Asymptotic expansion of Skorohod integrals

Asymptotic expansion of the distribution of a perturbation $Z_n$ of a Skorohod integral jointly with a reference variable $X_n$ is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale expansion. The second-order interpolation and Fourier inversion give asymptotic expansion of the expectation $E[f(Z_n,X_n)]$ for differentiable functions $f$ and also measurable functions $f$. In the latter case, the interpolation method connects the two non-degeneracies of variables for finite $n$ and $\infty$. Random symbols are used for expressing the asymptotic expansion formula. Quasi tangent, quasi torsion and modified quasi torsion are introduced in this paper. We identify these random symbols for a certain quadratic form of a fractional Brownian motion and for a quadratic from of a fractional Brownian motion with random weights. For a quadratic form of a Brownian motion with random weights, we observe that our formula reproduces the formula originally obtained by the martingale expansion.


Introduction
Asymptotic expansion of distributions is one of the fundamentals of theoretical statistics. Its applications spread over higher order approximation of probability distributions, theory of higher order asymptotic efficiency of estimators, prediction, information criteria for model selection, saddle point approximation, bootstrap and resampling methods, information geometry and so on. Bhattacharya and Rao [1] give an excellent exposition of the probabilistic aspects of asymptotic expansion for independent variables, while we omit a huge amount of literature on statistical applications of asymptotic expansion methods. Asymptotic expansion has a long history even for dependent models. A celebrated paper Götze and Hipp [4] is a compilation of the studies of asymptotic expansion for Markovian or nearly Markovian chains with mixing property. It was followed by Götze and Hipp [5], that applied their result to time series models more explicitly.
For functionals in stochastic analysis, there are two ways: martingale approach and mixing approach. Yoshida [26,27] gave asymptotic expansions for martingales with the Malliavin calculus on Wiener/Wiener-Poisson space and applied them to an ergodic diffusion, a volatility estimation over a finite time interval in central limit case, and a stochastic regression model with an explanatory process having long memory. Regularity of the distribution is critical to validate an asymptotic expansion. Thus, naturally the Malliavin calculus was used there to ensure a decay rate of characteristic type functionals. In connection, though the regularity problem does not occur there, Mykland [11] is a pioneering work on expansion of moments for a smooth function of a martingale. The mixing approach is more efficient if a sufficiently fast mixing property is available. Kusuoka and Yoshida [9] and Yoshida [28] developed asymptotic expansions for ǫ-Markov processes possessing a mixing property. The Malliavin calculus was used to estimate certain conditional characteristic functionals defined locally in time with the assistance of support theorems. See e.g. Yoshida [31] for an overview and references therein.
In the last three decades, along with the developments in statistics for high frequency data, stable limit theorems have attracted a lot of attention. Estimation of volatility from high frequency data under finite time horizon typically becomes non-ergodic statistics. Then, the asymptotic expansion of functionals of increments of stochastic processes is once again an issue after recent tremendous progresses in limit theorems in this area. Even though big data is available, the problem of microstructure noise motivates the use of asymptotic expansion. For example, Yoshida [30] extended [26] to martingales with mixed Gaussian limit * , Podolskij and Yoshida [24] derived a distributional asymptotic expansion of the p-variation of a diffusion process and Podolskij, Veliyev and Yoshida [23] gave an Edgeworth expansion for the pre-averaging estimator for a diffusion process sampled under microstructure noise.
Beyond semimartingales theory, special attention has been focused in recent years on limit theorems for objects in the Malliavin calculus. Nualart and Peccati [19] established the fourth moment theorem and characterized the central limit theorem for a sequence of multiple stochastic integrals of a fixed order. Nualart and Oriz-Latorre [18] extended the result in Nualart and Peccati [19]. Peccati and Tudor [21] presented necessary and sufficient conditions for the central limit theorem for vectors of multiple stochastic integrals and showed that componentwise convergence implies joint convergence. Nourdin, Nualart and Peccati [13] introduced an interpolation technique and proved quantitative stable limit theorems where the limit distribution is a mixture of Gaussian distributions. Power variation, stable convergence and Berry-Esseen type inequality are also in the scope of this trend.
Nourdin and Peccati [15] showed the asymptotic behavior of a weighted power variation processes associated with the so-called iterated Brownian motion. Corcuera, Nualart and Woerner [2] gave a mixture type central limit theorem for the power variation of a stochastic integral with respect to a fractional Brownian motion. Nourdin, Nualart and Tudor [14] derived central and non-central limit theorems for certain weighted power variations of the fractional Brownian motion. Nourdin [12] showed various asymptotic behavior of weighted quadratic and cubic variations of a fractional Brownian motion having a small Hurst index.
By connecting a martingale approach and deforming a nesting condition, Peccati and Taqqu [20] showed stable convergence of multiple Wiener-Itô integrals. Nourdin and Nualart [14] proved a central limit theorem for a sequence of multiple Skorohod integrals and applied it to renormalized weighted Hermite variations of the fractional Brownian motion. Related to stable convergence are Harnett and Nualart [6,7] on weak convergence of the Stratonovich integral with respect to a class of Gaussian processes.
Based on Stein's method, among many others, Nourdin and Peccati [16] presented a Berry-Esseen bound for multiple Wiener-Itô integrals, and Edan and Víquez [3] obtained central limit theorems with Wiener-Poisson space. Kusuoka and Tudor [10] proposed Stein's method for invariant measures of diffusions. An advantage of Stein's method is that it provides fairly explicit error bounds of approximation. The interpolation method recently introduced by Nourdin, Nualart and Peccati [13] keeps this merit.
After observing these developments, the aim of this paper is to derive asymptotic expansions for Skorohod integrals by means of the Malliavin calculus. It is worth recalling the terminology in the martingale expansion of [30] though our discussion will be apart from the martingale theory. For a sequence of continuous martingales M n = {M n t , t ∈ [0, 1]}, denote by C n = M n the quadratic variation of M n . When C n := C n 1 → p C ∞ as n → ∞, stable convergence of M n := M n 1 to a mixed normal limit M ∞ ∼ N (0, C ∞ ) usually takes place even if C ∞ is random. More precise an evaluation of the gap C n −C ∞ is necessary to go up to an asymptotic expansion.

