representation property of isonormal processes, one hasthat F ( X ) = X α : | α |≥ p b α H α ( X ) , G ( Y ) = X α : | α |≥ c α H α ( Y ) , with convergence in L (Ω). By virtue of the previous discussion, | E [ F ( X ) G ( Y )] | ≤ X α : | α |≥ p | b α c α || θ | | α | α ! ≤ | θ | p X α : | α |≥ | b α c α | α ! , and the conclusion follows from an application of the Cauchy-Schwarzinequality. (cid:3) We now turn to the proof of Theorem 6. Proof of Theorem 6. Without loss of generality, we assume that θ ∈ (0 , i = 1 , 2, we denote by π H i the orthogonal projection operatoronto H i . We will make use of the following estimate: for every g ∈ H with unit norm,(14) k π H ( g ) k ≤ θ, which follows from the relation k π H ( g ) k = |h g, π H ( g ) i| ≤ θ k π H ( g ) k . Now write H ⊕ H to indicate the direct sum of H and H . The keyof the proof is the definition of mappings τ : H → H , τ : H → H ,and τ : H → H ⊕ H given by g τ ( g ) ⊕ τ ( g ), with the followingtwo properties:(i) for h ∈ H and g ∈ H , h h, τ ( g ) i = θ − h h, g i ;(ii) τ verifies the isometric property: h τ ( g ) , τ ( k ) i H ⊕ H = h g, k i H ,for every g, k ∈ H .In order to define such a mapping τ , we first observe that, by virtue of(14), the positive self-adjoint and bounded operator U , from H intoitself, given by g U ( g ) = π H ( π H ( g )) , is such that k U k op = sup g,k ∈ H : k g k , k k k =1 |h U ( g ) , k i| ≤ θ sup g ∈ H : k g k =1 k π H ( g ) k , and therefore k U k op ≤ θ , by virtue of (14). This implies that theoperator V ( g ) := p Id − U/θ is well-defined. In particular one checksthat a mapping τ satisfying the two properties (i) and (ii) listed aboveis given by τ ( g ) = θ − π H ( g ) and τ ( g ) = V ( g ), for every g ∈ H . Wenow consider two auxiliary independent isonormal Gaussian processes Y, Z over H ⊕ H , and we set R := θY + √ − θ Z, in such a way that ATE FOR THE BREUER-MAJOR THEOREM 15 Y, R are rigidly correlated with parameter θ , in the sense made clearin the statement of Proposition 9. It is also easily verified that, by adirect covariance computation and with obvious notation, (cid:0) Y ( H ⊕ { } ) , R ( τ ( H )) (cid:1) law = ( X , X ) . To conclude the proof, we apply Proposition 9 as follows: | E [ F ( X ) F ( X )] | = | E [ F ( Y ( H ⊕ { } )) F ( R ( τ ( H )))] |≤ θ p V ar( F ( X )) / V ar( F ( X )) / , where we have used the fact that F ( Y ( H ⊕ { } )) has also Hermiterank p , as well as the relations V ar( F ( X )) = V ar( F ( Y ( H ⊕ { } ))) , and V ar( F ( X )) = V ar( F ( R ( τ ( H )))) . (cid:3) Acknowledgments . We are grateful to David Nualart for stimulatingdiscussions on this topic. 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Ivan Nourdin, Universit´e du Luxembourg, Unit´e de Recherche enMath´ematiques, Maison du Nombre, 6 avenue de la Fonte, L-4364 Esch-sur-Alzette, Grand Duch´e du Luxembourg E-mail address : [email protected] Giovanni Peccati, Universit´e du Luxembourg, Unit´e de Rechercheen Math´ematiques, Maison du Nombre, 6 avenue de la Fonte, L-4364Esch-sur-Alzette, Grand Duch´e du Luxembourg E-mail address : [email protected] Xiaochuan Yang, Universit´e du Luxembourg, Unit´e de Rechercheen Math´ematiques, Maison du Nombre, 6 avenue de la Fonte, L-4364Esch-sur-Alzette, Grand Duch´e du Luxembourg E-mail address ::