It is well known that (see [22, Chapter 6]) for a semimartingale X, the bracket [X,X] can be identified with [X,X]t=P-lim|π|→0∑tk∈π(Xtk−Xtk−1)2, where π={tk:0=t0<t1<⋯<tn=t} is a partition of the interval [0,t] , |π|=max{tk−tk−1:tk∈π} , and P-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 a rather noisy estimator. So one should seek for new estimators of the quadratic variation.

Dzhaparidze and Spreij [12] suggested another characterization of the bracket [X,X] . Let FX be the filtration of X, and τ be a finite stopping time. For λ∈R , define the periodogram Iτ(X;λ) of X at τ by (1) Iτ(X;λ):=|∫0τeiλsdXs|2=2Re∫0τ∫0teiλ(t−s)dXsdXt+[X,X]τ(by Itô formula). Let ξ be a symmetric random variable independent of the filtration FX with density gξ and real characteristic function φξ . For given L>0 0$]]>, define the randomized periodogram by (2) EξIτ(X;Lξ)=∫RIτ(X;Lx)gξ(x)dx. If the characteristic function φξ is of bounded variation, then Dzhaparidze and Spreij have shown that we have the following characterization of the bracket as L→∞ : (3) EξIτ(X;Lξ)→P[X,X]τ.

Recently, the convergence (3) is extended in [2] to some class of stochastic processes which contains nonsemimartingales in general. Let W={Wt}t∈[0,T] be a standard Brownian motion, and BH={BtH}t∈[0,T] be a fractional Brownian motion with Hurst parameter H∈(12,1) , independent of the Brownian motion W. Define the mixed Brownian–fractional Brownian motion Xt by Xt=Wt+BtH,t∈[0,T].

Given λ∈R , define the periodogram of X at T as (1), that is, IT(X;λ)=|∫0TeiλtdXt|2=|eiλTXT−iλ∫0TXteiλtdt|2=XT2+XT∫0Tiλ(eiλ(T−t)−e−iλ(T−t))Xtdt+λ2|∫0TeiλtXtdt|2.

Let (Ω˜,F˜,P˜) be another probability space. We identify the σ-algebra F with F⊗{ϕ,Ω˜} on the product space (Ω×Ω˜,F⊗F˜,P⊗P˜) . Let ξ:Ω˜→R be a real symmetric random variable with density gξ and independent of the filtration FX . For any positive real number L, define the randomized periodogram EξIT(X;Lξ) as in (2) by (5) EξIT(X;Lξ):=∫RIT(X;Lx)gξ(x)dx, where the term IT(X;Lx) is understood as before. Azmoodeh and Valkeila [2] proved the following: Theorem 1.

Assume that X is a mixed Brownian–fractional Brownian motion, EξIT(X;Lξ) be the randomized periodogram given by (5), and Eξ2<∞. Then, as L→∞ , we have EξIT(X;Lξ)⟶P[X,X]T.

Föllmer [13] obtained a pathwise calculus for continuous functions with finite quadratic variation. The next theorem essentially belongs to Föllmer. For a nice exposition and its use in finance, see Sondermann [24].

The rest of the section contains the essential elements of Gaussian analysis and Malliavin calculus that are used in this paper. See, for instance, Refs. [17, 18] for further details. In what follows, we assume that all the random objects are defined on a complete probability space (Ω,F,P) .

Let X={Xt}t∈[0,T] be a centered continuous Gaussian process on the interval [0,T] with X0=0 and continuous covariance function RX(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 ⟨1[0,t],1[0,s]⟩H=RX(t,s),s,t∈[0,T]. For example, when X is a Brownian motion, H reduces to the Hilbert space L2([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∈(12,1) (see [21]). The mapping 1[0,t]⟼Xt can be extended to a linear isometry between H and the Gaussian space H1 spanned by a 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], that is, it is a Gaussian family with covariance function E(X(φ1)X(φ2))=⟨φ1,φ2⟩H=∫[0,T]2φ1(s)φ2(t)dRX(s,t),φ1,φ2∈E, where dRX(s,t)=RX(ds,dt) stands for the measure induced by the covariance function RX on [0,T]2 . Let S be the space of smooth and cylindrical random variables of the form (7) F=f(X(φ1),…,X(φn)), where f∈Cb∞(Rn) (f and all its partial derivatives are bounded). For a random variable F of the form (7), we define its Malliavin derivative as the H -valued random variable DF= ∑i=1n∂f∂xi(X(φ1),…,X(φn))φi.

