Quantitative stable limit theorems on the Wiener space
aa r X i v : . [ m a t h . P R ] F e b The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2016
QUANTITATIVE STABLE LIMIT THEOREMS ONTHE WIENER SPACE
By Ivan Nourdin , David Nualart and Giovanni Peccati Universit´e de Lorraine, Kansas University and Universit´e du Luxembourg
We use Malliavin operators in order to prove quantitative stablelimit theorems on the Wiener space, where the target distribution isgiven by a possibly multidimensional mixture of Gaussian distribu-tions. Our findings refine and generalize previous works by Nourdinand Nualart [
J. Theoret. Probab. (2010) 39–64] and Harnett andNualart [ Stochastic Process. Appl. (2012) 3460–3505], and pro-vide a substantial contribution to a recent line of research, focussingon limit theorems on the Wiener space, obtained by means of theMalliavin calculus of variations. Applications are given to quadraticfunctionals and weighted quadratic variations of a fractional Brown-ian motion.
1. Introduction and overview.
Originally introduced by R´enyi in thelandmark paper [33], the notion of stable convergence for random variables(see Definition 2.2 below) is an intermediate concept, bridging convergencein distribution (which is a weaker notion) and convergence in probability(which is stronger). One crucial feature of stably converging sequences isthat they can be naturally paired with sequences converging in probability(see, e.g., the statement of Lemma 2.3 below), thus yielding a vast arrayof noncentral limit results—most notably convergence toward mixtures ofGaussian distributions. This last feature makes indeed stable convergenceextremely useful for applications, in particular to the asymptotic analysis offunctionals of semimartingales, such as power variations, empirical covari-ances, and other objects of statistical relevance. See the classical reference
Received May 2013; revised August 2014. Supported in part by the French ANR Grant ANR-10-BLAN-0121. Supported in part by NSF Grant DMS-12-08625. Supported in part by Grant F1R-MTH-PUL-12PAMP (PAMPAS), from LuxembourgUniversity.
AMS 2000 subject classifications.
Key words and phrases.
Stable convergence, Malliavin calculus, fractional Brownianmotion.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2016, Vol. 44, No. 1, 1–41. This reprint differs from the original in pagination andtypographic detail. 1
I. NOURDIN, D. NUALART AND G. PECCATI [11], Chapter VIII.5, as well as the recent survey [31], for a discussion ofstable convergence results in a semimartingale context.Outside the (semi)martingale setting, the problem of characterizing stablyconverging sequences is for the time being much more delicate. Within theframework of limit theorems for functionals of general Gaussian fields, a stepin this direction appears in the paper [28], by Peccati and Tudor, where itis shown that central limit theorems (CLTs) involving sequences of multipleWiener–Itˆo integrals of order ≥ L -closed Gaussian space generated bythe field itself (see [20], Chapter 6, for a general discussion of multidimen-sional CLTs on the Wiener space). Some distinguished applications of theresults in [28] appear, for example, in the two papers [1, 4], respectively, byCorcuera et al. and by Barndorff-Nielsen et al., where the authors establishstable limit theorems (toward a Gaussian mixture) for the power variationsof pathwise stochastic integrals with respect to a Gaussian process with sta-tionary increments. See [19] for applications to the weighted variations ofan iterated Brownian motion. See [2] for some quantitative analogues of thefindings of [28] for functionals of a Poisson measure.Albeit useful for many applications, the results proved in [28] do not pro-vide any intrinsic criterion for stable convergence toward Gaussian mixtures.In particular, the applications developed in [1, 4, 19] basically require thatone is able to represent a given sequence of functionals as the combination ofthree components—one converging in probability to some nontrivial randomelement, one living in a finite sum of Wiener chaoses and one vanishing inthe limit—so that the results from [28] can be directly applied. This is ingeneral a highly nontrivial task, and such a strategy is technically too de-manding to be put into practice in several situations (e.g., when the chaoticdecomposition of a given functional cannot be easily computed or assessed).The problem of finding effective intrinsic criteria for stable convergenceon the Wiener space toward mixtures of Gaussian distributions—without re-sorting to chaotic decompositions—was eventually tackled by Nourdin andNualart in [17], where one can find general sufficient conditions ensuringthat a sequence of multiple Skorohod integrals stably converges to a mixtureof Gaussian distributions. Multiple Skorohod integrals are a generalizationof multiple Wiener–Itˆo integrals (in particular, they allow for random inte-grands), and are formally defined in Section 2.1 below. It is interesting tonote that the main results of [17] are proved by using a generalization ofa characteristic function method, originally applied by Nualart and Ortiz-Latorre in [25] to provide a Malliavin calculus proof of the CLTs establishedin [26, 28]. In particular, when specialized to multiple Wiener–Itˆo integrals,the results of [17] allow to recover the “fourth moment theorem” by Nualart UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE and Peccati [26]. A first application of these stable limit theorems appearsin [17], Section 5, where one can find stable mixed Gaussian limit theoremsfor the weighted quadratic variations of the fractional Brownian motion(fBm), complementing some previous findings from [18]. Another class ofremarkable applications of the results of [17] are the so-called Itˆo formulaein law ; see [8, 9, 22, 23]. Reference [9] also contains some multidimensionalextensions of the abstract results proved in [17] (with a proof again basedon the characteristic function method). Further applications of these tech-niques can be found in [34]. An alternative approach to stable convergenceon the Wiener space, based on decoupling techniques, has been developedby Peccati and Taqqu in [27].One evident limitation of the abstract results of [9, 17] is that they donot provide any information about rates of convergence. The aim of thispaper is to prove several quantitative versions of the abstract results provedin [9, 17], that is, statements allowing one to explicitly assess quantities ofthe type | E [ ϕ ( δ q ( u ) , . . . , δ q d ( u d ))] − E [ ϕ ( F )] | , where ϕ is an appropriate test function on R d , each δ q i ( u i ) is a multiple Sko-rohod integral of order q i ≥
1, and F is a d -dimensional mixture of Gaussiandistributions. Most importantly, we shall show that our bounds also yieldnatural sufficient conditions for stable convergence toward F . To do this, wemust overcome a number of technical difficulties, in particular: • We will work in a general framework and without any underlying semi-martingale structure, in such a way that the powerful theory of stableconvergence for semimartingales (see again [11]) cannot be applied. • Although there are many versions of Stein’s method allowing one to dealwith general continuous non-Gaussian targets (see, e.g., [3, 5–7, 12, 13,32]), it seems that none of them can be reasonably applied to the limittheorems that are studied in this paper. Indeed, the above quoted con-tributions fall mainly in two categories: either those requiring that thedensity of the target distribution is explicitly known (and in this case theso-called “density approach” can be applied—see, e.g., [3, 5–7]), or thoserequiring that the target distribution is the invariant measure of some dif-fusion process (so that the “generator approach” can be used—see, e.g.,[12, 13, 32]). In both instances, a detailed analytical description of thetarget distribution must be available. In contrast, in the present paper weconsider limit distributions given by the law of random elements of thetype S · η = ( S η , . . . , S d η d ), where η = ( η , . . . , η d ) is a Gaussian vector,and S = ( S , . . . , S d ) is an independent random element that is suitablyregular in the sense of Malliavin calculus. In particular, in our framework no a priori knowledge of the distribution of S (and therefore of S · η ) is I. NOURDIN, D. NUALART AND G. PECCATI required. One should note that in [3] one can find an application of Stein’smethod to the law of random objects with the form Sη , where η is a one-dimensional Gaussian random variable and S has a law with a two-pointsupport (of course, in this case the density of Sη can be directly computedby elementary arguments).Our techniques rely on an interpolation procedure and on the use of Malli-avin operators. To our knowledge, the main bounds proved in this paper,that is, the ones appearing in Proposition 3.1, Theorems 3.4 and 5.1, arefirst ever explicit upper bounds for mixed normal approximations in a non-semimartingale setting.Note that, in our discussion, we shall separate the case of one-dimensionalSkorohod integrals of order 1 (discussed in Section 3) from the general case(discussed in Section 5), since in the former setting one can exploit someuseful simplifications, as well as obtain some effective bounds in the Wasser-stein and Kolmogorov distances. As discussed below, our results can be seenas abstract versions of classic limit theorems for Brownian martingales, suchas the ones discussed in [35], Chapter VIII.Although our results deal only with Skorohod integrals, they can be ap-plied in the context of Stratonovich integrals. In fact, the Stratonovich in-tegral can be expressed as a Skorohod integral plus a complementary termand in many problems this complementary term does not contribute to thelimit. Examples of this situation are the Itˆo formulas in law for differenttypes of Stratonovich integrals obtained by Harnett and Nualart in [8, 9]and the weak convergence of weighted variations established by Nourdinand Nualart in [17].To illustrate our findings, we provide applications to quadratic functionalsof a fractional Brownian motion (Section 3.3) and to weighted quadraticvariations (Section 6). The results of Section 3.3 generalize some previousfindings by Peccati and Yor [29, 30], whereas those of Section 6 complementsome findings by Nourdin, Nualart and Tudor [18].The paper is organized as follows. Section 2 contains some preliminarieson Gaussian analysis and stable convergence. In Section 3, we first deriveestimates for the distance between the laws of a Skorohod integral of order 1and of a mixture of Gaussian distributions (see Proposition 3.1). As a corol-lary, we deduce the stable limit theorem for a sequence of multiple Skorohodintegrals of order 1 obtained in [9], and we obtain rates of convergence inthe Wasserstein and Kolmogorov distances. We apply these results to a se-quence of quadratic functionals of the fractional Brownian motion. Section 4contains some additional notation and a technical lemma that are used inSection 5 to establish bounds in the multidimensional case for Skorohod in-tegrals of general orders. Finally, in Section 6 we present the applications ofthese results to the case of weighted quadratic variations of the fractional UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE Brownian motion. The Appendix contains some technical lemmas needed inSection 6.
2. Gaussian analysis and stable convergence.
In the next two subsec-tions, we discuss some basic notions of Gaussian analysis and Malliavincalculus. The reader is referred to the monographs [24] and [20] for anyunexplained definition or result.2.1.
Elements of Gaussian analysis.