The variable
• Cn= r −1 n (C n − C ∞ ) is called tangent, where r n is a positive number tending to zero as n → ∞. The effect of • Cn appears in the first order asymptotic expansion, and it gives everything in the classical case of constant C ∞ . On the other hand, if C ∞ is random, as it is the case of non-ergodic statistics, then the exponential local martingale e n t (z) = exp izM n + 2 −1 z 2 C n t is no longer a local martingale under the transformed measure exp − 2 −1 z 2 C ∞ dP/E[exp − 2 −1 z 2 C ∞ ]. This effect remains in the asymptotic expansion, called the torsion the exponential martingale suffers from. Two random symbols σ and σ are defined for tangent and torsion, respectively, and the asymptotic expansion formula is given in terms of the Gaussian density φ(z; 0, C ∞ ) with variance C ∞ and the adjoint operation of σ and σ. In this article, we will make an expansion formula for the Skorohod integral M n = δ(u n ) of u n in a similar way through certain random symbols. However, since we do not have any self-evident martingale structure, we introduce new random symbols called quasi tangent, quasi torsion and modified quasi torsion defined only by Malliavin derivatives of functionals.
We will take a Fourier analytic approach. It is because the formula is a perturbation of a Gaussian density and the actions of random symbols are simply expressed, as it was the case in classical theory. Moreover, if we extend such a result, the limit is possibly related to infinitely divisible distributions even if their mixture appears, and then formulation by random symbols seems natural from an operational point of view. We use an interpolation method in the frequency domain and expand a characteristic function of the interpolation. The second-order interpolation is provided to relate the distribution of M n with the random symbols that determine the expansion formula.
In this paper, we combine the interpolation method and the scheme originating from martingale expansion. Non-degeneracy of distributions plays an essential role to validate the asymptotic expansion of the expectation E[f (M n )] for measurable functions f . For that, it is necessary to connect the two non-degeneracies of M n and M ∞ . The interpolation method serves as a homotopy between the two random functions, just as the interpolation along the real time t was used in the martingale expansion [30]. We require only a local nondegeneracy of M n , not the full non-degeneracy. There is a big difference between them. For the latter, we need large deviation estimates and that plot often fails in practice. In parametric estimation, we quite often meet a situation where the estimator is not defined on the whole Ω but defined locally as a smooth functional. Then localization is inevitable.
Finally, related to this article, we mention a recent work by Tudor and Yoshida [25] on asymptotic expansion of multiple stochastic integrals.
The organization of this paper is as follows. We will work with the variable Z n defined in Section 2 as a perturbation of a Skorohod integral M n = δ(u n ) since such a stochastic expansion appears when statistical estimators are considered. A reference variable X n is also considered. This formulation is natural because Studentization is common in non-ergodic statistics, and also because the principal part of the normalized estimator is often expressed as the ratio of a Skorohod integral and Fisher information. Section 2 introduces the interpolation method and an expansion of a characteristic type functional along the interpolation, as well as notion of quasi tangent, quasi tosion and modified quasi torsion. Section 3 gives asymptotic expansion of E[f (Z n , X n )] for differentiable functions f . We compute the random symbols for a functional of a fractional Brownian motion in Section 4. Since the Skorohod integral generalizes the Itô integral, our formula should reproduce the same formula as that of [30] if applied to the quadratic form of a Brownian motion with random weights. We will see this in Section 5 but the derivation is more complicated than the direct use of the martingale expansion for the double Itô integrals. In Section 6, the random symbols are computed for a quadratic form of a fractional Brownian motion with random weights. Finally, Section 7 validates asymptotic expansions of E[f (Z n , X n )] for measurable functions f . As mentioned above, we carry out this task by using two nondegeneracies of the Malliavin covariances, with the help of the interpolation.

