Functional limit theorems for the fractional Ornstein-Uhlenbeck process

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the joint convergence to a limit with both Gaussian and non-Gaussian components. This is valid for any $L^2$ functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the H\"older topology. As an application we prove a `rough creation' result, i.e. the weak convergence of a family of random smooth curves to a non-Markovian random process with rough sample paths. This includes the second order problem and the kinetic fractional Brownian motion model.


Introduction
The functional limit theorem we study here lays the foundation for the fluctuation problem for a slow/fast system with the fast variable given by a family of non-strong mixing stochastic processes, this will be discussed in [GL ], see [GL ] for the preliminary version. A pivot theorem for obtaining effective dynamics for the slow moving particles in a fast turbulent environment are scaling limit theorems for the convergence of the following functionals X ε := X 1,ε , . . . , X N,ε , with weak convergence in C α ([0, T ], R), where T is some finite fixed time horizon, α(ε) a suitable scaling and G k : R → R. If y ε t = y t ε and y t is a strong mixing process, α(ε) = 1 √ ε and the limit is a Markov process, for details see e.g. the book [KLO ] and the references therein. For stochastic processes whose auto-correlation function does not decay sufficiently fast at infinity there is no reason to have the √ ε scaling or to obtain a diffusive limit. Furthermore, the scaling limit and the limit function may depend on the individual functions G k .
In this article we take y ε t to be the stationary and rescaled fractional Ornstein-Uhlenbeck process with Hurst parameter H, which, for H > 1 2 , exhibits long range dependence (LRD) and is not strong-mixing. Our interest for long range dependent / non-strong mixing noise comes the time series data of the river Nile. In a study of water flows of the Nile river, Hurst and his colleagues [HBS] observed long range time dependence and found that the time dependence varies proportionally to t H where H ∼ 0.73, by contrast, Brownian motions and stable processes have independent increments. Fractional Brownian motions (fBM) were then proposed by Benoit Mandelbrot and John Van Ness [MVN ] for modelling the Hurst phenomenon. A fBM is a continuous mean zero Gaussian process with stationary increments and covariance E(B t −B s ) 2 = |t−s| 2H , it is self-similar with similarity exponent H, and distinguished by the Gaussian property and stationary, but dependent increments. See e.g. [Mis ] and the reference therein for stochastic calculus for fBM's.
Self-similar processes appeared also in mathematically rigorous descriptions of critical phenomena and in renormalisation theory. In [Sin ], Sinai constructed non-Gaussian self-similar fields; while Dobrushin [Dob ] studied self-similar fields subordinated to self-similar Gaussian fields (multiple Wiener integrals). Those self-similar stochastic processes with stationary increments are a particular interesting class. When normalized to begin at 0, to have mean 0 and to have variance 1 at t = 1, they necessarily have the covariance 1 2 (t 2H + s 2H − |t − s| 2H ). Those of Gaussian variety are fBMs. Hermite processes are non-Gaussian self-similar processes with stationary increments and the above mentioned covariance. They appeared as scaling limits of functionals of long range dependent Gaussian processes, see [Ros ]. Jona-Lasinio was also concerned with the construction of a systematic theory of limit distributions for sums of 'strongly dependent' random variables for which the classical central limit theorems do not hold, [JL ], see also the book [EM ].
Let us first consider the convergence of one single component, the scalar case. The scaling constant α k (ε) depends on the function G k , and is a reflection of the self-similarity exponents of the limiting process. If α k (ε) = 1 √ ε , the limit of X k,ε is a Wiener process and the functional central limit theorem is expected to hold. Let µ denote the centred and normalized Gaussian measure on R and m k denote the Hermite rank of a centred function G k ∈ L 2 (µ), which is the smallest non zero term in its chaos expansion. Let (H, m) → H * (m) denote the function given by ( . ), which decreases with m. Then, the relevant scaling constants are given as below: See Lemma . for a preliminary computation indicating the scales. In the past the limit theorems for the non-Gaussian limits had been called non-central limit theorems, we use the terminology 'functional central limit theorems' for all cases.