Let H be a real separable infinite-dimensional Hilbert space. For any integer q ≥
1, we denote by H ⊗ q and H ⊙ q ,respectively, the q th tensor product and the q th symmetric tensor productof H . In what follows, we write X = { X ( h ) : h ∈ H } to indicate an isonormalGaussian process over H . This means that X is a centered Gaussian family,defined on some probability space (Ω , F , P ), with a covariance structuregiven by E [ X ( h ) X ( g )] = h h, g i H , h, g ∈ H . (2.1)From now on, we assume that F is the P -completion of the σ -field generatedby X . For every integer q ≥
1, we let H q be the q th Wiener chaos of X ,that is, the closed linear subspace of L (Ω) generated by the random vari-ables { H q ( X ( h )) , h ∈ H , k h k H = 1 } , where H q is the q th Hermite polynomialdefined by H q ( x ) = ( − q e x / d q dx q ( e − x / ) . We denote by H the space of constant random variables. For any q ≥ I q ( h ⊗ q ) = q ! H q ( X ( h )) provides a linear isometry between H ⊙ q (equipped with the modified norm √ q ! k · k H ⊗ q ) and H q [equipped with the L (Ω) norm]. For q = 0, we set by convention H = R and I equal to theidentity map.It is well known (Wiener chaos expansion) that L (Ω) can be decomposedinto the infinite orthogonal sum of the spaces H q , that is: any square inte-grable random variable F ∈ L (Ω) admits the following chaotic expansion: F = ∞ X q =0 I q ( f q ) , (2.2)where f = E [ F ], and the f q ∈ H ⊙ q , q ≥
1, are uniquely determined by F . Forevery q ≥
0, we denote by J q the orthogonal projection operator on the q thWiener chaos. In particular, if F ∈ L (Ω) is as in (2.2), then J q F = I q ( f q )for every q ≥ I. NOURDIN, D. NUALART AND G. PECCATI
Let { e k , k ≥ } be a complete orthonormal system in H . Given f ∈ H ⊙ p , g ∈ H ⊙ q and r ∈ { , . . . , p ∧ q } , the r th contraction of f and g is the elementof H ⊗ ( p + q − r ) defined by f ⊗ r g = ∞ X i ,...,i r =1 h f, e i ⊗ · · · ⊗ e i r i H ⊗ r ⊗ h g, e i ⊗ · · · ⊗ e i r i H ⊗ r . (2.3)Notice that f ⊗ r g is not necessarily symmetric. We denote its symmetriza-tion by f e ⊗ r g ∈ H ⊙ ( p + q − r ) . Moreover, f ⊗ g = f ⊗ g equals the tensor prod-uct of f and g while, for p = q , f ⊗ q g = h f, g i H ⊗ q . Contraction operators areuseful for dealing with products of multiple Wiener–Itˆo integrals.In the particular case where H = L ( A, A , µ ), with ( A, A ) is a measur-able space and µ is a σ -finite and nonatomic measure, one has that H ⊙ q = L s ( A q , A ⊗ q , µ ⊗ q ) is the space of symmetric and square integrable func-tions on A q . Moreover, for every f ∈ H ⊙ q , I q ( f ) coincides with the multipleWiener–Itˆo integral of order q of f with respect to X (as defined, e.g., in[24], Section 1.1.2) and (2.3) can be written as( f ⊗ r g )( t , . . . , t p + q − r )= Z A r f ( t , . . . , t p − r , s , . . . , s r ) × g ( t p − r +1 , . . . , t p + q − r , s , . . . , s r ) dµ ( s ) · · · dµ ( s r ) . Malliavin calculus.
Let us now introduce some elements of the Malli-avin calculus of variations with respect to the isonormal Gaussian process X . Let S be the set of all smooth and cylindrical random variables of theform F = g ( X ( φ ) , . . . , X ( φ n )) , (2.4)where n ≥ g : R n → R is a infinitely differentiable function with compactsupport, and φ i ∈ H . The Malliavin derivative of F with respect to X is theelement of L (Ω , H ) defined as DF = n X i =1 ∂g∂x i ( X ( φ ) , . . . , X ( φ n )) φ i . By iteration, one can define the q th derivative D q F for every q ≥
2, whichis an element of L (Ω , H ⊙ q ).For q ≥ p ≥ D q,p denotes the closure of S with respect to thenorm k · k D q,p , defined by the relation k F k p D q,p = E [ | F | p ] + q X i =1 E ( k D i F k p H ⊗ i ) . UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE The Malliavin derivative D verifies the following chain rule. If ϕ : R n → R is continuously differentiable with bounded partial derivatives and if F =( F , . . . , F n ) is a vector of elements of D , , then ϕ ( F ) ∈ D , and Dϕ ( F ) = n X i =1 ∂ϕ∂x i ( F ) DF i . We denote by δ the adjoint of the operator D , also called the divergenceoperator or Skorohod integral (see, e.g., [24], Section 1.3.2, for an explanationof this terminology). A random element u ∈ L (Ω , H ) belongs to the domainof δ , noted Dom δ , if and only if it verifies | E ( h DF, u i H ) | ≤ c u q E ( F )for any F ∈ D , , where c u is a constant depending only on u . If u ∈ Dom δ ,then the random variable δ ( u ) is defined by the duality relationship (called“integration by parts formula”): E ( F δ ( u )) = E ( h DF, u i H ) , (2.5)which holds for every F ∈ D , . The formula (2.5) extends to the multipleSkorohod integral δ q , and we have E ( F δ q ( u )) = E ( h D q F, u i H ⊗ q ) , (2.6)for any element u in the domain of δ q and any random variable F ∈ D q, .Moreover, δ q ( h ) = I q ( h ) for any h ∈ H ⊙ q .The following statement will be used in the paper, and is proved in [17]. Lemma 2.1.
Let q ≥ be an integer. Suppose that F ∈ D q, , and let u be a symmetric element in Dom δ q . Assume that, for any ≤ r + j ≤ q , h D r F, δ j ( u ) i H ⊗ r ∈ L (Ω , H ⊗ q − r − j ) . Then, for any r = 0 , . . . , q − , h D r F, u i H ⊗ r belongs to the domain of δ q − r and we have F δ q ( u ) = q X r =0 (cid:18) qr (cid:19) δ q − r ( h D r F, u i H ⊗ r ) . (2.7) [With the convention that δ ( v ) = v , v ∈ L (Ω) and D F = F , F ∈ L (Ω) .] For any Hilbert space V , we denote by D k,p ( V ) the corresponding Sobolevspace of V -valued random variables (see [24], page 31). The operator δ q iscontinuous from D k,p ( H ⊗ q ) to D k − q,p , for any p > k ≥ q ≥
1, that is, we have k δ q ( u ) k D k − q,p ≤ c k,p k u k D k,p ( H ⊗ q ) , (2.8) I. NOURDIN, D. NUALART AND G. PECCATI for all u ∈ D k,p ( H ⊗ q ), and some constant c k,p >
0. These estimates are con-sequences of Meyer inequalities (see [24], Proposition 1.5.7). In particular,these estimates imply that D q, ( H ⊗ q ) ⊂ Dom δ q for any integer q ≥ Dδ ( u ) = u + δ ( Du ) , (2.9)for any u ∈ D , ( H ). By induction, we can show the following formula forany symmetric element u in D j + k, ( H ⊗ j ) D k δ j ( u ) = j ∧ k X i =0 (cid:18) ki (cid:19) (cid:18) ji (cid:19) i ! δ j − i ( D k − i u ) . (2.10)Also, we will make sometimes use of the following formula for the varianceof a multiple Skorohod integral. Let u, v ∈ D q, ( H ⊗ q ) ⊂ Dom δ q be two sym-metric functions. Then E ( δ q ( u ) δ q ( v )) = E ( h u, D q ( δ q ( v )) i H ⊗ q )= q X i =0 (cid:18) qi (cid:19) i ! E ( h u, δ q − i ( D q − i v ) i H ⊗ q )(2.11) = q X i =0 (cid:18) qi (cid:19) i ! E ( D q − i u b ⊗ q − i D q − i v ) , with the notation D q − i u b ⊗ q − i D q − i v = ∞ X j,k,ℓ =1 h D q − i h u, ξ j ⊗ η ℓ i H ⊗ q , ξ k i H ⊗ q − i h D q − i h v, ξ k ⊗ η ℓ i H ⊗ q , ξ j i H ⊗ q − i , where { ξ j , j ≥ } and { η ℓ , ℓ ≥ } are complete orthonormal systems in H ⊗ q − i and H ⊗ i , respectively.The operator L is defined on the Wiener chaos expansion as L = P ∞ q =0 − qJ q ,and is called the infinitesimal generator of the Ornstein–Uhlenbeck semi-group . The domain of this operator in L (Ω) is the setDom L = ( F ∈ L (Ω) : ∞ X q =1 q k J q F k L (Ω) < ∞ ) = D , . There is an important relationship between the operators D , δ and L (see[24], Proposition 1.4.3). A random variable F belongs to the domain of L ifand only if F ∈ Dom( δD ) (i.e., F ∈ D , and DF ∈ Dom δ ), and in this case δDF = − LF. (2.12)
UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE Note also that a random variable F as in (2.2) is in D , if and only if P ∞ q =1 qq ! k f q k H ⊗ q < ∞ , and, in this case, E ( k DF k H ) = P q ≥ qq ! k f q k H ⊗ q . If H = L ( A, A , µ ) (with µ nonatomic), then the derivative of a random variable F as in (2.2) can be identified with the element of L ( A × Ω) given by D a F = ∞ X q =1 qI q − ( f q ( · , a )) , a ∈ A. (2.13)2.3. Stable convergence.
The notion of stable convergence used in thispaper is provided in the next definition. Recall that the probability space(Ω , F , P ) is such that F is the P -completion of the σ -field generated by theisonormal process X . Definition 2.2 (Stable convergence). Fix d ≥
1. Let { F n } be a sequenceof random variables with values in R d , all defined on the probability space(Ω , F , P ). Let F be a R d -valued random variable defined on some extendedprobability space (Ω ′ , F ′ , P ′ ). We say that F n converges stably to F , written F n st → F , if lim n →∞ E [ Ze i h λ,F n i R d ] = E ′ [ Ze i h λ,F i R d ] , (2.14)for every λ ∈ R d and every bounded F -measurable random variable Z .Choosing Z = 1 in (2.14), we see that stable convergence implies conver-gence in distribution. For future reference, we now list some useful propertiesof stable convergence. The reader is referred, for example, to [11], Chapter 4,for proofs. From now on, we will use the symbol P → to indicate convergencein probability with respect to P . Lemma 2.3.
Let d ≥ , and let { F n } be a sequence of random variableswith values in R d . F n st → F if and only if ( F n , Z ) law → ( F, Z ) , for every F -measurable ran-dom variable Z . F n st → F if and only if ( F n , Z ) law → ( F, Z ) , for every random variable Z belonging to some set Z = { Z α : α ∈ A } such that the P -completion of σ ( Z ) coincides with F . If F n st → F and F is F -measurable, then necessarily F n P → F . If F n st → F and { Y n } is another sequence of random elements, definedon (Ω , F , P ) and such that Y n P → Y , then ( F n , Y n ) st → ( F, Y ) . I. NOURDIN, D. NUALART AND G. PECCATI
The following statement (to which we will compare many results of thepresent paper) contains criteria for the stable convergence of vectors of mul-tiple Skorohod integrals of the same order. The case d = 1 was proved in[17], Corollary 3.3, whereas the case of a general d is dealt with in [9], Theo-rem 3.2. Given d ≥ µ ∈ R d and a nonnegative definite d × d matrix C , weshall denote by N d ( µ , C ) the law of a d -dimensional Gaussian vector withmean µ and covariance matrix C . Theorem 2.4.
Let q, d ≥ be integers, and suppose that F n is a se-quence of random variables in R d of the form F n = δ q ( u n ) = ( δ q ( u n ) , . . . ,δ q ( u dn )) , for a sequence of R d -valued symmetric functions u n in D q, q ( H ⊗ q ) .Suppose that the sequence F n is bounded in L (Ω) and that: h u jn , N mℓ =1 ( D a ℓ F j ℓ n ) ⊗ h i H ⊗ q converges to zero in L (Ω) for all integers ≤ j, j ℓ ≤ d , all integers ≤ a , . . . , a m , r ≤ q − such that a + · · · + a m + r = q , and all h ∈ H ⊗ r . For each ≤ i, j ≤ d , h u in , D q F jn i H ⊗ q converges in L (Ω) to a randomvariable s ij , such that the random matrix Σ := ( s ij ) d × d is nonnegativedefinite.Then F n st → F , where F is a random variable with values in R d and withconditional Gaussian distribution N d (0 , Σ) given X . Distances.