Perturbation of a Skorohod integral
Given a probability space (Ω, F , P ), we consider an isonormal Gaussian process W = {W(h), h ∈ H} on a real separable Hilbert space H. For any Hilbert space E, any real number p ≥ 1 and any integer k ≥ 1, we denote by D k,p (E) the Sobolev space of E-valued random variables which are k times differentiable in the sense of Malliavin calculus and the derivatives up to order k have finite moments of order p. We denote by D the derivative operator in the framework of Malliavin calculus. Its adjoint, denoted by δ, is called the divergence or the Skorokod integral. We refer to Nualart [17] for a detailed account on Malliavin calculus. We simply write D s,p for D s,p (R). Moreover we write D s,∞ (E) = ∩ p≥1 D s,p (E).
Consider random vectors W n (n ∈ N = N ∪ {∞}) and N n : Ω → R d (n ∈ N). For a sequence of positive numbers (r n ) n∈N tending to 0 as n → ∞, we will consider a perturbation Z n of M n given by The random matrix G ∞ will be the asymptotic random variance matrix of Z n . A reference variable is denoted by X n : Ω → R d1 , n ∈ N. We are interested in a higher-order approximation of the joint distribution of (Z n , X n ).
Remark 2.2. We could start with the decomposition Z n = M n + W ∞ + r nÑn of Z n , by takingÑ n = • W n +N n . This decomposition would be expected to slightly simplify the presentation but the complexity would be the same because, as a matter of fact, • W n and N n will always be treated as a set like inǦ (1) n andĜ (1) n defined below.
Consider a sequence ψ n ∈ D 1,p1 (R), and for a while we suppose that u n ∈ D 2,p (H⊗R d ), In the special case ψ n ≡ 1, we let p 1 = ∞.
Remark 2.3. The variable G n is different from C n = M n T in the martingale expansion since, in general, That is, Cn have different limits, in general. However, the limit G ∞ of G n may coincide with the limit C ∞ of C n . In short, we may have G ∞ = C ∞ , however, in general, In particular, • Gn may converge to 0.
The random tensor is called the quasi tangent (q-tangent), and the random tensor , is called the quasi torsion (q-torsion). Moreover, we call the random tensor ⊗3 , the modified quasi torsion (modified q-torsion). Then The derivative of ϕ n (θ; ψ n ) is computed as follows Applying the duality relationship between the Skorohod integral δ and the derivative operator D (we also call this duality relationship integration by parts (IBP) formula), yields This expression can be written as Thus, we obtained the following lemma.
In order to establish a second-order interpolation formula we need to further expand the last three summands in the right-hand side of (2.6). To do this, we will denote by G n any one of the terms G (1) n (θ; z, x), G (2) n (z) and Then, by Lemma 2.4, we have ϕ n θ; G n ψ n − ϕ n 0; G n ψ n = θ 0 ∂ θ1 ϕ n θ 1 ; G n ψ n dθ 1 where Therefore, once again by Lemma 2.4, we obtain where By (2.6) and (2.7), we can write ∂ θ ϕ n (θ; ψ n ) = ϕ n θ; Dψ n , u n [iz] H +θ ϕ n 0; G (2) n (z)ψ n + θϕ n 0; D(G (2) n (z)ψ n ), u n [iz] H + R (2) n (θ; z, x, G (2) n (z)) Then, integrating ∂ θ ϕ n (θ; ψ n ) and using the expression for R (3) n (z, x) and the decomposition (2.3), yields Consequently, Using the definition of Ψ(z, x) given in (2.2) and the decomposition of G (1) n given in (2.3), we obtain Suppose that there are random symbols S (3,0) , S , i.e., they are polynomials in (iz, ix) with coefficients in L 1 (Ω). Let However this does not mean that In fact, S (3,0) is not necessarily of third order in z, as we will see in this paper.
From the above argument, we obtain the following second-order interpolation formula.
n (f ) are expressed as a sum of bounded signed measures applied to the derivatives of f , Equation (3.7) holds for all Define the following random symbols Remark 3.2. As mentioned in Remark 2.5, the order of the random symbol S (3,0) (iz, ix) appearing as the limit of the corresponding sequence S (3,0) n (iz, ix) does not necessarily coincide with that of the latter because It is also the case for other symbols.
(ii) The following estimates hold: We say that a family of random symbols if the degrees of polynomials are bounded and the family {c λ k,m ; λ ∈ Λ} of tensor-valued random variables is uniformly integrable for every (k, m). Recall that β = max{7, β 0 }, where β 0 is the degree in (z, x) of the random symbol S. 1,n ; n ∈ N} are u.i. from (3.8) and (3.16). We consider first the case of ρ (8) n (f ), given by

By [B]
(iii) and the formula (3.2), we obtain Due to the uniformly integrability of the family {S (1,1) n ; n ∈ N}, we can write if we choose χ satisfying χ = 1 on a sufficiently large compact set. Combining (3.21) and (3.22), we obtain since the functionals in | | are equi-continuous in f in C β (Rď) and B β+1 is relatively compact in B β if the domain is restricted to supp(χ). Therefore we showed that ρ Similarly, we obtain the same estimate for ρ (3.19), and sup f ∈B 2 |ρ (3.16) and (3.19).

