Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian--fractional Brownian model

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter $H$ of the fractional part satisfies $H\in(3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in(1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in(1/2,3/4)$, the central convergence does not take place. We also provide Berry--Esseen estimates for the estimator.

2 Two methods for estimating quadratic variation 2.1 Using discrete observations: realized quadratic variation It is well-known that (see [21,Chapter 6]) for a semimartingale X, the bracket [X, X] can be identified with where π = {t k : 0 = t 0 < t 1 < · · · < t n = t} is a partition of the interval [0, t], |π| = max {t k − t k−1 : t k ∈ π}, and IP-lim means convergence in probability. Statistically speaking, the sums of squared increments (realized quadratic variation) is a consistent estimator for the bracket as the volume of observations tends to infinity. Barndorff-Nielsen and Shephard [3] studied precision of the realized quadratic variation estimator for a special class of continuous semimartingales. They showed that sometimes the realized quadratic variation estimator can be rather noisy estimator. So one should seek for new estimators of the quadratic variation.

Using continuous observations: randomized periodogram
Dzhaparidze and Spreij [12] suggested another characterization of the bracket [X, X]. Let IF X be the filtration of X and τ be a finite stopping time. For λ ∈ IR, define the periodogram I τ (X; λ) of X at τ by (2.1) Given L > 0 and ξ be a symmetric random variable with a density g ξ , real characteristic function ϕ ξ , and independent of the filtration IF X . Define the randomized periodogram by If the characteristic function ϕ ξ is of bounded variation, then Dzhaparidze and Spreij have shown that we have the following characterization of the bracket as L → ∞ IE ξ I τ (X; Lξ) Recently, the convergence (2.3) is extended in [2] to some class of stochastic processes which contains non-semimartingales in general. Let W = {W t } t∈[0,T ] is a standard Brownian motion and B H = {B H t } t∈[0,T ] is a fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1), independent of the Brownian motion W . Define the mixed Brownian-fractional Brownian motion X t by Remark 2.1. It is known that (see [9]) the process X is a (IF X , IP) semimartingale, if H ∈ ( 3 4 , 1), and for H ∈ 1 2 , 3 4 , X is not a semimartingale with respect to its own filtration IF X . Moreover in both cases we have If the partitions in (2.4) are nested, then the convergence can be strengthened to almost sure convergence. Hereafter, we always assume that the sequence of partitions are nested.
Assume (Ω,F ,P) be an another probability space and identify the σ-algebra F by F ⊗ {φ,Ω} on the product space (Ω ×Ω, F ⊗F, P ⊗P). Let ξ :Ω → IR be a real symmetric random variable with density g ξ , and independent of the filtration IF X . Define for any positive real number L the randomized periodogram by Azmoodeh and Valkeila in [2] proved the following.
Theorem 2.1. Assume that X is a mixed Brownian-fractional Brownian motion, IE ξ I T (X; Lξ) is the randomized periodogram given by (2.5) and Then, as L → ∞ we have 3 Stochastic analysis for Gaussian processes 3.1 Path-wise Itô formula Föllmer [13] obtained a path-wise calculus for continuous functions with finite quadratic variation. The next theorem is essentially due to Föllmer. For a nice exposition, and its use in finance, see Sondermann [23].
[23] Let X : [0, T ] → IR be a continuous process with continuous quadratic variation [X, X] t and F ∈ C 2 (IR). Then for any t ∈ [0, T ], the limit of the Riemann-Stieltjes sums exists almost surely. Moreover, we have The rest of the section contains the essential elements of Gaussian analysis and Malliavin calculus that are used in this paper. See for instance the references [17,18] for further details. In what follows, we assume that all the random objects are defined on a complete probability space (Ω, F, IP).