For future reference, we recall the definition of some use-ful distances between the laws of two real-valued random variables
F, G . • The
Wasserstein distance between the laws of F and G is defined by d W ( F, G ) = sup ϕ ∈ Lip(1) | E [ ϕ ( F )] − E [ ϕ ( G )] | , where Lip(1) indicates the collection of all Lipschitz functions ϕ withLipschitz constant less than or equal to 1. • The
Kolmogorov distance is d Kol ( F, G ) = sup x ∈ R | P ( F ≤ x ) − P ( G ≤ x ) | . • The total variation distance is d TV ( F, G ) = sup A ∈ B ( R ) | P ( F ∈ A ) − P ( G ∈ A ) | . • The
Fortet–Mourier distance is d FM ( F, G ) = sup ϕ ∈ Lip(1) , k ϕ k ∞ ≤ | E [ ϕ ( F )] − E [ ϕ ( G )] | . UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE Plainly, d W ≥ d FM and d TV ≥ d Kol . We recall that the topologies inducedby d W , d Kol and d TV , over the class of probability measures on the realline, are strictly stronger than the topology of convergence in distribution,whereas d FM metrizes convergence in distribution (see, e.g., [20], AppendixC, for a review of these facts).
3. Quantitative stable convergence in dimension one.
We start by fo-cussing on stable limits for one-dimensional Skorohod integrals of order one,that is, random variables having the form F = δ ( u ), where u ∈ D , ( H ).As already discussed, this framework permits some interesting simplifica-tions that are not available for higher order integrals and higher dimensions.Notice that any random variable F such that E [ F ] = 0 and E [ F ] < ∞ can be written as F = δ ( u ) for some u ∈ Dom δ . For example, we can take u = − DL − F , or in the context of the standard Brownian motion, we cantake u an adapted and square integrable process.3.1. Explicit estimates for smooth distances and stable CLTs.
The fol-lowing estimate measures the distance between a Skorohod integral of order1, and a (suitably regular) mixture of Gaussian distributions. In order todeduce a stable convergence result in the subsequent Corollary 3.2, we alsoconsider an element I ( h ) in the first chaos of the isonormal process X . Proposition 3.1.
Let F ∈ D , be such that E [ F ] = 0 . Assume F = δ ( u ) for some u ∈ D , ( H ) . Let S ≥ be such that S ∈ D , , and let η ∼ N (0 , indicate a standard Gaussian random variable independent of the underlyingisonormal Gaussian process X . Let h ∈ H . Assume that ϕ : R → R is C with k ϕ ′′ k ∞ , k ϕ ′′′ k ∞ < ∞ . Then | E [ ϕ ( F + I ( h ))] − E [ ϕ ( Sη + I ( h ))] |≤ k ϕ ′′ k ∞ E [2 |h u, h i H | + |h u, DF i H − S | ](3.1) + k ϕ ′′′ k ∞ E [ |h u, DS i H | ] . Proof.
We proceed by interpolation. Fix ε > S ε = √ S + ε .Clearly, S ε ∈ D , . Let g ( t ) = E [ ϕ ( I ( h ) + √ tF + √ − tS ε η )], t ∈ [0 , E [ ϕ ( F + I ( h ))] − E [ ϕ ( S ε η + I ( h ))] = g (1) − g (0) = R g ′ ( t ) dt .For t ∈ (0 , g ′ ( t ) = 12 E (cid:20) ϕ ′ ( I ( h ) + √ tF + √ − tS ε η ) (cid:18) F √ t − S ε η √ − t (cid:19)(cid:21) = 12 E (cid:20) ϕ ′ ( I ( h ) + √ tF + √ − tS ε η ) (cid:18) δ ( u ) √ t − S ε η √ − t (cid:19)(cid:21) I. NOURDIN, D. NUALART AND G. PECCATI = 12 E (cid:20) ϕ ′′ ( I ( h ) + √ tF + √ − tS ε η ) × (cid:18) √ t h u, h i H + h u, DF i H + √ − t √ t η h u, DS ε i H − S ε (cid:19)(cid:21) . Integrating again by parts with respect to the law of η yields g ′ ( t ) = 12 E [ ϕ ′′ ( I ( h ) + √ tF + √ − tS ε η )( t − / h u, h i H + h u, DF i H − S ε )]+ 1 − t √ t E [ ϕ ′′′ ( I ( h ) + √ tF + √ − tS ε η ) h u, DS i H ] , where we have used the fact that S ε DS ε = DS ε = DS . Therefore, | E [ ϕ ( I ( h ) + F )] − E [ ϕ ( I ( h ) + S ε η )] |≤ k ϕ ′′ k ∞ E [2 |h u, h i H | + |h u, DF i H − S − ε | ]+ k ϕ ′′′ k ∞ E [ |h u, DS i H | ] Z − t √ t dt, and the conclusion follows letting ε go to zero, because R
10 1 − t √ t dt = .The following statement provides a stable limit theorem based on Propo-sition 3.1. Corollary 3.2.
Let S and η be as in the statement of Proposition 3.1.Let { F n } be a sequence of random variables such that E [ F n ] = 0 and F n = δ ( u n ) , where u n ∈ D , ( H ) . Assume that the following conditions hold as n → ∞ : h u n , DF n i H → S in L (Ω) ; h u n , h i H → in L (Ω) , for every h ∈ H ; h u n , DS i H → in L (Ω) .Then F n st → Sη , and selecting h = 0 in (3.1) provides an upper bound for therate of convergence of the difference | E [ ϕ ( F n )] − E [ ϕ ( Sη )] | , for every ϕ ofclass C with bounded second and third derivatives. Proof.
Relation (3.1) implies that, if conditions 1–3 in the statementhold true, then | E [ ϕ ( F n + I ( h ))] − E [ ϕ ( Sη + I ( h ))] | → h ∈ H and every smooth test function ϕ . Selecting ϕ to be a complex exponentialand using point 2 of Lemma 2.3 yields the desired conclusion. (cid:3) Remark 3.3. (a) Corollary 3.2 should be compared with Theorem 2.4in the case d = q = 1 (which exactly corresponds to [17], Corollary 3.3). This UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE result states that, if (i) u n ∈ D , ( H ) and (ii) { F n } is bounded in L (Ω), thenit is sufficient to check conditions 1–2 in the statement of Corollary 3.2 forsome S in L (Ω) in order to deduce the stable convergence of F n to Sη .The fact that Corollary 3.2 requires more regularity on S , as well as theadditional condition 3, is compensated by the less stringent assumptions on u n , as well as by the fact that we obtain explicit rates of convergence for alarge class of smooth functions.(b) The statement of [17], Corollary 3.3, allows one also to recover amodification of the so-called asymptotic Knight Theorem for Brownian mar-tingales, as stated in [35], Theorem VIII.2.3. To see this, assume that X isthe isonormal Gaussian process associated with a standard Brownian mo-tion B = { B t : t ≥ } [corresponding to the case H = L ( R + , ds )] and alsothat the sequence { u n : n ≥ } is composed of square-integrable processesadapted to the natural filtration of B . Then, F n = δ ( u n ) = R ∞ u n ( s ) dB s ,where the stochastic integral is in the Itˆo sense, and the aforementionedasymptotic Knight theorem yields that the stable convergence of F n to Sη is implied by the following: (A) R t u n ( s ) ds P →
0, uniformly in t in compactsets and (B) R ∞ u n ( s ) ds → S in L (Ω).3.2. Wasserstein and Kolmogorov distances.
The following statementprovides a way to deduce rates of convergence in the Wasserstein and Kol-mogorov distance from the previous results.
Theorem 3.4.
Let F ∈ D , be such that E [ F ] = 0 . Write F = δ ( u ) for some u ∈ D , ( H ) . Let S ∈ D , , and let η ∼ N (0 , indicate a standardGaussian random variable independent of the isonormal process X . Set ∆ = 3 (cid:18) √ π E [ |h u, DF i H − S | ] + √ E [ |h u, DS i H | ] (cid:19) / × max (cid:26) √ π E [ |h u, DF i H − S | ] + √ E [ |h u, DS i H | ] , (3.2) r π (2 + E [ S ]) + E [ | F | ] (cid:27) / . Then d W ( F, Sη ) ≤ ∆ . Moreover, if there exists α ∈ (0 , such that E [ | S | − α ] < ∞ , then d Kol ( F, Sη ) ≤ ∆ α/ ( α +1) (1 + E [ | S | − α ]) . (3.3) Remark 3.5.
Theorem 3.4 is specifically relevant whenever one dealswith sequences of random variables living in a finite sum of Wiener chaoses.Indeed, in [21], Theorem 3.1, the following fact is proved: let { F n : n ≥ } be I. NOURDIN, D. NUALART AND G. PECCATI a sequence of random variables living in the subspace L pk =0 H k , and assumethat F n converges in distribution to a nonzero random variable F ∞ ; then,there exists a finite constant c > n ) such that d TV ( F n , F ∞ ) ≤ cd FM ( F n , F ∞ ) / (1+2 p ) ≤ cd W ( F n , F ∞ ) / (1+2 p ) , (3.4) n ≥ . Exploiting this estimate, and in the framework of random variables with afinite chaotic expansion, the bounds in the Wasserstein distance obtainedin Theorem 3.4 can be used to deduce rates of convergence in total varia-tion toward mixtures of Gaussian distributions. The forthcoming Section 3.3provides an explicit demonstration of this strategy, as applied to quadraticfunctionals of a (fractional) Brownian motion.
Proof of Theorem 3.4.
It is divided into two steps.