A functional of a fractional Brownian motion
In this section, we shall consider a functional of a fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1) on the time interval [0, 1]. The fBm is a centered Gaussian process B = {B t , t ∈ [0, 1]} defined on a probability space (Ω, F , P ) with covariance function The process B is a standard Brownian motion for H = 1 2 . Denote by E the set of step functions on [0, 1]. Then it is possible to introduce an inner product in E such that Let · H = ·, · 1/2 H . Hilbert space H is defined as the closure of E with respect to · H . It is known that the mapping 1 [0,t] → B t can be extended to a linear isometry between H and the Gaussian space spanned by B in L 2 = L 2 (Ω, F , P ). We denote this isometry by φ → B(φ). The process {B(φ), φ ∈ H} is an isonormal Gaussian process. We refer to [17] for a detailed account on the basic properties of the fBm. Assume again that F is the σ-field generated by B.
In the case H > 1 2 , the space H contains the linear space |H| of all measurable functions ϕ : In this case, the inner product ϕ, φ H is represented by The following lemma provides useful formulas for the inner product in the Hilbert space H.
Then for any piecewise continuous function ϕ on [0, 1], Proof. Approximating ϕ by step functions we can derive (i). For (ii), using (i) we can write Simple calculus with (i) gives (iii).
. We will consider the sequence of random variables Z n = δ(u n ), n ≥ 1, where The following provides the convergence in law of the sequence Z n . For the Brownian motion case, this result goes back to Peccati and Yor [22]. For H > 1 2 , it was proved by Peccati and Tudor [20] and a rate of convergence in the total variation distance was established in [13]. For H < 1 2 , the convergence in law of Z n it is a consequence of our asymptotic expansion proved below.
In the setting of Section 2, the variables are now Z n = M n , W n = W ∞ = 0, X n = X ∞ = 0, ψ n = 1 and We are interested in investigating the asymptotic behavior of the three basic terms: modified quasi torsion, quasi tangent and quasi torsion. We denote by C H a generic constant depending on H, that may vary in different lines.
Consider first the case of the modified quasi torsion.
. Therefore, As a consequence, taking r n = n −1/2 , we obtain As a consequence, we obtain the following results: If H > 1 2 we take r n = n H−1 and If H < 1 2 we take r n = n −H and Notice that the limit is discontinuous at H = 1 2 . With there preliminaries, we can now proceed to deduce the asymptotic expansion for E[f (Z n )] without classification for H.

Brownian motion case
We will analyze the asymptotic behavior of the quasi tangent and the quasi torsion, which are the main ingredients in the asymptotic expansions.

Quasi tangent
Let us now establish the asymptotic behavior of the quasi tangent, defined by We have, for s ∈ [0, 1], Therefore, Then, The term Z n,1 belongs to the second Wiener chaos and it can be written as a double stochastic integral: It is not difficult to check that n 2 f n 2 H ⊗2 converges to a constant as n tends to infinity. Therefore, Z n,1 2 = O(n −1 ). Clearly, we also have Z n,2 2 = O(n −1 ). Finally, Z n,3 2 = O(n −3/2 ). Consequently, √ nG (2) n 2 = O(n −1/2 ), and hence the effect of the quasi tangent disappears in the limit, that is, S

Quasi torsion
Let us first recall the definition of the quasi torsion Since G (2) n is in the second chaos, it follows that √ n DG

Asymptotic expansion
From the computations in Sections 4.
Recall that in that case r n = n H−1 .

Quasi tangent
We are going to establish the convergence in law of the tangent and show that it does not contribute to the asymptotic expansion. We have, for s ∈ [0, 1], Therefore, 9) and the quasi tangent qTan is given by and where ζ is a N (0, 1)-random variable independent of B. On the other hand, where ζ is a N (0, 1)-random variable independent of B. As a consequence, taking into account that H > 1 2 , we obtain (4.14) Proof: We first show (4.12). We can write The term A n,2 is a deterministic term bounded by Cn −1 , therefore it does not contribute to the limit. In order to handle the term A n,1 we make the decomposition We claim that the product (B s − B 1 )(B t − B 1 ) does not contribute to the limit of n H A n,1 . In fact, By Minkowski's inequality, the expectation of this quantity is estimated as follows Therefore, it suffices to consider the term and to show that n H A n,3 converges in law to a Gaussian random variable with mean zero and variance σ 2 This a consequence of the following two facts: The proof of (i) is based on the computation of the limit of the following quantity This limit can be evaluated using the change of variables s n+1 = y 1 and t n+1 2 = y 2 , which leads to the representation (4.10) of σ 2 H,1 . The proof of (ii) can be done in a similar way. This concludes the proof of (4.12).
For (4.13), we can write We first show that Φ n,1 does not contribute to the limit: Then, taking expectation and using Minkowski's inequality, we get and n 1−H Φ n,1 2 converges to zero as n tends to infinity. Finally, it suffices to consider the term We claim that n 1−HΦ n,2 converges in law to a Gaussian random variable with zero mean and variance σ 2 This is a consequence of the following two facts: (ii) E(n 1−HΦ n,2 B t ) → 0, for any t ∈ [0, 1].
We first show (i): Using the change of variable t n+1 we can show that this quantity converges to σ 2 H,2 given in (4.11). The proof of (ii) can be done in a similar way.
In spite of the preceding proposition, the quasi tangent does not contribute to the asymptotic expansion derived in the last section. In fact, the convergence (4.14), together with uniform integrability, gives lim n→∞ E[Ψ(z) qTan] = 0, that is, S (2,0) 0 = 0. More strongly, using the duality relationship between the Skorohod integral and the derivative operator (IBP formula), we can show this fact directly for Φ n,2 as follows: By (4.14), qTan never converges to zero in probability. Thus DqTan potentially has some effect at the rate n 1−H .