The intuition for this comes from its counter part for sequences. If Y n is a mean zero, stationary, and strong mixing sequence, such that σ 2 . If Y n is not strong mixing, this CLT may fail. Indeed, if X n is a stationary mean zero variance 1 Gaussian sequence with auto-correlation r(n) ∼ cn −γ for some γ ∈ (0, 1) and c ∈ R (allowing for negative correlations), G a centred function with Hermite rank m ≥ 1, and A(n) a sequence such that k=1 G(X n ) is expected to converge in finite dimensional distributions. The scaling constant A(n) is of the order n 1− 1 2 γm in the long range dependent case, of order √ n in the short range dependent case, and of order √ n ln n for the borderline case, see [BM , BT ]. By long range dependence, we mean ∞ n=1 |r(n)| = +∞. The limit process, lim n→∞ z n , is a Wiener process for fast decaying correlations, i.e. in case γ ∈ ( 1 m , 1), [BM ]. In the borderline case, γ = 1 m , the scaling limit is also a Wiener process. However if γ ∈ (0, 1 m ) the correlations fail to decay sufficiently fast, the scaling limit is a Hermite process in the m-th chaos, [Dob , BT ]. The first convergence to a non-Gaussian similar process was shown in [Ros ] where the aim was to construct a not strong mixing sequence of random variables, he achieved this by showing the sequence of random variables has a non-Gaussian scaling limit which is now known as the Rosenblatt process. In [BT ] vector valued combinations of short and long range dependent sums were studied, however, the limit of each component is assumed to be moment determinate (which can only happen when they are in the L 2 chaos expansion of order less or equal to 2). This is due to a restriction in the asymptotic independence result in [NR ], which was extended in [NNP ].
We return to the continuous functional limit theorems. For the scalar case, the continuous version CLT for γ ∈ ( 1 m , 1) was obtained in [BH ], the borderline case γ = 1 m in [BC ]. These are shown for the convergence in finite dimensional distribution and for G to be a centred L 2 function. They also obtained uniform convergences in the continuous topology for a restrictive class of functions G (assuming sufficiently fast decay of the coefficients in the Wiener chaos expansion). This was extended in [NNZ ] to vector valued X ε , when each component of X ε falls in the Brownian case, with convergence understood in the sense of finite dimensional distributions. The result in [NNZ ] was improved in [CNN ], where the fast chaos decay restrictions on G k , for G k ∈ L p for p > 2, are removed with techniques from Malliavin calculus. In the continuous long range dependent case Taqqu, [Taq ], obtained convergence in the continuous topology. These results, although fragmented (in some regimes these are only known for scalar valued processes or only at the level of sequences), provide a fairly good picture of what is going on.
There exists however no vector valued functional limit theorem with joint convergence, when the scaling limit of the components are mixed, in this article we provide a complete description for the joint convergence of {X k,ε } for G k ∈ L 2 (µ). We have a functional limit theorem for vector valued processes whose components may exhibit both short and long range dependence. For G k satisfying a stronger integrability condition, we can also show weak convergence in the C α ([0, T ], R d )-topology and for each fixed time in L 2 for the low Hermite rank case, which already have interesting applications. Furthermore, they are the basis for the convergence in a suitable rough topology, which due to the change of the nature of the problem will appear in [GL ] where rough path theory is used to study slow/fast systems, leading to 'rough creation' / 'rough homogenization' in which the effective limit is not necessarily a Markov process.