Step Wasserstein distance . Let ϕ : R → R be a function of class C which is bounded together with all its first three derivatives. For any t ∈ (0 , ϕ t ( x ) = Z R ϕ ( √ ty + √ − tx ) dγ ( y ) , where dγ ( y ) = √ π e − y / dy denotes the standard Gaussian measure. Then,we may differentiate and integrate by parts to get ϕ ′′ t ( x ) = 1 − t √ t Z R yϕ ′ ( √ ty + √ − tx ) dγ ( y )= 1 − tt Z R ( y − ϕ ( √ ty + √ − tx ) dγ ( y )and ϕ ′′′ t ( x ) = (1 − t ) / t Z R ( y − ϕ ′ ( √ ty + √ − tx ) dγ ( y ) . Hence, for 0 < t < k ϕ ′′ t k ∞ ≤ − t √ t k ϕ ′ k ∞ Z R | y | dγ ( y ) ≤ r π k ϕ ′ k ∞ t (3.5)and k ϕ ′′′ t k ∞ ≤ (1 − t ) / t k ϕ ′ k ∞ Z R | y − | dγ ( y )(3.6) ≤ k ϕ ′ k ∞ t sZ R ( y − dγ ( y ) = √ k ϕ ′ k ∞ t . UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE Taylor expansion gives that | E [ ϕ ( F )] − E [ ϕ t ( F )] | ≤ Z R E [ | ϕ ( √ ty + √ − tF ) − ϕ ( √ − tF ) | ] dγ ( y )+ E [ | ϕ ( √ − tF ) − ϕ ( F ) | ] ≤ k ϕ ′ k ∞ √ t Z R | y | dγ ( y ) + k ϕ ′ k ∞ |√ − t − | E [ | F | ] ≤ √ t k ϕ ′ k ∞ (cid:26)r π + E [ | F | ] (cid:27) . Here, we used that |√ − t − | = t/ ( √ − t + 1) ≤ √ t . Similarly, | E [ ϕ ( Sη )] − E [ ϕ t ( Sη )] | ≤ √ t k ϕ ′ k ∞ (cid:26)r π + E [ | Sη | ] (cid:27) = r π √ t k ϕ ′ k ∞ { E [ S ] } . Using (3.1) with (3.5)–(3.6) together with the triangle inequality and theprevious inequalities, we have | E [ ϕ ( F )] − E [ ϕ ( Sη )] |≤ √ t k ϕ ′ k ∞ (cid:18)r π { E [ S ] } + E [ | F | ] (cid:19) (3.7) + k ϕ ′ k ∞ t (cid:26) √ π E [ |h u, DF i H − S | ] + √ E [ |h u, DS i H | ] (cid:27) . Set Φ = r π { E [ S ] } + E [ | F | ]and Φ = 1 √ π E [ |h u, DF i H − S | ] + √ E [ |h u, DS i H | ] . The function t
7→ √ t Φ + t Φ attains its minimum at t = ( Φ ) / . Then,if t ≤ t = t and if t > t = 1. With these choiceswe obtain | E [ ϕ ( F )] − E [ ϕ ( Sη )] | (3.8) ≤ k ϕ ′ k ∞ Φ / max((2 − / + 2 / )Φ / , / ) ≤ k ϕ ′ k ∞ ∆ . This inequality can be extended to all Lispchitz functions ϕ , and this im-mediately yields that d W ( F, Sη ) ≤ ∆. I. NOURDIN, D. NUALART AND G. PECCATI
Step Kolmogorov distance . Fix z ∈ R and h >
0. Consider the function ϕ h : R → [0 ,
1] defined by ϕ h ( x ) = ( , if x ≤ z ,0 , if x ≥ z + h ,linear , if z ≤ x ≤ z + h ,and observe that ϕ h is Lipschitz with k ϕ ′ h k ∞ = 1 /h . Using that ( −∞ ,z ] ≤ ϕ h ≤ ( −∞ ,z + h ] as well as (3.8), we get P [ F ≤ z ] − P [ Sη ≤ z ] ≤ E [ ϕ h ( F )] − E [ ( −∞ ,z ] ( Sη )]= E [ ϕ h ( F )] − E [ ϕ h ( Sη )] + E [ ϕ h ( Sη )] − E [ ( −∞ ,z ] ( Sη )] ≤ ∆ h + P [ z ≤ Sη ≤ z + h ] . On the other hand, we can write P [ z ≤ Sη ≤ z + h ]= 1 √ π Z R e − x / [ z,z + h ] ( sx ) dP S ( s ) dx = 1 √ π (cid:18)Z R + dP S ( s ) Z ( z + h ) /sz/s e − x / dx + Z R − dP S ( s ) Z z/s ( z + h ) /s e − x / dx (cid:19) ≤ | h | α √ π Z R | s | − α dP S ( s ) (cid:18)Z R e − x / (2(1 − α )) dx (cid:19) − α ≤ | h | α E [ | S | − α ] , because ( R R e − x / (2(1 − α )) dx ) − α = ( √ − α R R e − y / dy ) − α ≤ √ π , so that P [ F ≤ z ] − P [ Sη ≤ z ] ≤ ∆ h + | h | α E [ | S | − α ] . Hence, by choosing h = ∆ / ( α +1) , we get that P [ F ≤ z ] − P [ Sη ≤ z ] ≤ ∆ α/ ( α +1) (1 + E [ | S | − α ]) . We prove similarly that P [ F ≤ z ] − P [ Sη ≤ z ] ≥ − ∆ α/ ( α +1) (1 + E [ | S | − α ]) , so the proof of (3.3) is done. (cid:3) UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE Quadratic functionals of Brownian motion and fractional Brownianmotion.
We will now apply the results of the previous sections to somenonlinear functionals of a fractional Brownian motion with Hurst parameter H ≥ . Recall that a fractional Brownian motion (fBm) with Hurst parame-ter H ∈ (0 ,
1) is a centered Gaussian process B = { B t : t ≥ } with covariancefunction E ( B s B t ) = ( t H + s H − | t − s | H ) . Notice that for H = the process B is a standard Brownian motion. Wedenote by E the set of step functions on [0 , ∞ ). Let H be the Hilbert spacedefined as the closure of E with respect to the scalar product h [0 ,t ] , [0 ,s ] i H = E ( B s B t ) . The mapping [0 ,t ] → B t can be extended to a linear isometry between theHilbert space H and the Gaussian space spanned by B . We denote thisisometry by φ → B ( φ ). In this way, { B ( φ ) : φ ∈ H } is an isonormal Gaussianprocess. In the case, H > , the space H contains all measurable functions ϕ : R + → R such that Z ∞ Z ∞ | ϕ ( s ) || ϕ ( t ) || t − s | H − ds dt < ∞ , and in this case if ϕ and φ are functions satisfying this integrability condi-tion, h ϕ, φ i H = H (2 H − Z ∞ Z ∞ ϕ ( s ) φ ( t ) | t − s | H − ds dt. (3.9)Furthermore, L /H ([0 , ∞ )) is continuously embedded into H . In what fol-lows, we shall write c H = p H (2 H − H − , H > / , (3.10)and also c / := lim H ↓ / c H = √ .The following statement contains explicit estimates in total variation forsequences of quadratic Brownian functionals converging to a mixture ofGaussian distributions. It represents a significant refinement of [29], Propo-sition 2.1 and [27], Proposition 18. Theorem 3.6.
Let { B t : t ≥ } be a fBm of Hurst index H ≥ . Forevery n ≥ , define A n := n H Z t n − ( B − B t ) dt. I. NOURDIN, D. NUALART AND G. PECCATI As n → ∞ , the sequence A n converges stably to Sη , where η is a randomvariable independent of B with law N (0 , and S = c H | B | . Moreover, thereexists a constant k (independent of n ) such that d TV ( A n , Sη ) ≤ kn − (1 − H ) / , n ≥ . The proof of Theorem 3.6 is based on the forthcoming Proposition 3.7and Proposition 3.8, dealing with the stable convergence of some auxiliarystochastic integrals, respectively in the cases H = 1 / H > /
2. Noticethat, since lim H ↓ / c H = c / = √ , the statement of Proposition 3.7 can beregarded as the limit of the statement of Proposition 3.8, as H ↓ . Proposition 3.7.
Let B = { B t : t ≥ } be a standard Brownian motion.Consider the sequence of Itˆo integrals F n = √ n Z t n B t dB t , n ≥ . Then the sequence F n converges stably to Sη as n → ∞ , where η is a randomvariable independent of B with law N (0 , and S = | B |√ . Furthermore, wehave the following bounds for the Wasserstein and Kolmogorov distances d Kol ( F n , Sη ) ≤ C γ n − γ , for any γ < , where C γ is a constant depending on γ , and d W ( F n , Sη ) ≤ Cn − / , where C is a finite constant independent of n . Proof.
Taking into account that the Skorohod integral coincides withthe Itˆo integral, we can write F n = δ ( u n ), where u n ( t ) = √ nt n B t [0 , ( t ).In order to apply Theorem 3.4, we need to estimate the quantities E ( |h u n ,DF n i H − S | ) and E ( |h u n , DS i H | ). We recall that H = L ( R + , ds ). For s ∈ [0 , D s F n = √ ns n B s + √ n Z s t n dB t . As a consequence, h u n , DF n i H = n Z s n B s ds + n Z s n B s (cid:18)Z s t n dB t (cid:19) ds. UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE From the estimates, E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) n Z s n B s ds − B (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ n Z s n E ( | B s − B | ) ds + (cid:12)(cid:12)(cid:12)(cid:12) n n + 1 − (cid:12)(cid:12)(cid:12)(cid:12) ≤ n Z s n √ − s ds + 12(2 n + 1) ≤ n √ n + 1 sZ s n (1 − s ) ds + 12(2 n + 1) ≤ √ n + 14 n and nE (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z s n B s (cid:18)Z s t n dB t (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ n √ n + 1 Z s n +1 / p − s n +1 ds ≤ n ( n + 3 / √ n + 1 ≤ √ n , we obtain E ( |h u n , DF n i H − S | ) ≤ √ √ n + 14 n . (3.11)On the other hand, |h u n , DS i H | = √ nE (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) B Z s n B s ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ √ nn + 3 / ≤ √ n . (3.12)Notice that E ( | F n | ) ≤ √ n √ n + 2 ≤ √ . (3.13)Therefore, using (3.11), (3.12) and (3.13) and with the notation of Theo-rem 3.4, for any constant C < C , where C = 3 (cid:18) √ π (cid:18) √ (cid:19) + √ (cid:19) / (cid:18)r π (cid:18) √ π + 1 √ (cid:19)(cid:19) / , there exists n such that for all n ≥ n we have ∆ ≤ Cn − / . Therefore, d W ( F n , Sη ) ≤ Cn − / for n ≥ n . Moreover, E [ | S | − α ] < ∞ for any α < d Kol ( F n , Sη ) ≤ C γ n − γ , for any γ < . This completes the proof of the proposition. (cid:3) As announced, the next result is an extension of Proposition 3.7 to thecase of the fractional Brownian motion with Hurst parameter
H > . I. NOURDIN, D. NUALART AND G. PECCATI
Proposition 3.8.
Let B = { B t : t ≥ } be fractional Brownian motionwith Hurst parameter H > . Consider the sequence of random variables F n = δ ( u n ) , n ≥ , where u n ( t ) = n H t n B t [0 , ( t ) . Then, the sequence F n converges stably to Sη as n → ∞ , where η is a randomvariable independent of B with law N (0 , and S = c H | B | . Furthermore,we have the following bounds for the Wasserstein and Kolmogorov distances d Kol ( F n , Sη ) ≤ C γ,H n − γ , for any γ < − H , where C γ,H is a constant depending on γ and H , and d W ( F n , Sη ) ≤ C H n − (1 − H ) / , where C H is a constant depending on H . Proof.