Quasi torsion
By (4.9), we have Notice that which implies that u n H p is uniformly bounded for any p ≥ 2. On the other hand, the computations in the previous section imply D u n 2 and this term does not contribute to the limit. Consider the term DΦ n , u n H . Using the decomposition (4.15), we can write DΦ n , u n H = DΦ n,1 , u n H + DΦ n,2 , u n H .
The term DΦ n,1 , u n H does not contribute to the limit since Φ n,1 is in the second chaos and n 1−H Φ n,1 2 → 0.
As for DΦ n,2 , u n H , we can write DΦ n,2 , u n H = DB 1 , u n HΦn,2 + B 1 DΦ n,2 , u n H , whereΦ n,2 is defined in (4.16). The term DB 1 , u n HΦn,2 does not contribute to the limit at the rate n 1−H in L p , p ≥ 2 due to the computations in the previous section. On the other hand, for the second term we can write With the change of variables t n+1 = x, θ n+1 = y, ξ n+1 = z, we obtain On the other hand, it is easy to check that ∆ n p ≤ Cn −1 , so this term does not contribute to the limit. In conclusion, taking into account (4.6), the quasi torsion qTor = n 1−H D DZ n , u n H , u n H converges in L p to 3c 2 H HB 2 1 for all p ≥ 2. In other words, S (3,0) = c 2 H HB 2 1 for H > 1/2, which is discontinuous at H = 1/2. In this way, we obtain the expansion   Proof: We can write We have As a consequence, where c H is the constant introduced in (4.4).

Quasi tangent
Recall that r n = n −H and the quasi tangent is defined by We know that The inner product in the Hilbert space H is more involved than in the case H > 1 2 , and it is convenient to rewrite the stochastic integral 1 t s n dB s using integration by parts: Substituting (4.19) into (4.18) yields Actually, we can combine the second and third terms and last two terms as follows: The dominant term in the limit will be A 1,n , which can be expressed as and converges to G ∞ = HΓ(2H)B 2 1 as n tends to infinity by Lemma 4.4. It is not difficult to check, using formulas (4.2) and (4.3), that the other two terms converge to zero in L 2 as n tends to infinity.
We are going to show that the quasi tangent does not contribute to the asymptotic expansion. From the preceding computations, we deduce We examine each term of this expression as follows: (i) For the first term, by Lemma 4.4, we havẽ which converges to zero. In fact, integrating by parts, the factor B t − B 1 produces a term of the form |t − 1| 2H due to Lemma 4.1 (iii), and then we have hence the first term on the right-hand side of (4.20) converges to 0. For two sequences numbers a n and b n , a n < ∼ b n means that there exists a positive constant C independent of n such that a n ≤ Cb n for all n ∈ N. For the second term we apply the integration-by-parts formula to B t − B 1 as well as Lemma 4.1 (iii) and Lemma 4.5 below to obtain the bound Therefore the second term on the right-hand side of (4.20) converges to 0, which proves (4.21). Lemma 4.5. Let α, β, µ ∈ (−1, ∞) and ν ∈ [0, ∞). Let B(α, β, µ, ν) = B(µ + ν + β + 2, α + 1)B(β + 1, ν + 1) + 1 β + 1 B(µ + 1, α + β + 2).

Then
(ii) For fixed α and β, it holds that Proof. First, Next,

Property (i) follows from these inequalities and (ii) is obvious.
(iii) The third term also does not produce contribution. By In this way, we have proved that qTan has no contribution in the limit, that is, S (2,0) 0 = 0.

Quasi torsion
The quasi torsion can be written as Let us show that n H DG Thus we have proved that n H DG (2) n , u n H p = o(1). Therefore, taking into account (4.7), the quasi torsion qTor converges in L p to 2HΓ(2H)c 2 H B 2 1 for all p ≥ 2. In other words, S (3,0) = 2 3 HΓ(2H)c 2 H B 2 1 for H < 1/2.

