1 Introduction
Let
be a complete probability space. Consider the stochastic process
defined on of the form(1.1) 
where is a fractional Brownian motion (fBm) with Hurst parameter , is a continuous stochastic process and
is a random variable. The stochastic process in (
1.1) is, for example, used for the logvolatility process and such volatility models recently attract much attention from researchers in mathematical finance and financial econometrics and practitioners in the financial industry, e.g. see [11], [2] and [9] for details. The aim of the present paper is to investigate an asymptotic distribution of a quasilikelihoodtype estimator of the parameter based on the highfrequency data with when the sample size goes to infinity.First, note that is the only parameter to be estimated in (1.1) because we can not consistently estimate drift parameters even if a sample path of is continuously observed. Under highfrequency asymptotics, i.e. as , we can show
(1.2) 
and, thanks to the selfsimilarity property of the fBm, we have
(1.3) 
where means that the equality holds in law. If we know the Hurst parameter , then the volatility is the only parameter to be estimated and it is wellknown that the QMLE (QuasiMaximum Likelihood Estimator) of based on the local Gaussian approximation (1.2), which is same as the quadratic variation of with equidistant sampling since the fBm has the independent increments property when , is consistent and asymptotically normal as under some mild technical assumptions of , e.g. see [6]. Then the drift term can be seen as a nuisance parameter because the QMLE of can be computed without identifying the drift term and its asymptotic distribution does not depend on the drift term.
Thanks to (1.2) and (1.3), even if is unknown, it would be possible to consistently estimate the parameter without identifying the drift term under highfrequency asymptotics using a quasilikelihoodtype estimator based on the local Gaussian approximation (1.2). On the other hand, it is unclear whether an asymptotic distribution of the quasilikelihoodtype estimator does not depend on the drift term because (1.2) and (1.3) imply that it becomes more difficult to distinguish between the noise and drift terms of from highfrequency data when approaches .
Recently, [10] proposed an estimator of the parameter using the changeoffrequency method and proved its asymptotic normality property for all with under the technical condition for some satisfying if . We remark that the condition assumed in the case is not standard because it implies the length of the observation period converges to zero. Therefore, it is not trivial whether their proposed estimator enjoys the same asymptotic normality property when under the condition . Moreover, their proposed estimator is, of course, not optimal because the covariance structure of the noise is not used in their estimation procedure.
In the present paper, we propose an estimator of the parameter combining the following two ideas in the similar way to [9]: (1) the local Gaussian approximation (1.2) and (2) the frequency domain approximation, the socalled Whittle approximation, of the quasilikelihood function, see Section 3.1 for details. Then we can easily prove the consistency of the proposed estimator in the similar way to Theorem 2.8 of [9]. Our contribution in the present paper is to prove that the proposed estimator enjoys the asymptotic normality property for all even when as , which implies the technical condition of assumed in [10] is not essential to derive asymptotic distributions of estimators of , and an asymptotic distribution of the proposed estimator does not depend on the drift term for all , at least, when is . constant.
The present paper is organized as follows. We summarize preliminary results and notation in Section 2. In Section 3, our proposed estimator is defined and a main theorem in the present paper is given. The main theorem is proven in Section 4 and a preliminary lemma used in the proof of the main theorem is proven in Section 5.
2 Preliminary Results
2.1 Fractional Brownian Motion
A centered continuous Gaussian process , defined on a complete probability space , is called a fractional Brownian motion (fBm) with Hurst parameter if a.s. and it satisfies the following scaling property:
(2.1) 
From (2.1), it is obvious that the fBm has the stationary increments and selfsimilarity properties. Moreover, it is wellknown that the spectral density function of the stationary increments process is given by
(2.2) 
with , e.g. see [14].
2.2 Notation
Consider the parameter space , where is a compact set of . Denote by the true value of the parameter . Let be the sample size and be the length of sampling intervals. Denote by and
Moreover, set and .
Let . For an integrable function , the th Fourier coefficient of is defined by
We denote by the Toeplitz matrix whose element is given by for each . Thanks to the selfsimilarity property of the fBm, we can write with . Define by
Set and for . Finally, denotes the convergence in law under the probability measure .
3 Main Result
3.1 QuasiWhittle Likelihood Estimator (QWLE)
In the present paper, we propose the estimator defined by
(3.1) 
We call the QWLE (QuasiWhittle Likelihood Estimator) in the following. In the rest of this subsection, we make several remarks on the proposed estimator in order.
Remark 3.1 (Local Gaussian Approximation and Whittle Approximation).
Thanks to (1.2) and (1.3), it would be possible that
is approximated by the Gaussian vector
in some suitable sense under highfrequency asymptotics. Therefore, we use the likelihood function of as a quasilikelihood function of . Actually, we utilize an approximate likelihood function of in the frequency domain, the socalled Whittle likelihood function, as a quasilikelihood function of because the quasimaximum likelihood estimator is computationally infeasible when the sample size is quite large due to the high computational cost of the inverse and determinant of .In the rest of this remark, we explain why the estimator is defined by (3.1). Set and . First note that is the spectral density function of the stationary Gaussian sequence with respect to the reparameterized parameter . Since we have
(3.2) 
for all , the quasiWhittle likelihood function of with respect to the reparameterized parameter is defined by
Since is a minimizer of the quasiWhittle likelihood function on for each , the estimator can be defined as (3.1) using the estimator and the relation .
Remark 3.2 (Reparameterization).
Under highfrequency observations, the effects of and fuse in the limit and the asymptotic Fisher information matrix when a “diagonal” ratematrix is used becomes singular due to the selfsimilarity property of the fractional Gaussian noise. As a result, it is necessary to reparametrize the parameter in order to obtain a limit theorem of an estimator. See [3] and [7] for more details.
In the rest of this remark, we briefly explain of our strategy to prove asymptotic properties of the QWLE . First note that is also a minimizer of the function
with respect to on so that the random variable with is a minimizer of the function
(3.3) 
with respect to on . Note that is not an estimation function since the true value is used in its definition. It plays, however, a similar role to the usual estimation function in proofs of asymptotic properties of the QWLE because is the quasiWhittle likelihood function of the suitably rescaled random vector and the quasispectral density function which appears in no longer depends on the asymptotic parameter . Therefore, it would be possible that the random variable converges to in some suitable sense as and asymptotic properties of the estimator can be proven using the convergence of . See Section 5.2 of [7], Section 4 of [9] and Section 4.2 for more details.
Remark 3.3 (Implementation).
In this remark, we briefly explain how to efficiently implement the QWLE. First note that we can write
(3.4) 
where is the periodogram defined by
Then the Riemann approximation of the integral (3.4) gives
(3.5) 
and the sum in (3.5
) can be effectively computed using the fast Fourier transform algorithm. Note that the series appears in the function
can be accurately and efficiently computed using the approximation method proposed by [13]. See also [7] and its supplementary article [8] for more details.3.2 Asymptotic Normality Property of QWLE
First, we introduce a class of sequences of nondiagonal rate matrices which plays a key role to prove an asymptotic normality property of QWLE with a nondegenerate asymptotic variancecovariance matrix.
Assumption 3.4.
Assume a sequence of matrices and a matrix of the forms
satisfy the following properties for each :