Application. Consider the second order equation on R: Taking ε → 0, does x ε t converge? In case H = 1 2 and f = 1, this is essentially the Kramer-Smolouchowski limit (this is also called the kinetic Brownian motion model). For H = 1 2 and for f = 1 this was shown in [BT , Zha , ATH ] to converge to a fBM, see also [FGL ] for the case with a magnetic field. Given H > 1 3 and f ∈ C 3 b (for any H if f = 1), we can show x ε t converges to the solution of the equatioṅ x t = f (x t ) dB H t with initial value x 0 where the integral in the differential equation is interpreted as a Riemann-Stieltjes integral. Furthermore we obtain the following bound in C γ ′ where 0 < γ ′ < γ < H: This computation is straightforward, see Propositions . and . for detail.
With the functional limit theorem below, Theorem A, we can conclude also the convergence of solutions of the equations, for h ∈ C 2 b (R d , R d ) and g ∈ C b (R, R), either to the solution of the equation where Z H * (m),m t is a Hermite process, or to the solution to the Stratonovich stochastic differential equation where W t is a standard Wiener process (given enough integrability on G). Here c is a specific constant (c.f. equation ( . ) depending on G arising from the homogenization procedure. For the above we follow [CNN ] and use Malliavin calculus to obtain suitable moment bounds on t 0 G(y ε s )ds. These results appeared in the previous version of the current paper [GL ]. Equations driven by fractional Brownian motions are also studied for the averaging regime, see [HL ] and [FK ]. A fluctuation theorem around the effective average was obtained [BGS ].
Main Results We denote our underlying probability space by (Ω, F , P). Let µ denote the standard Gaussian distribution and we choose σ such that the stationary scaled fOU process, to be defined below, satisfies y ε t ∼ µ. Let {H m , m ≥ 0} be the orthogonal Hermite polynomials on L 2 (µ), such that they have leading coefficient 1 and L 2 (µ) norm √ m!. Given G ∈ L 2 (µ), then it posses an expansion of the form A function G is centred if and only if c 0 = 0. The smallest m with c m = 0 is called the Hermite rank of G. In case the correlations of y ε t do not decay sufficiently fast the path integral α(ε) t 0 G(y ε s )ds ought to be approximated by that of the first term of its Wiener chaos expansion. By orthogonality of the H m 's it is sufficient to study the asymptotics of α(ε) t 0 H m (y ε s )ds to deduce α(ε). Although the solutions to the fOU equation converge exponentially fast to each other, their autocorrelation function decays only algebraically. The indicator for the behaviour of α(ε) t 0 H m (y ε s )ds turns out to be and the self-similarity exponent of the limiting process is determined by α(ε, H * (m)). For large m, the limit will be a Wiener process, and otherwise the limit Z t should have the scaling property: ε H * (m) Z t ε ∼ Z t . Indeed, Z t are the self-similar Hermite processes. To state the functional limit theorem concisely, we make the following convention, Convention . Given a collection of functions (G k ∈ L 2 (µ), k ≤ N ), we will label the high rank ones first, so the first n functions satisfy H * (m k ) ≤ 1 2 , where n ≥ 0, and the remaining satisfy H * (m k ) > 1 2 .
Theorem A Let y ε be the stationary solution to the scaled fractional Ornstein-Uhlenbeck equation ( . ) with standard Gaussian distribution µ. Let G k : R → R be centred functions in L 2 (µ) with Hermite ranks m k . Write Then, the following holds: . (a) There exist stochastic processes X W = (X 1 , . . . , X n ) and be the Hermite processes, represented by ( . ). Then, where, We emphasize that the Wiener process W t defining the Hermite processes is the same for every k (c.f. equation ( . )), which is in addition independent ofŴ t .
• High rank case.
Remark . The case H = 1 2 is classical, and is not of interest here. In this case the result is independent of the Hermite rank and the scaling is given by α(ε) = 1 √ ε , due to the exponential decay of correlations.
An immediate application is the following rough homogenisation theorem for a toy model: . Fix a finite time T and consideṙ .
to the solution to the Young differential equation We also take the liberty to point out an intermediate result on the joint convergence of stochastic processes in finite L 2 chaos, for it maybe of service. The proof is a slight modification of results in [UZ , NNP ].
, . . . , I qn (g ε n )), where I q denotes the Wiener integral of order q. Suppose that for every i, j, and any 1 ≤ r ≤ q i : where U and V are taken to be independent random variables.