Asymptotic expansion
In this way, we obtain the expansion

Quadratic form of a Brownian motion with predictable weights
In this section, we consider a quadratic form of a Brownian motion with predictable weights and show that the asymptotic expansion formula for the Skorohod integral reproduces the results obtained in [30,29]. where t j = j/n, a t = a(B t ) and a is an infinitely differentiable function with derivatives of moderate growth (g has moderate growth if |g(x)| ≤ c exp(α|x|) for some constants c > 0 and 0 ≤ α < 2). That is, Z n = δ(u n ) with

Quadratic form with random weights and H-derivatives
where I j = [t j−1 , t j ). In this situation Z n = M n and we have W n = W ∞ = 0, N n = 0, X n = X ∞ = 0 and ψ = 1. Let We have It is known that in this example, G ∞ = 1 2 1 0 a 2 t dt and r n = n −1/2 . Then,

Quasi torsion
We shall study the asymptotic behavior of the eight terms appearing in the expression of DZ n , u n H , u n H corresponding to (5.2).
(i) The first term is We investigate the rate of E[Ψ(z, x)I 1 ]. The factor n 1.5 comes from three √ n. It suffices to consider the terms for which k ∨ ℓ < j; otherwise the term vanishes due to D s D t a tj−1 . The number of terms in the sum j is of order n 1 . The number of terms in the sum k.ℓ for k = ℓ and k ∨ ℓ < j is O(n 1 ), and each B t − B t k−1 (= B s − B t ℓ−1 ) or its H-derivative contributes O(n −0.5 ) in L p -norm. By the IBP formula for q j we get a factor n −2 . So that the partial sum in E[Ψ(z, x)I 1 ] for k = ℓ is O(n −1.5 ) since both ds and dt-integrals give O(n −1 ). For the partial sum in E[Ψ(z, x)I 1 ] for k = ℓ is also O(n −1.5 ), since the consecutive IBP formulas (i.e., duality) for B t − B t k−1 and B s − B t ℓ−1 gives the rate O(n −2 ). Thus, we obtain E[Ψ(z, x) is negligible in the expansion. Table 1 summarizes how the orders of the partial sums were obtained.
The second term can be written as Only terms with k < j = ℓ remain due to the product 1 I k (t)D t a tj−1 1 Ij (s)1 I ℓ (s). Table 2 shows E[Ψ(z, x)I 2 ] = O(n −0.5 ), as explained more precisely below. Table 2: The contribution of √ nE[Ψ(z, x)I 2 ] is evaluated as follows. A n ≡ a B n means A n − B n = o(1) as n → ∞. By Itô's formula, for s ∈ I j . As already mentioned, only the terms with k < j = ℓ contribute the result. Applying the IBP formula for the first two terms of the right-hand side of (5.3), we obtain Therefore, The third term is given by By symmetry, it is easy to see , and hence the limit is the same as (5.4).
(iv) Consider now the fourth term given by  Table 3: The contribution of this term is given by (v) The fifth term is Notice that I 5 looks like I 1 but they are slightly different from each other. Only the terms satisfying ℓ < k < j remain due to D t a tj−1 and D s a t k−1 . According to Table 4, we see √ nE[Ψ(z, x)I 5 ] = O(n −1 ) and is negligible.
The sixth term is given by Thanks to the product 1 [t k−1 ,t] (s)1 I k (t)1 I ℓ (s)D t a tj−1 , only the terms satisfying k = ℓ < j remain. By table 5 below √ nE[Ψ(z, x)I 6 ] = O(n −1 ) and this term is negligible.
(vii) Consider the seventh term given by Due to the product 1 I ℓ (s)D s a t k−1 1 Ij (t)1 I k (t), only the terms satisfying ℓ < k = j contribute to the sum. Then it turns out that I 7 is the same as and the limit is given by (5.4).
(viii) Finally, the last term is It suffices to consider the case j = k = ℓ. Table 3 says that √ nE[Ψ(z, x)I 8 ] contributes to the limit.
More precisely, following a procedure quite similar to that of I 4 , we obtain

Now, from (i)-(viii) and
we obtain It should be remarked that the three terms on the right-hand side of the above equation correspond to C 2 , C 3 and C 1 of [30], pp. 917-918, respectively. We remark that two random symbols with the same adjoint action are considered equivalent. Obviously S (2,0) = S (1,1) = S (1,0) = S (0,1) = 0 in the present situation, and moreover we will show that S (2,0) 0 = 0 in Section 5.3. Consequently, This random symbol is equivalent to the full random symbol σ(iz, ix) of [30], p. 918 with a replaced by a/2. For the quadratic form of a Brownian motion, S (3,0) provides both the adapted random symbol and the anticipative random symbol, in other words, the quasi torsion includes the tangent as well as the torsion. In this way, we found that the quasi torsion reproduces the asymptotic expansion of the quadratic form of a Brownian motion.

Quasi tangent
For the quasi tangent, we have and We shall investigate these terms.
(i) For G 1 , Table 7 shows √ nE[Ψ(z, x)G 1 ] = O(n −0.5 ) and it is negligible in the asymptotic expansion. Table 7:  Cn of [30] and it has non-trivial limit distribution though the expectation √ nE[Ψ(z, x)G ′ 2 ] asymptotically vanishes. We see G ′′ 2 of of O(n −1 ) in L 2 , and it is also negligible. Consequently, G 2 is negligible in the asymptotic expansion.  Table 9 shows √ nE[Ψ(z, x)G 3 ] = O(n −0.5 ) and we can neglect it. Table 9: As a consequence of these observations, S (2,0) 0 = 0, i.e., the quasi tangent has no effect in the asymptotic expansion. However, the effect of the tangent already appeared in that of the quasi torsion.
6 Quadratic form of a fractional Brownian motion with random weights