Let us compute D s F n = n H s n B s + n H Z s t n dB t . As a consequence, h u n , DF n i H = k u n k H + n H (cid:28) u n , Z · t n dB t (cid:29) H . As in the proof of Proposition 3.7, we need to estimate the following quan-tities: ε n = E ( |k u n k H − S | )and δ n = E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) n H (cid:28) u n , Z · t n dB t (cid:29) H (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . We have, using (3.9), ε n ≤ H (2 H − E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) n H Z Z t s n t n B s B t ( t − s ) H − ds dt − Γ(2 H − B (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ H (2 H − n H E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) Z Z t s n t n [ B s B t − B ]( t − s ) H − ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + H (2 H − (cid:12)(cid:12)(cid:12)(cid:12) n H Z Z t s n t n ( t − s ) H − ds dt − Γ(2 H − (cid:12)(cid:12)(cid:12)(cid:12) = a n + b n . UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE We can write for any s ≤ tE ( | B s B t − B | ) = E ( | B s B t − B s B + B s B − B | ) ≤ (1 − t ) H + (1 − s ) H ≤ − s ) H . Using this estimate, we get a n ≤ H (2 H − n H Z Z t s n t n (1 − s ) H ( t − s ) H − ds dt. For any positive integers n, m set ρ n,m = Z Z t s n t m ( t − s ) H − ds dt = Γ( n + 1)Γ(2 H − n + 2 H )( n + m + 2 H ) . (3.14)Then, by H¨older’s inequality, a n ≤ H (2 H − n H ρ − Hn,n (cid:18)Z Z t s n t n (1 − s )( t − s ) H − ds dt (cid:19) H = 4 H (2 H − n H ρ − Hn,n ( ρ n,n − ρ n +1 ,n ) H . Taking into account that ρ n,n − ρ n +1 ,n = Γ( n + 1)( n (2 H + 1) + 4 H )Γ( n + 2 H )(2 n + H )( n + 2 H )(2 n + 1 + 2 H ) , and using Stirling’s formula, we obtain that ρ n,n is less than or equal toa constant times n − H and ρ n,n − ρ n +1 ,n is less than or equal to a con-stant times n − H − . This implies that a n ≤ C H n − H , for some constant C H depending on H .For the term b n , using (3.14) we can write b n = H (2 H − H − (cid:12)(cid:12)(cid:12)(cid:12) n H Γ( n + 1)Γ( n + 2 H )(2 n + 2 H ) − (cid:12)(cid:12)(cid:12)(cid:12) , which converges to zero, by Stirling’s formula, at the rate n − .On the other hand, δ n = H (2 H − n H E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z Z s n B s (cid:18)Z t r n dB r (cid:19) | t − s | H − ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (3.15) ≤ H (2 H − n H Z Z s n + H (cid:20) E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z t r n dB r (cid:12)(cid:12)(cid:12)(cid:12) (cid:19)(cid:21) / | t − s | H − ds dt. We can write, using the fact that L /H ([0 , ∞ )) is continuously embeddedinto H , E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z t r n dB r (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) ≤ C H (cid:18)Z t r n/H dr (cid:19) H ≤ C H ( n/H + 1) H . (3.16) I. NOURDIN, D. NUALART AND G. PECCATI
Substituting (6.13) into (6.14) we obtain δ n ≤ C H n H − , for some constant C H , depending on H . Thus, E ( |h u n , DF n i H − S | ) ≤ C H n H − . Finally, E ( |h u n , DS i H | ) = n H E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z Z s n B s | t − s | H − ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ n H (cid:12)(cid:12)(cid:12)(cid:12)Z Z s n + H | t − s | H − ds dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C H n H − . Notice that in this case E ( |h u n , DF n i H − S | ) converges to zero faster than E ( |h u n , DS i H | ). As a consequence, ∆ ≤ C H n ( H − / , for some constant C H and we conclude the proof using Theorem 3.4. (cid:3) Proof of Theorem 3.6.
Using Itˆo’s formula (in its classical form for H = , and in the form discussed, e.g., in [24], pages 293–294, for the case H > ) yields that ( B − B t ) = δ ( B · [ t, ( · )) + (1 − t H )[note that δ ( B · [ t, ( · )) is a classical Itˆo integral in the case H = ]. Inter-changing deterministic and stochastic integration by means of a stochasticFubini theorem yields therefore that A n = F n + H n H H + n . In view of Propositions 3.7 and 3.8, this implies that A n converges in distri-bution to Sη . The crucial point is now that each random variable A n belongsto the direct sum H ⊕ H : it follows that one can exploit the estimate (3.4)in the case p = 2 to deduce that there exists a constant c such that d TV ( A n , Sη ) ≤ cd W ( A n , Sη ) / ≤ c ( d W ( F n , Sη ) + d W ( A n , F n )) / , where we have applied the triangle inequality. Since (trivially) d W ( A n , F n ) ≤ H n H H + n < n H − , we deduce the desired conclusion by applying the estimatesin the Wasserstein distance stated in Propositions 3.7 and 3.8. (cid:3)
4. Further notation and a technical lemma.
A technical lemma.
The following technical lemma is needed in thesubsequent sections.
UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE Lemma 4.1.
Let η , . . . , η d be a collection of i.i.d. N (0 , random vari-ables. Fix α , . . . , α d ∈ R and integers k , . . . , k d ≥ . Then, for every f : R d → R of class C ( k,...,k ) (where k = k + · · · + k d ) such that f and all its partialderivatives have polynomial growth, E [ f ( α η , . . . , α d η d ) η k · · · η k d d ]= ⌊ k / ⌋ X j =0 · · · ⌊ k d / ⌋ X j d =0 d Y l =1 (cid:26) k l !2 j l ( k − j l )! j ! α k l − j l (cid:27) × E (cid:20) ∂ k + ··· + k d − j + ··· + j d ) ∂x k − j · · · ∂x k d − j d d f ( α η , . . . , α d η d ) (cid:21) . Proof.
By independence and conditioning, it suffices to prove the claimfor d = 1, and in this case we write η = η , k = k , and so on. The decom-position of the random variable η k in terms of Hermite polynomials is givenby η k = ⌊ k/ ⌋ X j =0 k !2 j ( k − j )! j ! H k − j ( η ) , where H k − j ( x ) is the ( k − j )th Hermite polynomial. Using the relation E [ f ( αη ) H k − j ( η )] = α k − j E [ f ( k − j ) ( αη )], we deduce the desired conclusion. (cid:3) Notation.
The following notation is needed in order to state ournext results. For the rest of this section, we fix integers m ≥ d ≥ ψ : R m × d → R : ( y , . . . , y m ; x , . . . , x d ) ψ ( y , . . . , y m ; x , . . . , x d ) . (4.1)Here, the implicit convention is that, if m = 0, then ψ does not depend on( y , . . . , y m ). We also write ψ x k = ∂∂x k ψ, k = 1 , . . . , d. (ii) For every integer q ≥
1, we write A ( q ) = A ( q ; m, d ) (the dependenceon m, d is dropped whenever there is no risk of confusion) to indicate thecollection of all ( m + q (1 + d ))-dimensional vectors with nonnegative integerentries of the type α ( q ) = ( k , . . . , k q ; a , . . . , a m ; b ij , i = 1 , . . . , q, j = 1 , . . . , d ) , (4.2) I. NOURDIN, D. NUALART AND G. PECCATI verifying the set of Diophantine equations k + 2 k + · · · + qk q = q,a + · · · + a m + b + · · · + b d = k ,b + · · · + b d = k , · · · b q + · · · + b qd = k q . (4.3)(iii) Given q ≥ α ( q ) as in (4.2), we define C ( α ( q ) ) := q ! Q qi =1 i ! k i Q ml =1 a l ! Q qi =1 Q dj =1 b ij ! . (4.4)(iv) Given a smooth function ψ as in (4.1) and a vector α ( q ) ∈ A ( q ) asin (4.2), we set ∂ α ( q ) ψ := ∂ k + ··· + k d ∂y a · · · ∂y a m m ∂x b + ··· + b q · · · ∂x b d + ··· + b qd d ψ. (4.5)The coefficients C ( α ( q ) ) and the differential operators ∂ α ( q ) , defined respec-tively in (4.4) and (4.5), enter the generalized Faa di Bruno formula (asproved, e.g., in [14]) that we will use in the proof of our main results.(v) For every integer q ≥
1, the symbol B ( q ) = B ( q ; m, d ) indicates theclass of all ( m + q (1 + 2 d ))-dimensional vectors with nonnegative integerentries of the type β ( q ) = ( k , . . . , k q ; a , . . . , a m ; b ′ ij , b ′′ ij , i = 1 , . . . , q, j = 1 , . . . , d ) , (4.6)such that α ( β ( q ) ) := ( k , . . . , k q ; a , . . . , a m ; b ′ ij + b ′′ ij , i = 1 , . . . , q, j = 1 , . . . , d ) , (4.7)is an element of A ( q ), as defined at point (ii). Given β ( q ) as in (4.6), we alsoadopt the notation | b ′ | := q X i =1 d X j =1 b ′ ij , | b ′′ | := q X i =1 d X j =1 b ′′ ij , (4.8) | b ′′• j | := q X i =1 b ′′ ij , j = 1 , . . . , d. (vi) For every β ( q ) ∈ B ( q ) as in (4.6) and every ( l , . . . , l d ) such that l s ∈ { , . . . , ⌊| b ′′• s | / ⌋} , s = 1 , . . . , d , we set W ( β ( q ) ; l , . . . , l d )(4.9) := C ( α ( β ( q ) )) q Y i =1 d Y j =1 (cid:18) b ′ ij + b ′′ ij b ′ ij (cid:19) d Y s =1 | b ′′• s | !2 l s ( | b ′′• s | − l s )! l s ! , UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE where C ( α ( β ( q ) )) is defined in (4.4), and ∂ ( β ( q ) ; l ,...,l d ) ⋆ := ∂ α ( β ( q ) ) ∂ | b ′′ |− l + ··· + l d ) ∂x | b ′′• |− l · · · ∂x | b ′′• d |− l d d , (4.10)where α ( β ( q ) ) is given in (4.7), and ∂ α ( β ( q ) ) is defined according to (4.5).(vii) The Beta function B ( u, v ) is defined as B ( u, v ) = Z t u − (1 − t ) v − dt, u, v > .
5. Bounds for general orders and dimensions.
A general statement.
The following statement contains a generalupper bound, yielding stable limit theorems and associated explicit rates ofconvergence on the Wiener space.
Theorem 5.1.
Fix integers m ≥ , d ≥ and q j ≥ , j = 1 , . . . , d . Let η = ( η , . . . , η d ) be a vector of i.i.d. N (0 , random variables independentof the isonormal Gaussian process X . Define ˆ q = max j =1 ,..,d q j . For every j = 1 , . . . , d , consider a symmetric random element u j ∈ D q, q ( H q j ) , andintroduce the following notation: • F j := δ q j ( u j ) and F := ( F , . . . , F d ) ; • ( S , . . . , S d ) is a vector of real-valued elements of D ˆ q, q , and S · η := ( S η , . . . , S d η d ) . Assume that the function ϕ : R m × d → R admits continuous and bounded par-tial derivatives up to the order q + 1 . Then, for every h , . . . , h m ∈ H , | E [ ϕ ( X ( h ) , . . . , X ( h m ); F )] − E [ ϕ ( X ( h ) , . . . , X ( h m ); S · η )] |≤ d X k,j =1 (cid:13)(cid:13)(cid:13)(cid:13) ∂ ∂x k ∂x j ϕ (cid:13)(cid:13)(cid:13)(cid:13) ∞ E [ |h D q k F j , u k i H ⊗ qk − j = k S j | ](5.1) + 12 d X k =1 X β ( qk ) ∈ B ( q k ) ⌊| b ′′• | / ⌋ X l =0 · · · ⌊| b ′′• d | / ⌋ X l d =0 c W ( β ( q k ) ; l , . . . , l d )(5.2) × k ∂ ( β ( qk ) ; l ,...,l d ) ⋆ ϕ x k k ∞ × E " d Y s =1 S | b ′′• s |− l s × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* u k , h ⊗ a ⊗ · · · ⊗ h ⊗ a m m q k O i =1 d O j =1 { ( D i F j ) ⊗ b ′ ij ⊗ ( D i S j ) ⊗ b ′′ ij } + H ⊗ qk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , I. NOURDIN, D. NUALART AND G. PECCATI where we have adopted the same notation as in Section 4.2, with the followingadditional conventions: (a) B ( q ) is the subset of B ( q ) composed of those β ( q k ) as in (4.6) such that b ′ qj = 0 for j = 1 , . . . , d , (b) c W ( β ( q k ) ; l , . . . , l d ) := W ( β ( q k ) ; l , . . . , l d ) × B ( | b ′ | / / | b ′′ | / , where B is the Beta func-tion. Case m = 0 , d = 1 . Specializing Theorem 5.1 to the choice of pa-rameters m = 0, d = 1 and q ≥ Proposition 5.2.