Isonormal Gaussian processes derived from covariance functions
Let X = {X t } t∈[0,T ] be a centered, continuous Gaussian process on interval [0, T ] with X 0 = 0 and a continuous covariance function R X (s, t). We assume that F is generated by X. Denote by E the set of real valued step functions on [0, T ] and let H be the Hilbert space defined as the closure of E with respect to the scalar product For example, when X is a Brownian motion, then H reduces to Hilbert space L 2 ([0, T ], dt). However, in general H is not a space of functions. For example, when X is a fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1) (see [20]). The mapping 1 [0,t] −→ X t can be extended to a linear isometry between H and the Gaussian space H 1 spanned by Gaussian process X. We denote this isometry by ϕ −→ X(ϕ), and {X(ϕ); ϕ ∈ H} is an isonormal Gaussian process in the sense of [18, Definition 1.1.1], i.e. it is a Gaussian family with covariance function Let S be the space of smooth and cylindrical random variables of the form F = f (X(ϕ 1 ), . . . , X(ϕ n )), (3.2) where f ∈ C ∞ b (R n ) (f and all its partial derivatives are bounded). For a random variable F of the form (3.2), we define its Malliavin derivative as the H-valued random variable By iteration, one can define the mth derivative D m F , which is an element of L 2 (Ω; H ⊗m ), for every m ≥ 2. For m ≥ 1, D m,2 denotes the closure of S with respect to the norm · m,2 , defined by the relation Let δ be the adjoint of the operator D, also called the divergence operator. A random element u ∈ L 2 (Ω, H) belongs to the domain of δ, denoted Dom(δ), if and only if it verifies for any F ∈ D 1,2 , 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 which holds for every F ∈ D 1,2 . The divergence operator δ is also called the Skorohod integral because when the Gaussian process X is a Brownian motion, it coincides with the anticipating stochastic integral introduced by Skorohod in [18]. We will make use of the notation δ(u) = T 0 u t δX t .
For every q ≥ 1, the symbol H q stands for the qth Wiener chaos of X, defined as the closed linear subspace of L 2 (Ω) generated by the family {H q (X(h)) : h ∈ H, h H = 1}, where H q is the qth Hermite polynomial, defined as follows: We write by convention H 0 = R. For any q ≥ 1, the mapping I X q (h ⊗q ) = H q (X(h)) can be extended to a linear isometry between the symmetric tensor product H ⊙q (equipped with the modified norm √ q! · H ⊗q ) and the qth Wiener chaos H q . For q = 0, we write by convention I X 0 (c) = c, c ∈ R. For any h ∈ H ⊙q , the random variable I X q (h) is customarily called a multiple Wiener integral of order q. A crucial fact is that, when H = L 2 (A, A, ν), where ν is a σ-finite and non-atomic measure on the measurable space (A, A), then H ⊙q = L 2 s (ν q ), where L 2 s (ν q ) stands for the subspace of L 2 (ν q ) composed of the symmetric functions. Moreover, for every h ∈ H ⊙q = L 2 s (ν q ), the random variable I X q (h) coincides with the q-fold multiple Wiener-Itô integral of h with respect to the centered Gaussian measure (with control ν) generated by X (see [18]). We will also make use of the following central limit theorem for sequences living in a fixed Wiener chaos (see [19]).
Then, as n → ∞, the following asymptotic statements are equivalent: To obtain Berry-Esseen type estimate, we shall use the following result from [17, Corollary 5.2.10].

Isonormal Gaussian process associated with two Gaussian processes
In this subsection, we briefly describe how two Gaussian processes can be embedded into an isonormal Gaussian process. Let X 1 and X 2 be two independent centered, continuous Gaussian processes, with X 1 (0) = X 2 (0) = 0 and continuous covariance functions R X 1 and R X 2 respectively. Assume that H 1 and H 2 denote the associated Hilbert spaces as explained in Subsection 3.2. The appropriate setẼ of elementary functions is the set of the functions that can be written as where ϕ 1 , ϕ 2 ∈ E, and δ ij is the Kronecker's delta. On the setẼ, we define the following inner product (3.5) Let H denote the Hilbert space which is the completion ofẼ with respect to the above inner product. Notice that Now, for any ϕ ∈Ẽ, we define X(ϕ) := X 1 (ϕ(·, 1)) + X 2 (ϕ(·, 2)). Using the independence between X 1 and X 2 , one infers that IE (X(ϕ)X(ψ)) = ϕ 1 , ψ H for all ϕ, ψ ∈Ẽ. Hence, the mapping X can be extended to an isometry on H, and therefore {X(h), h ∈ H} defines an isonormal Gaussian process associated to the Gaussian processes X 1 and X 2 .