Preliminaries
We take the Hermite polynomials of degree m to be

. Hermite processes
The rank 1 Hermite processes Z H,1 are fractional BMs, the formulation ( . ) below is exactly the Mandelbrot-Van Ness representation for a fBM.
Definition . Let m ∈ N withĤ(m) > 1 2 . The class of Hermite processes of rank m is given by the following mean-zero processes, where the constant K(H, m) is chosen so their variances are 1 at t = 1.
The integration in ( . ) is understood as a multiple Wiener-Itô integral (over the region R n without the diagonal). NoteĤ(1) = H. By the properties of Wiener integrals, two Hermite processes Z H,m and Z H ′ ,m ′ , defined by the same Wiener process, are uncorrelated if m = m ′ . The Hermite processes have stationary increments and finite moments of all orders with covariance .

Remark .
In some literature, see e.g. [MT ] where further details on Hermite processes can also be found, the Hermite processes are defined with a different exponent as below: The two notions are related by ( . ) We refer to [PT , Sam , CKM ] for detailed studies of fBM's which are used in this article.

. Fractional Ornstein-Uhlenbeck processes
Let us normalise the fractional Brownian motion, so that B H 0 = 0 and E(B H 1 ) 2 = 1. Disjoint increments of B H t have a covariance of the form: We define the stationary fractional Ornstein-Uhlenbeck processes to be y It is the solution of the following Langevin equation: We take y ε t , the fast or rescaled fOU, to be the stationary solution of Observe that y ε · and y · ε have the same distributions, In particular, both y t and y ε t are Hölder continuous with Hölder exponents γ ∈ (0, H). Let us denote their correlation functions by ̺ and ̺ ε : ̺(s, t) := E(y s y t ), ̺ ε (s, t) := E(y ε s y ε t ). Let ̺(s) = E(y 0 y s ) for s ≥ 0 and extended to R by symmetry, then ̺(s, t) = ̺(t − s) and similarly for ̺ ε . For H > 1 2 , the set of functions for which Wiener integrals are defined include L 2 functions and so ̺ posses an analytical expression. Indeed, since is integrable, and therefore we may use the Wiener isometry to compute the covariances For u > 0, we set Using this, the following correlation decay was shown in [CKM ].
Hence, ∞ 0 ̺ m (s)ds is finite if and only if H * (m) < 1 2 , or H = 1 2 and m ∈ N, as in the latter the usual OU process admits exponential decay of correlations.
Lemma . Let H ∈ (0, 1) \ { 1 2 }, fix a finite time horizon T , then for t ∈ [0, T ] the following holds uniformly for ε ∈ (0, 1 2 ]: ( . ) Note, if H = 1 2 , for and any m ∈ N, the bound is always t Proof. We first observe that By a change of variables and using estimate ( . ) on the decay of the auto correlation function, To complete the proof we observe that by a simple change of variables, Next, we recall the Garsia-Rodemich-Romsey-Kolmogorv inequality.

Lemma . [Garsia-Rodemich-Romsey-Kolmogorov inequality] Let
See [FV , section A. ] in the Appendix, as well as [SV ], for a proof. As a consequence of this inequality one obtains the following theorem.
Lemma . For any γ ∈ (0, H), p > 1, the following estimates hold: Proof. We use the fractional Ornstein-Uhlenbeck equation to obtain E|y s − y r | 2 (s − r) 2 E|y 1 | 2 + q|s − r| 2H . Using the stationarity of y t , one also has E|y s − y r | 2 ≤ 2E|y 1 | 2 = 2. Since for Gaussian random variables the L 2 norm controls the L p norm we have Thus, by symmetry and a change of variables,

Applications . The second order problem
If x is a stochastic process, we write x s,t = x t − x s .
Proof. Set v ε t = ε H−1 y ε t , then, v ε t solves the following equation Using the equation for v ε t we have Therefore, for any p > 1, In the last step we used the stationarity of y ε t . Thus, applying Kolmogorov's theorem . to X ε −σB H , we see that the following holds for any t ∈ [0, T ], p > 1 and any γ ′ < γ hence, the claim follows.
As an application we consider the following slow/fast system, To describe its limit, we review the concept of Young integrals. If f, g : [0, T ] → R with f ∈ C α and g ∈ C β such that α + β > 1, then the Riemann-Stieljes integral makes sense, and For details see [You ]; this integral is called a Young integral. Since Young integrals have the regularity of its integrand, for H > 1 2 the equationẋ t = f (x t ) dB H t makes sense. In [Lyo ], it was shown that if f ∈ C 3 b , then the equation has a unique global solution from each initial value. This type of equations are Young equation, the simplest type of rough differential equation. The notation C 3 b denotes the space of bounded functions such that their first three derivatives are bounded.