Suppose that u ∈ D q, q ( H q ) is symmetric. Let F = δ q ( u ) . Let S ∈ D q, q , and let η ∼ N (0 , indicate a standard Gaussian ran-dom variable, independent of the underlying isonormal process X . Assumethat ϕ : R → R is C q +1 with k ϕ ( k ) k ∞ < ∞ for any k = 0 , . . . , q + 1 . Then | E [ ϕ ( F )] − E [ ϕ ( Sη )] |≤ k ϕ ′′ k ∞ E [ |h u, D q F i H ⊗ q − S | ]+ X ( b ′ ,b ′′ ) ∈Q ,b ′ q =0 ⌊| b ′′ | / ⌋ X j =0 c q,b ′ ,b ′′ ,j k ϕ (1+ | b ′ | +2 | b ′′ |− j ) k ∞ × E [ S | b ′′ |− j × |h u, ( DF ) ⊗ b ′ ⊗ · · · ⊗ ( D q − F ) ⊗ b ′ q − ⊗ ( DS ) ⊗ b ′′ ⊗ · · · ⊗ ( D q S ) ⊗ b ′′ q i H ⊗ q | ] , where Q is the set of all pairs of q -ples b ′ = ( b ′ , b ′ , . . . , b ′ q ) and b ′′ = ( b ′′ , . . . , b ′′ q ) of nonnegative integers satisfying the constraint b ′ + 2 b ′ + · · · + qb ′ q + b ′′ +2 b ′′ + · · · + qb ′′ q = q . The constants c q,b ′ ,b ′′ ,j are given by c q,b ′ ,b ′′ ,j = 12 B ( | b ′ | / / , | b ′′ | / × q Y i =1 (cid:18) b i b ′ i (cid:19) × | b ′′ | !2 j ( | b ′′ | − j )! j ! × q ! Q qi =1 i ! b i b i ! , where b = b ′ + b ′′ . In the particular case q = 2 we obtain the following result. Proposition 5.3.
Suppose that u ∈ D , ( H ) is symmetric. Let F = δ ( u ) . Let S ∈ D , , and let η ∼ N (0 , indicate a standard Gaussian random UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE variable, independent of the underlying isonormal process X . Assume that ϕ : R → R is C with k ϕ ( k ) k ∞ < ∞ for any k = 0 , . . . , . Then | E [ ϕ ( F )] − E [ ϕ ( Sη )] |≤ k ϕ ′′ k ∞ E [ |h u, D F i H ⊗ − S | ]+ C max ≤ i ≤ k ϕ ( i ) k ∞ ( E [ |h u, ( DF ) ⊗ i H ⊗ | ] + E [ S |h u, DF ⊗ DS i H ⊗ | ]+ E [( S + 1) |h u, ( DS ) ⊗ i H ⊗ | ]+ E [ S |h u, D S i H ⊗ | ]) , where C = B ( , ) + B ( ,
1) + B ( , . Taking into account that DS = 2 SDS and D S = 2 DS ⊗ DS + 2 SD S ,we can write the above estimate in terms of the derivatives of S , which ishelpful in the applications. In this way, we obtain | E [ ϕ ( F )] − E [ ϕ ( Sη )] |≤ k ϕ ′′ k ∞ E [ |h u, D F i H ⊗ − S | ]+ C max ≤ i ≤ k ϕ ( i ) k ∞ ( E [ |h u, ( DF ) ⊗ i H ⊗ | ] + E [ |h u, DF ⊗ DS i H ⊗ | ](5.3) + E [( S − + 1) |h u, ( DS ) ⊗ i H ⊗ | ]+ E [ |h u, D S i H ⊗ | ]) . Notice that a factor S − appears in the right-hand side of the above inequal-ity.5.3. Case m > , d = 1 . Fix q ≥
1. In the case m > d = 1, the class B ( q ) is the collection of all vectors with nonnegative integer entries of thetype β ( q ) = ( a , . . . , a m ; b ′ , b ′′ , . . . , b ′ q , b ′′ q ) verifying a + · · · + a m + ( b ′ + b ′′ ) + · · · + q ( b ′ q + b ′′ q ) = q, whereas B ( q ) is the subset of B ( q ) verifying b ′ q = 0. Specializing Theo-rem 5.1 yields upper bounds for one-dimensional σ ( X )-stable convergence. Proposition 5.4.
Suppose that u ∈ D q, q ( H q ) is symmetric, select h , . . . , h m ∈ H , and write X = ( X ( h ) , . . . , X ( h m )) . Let F = δ q ( u ) . Let S ∈ D q, q , and let η ∼ N (0 , indicate a standard Gaussian random variable,independent of the underlying Gaussian field X . Assume that ϕ : R m × R → R : ( y , . . . , y m , x ) ϕ ( y , . . . , y m , x ) I. NOURDIN, D. NUALART AND G. PECCATI admits continuous and bounded partial derivatives up to the order q + 1 .Then | E [ ϕ ( X , F )] − E [ ϕ ( X , Sη )] |≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂ ∂x ϕ (cid:13)(cid:13)(cid:13)(cid:13) ∞ E [ |h u, D q F i H ⊗ q − S | ]+ 12 X β ( q ) ∈ B ( q ) ⌊| b ′′ | / ⌋ X j =0 c W ( β ( q ) , j ) (cid:13)(cid:13)(cid:13)(cid:13) ∂ | a | ∂y a · · · ∂y a m m ∂ | b ′ | +2 | b ′′ |− j ∂x | b ′ | +2 | b ′′ |− j ϕ (cid:13)(cid:13)(cid:13)(cid:13) ∞ × E " S | b ′′ |− j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* u, h ⊗ a ⊗ · · ·⊗ h ⊗ a m m q O i =1 { ( D i F ) ⊗ b ′ i ⊗ ( D i S ) ⊗ b ′′ i } + H ⊗ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where | a | = a + · · · + a m . Proof of Theorem 5.1.
The proof is based on the use of an interpo-lation argument. Write X = ( X ( h ) , . . . , X ( h m )) and g ( t ) = E [ ϕ ( X ; √ tF + √ − tS · η )], t ∈ [0 , E [ ϕ ( X ; F )] − E [ ϕ ( X ; Sη )] = g (1) − g (0) = R g ′ ( t ) dt . For t ∈ (0 , F or to η , we get g ′ ( t ) = 12 d X k =1 E (cid:20) ϕ x k ( X ; √ tF + √ − tS · η ) (cid:18) F k √ t − S k η k √ − t (cid:19)(cid:21) = 12 d X k =1 E (cid:20) ϕ x k ( X ; √ tF + √ − tS · η ) (cid:18) δ q k ( u k ) √ t − S k η k √ − t (cid:19)(cid:21) = 12 √ t d X k =1 E [ h D q k ϕ x k ( X ; √ tF + √ − tS · η ) , u k i H ⊗ qk ] − d X k =1 E (cid:20) ∂ ∂x k ϕ ( X ; √ tF + √ − tS · η ) S k (cid:21) . Using the Faa di Bruno formula for the iterated derivative of the compositionof a function with a vector of functions (see [14], Theorem 2.1), we infer that,for every k = 1 , . . . , d , h D q k ϕ x k ( X ; √ tF + √ − tS · η ) , u k i H ⊗ qk UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE = X α ( qk ) ∈ A ( q k ) C ( α ( q k ) ) ∂ ( α ( qk ) ) ϕ x k ( X ; √ tF + √ − tS · η )(5.4) × * h ⊗ a ⊗ · · · ⊗ h ⊗ a m m q k O i =1 d O j =1 ( D i ( √ tF j + √ − tS j η j )) ⊗ b ij , u k + H ⊗ qk . For every i = 1 , . . . , q k , every j = 1 , . . . , d and every symmetric v ∈ H ⊗ b ij , wehave h ( D i ( √ tF j + √ − tS j η j )) ⊗ b ij , v i H ⊗ bij = b ij X u =0 (cid:18) b ij u (cid:19) t u/ (1 − t ) ( b ij − u ) / η ( b ij − u ) (5.5) × h ( D i F j ) ⊗ u ⊗ ( D i S j ) ⊗ ( b ij − u ) , v i H ⊗ bij . Substituting (5.5) into (5.4), and taking into account the symmetry of u k ,yields E [ h D q k ϕ x k ( X ; √ tF + √ − tS · η ) , u k i H ⊗ qk ]= X β ( qk ) ∈ B ( q k ) C ( α ( q k ) ) t | b ′ | / (1 − t ) | b ′′ | / q k Y i =1 d Y j =1 (cid:18) b ′ ij + b ′′ ij b ′ ij (cid:19) × E " ∂ α ( β ( qk ) ) ϕ x k ( X ; √ tF + √ − tS · η ) d Y j =1 η | b ′′• j | j × * u k , h ⊗ a ⊗ · · · ⊗ h ⊗ a m m q k O i =1 d O j =1 { ( D i F j ) ⊗ b ′ ij ⊗ ( D i S j ) ⊗ b ′′ ij } + H ⊗ qk . Notice that if β ( q k ) does not belong to B ( q k ), then b ′ q k l ≥ l = 1 , . . . , d . Taking into account the relations (4.3) this implies that b ′ q k l = 1, b ′ q k j = 0 for all j = l , k q k = 1 and all the other entries of β ( q k ) mustbe equal to zero. In this way, the above sum can be decomposed as follows: X β ( qk ) ∈ B ( q k ) C ( α ( q k ) ) t | b ′ | / (1 − t ) | b ′′ | / q k Y i =1 d Y j =1 (cid:18) b ′ ij + b ′′ ij b ′ ij (cid:19) × E " ∂ α ( β ( qk ) ) ϕ x k ( X ; √ tF + √ − tS · η ) d Y j =1 η | b ′′• j | j × * u k , h ⊗ a ⊗ · · · ⊗ h ⊗ a m m q k O i =1 d O j =1 { ( D i F j ) ⊗ b ′ ij ⊗ ( D i S j ) ⊗ b ′′ ij } + H ⊗ qk I. NOURDIN, D. NUALART AND G. PECCATI + d X l =1 √ tE (cid:20) ∂ ∂x k ∂x l ϕ ( X ; √ tF + √ − tS · η ) h D q k F l , u k i H ⊗ qk (cid:21) := D ( k, t ) + F ( k, t ) . Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ t d X k =1 F ( k, t ) − d X k =1 E (cid:20) ∂ ∂x k ϕ ( X ; √ tF + √ − tS · η ) S k (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (5 . , the theorem is proved once we show that d X k =1 Z √ t | D ( k, t ) | dt is less than the sum in (5.2). Using the independence of η and X , condition-ing with respect to X and applying Lemma 4.1 yields E " ∂ α ( β ( qk ) ) ϕ x k ( X ; √ tF + √ − tS · η ) d Y j =1 η | b ′′• j | j × * u k , h ⊗ a ⊗ · · · ⊗ h ⊗ a m m q k O i =1 d O j =1 { ( D i F j ) ⊗ b ′ ij ⊗ ( D i S j ) ⊗ b ′′ ij } + H ⊗ qk = ⌊| b ′′• | / ⌋ X l =0 · · · ⌊| b ′′• d | / ⌋ X l d =0 d Y s =1 | b ′′• s | !2 l s ( | b ′′• s | − l s )! l s ! × E "* u k , h ⊗ a ⊗ · · · ⊗ h ⊗ a m m q k O i =1 d O j =1 { ( D i F j ) ⊗ b ′ ij ⊗ ( D i S j ) ⊗ b ′′ ij } + H ⊗ qk × d Y s =1 S | b ′′• s |− l s ∂ ( β ( qk ) ; l ,...,l d ) ⋆ ϕ x k ( X ; √ tF + √ − tS · η ) . Then, estimating the term ∂ ( β ( qk ) ; l ,...,l d ) ⋆ ϕ x k ( X ; √ tF + √ − tS · η ) by k ∂ ( β ( qk ) ; l ,...,l d ) ⋆ ϕ x k k ∞ , which does not depend on t , and using the equation Z √ t t | b ′ | / (1 − t ) | b ′′ | / dt = B ( | b ′ | / / , | b ′′ | / , we obtain the desired estimate. (cid:3) UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE
6. Application to weighted quadratic variations.
In this section, we ap-ply the previous results to the case of weighted quadratic variations of frac-tional Brownian motion. Let us introduce first some notation.Given a measurable function f : R → R , an integer N ≥ p ≥ k f k N,p = N X i =0 sup ≤ t ≤ k f ( i ) k L p ( R ,γ t ) , (6.1)where γ t is the normal distribution N (0 , t ).We say that a function f : R → R has moderate growth if there exist posi-tive constants A , B and α < x ∈ R , | f ( x ) | ≤ A exp( B | x | α ).Notice that the seminorm (6.1) is finite if f and all its derivatives up to theorder N have moderate growth.Consider a fractional Brownian motion B = { B t : t ∈ [0 , } with Hurstparameter H ∈ (0 , B is a zero mean Gaussian process with co-variance E ( B t B s ) = ( t H + s H − | t − s | H ). The process B can be extendedto an isonormal Gaussian process indexed by the Hilbert space H , which isthe closure of the set of simple functions on [0 ,
1] with respect to the innerproduct h [0 ,t ] , [0 ,s ] i H = E ( B t B s ). We refer the reader to the basic references[16, 24] for a detailed account on this process. We denote by ρ H ( k ) = ( | k + 1 | H + | k − | H − | k | H ) , k ∈ Z , (6.2)the covariance function of the stationary sequence { B ( k + 1) − B ( k ) : k ≥ } .We consider the uniform partition of the interval [0 , n ≥ k = 0 , . . . , n − B k/n = B ( k +1) /n − B k/n , δ k/n = [ k/n, ( k +1) /n ] and ε k,n = [0 ,k/n ] . We will also make use of the notation β j,k = h δ j/n , δ k/n i H and α j,t = h δ j/n , [0 ,t ] i H , for any t ∈ [0 ,
1] and j, k = 0 , . . . , n − f : R → R , we define u n = n H − / n − X k =0 f ( B k/n ) δ ⊗ k/n . We are interested in the asymptotic behavior of the weighted quadraticfunctionals F n = n H − / n − X k =0 f ( B k/n )[(∆ B k/n ) − n − H ](6.3) = n H − / n − X k =0 f ( B k/n ) I ( δ ⊗ k/n ) . It is known (see, e.g., [15, 17, 18]) that for < H < , F n converges inlaw to a mixture of Gaussian distributions. When the Hurst parameter H I. NOURDIN, D. NUALART AND G. PECCATI is not in this range, a different phenomenon occurs, as it was observed byNourdin in [15]. More precisely, for
H < , n H − / F n converges in L (Ω)to R f ′′ ( B s ) ds , whereas for H > , n / − H F n converges in L (Ω) to R f ( B s ) dZ s , where Z is the Rosenblatt process (see [15, 18]). In the criticalcase H = , there is convergence in law to a linear combination of the limitsin the cases H < and < H < , and in the critical case H = there isconvergence in law with an additional logarithmic factor (see [15, 18]).In view of these results, we will focus on the case < H < , although ourresult could easily be extended to the limit case H = . Outside the interval[ , ] the convergence is in L (Ω) and our methodology does not seem to bewell suited to study the rate of convergence. Applying the general approachdeveloped in previous sections, we are able to show the following rate ofconvergence in the asymptotic behavior of F n , in the case H ∈ ( , ). Thisrepresents a quantitative version of the convergence in law proved in [18]. Proposition 6.1.
Assume that the Hurst index H of B belongs to ( , ) . Consider a function f : R → R of class C such that f and its first derivatives have moderate growth. Suppose in addition that E [( R f ( B s ) ds ) − α ] < ∞ for some α > . Consider the sequence of ran-dom variables F n defined by (6.3). Set S = q σ H R f ( B s ) ds , with σ H = P ∞ k = −∞ ρ H ( k ) , where ρ H is defined in (6.2). Then, for any function ϕ : R → R of class C with k ϕ ( k ) k ∞ < ∞ for any k = 0 , . . . , we have | E [ ϕ ( F n )] − E [ ϕ ( Sη )] | ≤ C f,H max ≤ i ≤ k ϕ ( i ) k ∞ n − ( | H − / |∧| H − / | ) , (6.4) where η is a standard normal variable independent of B . The constant C f,H has the form C f,H = C H max(1 , k f k , , (1 + | E [ S − α ] | /α k f k , β )) , where C H depends on H and α + β = 1 . Proof.
Along the proof C will denote a generic constant that mightdepend on H .Notice first that the random variable F n does not coincide with δ ( u n ),except in the case H = . For this reason, we define G n = δ ( u n ), and showthe following estimate for the difference F n − G n : E [ | F n − G n | ] ≤ C k f k , n − ( | H − / |∧| H − / | ) . (6.5)To show (6.5), we first apply Lemma 2.1 and we obtain F n − G n = n H − / n − X k =0 δ ( f ′ ( B k/n ) δ k/n ) α k,k/n + n H − / n − X k =0 f ′′ ( B k/n ) α k,k/n . UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE Using the equality δ ( f ′ ( B k/n ) δ k/n ) = f ′ ( B k/n ) I ( δ k/n ) − f ′′ ( B k/n ) α k,k/n , yields F n − G n = 2 n H − / n − X k =0 f ′ ( B k/n ) I ( δ k/n ) α k,k/n − n H − / n − X k =0 f ′′ ( B k/n ) α k,k/n := 2 M n − R n . Point (a) of Lemma A.1 implies | α k,k/n | ≤ n − (2 H ) ∧ and we can write E [ | R n | ] ≤ k f k , n / H − (4 H ∧ . (6.6)On the other hand, E [ M n ] = n H − n − X j,k =0 E [ f ′ ( B j/n ) f ′ ( B k/n ) I ( δ j/n ) I ( δ k/n )] α j,j/n α k,k/n , and using the relation I ( δ j/n ) I ( δ k/n ) = I ( δ j/n e ⊗ δ k/n ) + h δ j/n , δ k/n i H the duality relationship (2.5) yields E [ M n ] ≤ k f k , n H − n − X j,k =0 [ | β j,k | + | α j,j/n α k,k/n | + | α j,k/n α k,j/n | ] × | α j,j/n α k,k/n | . Finally, applying points (a) and (c) of Lemma A.1, we obtain, E [ M n ] ≤ C k f k , n H − ( n (1 − H ) ∨ + n − (4 H ∧ ) n − (4 H ∧ . (6.7)If H < , we obtain a rate of the form n − H and if H ≥ we obtain thebound n H − . Then the estimates (6.6) and (6.7) imply (6.5).Taking into account the estimate (6.5), the estimate (6.4) will follow from(5.3), provided we show the following inequalities for some constant C de-pending on H and for any β > E ( |h u n , D G n i H ⊗ − S | ) ≤ C k f k , n − ( | H − / |∧| H − / | ) , (6.8) E ( |h u n , DG ⊗ n i H ⊗ | ) ≤ C k f k , n − ( | H − / |∧| H − / | ) , (6.9) kh u n , D ( S ) ⊗ i H ⊗ k L β (Ω) ≤ C k f k , β n − ( | H − / |∧| H − / | ) , (6.10) E ( |h u n , D ( S ) i H ⊗ | ) ≤ C k f k , n − ( | H − / |∧| H − / | ) , (6.11) E ( |h u n , DG n ⊗ D ( S ) i H ⊗ | ) ≤ C k f k , n − ( | H − / |∧| H − / | ) . (6.12) I. NOURDIN, D. NUALART AND G. PECCATI
The derivatives S are given by the following expressions: D ( S ) = 2 σ H Z ( f f ′ )( B s ) [0 ,s ] ds,D ( S ) = 2 σ H Z ( f ′ + f f ′′ )( B s ) [0 ,s ] ds. On the other hand, applying formula (2.10) we obtain the following expres-sions for the derivatives of G n DG n = δ ( u n ) + δ ( Du n ) ,D G n = u n + 2 δ ( Du n ) + δ ( D u n ) . We are now ready to prove (6.8)–(6.12). The proof will be based on theestimates obtained in Lemma A.2 of the Appendix. (cid:3)
Proof of (6.8).
We have |h u n , D G n i H ⊗ − S |≤ |k u n k H ⊗ − S | + 2 |h u n , δ ( Du n ) i H ⊗ | + |h u n , δ ( D u n ) i H ⊗ | =: | A n | + 2 | B n | + | C n | . To estimate E [ | A n | ], we write k u n k H ⊗ = n H − n − X j,k =0 f ( B j/n ) f ( B k/n ) β j,k = 1 n n − X j,k =0 f ( B j/n ) f ( B k/n ) ρ H ( k − j ) = 1 n n − X p = − n +1 ( n − ∧ ( n − − p ) X j =0 ∨− p f ( B j/n ) f ( B ( j + p ) /n ) ρ H ( p ) . If we replace f ( B ( j + p ) /n ) by f ( B j/n ) we make an error in expectation of( p/n ) H , so this produces a total error of n − H . On the other hand, the se-quence P | p | >n ρ H ( p ) converges to zero at the rate n H − . As a consequence, E [ | A n | ] ≤ C ( k f k , n − H + k f k , n H − )+ σ H E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X k =0 f ( B k/n ) − Z f ( B s ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It remains to estimate1 n n − X k =0 f ( B k/n ) − Z f ( B s ) ds = n − X k =0 Z ( k +1) /nk/n [ f ( B k/n ) − f ( B s )] ds. UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE Using that E [ | f ( B k/n ) − f ( B s ) | ] ≤ C k f k , n − H for s ∈ [ k/n, ( k + 1) /n ], weobtain: E [ | A n | ] ≤ C ( k f k , n − H + k f k , n H − ) . (6.13)For the term B n we can write, using (A.2) and Meyer’s inequalities: E [ | B n | ] ≤ n H − / n − X k =0 E [ | f ( B k/n ) δ ( D k/n ( u n ⊗ δ k/n )) | ](6.14) ≤ C k f k , n H − (3 H ∧ / . The term C n is handled in the same way, by using Meyer’s inequalities andpoint (d) of Lemma A.1: E [ | C n | ] ≤ n H − / n − X k =0 E [ | f ( B k/n ) δ ( D k/n,k/n u n ) | ](6.15) ≤ Cn H − / k f k , n − X j,k =0 β j,k ≤ Cn / − H k f k , . Then (6.8) follows from (6.13), (6.14) and (6.15). (cid:3)
Proof of (6.9).