Malliavin calculus with respect to (mixed Brownian) fractional Brownian motion
The fractional Brownian motion B H = {B H t } t∈R with Hurst parameter H ∈ (0, 1) is a zero mean Gaussian process with covariance function Let H denote the Hilbert space associated to the covariance function R H , see Subsection 3.2. It is well known that for H = 1 If H > 1 2 , then for any ϕ, ψ ∈ |H|, we have The following proposition establishes the link between pathwise integral and Skorokhod integral in Malliavin calculus associated to fractional Brownian motion, and will play an important role in our analysis.
Then u is pathwise integrable, and we have For further uses, we also need the following ancillary facts related to the isonormal Gaussian process derived from covariance function of the mixed Brownian fractional Brownian motion. Assume that X = W + B H stands for a mixed Brownian fractional Brownian motion with H > 1 2 . We denote by H the Hilbert space associated to the covariance function of the process X with the inner product ·, · H . Then a direct application of relation (3.5) and Proposition 3.1 yields the following facts. We recall that the notations I X 1 and I X 2 in what follows stands for multiple Wiener integrals of order 1 and 2 with respect to isonormal Gaussian process X, see Subsection 3.2. Moreover,

Main results
Throughout this section, we assume that X = W + B H stands for a mixed Brownian fractional Brownian motion with H > 1 2 , unless otherwise mentioned. We denote by H the Hilbert space associated to process X with the inner product ·, · H .

Central limit theorem
We start with the following fact which is one of our key ingredients.
where the iterated stochastic integral in the right hand side is understood in path-wise way, i.e. as the limit of the Riemann-Stieltjes sums, see Subsection 3.1.
Our first aim is to transform the pathwise integral in (4.1) into the Skorokhod integral. This is the topic of the next Lemma.
where the stochastic integral in the right hand side is the Skorokhod integral with respect to mixed Brownian-fractional Brownian motion X, and D (B H ) denote the Malliavin derivative operator with respect to the fractional Brownian motion B H .
Proof. First, note that Moreover, IE( where we have used the independence between W and B H , Proposition 3.2 and the fact that for adapted integrands, the Skorohod integral coincides with the Itô integral. To finish the proof, we use the very definition of Skorohod integral, relation (3.3) to obtain that We will also pose the following assumption for characteristic function ϕ ξ of the symmetric random variable ξ.
We continue with the following technical lemma which in fact provide the correct normalization for our central limit theorems. where σ 2 T := 2 T ∞ 0 ϕ 2 ξ (x)dx and is independent of the Hurst parameter H.

Remark 4.2.
We point it out that the variance σ 2 T in Lemma 4.3 is finite. This is a simple consequence of the Assumption 4.1 and the fact that the characteristic function ϕ ξ is bounded by one over the real line.
Proof. Throughout the proof C denotes unimportant constant depending on T and H which may vary from line to line. First, note that clearly ψ L ∈ H ⊗2 , since ψ L is a bounded function. In order to prove (4.2), we show that as L → ∞, we have Next, by applying Lemma 3.1, we obtain ψ L 2

4)
First, we show that A 1 ∼ 1 L . By change of variables y = L T s and x = L T t, we obtain Now, by applying L'Hopital's rule and some elementary computations, we obtain that We also apply the following proposition. The proof is rather technical and it is postponed to Appendix A.
Then, for any value H ∈ 1 2 , 1 , as L → ∞, we have Our main theorem is the following.
Theorem 4.1. Assume that the characteristic function ϕ ξ of the symmetric random variable ξ satisfies in assumption 4.1 and let σ 2 T be given by 4.2. Then, as L → ∞, we have the following asymptotic statements: 3 4 , then for every ǫ > 0 Proof. First that by applying Lemmas 4.1 and 4.2, we can write Consequently, we obtain Now, thanks to Proposition 4.1, for any value of H ∈ 1 2 , 1 , we have where σ 2 H is given by (4.2). Hence, it remains to study the term A 2 . Using change of variables y = L T s and x = L T t, we obtain where by the L'Hopital's rule, we obtain Note also that the integral in the right hand side of the above identity is finite by Assumption 4.1. Consequently, we obtain (4.6) Therefore, lim which converges to zero for H ∈ 3 4 , 1 , and proving the item 1 of the claim. Similarly, for H = 3 4 , we obtain lim which proves the item 2 of the claim. Finally, for the item 3, from (4.6), we infer that for every γ < 2H − 1, as L → ∞, Furthermore, for term I X 2 (ψ L ), we obtain as L → ∞. This is because, when H < 3 4 implies γ − 1 2 < 2H − 3 2 < 0 and moreover √ L I X 2 (ψ L ) law → N (0, 1) and L γ− 1 2 → 0.
Corollary 4.1. When X = W is a standard Brownian motion, i.e. when the fractional Brownian motion part drops, then with similar arguments as in Theorem 4.1, we obtain