By iteration, the mth derivative DmF∈L2(Ω;H⊗m) is defined for every m≥2 . For m≥1 , Dm,2 denotes the closure of S with respect to the norm ‖·‖m,2 , defined by the relation ‖F‖m,22=E[|F|2]+ ∑i=1mE(‖DiF‖H⊗i2). Let δ be the adjoint of the operator D, also called the divergence operator. A random element u∈L2(Ω,H) belongs to the domain of δ, denoted Dom(δ) , if and only if it satisfies |E⟨DF,u⟩H|≤cu‖F‖L2 for any F∈D1,2 , where cu is a constant depending only on u. If u∈Dom(δ) , then the random variable δ(u) is defined by the duality relationship (8) E(Fδ(u))=E⟨DF,u⟩H, which holds for every F∈D1,2 . The divergence operator δ is also called the Skorokhod integral because when the Gaussian process X is a Brownian motion, it coincides with the anticipating stochastic integral introduced by Skorokhod [18]. We denote δ(u)=∫0TutδXt .

For every q≥1 , the symbol Hq stands for the qth Wiener chaos of X, defined as the closed linear subspace of L2(Ω) generated by the family {Hq(X(h)):h∈H,‖h‖H=1} , where Hq is the qth Hermite polynomial defined as (9) Hq(x)=(−1)qex22dqdxq(e−x22). We write by convention H0=R . For any q≥1 , the mapping IqX(h⊗q)=Hq(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 Hq . For q=0 , we write by convention I0X(c)=c , c∈R . For any h∈H⊙q , the random variable IqX(h) is called a multiple Wiener–Itô integral of order q. A crucial fact is that if H=L2(A,A,ν) , where ν is a σ-finite and nonatomic measure on the measurable space (A,A) , then H⊙q=Ls2(νq) , where Ls2(νq) stands for the subspace of L2(νq) composed of the symmetric functions. Moreover, for every h∈H⊙q=Ls2(νq) , the random variable IqX(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 use the following central limit theorem for sequences living in a fixed Wiener chaos (see [20, 19]).

To obtain Berry–Esseen-type estimate, we shall use the following result from [17, Corollary 5.2.10]. Theorem 4.

Let {Fn}n≥1 be a sequence of elements in the second Wiener chaos such that E(Fn2)→σ2 and Var‖DFn‖H2→0 as n→∞ . Then, Fn→lawZ∼N(0,σ2) , and supx∈R|P(Fn<x)−P(Z<x)|≤2E(Fn2)Var‖DFn‖H2+2|E(Fn2)−σ2|max{E(Fn2),σ2}.

In this subsection, we briefly describe how two Gaussian processes can be embedded into an isonormal Gaussian process. Let X1 and X2 be two independent centered continuous Gaussian processes with X1(0)=X2(0)=0 and continuous covariance functions RX1 and RX2 , respectively. Assume that H1 and H2 denote the associated Hilbert spaces as explained in Section 3.2. The appropriate set E˜ of elementary functions is the set of the functions that can be written as φ(t,i)=δ1iφ1(t)+δ2iφ2(t) for (t,i)∈[0,T]×{1,2} , where φ1,φ2∈E , and δij is the Kronecker’s delta. On the set E˜ , we define the inner product (10) ⟨φ,ψ⟩H˜:=⟨φ(·,1),ψ(·,1)⟩H1+⟨φ(·,2),ψ(·,2)⟩H2=∫[0,T]2φ(s,1)ψ(t,1)dRX1(s,t)+∫[0,T]2φ(s,2)ψ(t,2)dRX2(s,t), where dRXi(s,t)=RXi(ds,dt),i=1,2 .