Weighted quadratic variation
Let B = {B t , t ∈ [0, 1]} be a fractional Brownian motion with Hurst parameter H ∈ ( 1 4 , 3 4 ). We are interested in the following sequence of weighted quadratic variations: where t j = j/n, a t = a(B t ) and a is a function such that a and all its derivatives up to some order N have moderate growth. We use the notation ∆B j,n = B j/n − B (j−1)/n . It is known (see, for instance [14,13]) that for this example the limit variance G ∞ is given by Therefore, we obtain the decomposition In this example, we take W n = W ∞ = 0, X n = X ∞ = 0 and ψ = 1. We are going to study the quasi torsion and the quasi tangent of the Skorohod integral M n = δ(u n ). In this example there will be also a contribution to the asymptotic expansion coming from the perturbation term N n . The scaling factor r n will be taken as r n = n 2H− 3 2 when H ∈ ( 1 2 , 3 4 ), and r n = n 1 2 −2H when H ∈ 1 4 , 1 2 , respectively. This choice of r n is motivated by the rate of convergence [13] for ϕ ∈ C 5 b (R), where ζ is N (0, 1).

Quasi torsion
We recall that qTor = r −1 n D DM n , u n H , u n H .
(A) We first study the contribution of the term DΦ n,1 , u n H in the asymptotic expansion. We have where the product A * B means that whenever we found repeated variables in A and B, we compute the corresponding inner product in H. We have a total of eight terms, that we denote by We are interested in the asymptotic behavior of r −1 n E[Ψ(z)I i ] for i = 1, . . . , 8. (i) The first term is where we have used the notation α t,k = 1 [0,t] , 1 I k H . We can make the decomposition where β k,ℓ = 1 I k , 1 I ℓ H . Integrating by parts shows that the contribution of I 2 (1 I k ⊗ 1 I ℓ ) is of order lower than that of β k,ℓ . In this way, it suffices to consider the term In this case, the factors of the above expressions have the following orders of convergence: • First factor: n 6H− 3 2 • The IPB formula for q j produces a factor n −2(2H∧1) , due to part (a) of Lemma 6.2 below.
• Finally we get a factor n from the sum in j.
(ii) The second term is equal to where β j,ℓ = 1 Ij , 1 I ℓ H . As before, we can replace the product ∆B j,n ∆B ℓ,n by β j,ℓ , and we have to deal with the term We get the following contributions: • First factor: n 6H− 3 2 • The IPB formula for ∆B k,n produces a factor n −(2H∧1) , due to part (a) of Lemma 6.2 below.
Therefore, the order of this term is , which in the Brownian case gives n − 1 2 . From this result we deduce that this term will not contribute to the asymptotic expansion if H < 1 2 . The contribution of n as n tends to infinity. We can write Using (6.4) and (6.3), we obtain where α H = H(2H − 1) and (iii) By symmetriy, the third term is analogous to the second one and produces the same contribution.
(iv) The fourth term is given by We can make the decomposition ∆B k,n ∆B ℓ,n = I 2 (1 I k ⊗ 1 I ℓ ) + β k,ℓ .
By integration by parts the contribution of I 2 (1 I k ⊗ 1 I ℓ ) is of lower order than that of β k,ℓ . In this way, it suffices to consider the term a tj−1 a t k−1 a t ℓ−1 β j,k β j,ℓ β k,ℓ .
From part (f) of Lemma 6.2, we see that this term is bounded in L p by n 2H− 3 2 if H > 1 2 and by n − 1 2 if H < 1 2 . This clearly implies that lim n→∞ n 1 2 −2H I 4 = 0 (6.5) if H < 1 2 , in L p for all p ≥ 2. On the other hand, we claim that, if H > 1 2 we also have the following convergence is L p for all p ≥ 2, and, as a consequence, this term produces no contribution. Proof of (6.6): We need to show that and it suffices to show that By the inequality sup i≥1 |ρ(i)|/i 2H−2 < ∞, we have The last term is O(n 4H−3 ) when 2/3 ≤ H < 3/4, and O(n 1−2H ) when 1/2 < H < 2/3. This gives (6.7) and hence (6.6).
(v) The fifth term is As before, we replace ∆B k,n ∆B ℓ,n by β k,ℓ and it suffices to study the term We get the following contributions: • The first factor n 6H− 3 2 .
(vi) The sixth term is We get the following contributions: • The first factor n 6H− 3 2 .
We get the same order as for the fifth term and no contribution.
(vii) The seventh term is given by Replacing ∆B j,n ∆B k,n by β j,k it suffices to consider the term We get the following contributions: • The first factor n 6H− 3 2 .
(viii) The eighth term is given by Replacing ∆B j,n ∆B ℓ,n by β j,ℓ , is suffices to consider the reduced term By Lemma 6.2 part (f), this term is of order n 2H− 3 2 −(2H∧1) . Its contribution is the same as that of term (iv). In conclusion, we obtain the following results on the random symbol S (3,0) (iz): Case H > 1 2 We have proved that Taking into account that α H t 0 |r − s| 2H−2 ds = ∂RH ∂r (t, r), we can write the above expression as follows: Case H < 1 2 We have obtained, that in this case, S 3 (iz) = 0.