The Berry-Esseen estimates
As a consequence of the proof of Theorem 4.1, we obtain also the following Berry-Esseen bound for the semimartingale case. , where the variance σ 2 T is given by (4.2). Then there exists a constant C (independent of L) such that for sufficiently large L, we have Proof. By proof of Theorem 4.1, we have . Now, we know that the deterministic term A 2 converges to zero with the rate L Finally, using the notations of the proof of Lemma 4.3, we have To complete the proof, we have This gives us Now, the claim follows by an application of Theorem 3.3.

Remark 4.4.
Consider the case X = W , i.e. X is a standard Brownian motion. In this case, the correction term A 2 in the proof of Theorem 4.1 disappears, and we have Furthermore, by applying L'Hopital's rule twice and some elementary computations, it can be shown that Consequently, in this case, we obtain the Berry-Esseen bound which is in fact better in many cases of interest. For example, if ϕ ξ admits an exponential decay, then we obtain ρ(L) ≤ e −cL for some constant c.

A Proof of Proposition 4.1
Denote F L = I X 2 (ψ L ), and note that by definition ofψ L , we have IE(F 2 L ) = 1. Hence, it is sufficient to prove that, as L → ∞, we have Now, using definition of Malliavin derivative, we get For the rest of the proof C denotes unimportant constants and may vary from line to line. Furthermore, we also use the short notation where δ 0 denotes the Kronecker delta function, to denote the measure associated to the Hilbert space H generated by the mixed Brownian fractional Brownian motion X. Furthermore, without loss of generality, we assume that ϕ ξ ≥ 0. Indeed, otherwise we simply approximate the integral by taking absolute values inside the integral which is consistent with Assumption 4.1. Now we have Next using the multiplication formula for multiple Wiener integrals, we see that where the term J 1 is deterministic and J 2 has expectation zero. Hence, we need to show that Therefore, by applying Fubini's Theorem, it suffices to show that, as L → ∞, we have First, using isometry (iii) [18, page 9] , we get that By plugging into (A.2), we obtain that it suffices to have The rest of the proof is based on similar arguments as the proof of Lemma 4.3. Indeed, again by symmetric property of measures K(dx, dy) and functions ϕ ξ (L|u 1 −·|)⊗ϕ ξ (L|v 1 −·|), we obtain five different terms, denoted by A k , k = 1, 2, 3, 4, 5, of forms Next, we prove that A 3 ≤ CL −3 . First by change of variables, we obtain Note that Assumption 4.1 implies that where again the constant C is independent of L and y. Moreover, we have ϕ ξ (T |u − v|) ≤ 1 for any u, v ∈ IR. Hence, we can estimate To conclude, treating A 1 , A 2 , A 4 and A 5 similarly, we deduce that which, by change of variable, leads to ϕ ξ (T |t − u|)ϕ ξ (T |s − u|)|t − s| 2H−2 dtdsdu, ϕ ξ (T |t − u|)ϕ ξ (T |s − v|)|t − s| 2H−2 |v − u| 2H−2 dudvdtds.
We begin with the term A 2 . Denotẽ First, we analysis the term J 1 . Similarly to Appendix A, we assume that ϕ ξ ≥ 0. Hence, we have Now, it is straightforward to show that J 2,1 + J 2,2 ≤ C. Consequently, as L → ∞, we obtain For the term J 2,3 , we write Hence, by L'Hopital's rule, we have L −2H J 2,3 ∼ L 1−2H ϕ ξ (T L). On the other hand, we have ϕ ξ (T L) = o L 2H−2 , since ϕ is integrable by Assumption 4.1. Hence L −2H J 2,3 = o L −1 , which shows that lim L→∞ LA 2 = 0. Consequently, A 2 does not affect to the variance. The term A 3 is easier and it can be treated with similar elementary computations together with L'Hopital's rule. As a consequence, we obtain A 3 ∼ L −2H , so that lim L→∞ LA 3 = 0. Hence A 3 does not affect to the variance either which justifies (4.2).