Proposition . Let
to the solution of the rough differential equation: Proof. The idea is to consider equation ( . ) as a Young/rough differential equation. In case H ∈ ( 1 2 , 1), we can rewrite our equation asẋ ε t = f (x ε t )dX ε t , where X ε = ε H−1 t 0 y ε s ds is as in the previous lemma. Young's continuity theorem, see Theorem . , states that x ε converges weakly provided X ε converges weakly in a Hölder space of regularity greater than 1 2 .
For H ∈ ( 1 3 , 1 2 ) we need to rewrite our equation into a rough differential equation, where X ε is given by X ε enhanced with its canonical lift As we restrict ourselves to one dimension we obtain, by symmetry, X ε s,t = 1 2 (X ε s,t ) 2 , hence, X ε converges to a fBm σB H enhanced with B H s,t = 1 2 σB H s,t 2 . As the solution map, Φ, to a RDE satisfies, see [FH ] or Theorem . below, where ̺ γ denotes the inhomogeneous rough path norm of regularity γ and B H = (σB H , 1 2 σB H s,t 2 ). Thus, the L p convergence of the solutions follows from the L p convergence of the drivers, hence, we can conclude the proof by Proposition . .

Remark .
. For H < 1 2 and m = 1, Theorem A appears to contradict with Proposition . ; in the first we claim the limit is a Brownian motion, whereas in the second we claim that it is a fractional Brownian motion. Both results are correct and can be easily explained. It lies in the fact that R ̺(s) ds vanishes if H < 1 2 , and so the Brownian motion limit is degenerate. Since according to [CKM ], and by the decay estimate from ( . ), ̺ is integrable, s(λ) is the value at zero of the inverse Fourier transform of ̺(s), which is up to a multiplicative constant |λ| 1−2H 1+λ 2 . This is also the spectral density of y t and has value 0 at 0. This means we have scaled too much and the correct scaling is to multiply the integral t 0 y ε s ds by ε H−1 in which case we obtain a fBm as limit. . For m > 1 and H < 1 2 the Wiener limit is not trivial. Indeed,

. The d fluctuation problem
In this section we give an application of Theorem A. Given a function g ∈ L 1 (µ) we denoteḡ = R g(y)µ(dy).

Proof.
A stationary Gaussian process is ergodic if its spectral measure has no atom, see [CFS , Sam ]. The spectral measure F of a stationary Gaussian process is obtained from Fourier transforming its correlation function and ̺(λ) = R e iλx dF (x). According to [CKM ]: so the spectral measure is absolutely continuous with respect to the Lebesgue measure with spectral density, up to a non-zero constant, given by s(x) = |x| 1−2H 1+x 2 . Since t 0 g(y ε t )dt equals ε t ε 0 g(y s )ds in law, the former converges in law to tḡ by Birkhoff's ergodic theorem. The claim now follows as weak convergence to a constant implies convergence in probability.
In the following proof we will need the following theorem from Young/rough path theory for details we refer to [FH , FV , Lyo , LCL ]. We denote the space of rough paths of regularity β by C β .
has a unique solution which belongs to C β∧γ . Furthermore, the solution map Φ f,h :

the first component is the initial condition and the second and third components the drivers, is continuous.
Given a centred function G ∈ L 2 (µ), with chaos expansion G = ∞ k=m c k H k and let c ≥ 0 be given by ( . ) , G ∈ C(R, R) satisfying Assumption . and g ∈ C b (R; R). Let α(ε) = α(ε, H * (m)). Fix a finite time T and consideṙ . If H * (m) > 1 2 , x ε t converges weakly in C γ ([0, T ], R d ) to the solution to the Young differential equation Proof. As in Proposition . we can rewrite our equations as Young/rough differential equations and therefore reduce our analysis to the drivers α(ε) t 0 G(y ε s )ds, t 0 g(y ε s )ds . By Theorem A, α(ε) t 0 G(y ε s )ds converges in finite dimensional distributions either to a Wiener or a Hermite process. By Lemma . , t 0 g(y ε s )ds converges in probability to the deterministic path tḡ. Hence, α(ε) t 0 G(y ε s )ds, t 0 g(y ε s )ds converges jointly in finite dimensional distributions. Furthermore, t 0 g(y ε s )ds ∞ ≤ t g ∞ , this combined with the moment bounds obtained in Theorem A enables us to apply Theorem . to conclude the proof.
Remark . The constant c could be 0, for further details see Remark . .