as ,

as ,

as ,

as ,

for each ,

.
Then we can prove a main theorem in the present paper as follows.
Theorem 3.5.
Consider a sequence of rate matrices satisfying Assumption 3.4. Assume is an interior point of . Then we obtain the following result:

The sequence of the QWLEs is (weakly) consistent as .

If is identically equal to a measurable random variable , then the sequence of the QWLEs satisfies the following asymptotic normality property:
(3.6) where is the positive definite matrix defined by
Several examples of satisfying Assumption 3.4 can be found in [7]. Particular choices of imply that the convergence rates of and are and respectively. See [7] for details.
Remark 3.6.
In the case , i.e. , Theorem 3 of [7] proved that the Whittle estimator, defined in the same way as (3.1), has the same asymptotic distribution as under highfrequency asymptotics. Therefore, Theorem 3.5 (2) implies that the asymptotic distribution of the QWLE does not depend on the drift term, at least, when is constant. We will investigate asymptotic properties of the QWLE when is not constant in the future work.
4 Proof of Theorem 3.5
4.1 Preliminary Lemma
Before proving Theorem 3.5, we prepare the following lemma.
Lemma 4.1.
For any , and , we have
(4.1)  
(4.2) 
4.2 Proof of Theorem 3.5
First, note that the consistency of the QWLE can be proven in the similar way to the proof of Theorem 2.8 of [9]. In the following, we prove only the asymptotic normality property of the QWLE. In the similar way to the proof of Theorem 2.12 of [9] and the proof of Theorem 3 of [7], the asymptotic normality property of the QWLE follows once we have proven
(4.3) 
where the function is defined by (3.3) and
Now we introduce notation used in the proof. Define by
for . Set and for . By a straightforward calculation, we can write
Moreover, in the similar way to the proof of Theorem 2 of [5], we can prove
Therefore, in order to prove (4.3), it suffices to prove
(4.4) 
for any and . We prove (4.4) in the rest of the proof. Since and , we can write
(4.5) 
Moreover, we have
(4.6) 
Therefore (4.4) follows from (4.5), (4.6) and Lemma 4.1. This completes the proof.
5 Proof of Lemma 4.1
5.1 Notation
Suppose is a realvalued matrix. Define the operator norm of by
and the Frobenius norm of by
In the present paper, we use the following wellknown properties:

.

and .
5.2 Preliminary Lemma
Before proving Lemma 4.1, we prove the following preliminary lemma.
Lemma 5.1.
For any and ,
Proof.
First note that we can write
Then the conclusion follows from Theorem 3.1 of [15]. This completes the proof. ∎
5.3 Proof of in the case
5.4 Proof of in the case
First we can show
Then the conclusion follows from in the case and Lemma 2 in the full version of [12]. This completes the proof.
5.5 Proof of
First we introduce notation used in the proof. For , set ,
Note that follows once we have proven that
(5.1) 
holds for any and each because we can show
for any and using Theorem 5.2 of [1] and Lemma 2 in the full version of [12] in the similar way to the proof of Lemma 5.4 (d) of [4]. In the rest of the proof, we will prove (5.1). First we can write