We have h u n , DG ⊗ n i H ⊗ = h u n , δ ( u n ) ⊗ δ ( u n ) i H ⊗ + 2 h u n , δ ( u n ) ⊗ δ ( Du n ) i H ⊗ + h u n , δ ( Du n ) ⊗ δ ( Du n ) i H ⊗ =: A n + 2 B n + C n . For the term A n we have, applying H¨older’s and Meyer’s inequalities andthe estimate (A.1), E [ | A n | ] ≤ n H − / n − X k =0 E [ | f ( B k/n )( δ ( u n ⊗ δ k/n )) | ] ≤ C k f k , n H − / − (2 H ∧ . Similarly, using H¨older’s and Meyer’s inequalities and the estimates (A.1)and (A.3) yields E [ | B n | ] ≤ n H − / n − X k =0 E [ | f ( B k/n ) δ ( u n ⊗ δ k/n ) δ ( D k/n u n ) | ] ≤ C k f k , n H − (3 H ∧ / . I. NOURDIN, D. NUALART AND G. PECCATI
Finally, using again H¨older’s and Meyer’s inequalities and the estimate (A.1)yields E [ | C n | ] ≤ n H − / n − X k =0 E [ | f ( B k/n )( δ ( D k/n u n )) | ] ≤ C k f k , n H − / − (2 H ∧ . (cid:3) Proof of (6.10).
We have h u n , D ( S ) ⊗ i H ⊗ = 16 n H − / n − X k =0 f ( B k/n ) Z Z ( f f ′ )( B s )( f f ′ )( B t ) α k,t α k,s ds dt. Then, we can write, using points (a) and (b) of Lemma A.1, E [ |h u n , D ( S ) ⊗ i H ⊗ | ] ≤ C k f k , β n H − / sup s,t ∈ [0 , n − X k =0 | α k,t α k,s |≤ C k f k , β n H − / − (2 H ∧ , for any β ≥ (cid:3) Proof of (6.11).
We have h u n , D ( S ) i H ⊗ = 4 n H − / n − X k =0 f ( B k/n ) Z ( f ′ + f f ′′ )( B s ) α k,t ds. As a consequence, applying points (a) and (b) of Lemma A.1 yields E [ |h u n , D ( S ) i H ⊗ | ] ≤ C k f k , n H − / sup s ∈ [0 , n − X k =0 α k,s ≤ C k f k , n H − / − (2 H ∧ . (cid:3) Proof of (6.12).
We have h u n , DG n ⊗ D ( S ) i H ⊗ = h u n , δ ( u n ) ⊗ D ( S ) i H ⊗ + h u n , δ ( Du n ) ⊗ D ( S ) i H ⊗ =: A n + B n . For the term A n we can write, applying H¨older’s and Meyer’s inequalitiesand the estimate (A.1), E [ | A n | ] ≤ n H − / n − X k =0 E [ | f ( B k/n ) δ ( u n ⊗ δ k/n ) D k/n ( S ) | ] ≤ C k f k , n H − / − (2 H ∧ . UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE For the term A n we can write, applying H¨older’s and Meyer’s inequalitiesand the estimate (A.1), E [ | A n | ] ≤ n H − / n − X k =0 E [ | f ( B k/n ) δ ( D k/n u n ) D k/n ( S ) | ] ≤ C k f k , n H +1 / − (4 H ∧ . This completes the proof of Proposition 6.1. (cid:3)
Remark 6.2.
Note that the exponent in the rate δ = − ( | H − | ∧| H − | ) is minimum when H = with δ = − . On the other hand, itbecomes worst when H goes away from either from below or from above,and it converges to zero as H tends to or . This is natural in viewof the limit results for the weighted quadratic variations obtained in [15,18]. This phenomenon has not been observed in other asymptotic problems,such as the rate of convergence for Euler-type numerical approximations ofstochastic differential equations, where the rate − (2 H − ) improves when H increases from up to (see [10]). Remark 6.3.
In the case H = , the process B is a Brownian motion,and it has independent increments. As consequence β j,k = 0 for j = k . More-over, F n = G n . Therefore, the estimate (6.4) can be replaced by | E [ ϕ ( F n )] − E [ ϕ ( Sη )] | ≤ C f max ≤ i ≤ k ϕ ( i ) k ∞ n − / , where S = 2 R f ( B s ) ds . Remark 6.4.
The extension to weighted power variations of any orderor to Euler numerical schemes for stochastic differential equations drivenby a fractional Brownian motion seems more involved. In the case of Eulernumerical schemes, the results that could be obtained applying the method-ology developed in this paper would lead to a precise analysis of the rateof convergence of the error to a particular distribution, which is usually amixture of Gaussian laws. That is, we would be able to establish how close isthe error to a limit distribution in terms of a distance between probabilitiesdefined by means of regular functions.APPENDIXIn this section, we will show two technical lemmas that play a fundamentalrole in the analysis of the asymptotic quadratic variation of the fractionalBrownian motion. The notation in both lemmas is taken from Section 6. I. NOURDIN, D. NUALART AND G. PECCATI
Lemma A.1.
Let < H < and n ≥ . We have, for some constant C H : (a) | α k,t | ≤ n − (2 H ∧ for any t ∈ [0 , and k = 0 , . . . , n − . (b) sup t ∈ [0 , P n − k =0 | α k,t | ≤ C H . (c) P n − k,j =0 | β j,k | ≤ C H n (1 − H ) ∨ . (d) If H < , then P n − k,j =0 β j,k ≤ C H n − H . (e) P n − k,j =0 | β k,l β j,l | ≤ C H n − (4 H ∧ for any l = 0 , . . . , n − . (f) If H < , then P n − k,j =0 | β k,l , β j,l β j,k | ≤ C H n − H − (2 H ∧ for any l =0 , . . . , n − . Proof.
Parts (a), (c) and (d) are contained in Lemmas 5 and 6 of [18].Part (b) has been proved in Lemma 5.1 of [17] in the case
H < and theproof actually works for any H ∈ (0 , n − X k,j =0 | β k,l β j,l | = 14 n − H n − X k,j =0 | ρ H ( k − l ) ρ H ( j − l ) | , and the fact that the series P p ∈ Z | ρ H ( p ) | is convergent if 0 < H ≤ and itdiverges at the rate n H − if H > . Finally, to prove (f) we write, usingYoung’s inequality, n − X k,j =0 | β k,l β j,l β j,k | = 18 n − H n − X k,j =0 | ρ H ( k − l ) ρ H ( j − l ) ρ H ( j − k ) |≤ n − H (cid:18)X p ∈ Z ρ H ( p ) (cid:19) n X p = − n | ρ H ( p ) | ! ≤ C H n − H − (2 H ∧ , where we have exploited the fact that P p ∈ Z ρ H ( p ) is convergent (because H < ), together with the asymptotic behavior of the mapping n P np = − n | ρ H ( p ) | . (cid:3) The next lemma provides some technical estimates.
Lemma A.2.
For any integer M ≥ and any real number p > , thereexists a constant C depending on M, p and the Hurst parameter H suchthat: k u n ⊗ δ k/n k M,p ≤ C k f k M,p n − / − ( H ∧ / , (A.1) k D k/n ( u n ⊗ δ k/n ) k M,p ≤ C k f k M +1 ,p n − / − (3 H ∧ / , (A.2) k D k/n u n k M,p ≤ C k f k M +1 ,p n − (2 H ∧ , (A.3) UANTITATIVE STABLE LIMIT THEOREMS ON THE WIENER SPACE where D k/n F means h DF, δ k/n i H , for a given random variable F . Proof.
In order to show the first estimate, we can write, for any integer0 ≤ m ≤ M , D m ( u n ⊗ δ k/n ) = n H − / n − X j =0 f ( m ) ( B j/n ) β k,j δ j/n e ⊗ ε ⊗ mj/n . Then, using points (a), (e) and (f) of Lemma A.1 we obtain( E [ k D m ( u n ⊗ δ k/n ) k p H ⊗ ( m +1) ]) /p ≤ Cn H − / k f k m,p × n − X j,j ′ =0 | β k,j β k,j ′ h δ j/n e ⊗ ε ⊗ mj/n , δ j ′ /n e ⊗ ε ⊗ mj ′ /n i H ⊗ ( m +1) | ! / ≤ C k f k m,p n H − / n − X j,j ′ =0 | β k,j β k,j ′ | ( | β j,j ′ | + | α j,j ′ /n α j ′ ,j/n | ) ! / ≤ C k f k m,p n H − / ( n − H − ( H ∧ / + n − (4 H ∧ ) ≤ C k f k m,p n − / − ( H ∧ / , which shows (A.1).To show the second estimate, we can write, for any integer 0 ≤ m ≤ M , D m D k/n ( u n ⊗ δ k/n ) = n H − / n − X j =0 f ( m +1) ( B j/n ) β k,j α k,j/n δ j/n e ⊗ ε ⊗ mj/n . Then, using points (a), (e) and (f) of Lemma A.1 we obtain( E [ k D m D k/n ( u n ⊗ δ k/n ) k p H ⊗ ( m +1) ]) /p ≤ Cn H − / k f k m +1 ,p × n − X j,j ′ =0 | β k,j α k,j/n β k,j ′ α k,j ′ /n h δ j/n e ⊗ ε ⊗ mj/n , δ j ′ /n e ⊗ ε ⊗ mj ′ /n i H ⊗ ( m +1) | ! / ≤ Cn H − / − (2 H ∧ k f k m +1 ,p × n − X j,j ′ =0 | β k,j β k,j ′ | ( | β j,j ′ | + | α j,j ′ /n α j ′ ,j/n | ) ! / ≤ C k f k m +1 ,p n H − / − (2 H ∧ ( n − H − ( H ∧ / + n − (4 H ∧ ) I. NOURDIN, D. NUALART AND G. PECCATI ≤ C k f k m +1 ,p n − / − (3 H ∧ / , and (A.2) follows.Finally, for the estimate (A.3) we can write D m D k/n u n = n H − / n − X j =0 f ( m +1) ( B j/n ) α k,j/n δ ⊗ j/n e ⊗ ε ⊗ mj/n , which implies, using points (a), (c) and (d) of Lemma A.1,( E [ k D m D k/n u n k p H ⊗ ( m +2) ]) /p ≤ Cn H − / k f k m +1 ,p × n − X j,j ′ =0 | α k,j/n α k,j ′ /n h δ ⊗ j/n e ⊗ ε ⊗ mj/n , δ ⊗ j ′ /n e ⊗ ε ⊗ mj ′ /n i H ⊗ ( m +2) | ! / ≤ Cn H − / − (2 H ∧ k f k m +1 ,p × n − X j,j ′ =0 ( β j,j ′ + | β j,j ′ α j,j ′ /n α j ′ ,j/n | + α j,j ′ /n α j ′ ,j/n ) ! / ≤ C k f k m +1 ,p n H − / − (2 H ∧ × ( n / − H + n [(1 / − H ) ∨ − (2 H ∧ + n − (4 H ∧ ) . This shows (A.3) and the proof of the lemma is complete. (cid:3)
Acknowledgments.
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I. NourdinG. PeccatiFSTC—UR en Math´ematiquesUniversit´e du Luxembourg6 rue Richard Coudenhove-KalergiLuxembourg City 1359LuxembourgE-mail: [email protected]@gmail.com