Proof of Theorem A
We first establish the L 2 (Ω) convergence of X ε t = α(ε) t 0 G(y ε s )ds, where G = ∞ k=m c k H k has low Hermite rank, followed by a reduction theorem. We then prove moment bounds and conclude the proof of Theorem A.

. Preliminary lemmas
The basic scalar valued functional limit theorem, for low rank Hermite functions, was proved in [Taq ] for ε H * (m) t ε 0 G(X s )ds with X t = R p(t − ξ)dW ξ a moving average, where in order to prove convergence one uses the self-similarity of a Wiener process leading to weak convergence as this equivalence of course is only in law. Nevertheless, in our case we can choose a properly scaled fast variable and write, y ε t = Rĝ ( t−ξ ε )dW ξ for a functionĝ, and thus avoid using self-similarity. The key idea is to write a Wiener integral representation beginning with and . This can be obtained by applying the integral representation for fBM's: and by repeated applications of integration by parts (to the Young integrals): One may also use the following, see [PT ], taking f ∈ L 1 ∩ L 2 : Proof. This can be shown by applying [Taq , Theorem . ], where weak convergence is obtained. With a small modification and using [Taq , Lemma . , Lemma . ] directly we obtain the L 2 (Ω) convergence: using the Wiener integral representation of the Hermite processes this is equivalent, by a multiple Wiener-Itô isometry, to Examining Taqqu's proof, we note that in fact the L 2 convergence of ( . ) is obtained under the following conditions.
Lemma . Let G ∈ L 2 (µ) be a centred function with Hermite rank m satisfying H * (m) > 1 2 . Let H ∈ ( 1 2 , 1). Then the following statements hold for the stationary scaled fOU process y ε s . Fix t > 0, then, Proof. For G = H m the claim has already been shown in Lemma . . To conclude the claim in the case of a general G, we compute, . This finishes the proof.
The fact that only the first term in the chaos expansion gives a contributions is in the literature often called a reduction lemma. In the high Hermite rank case however it is not possible to restrict one's analysis to a pure Hermite polynomial, but as the next lemma shows finite linear combinations are indeed sufficient. To make the application later on easier we directly prove it in the multi-dimensional case.

Lemma . (Reduction Lemma)
Fix H ∈ (0, 1) \ { 1 2 }. For M ∈ N, define the truncated functions: Proof. Firstly, Using properties of the Hermite polynomials we obtain Let {t γ k,l , k ≤ N, l ≤ A} be a sequence of positive numbers. Now, by the triangle inequality, With Theorem . in [Bil ] this proves the claim.