Perturbation term
In this subsection we study the contribution of the term N n to the asymptotic expansion. More precisely we will compute the random symbol G 1 (iz). The action of this symbol is defined by Integrating by parts yields We know that Expanding the square (j 2H − (j − 1) 2H − 1) 2 it turns out that the terms −2(j 2H − (j − 1) 2H ) and 1 do not tribute to the limit. So, for G 1,n it suffices to consider the term The term G 2,n converges to zero because, by (6.8) and Lemma 6.2 (b), we can write Therefore, we have proved that (ii) Case H < 1 2 . In that case, We proceed as before, but in that case the dominating term in α 2 tj−1,j in G 1,n is the constant 1 and we obtain We can also show that lim n→∞ E[Ψ(z) DN n , u n H ] = 0.
Remark 6.1. The functional Ψ(z) should be replaced by Ψ(z)ψ n when we need a truncation ψ n . However, the above arguments are essentially unchanged because 1 − ψ n ℓ,p would converge to zero much faster than the total error we found.

Asymptotic expansion for measurable functions
Let ℓ =ď+8 and denote by β x the maximum degree in x of S. We denote by σ F the Malliavin covariance matrix of a multivariate functional F and write ∆ F = det σ F . Let where [x] is the maximum integer not larger than x. We consider the following condition.
Denote by φ(z; µ, Σ) the density function of the normal distribution with mean vector µ and covariance matrix Σ. We write S n = 1 + r n S. Define the function p n (z, x) by where δ x (X ∞ ) is Watanabe's delta function, i.e., the pull-back of the delta function δ x by X ∞ . See [8] for the notion of generalized Wiener functionals and Watanabe's delta function. The operation of the adjoint ς(∂ z , ∂ x ) * for a random polynomial symbol ς(iz, ix) = α c α (iz, ix) α is defined by The function p n (z, x) is well defined under [C]. Given positive numbers M and γ, denote by E(M, γ) the set of measurable functions f : Rď → R satisfying |f (z, x)| ≤ M (1 + |z| + |x|) γ for all (z, x) ∈ Rď. We intend to approximate the joint distribution of (Z n , X n ) by the density function p n (z, x). The error of the approximation is evaluated by the supremum of ForŽ n = (Z n , X n ), we writeŽ α n = Z α1 n X α2 n for α = (α 1 , α 2 ) ∈ Z d + × Z d1 for z ∈ R d and x ∈ R d1 . Define g α n (z, x) by if the integral exists.
The functional ψ n is re-defined by ψ n = ψ(ξ n ) with ξ n = 3s n 2s n + 12∆ n + e n s 2 n + f n ∆ 2 X∞ (7.36) for ∆ n = ∆ (Mn+W∞,X∞) this time. The functional f n is defined as before, and e n will be specified in the proof of the following lemma. Proof. The plot of the proof is quite similar to that of Lemma 7.3, however some modifications are necessary. Let ǫ be a positive number. We may assume that r n < 1 for all n ∈ N. Instead of (7.13) and (7.14), we will use the non-degeneracy of the forms sup θ∈( < ∞ (7.37) and sup θ∈(0, < ∞ (7.38) for every p > 1 and a suitably differentiable functional Ξ. For the same reason as before, we have ϕ n (θ, z, x; Ξ) < ∼ |(z, x)| −k uniformly in θ ∈ ( √ 1 − r ǫ n , 1], (z, x) ∈ Rď and n ∈ N. For θ ∈ (0, √ 1 − r ǫ n ], we will useď + β x + 1 IBP below, just like before, and this procedure gives some power of |z|. To cancel the power of |z| (including z's come from random polynomials when we use it), we attach the factor (1 − θ 2 ), and then a power of (1 − θ 2 ) −1 appears. We can replace it by r −ǫL n , where L is a definite number. Thus what we obtained is sup n sup θ∈(0,1) sup (z,x)∈Rď r ǫL n |(z, x)| k ϕ n (θ, z, x; Ξ) < ∞. (7.39) We need non-degeneracy (7.37) and (7.38) to apply the estimate (7.39). For our purposes, when the functional Ξ has ψ n or its derivative of certain order, it is sufficient to show non-degeneracy ofM n (θ) and X n (θ) under truncation by ψ n with ξ n of (7.36). We make ǫ sufficiently small. Then it is easy to see ∆M n (θ) = ∆ (Mn+W∞,X∞) + r ǫ/2 n d * * n (θ) for some functional d * * n (θ) such that sup θ∈( √ 1−r ǫ n ,1],n∈N r −ǫ/2 n d n (θ) ď +6,p < ∞ for every p > 1. Define e n as before with the coefficients of d * * n (θ). Then we see (7.17) holds and ∆M n (θ) and ∆ Xn(θ) have uniform nondegeneracy under ψ n , as before.
We remark that the factor x does not emerge but some product of z can newly appear though cancelled by the exponential. So for every α ∈ Zď + . Let ς be any random symbol, like S (3,0) , that appears in the r n -order term of S n . We apply d + β x + 1 times IBP with respect to X ∞ to \ ∂ α E Ψ(z, x)ς(iz, ix) , and next use the Gaussianity of Ψ in z to show