Let H denote the Hilbert space that is the completion of E˜ with respect to the inner product (10). Notice that H≅H1⊕H2 , where H1⊕H2 is the direct sum of the Hilbert spaces H1 and H2 , that is, it is a Hilbert space consisting of elements of the form of ordered pairs (h1,h2)∈H1×H2 equipped with the inner product ⟨(h1,h2),(g1,g2)⟩H1⊕H2:=⟨h1,g1⟩H1+⟨h2,g2⟩H2 .

Now, for any φ∈E˜ , we define X(φ):=X1(φ(·,1))+X2(φ(·,2)) . Using the independence of X1 and X2 , we infer that E(X(φ)X(ψ))=⟨φ1,ψ⟩H for all φ,ψ∈E˜ . 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 X1 and X2 .

The fractional Brownian motion BH={BtH}t∈R with Hurst parameter H∈(0,1) is a zero-mean Gaussian process with covariance function (11) E(BtHBsH)=RH(s,t)=12(|t|2H+|s|2H−|t−s|2H). Let H denote the Hilbert space associated to the covariance function RH ; see Section 3.2. It is well known that for H=12 , we have H=L2([0,T]) , whereas for H>12 \frac{1}{2}$]]>, we have L2([0,T])⊂L1H([0,T])⊂|H|⊂H , where |H| is defined as the linear space of measurable functions φ on [0,T] such that ‖φ‖|H|2:=αH∫0T∫0T|φ(s)||φ(t)||t−s|2H−2dsdt<∞, where αH=H(2H−1) .

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.

For further use, we also need the following ancillary facts related to the isonormal Gaussian process derived from the covariance function of the mixed Brownian–fractional Brownian motion. Assume that X=W+BH stands for a mixed Brownian–fractional Brownian motion with H>12 \frac{1}{2}$]]>. We denote by H the Hilbert space associated to the covariance function of the process X with inner product ⟨·,·⟩H . Then a direct application of relation (10) and Proposition 1 yields the following facts. We recall that in what follows the notations I1X and I2X stand for multiple Wiener integrals of orders 1 and 2 with respect to isonormal Gaussian process X; see Section 3.2. Lemma 1.

For any φ1,φ2,ψ1,ψ2∈L2([0,T]) , we have E(I1X(φ)I1X(ψ))=⟨φ,ψ⟩H=∫0Tφ(t)ψ(t)dt+αH∫0T∫0Tφ(s)ψ(t)|t−s|2H−2dsdt. Moreover, E(I2X(φ1⊗φ2)I2X(ψ1⊗ψ2))=2⟨φ1⊗φ2,ψ1⊗ψ2⟩H⊗2=∫[0,T]2φ1(s1)ψ1(s1)φ2(s2)ψ2(s2)ds1ds2+αH∫[0,T]3φ1(s1)ψ1(s1)φ2(s2)ψ2(t2)|t2−s2|2H−2ds1ds2dt2+αH∫[0,T]3φ1(s1)ψ1(t1)φ2(s1)ψ2(s1)|t1−s1|2H−2ds1dt1ds1+αH2∫[0,T]4φ1(s1)ψ1(t1)φ2(s2)ψ2(t2)×|t1−s1|2H−2|t2−s2|2H−2ds1dt1ds2dt2.

We start with the following fact, which is one of our key ingredients.

Our first aim is to transform the pathwise integral in (13) into the Skorokhod integral. This is the topic of the next lemma.

We will also pose the following assumption for characteristic function φξ of a symmetric random variable ξ.

We continue with the following technical lemma, which in fact provides a correct normalization for our central limit theorems.

We also apply the following proposition. The proof is rather technical and is postponed to Appendix A.

Our main theorem is the following. Theorem 5.

Assume that the characteristic function φξ of a symmetric random variable ξ satisfies Assumption 1 and let σT2 be given by (14). Then, as L→∞ , we have the following asymptotic statements: 1.

if H∈(34,1) , then L(EξIT(X;Lξ)−[X,X]T)⟶lawN(0,σT2).