. Moment bounds
We will use some results from Malliavin Calculus. Let x s be a stationary Gaussian process with β(s) = E(x s x 0 ), such that β(0) = 1. As a real separable Hilbert space we use H = L 2 (R + , ν) where for a Borel-set Borel subsets of R + . If F denotes the σ-field generated by x, then any F -measurable L 2 (Ω) function F has the chaos expansion: . This is due to the fact that L 2 (Ω) = ∞ m=0 H m where H m is the closed linear space generated by {H m (W (h)) : h L 2 = 1}, H m are the m-th Hermite polynomials, and H m = I m (L 2 sym (R m + )). The last fact is due to H m (W (h)) = I m (h ⊗ m ). In the following D k,p (H ⊗m ) denotes the closure of Malliavin smooth random variables under the following Lemma . (Meyer's inequality) [NP ] Let δ denote the divergence operator. Then for u ∈ D k,p (H ⊗m ), ( . ) Lemma .
[CNN ] If G : R → R is a function of Hermite rank m, then G has the following multiple Wiener-Itô-integral representation: where G m has the following properties: In the lemma below we estimate the moments of t 0 G(x r ε )dr, where we need the multiple Wiener-Itô-integral representation above to transfer the correlation function to L 2 norms of indicator functions. We use an idea from [CNN ] for the estimates below.
Lemma . Let x t = W ([0, t]) be a stationary Gaussian process with correlation β(t) = E(x t x 0 ) , stationary distribution µ and H the L 2 space over R + with measure β(r)dr. If G is a function of Hermite rank m and G ∈ L p (µ), for p > 2, then, For the stationary scaled fractional OU process y ε t , we have in particular, Proof. We first use Lemma . and then apply Meyer's inequality from Lemma . to obtain .
Here for G m is as given in Lemma . and G (k) m denotes its k-th order Malliavin derivative. We estimate the individual terms using the linearity of the inner product and the isometry 1 [0,r] , Using Minkowski's inequality we obtain We then estimate E|G , concluding ( . ). We finally apply Lemma . to conclude ( . ). Now we are ready to prove our main theorem.
These results benefit from the insights of Üstünel-Zakai [UZ ] on the independence of two iterated integrals I p (f ) and I q (g). They are independent if and only if the -contraction between f and g vanishes almost surely with respect to the Lebesgue measure. An asymptotic independence result follows as below, Lemma . [NNP , Thm. . ] Let F ε = I p (f ε ) and G ε = I q (g ε ), where f ε ∈ L 2 (R p ) and g ε ∈ L 2 (R q ) . Then, Cov (F ε ) 2 , (G ε ) 2 → 0 is equivalent to f ε ⊗ r g ε → 0, for 1 ≤ r ≤ p ∧ q.
It is also known that if two integrals I p (f ) and I q (g) are independent, then their Malliavin derivatives are orthogonal, see [UZ ]. This explains why Malliavin calculus comes into prominent play, which has been developed to its perfection in [NNP , Lemma . ]. Given a smooth test function ϕ we define, where the sum runs over multi-indices k = (k 1 , . . . , k m ). Let L = −δD and throughout this section f i : R pi → R and g : R q → R denote symmetric functions.

Lemma .
[NNP ] Let q ≤ p i , g ∈ L 2 (R q ), G = I q (g), f i ∈ L 2 (R pi ), and F i = I pi (f i ) with E(F 2 i ) = 1. Set F = (F 1 , . . . , F m ) and let θ be a smooth test function. Then, where c is a constant depending on F L 2 , G L 2 , and q, m, p 1 , . . . , p m .
The final piece of the puzzle is the observation that the defect in being independent is quantitatively controlled by the covariance of the squares of the relative components. The following is from [NNP ], our only modification is to take G to be vector valued. Let g i : R qi → R be symmetric functions.
Lemma . Given F = (I p1 (f 1 ), . . . I pm (f m )) and G = (I q1 (g 1 ), . . . , I qn (g n )) such that p k ≥ q l for every pair of k, l. Then, for all test functions ϕ and ψ, the following holds for some constant c, depending on F L 2 , G L 2 , and m, n, p 1 , . . . , p m , q 1 , . . . , q n , Multiplying both sides by ψ(G), taking expectations and using integration by parts we obtain To conclude, apply to each summand Lemma . with θ = ∂ j ϕ and G = G i .
Proof. Just combine Lemma . and Lemma . .
Finally, we finish the proof of Proposition . .

Proof.
Since (F ε , G ε ) is bounded in L 2 (Ω) it is tight. Now choose a weakly converging subsequence (F n , G n ) with limit denoted by (X, Y ). Let ϕ and ψ be smooth test functions, then by Lemma . and the bounds on ϕ, ψ, we pass to the limit under the expectation sign and obtain E(ϕ(X)ψ(Y )) = E(ϕ(X))E(ψ(Y )).
Thus every limit measure is the product measure determined by U and V , hence, (F ε , G ε ) converges as claimed.