Estimation of spectral functionals for Levy-driven continuous-time linear models with tapered data

The paper is concerned with the nonparametric statistical estimation of linear spectral functionals for Levy-driven continuous-time stationary linear models with tapered data. As an estimator for unknown functional we consider the averaged tapered periodogram. We analyze the bias of the estimator and obtain sufficient conditions assuring the proper rate of convergence of the bias to zero, necessary for asymptotic normality of the estimator. We prove a a central limit theorem for a suitable normalized stochastic process generated by a tapered Toeplitz type quadratic functional of the model. As a consequence of these results we obtain the asymptotic normality of our estimator.


Introduction
In spectral analysis of time series the data are frequently tapered before calculating the statistics of interest. Instead of the original data {X(t), 0 ≤ t ≤ T } the tapered data {h(t)X(t), 0 ≤ t ≤ T } with the data taper h(t) are used for all further calculations. Benefits of tapering the data have been widely reported in the literature. For example, data-tapers are introduced to reduce leakage effects, especially in the case when the spectrum of the model contains high peeks. Other application of data-tapers is in situations in which some of the data values are missing. Also, the use of tapers leads to the bias reduction, which is especially important when dealing with spatial data. In this case, the tapers can be used to fight the so-called "edge effects".
In this paper, we study the problem of nonparametric estimation of linear spectral functionals based on tapered data, in the case where the underlying model is a Lévy-driven continuous-time stationary linear process with possibly unbounded or vanishing spectral density function.
Notice that the covariance function r(t) of X(t), which is an even function (r(−t) = r(t)), is given by (1.2) and it possesses the spectral density The functions r(t) and f (λ) are connected by the Fourier integral: The function a(·) plays the role of a time-invariant filter. Processes of the form (1.1) appear in many fields of science (economics, finance, physics, etc.), and cover a large class of popular models in continuoustime time series modeling. For instance, the so-called continuous-time autoregressive moving average (CARMA) models, which are the continuous-time analogs of the classical autoregressive moving average (ARMA) models in discretetime case, are of the form (1.1) and play a central role in the representations of continuous-time stationary time series (see, e.g., Brockwell [8]).
The nonparametric estimation problem. Let {X(t), t ∈ R} be a centered stationary process with an unknown spectral density f (λ), λ ∈ R. We assume that f (λ) belongs to a given (infinite-dimensional) class F ⊂ L p := L p (R) (p ≥ 1) of spectral densities possessing some specified smoothness properties. The problem is to estimate the value J(f ) of a given functional J(·) at an unknown "point" f ∈ F on the basis of the observed data {X(t), 0 ≤ t ≤ T }, and investigate the asymptotic (as T → ∞) properties of the suggested estimators, depending on the dependence structure of the model X(t) and smoothness structure of the "parametric" set F.
This problem for discrete time stationary Gaussian processes has been considered in a number of papers. We cite merely the papers Dahlhaus and Wefelmeyer [12], Ginovyan [15,19], and Ibragimov and Khas'minskii [26,28], where can be found additional references.
For continuous time stationary Gaussian processes the problem was studied in Ginovyan [16,17,18,20,21], where efficient nonparametric estimators for linear and some nonlinear smooth spectral functionals were constructed and asymptotic bounds for minimax mean square risks of these estimators were obtained.
The problem of construction of consistent and asymptotically normal nonparametric estimators for linear and some nonlinear smooth spectral functionals in the case where the underlying model X(t) is a Lévy-driven continuous-time stationary linear process defined by (1.1) with possibly unbounded or vanishing spectral density function has been studied in Ginovyan and Sahakyan [23].
In this paper we are interested in nonparametric estimation of spectral functionals J(f ) based on tapered data: where h T (t) := h(t/T ) with a taper function h(·) satisfying assumption (T) below.
We assume that the estimand functional J(f ) is linear and continuous in L p (R), p > 1. Then J(f ) admits the representation where g(λ) ∈ L q (R), 1/p + 1/q = 1.
The estimator. As an estimator for functional J(f ), given by (1.6), we consider the averaged periodogram (or a simple "plug-in" statistic) based on the tapered data (1.5). To define the estimator, we first introduce some notation. Denote by H k,T (λ) the continuous-time tapered Dirichlet type kernel, defined by We set 8) and assume that H 2 = 0. Define the finite Fourier transform of the tapered data (1.5): (1.9) and the tapered continuous periodogram I h T (λ) of the process X(t): where Notice that for non-tapered case (h(t) = 1), we have C T = 2πT . An estimator J h T for functional (1.6) based on the tapered data (1.5) is defined to be the averaged tapered periodogram (or a simple "plug-in" statistic) defined by where C T is as in (1.11), and b(t) is the Fourier transform of function g(λ): (1.13) We will refer to g(λ) and to its Fourier transform b(t) := g(t) as a generating function and generating kernel for the functional J h T , respectively. Notation. Given numbers p ≥ 1, 0 < α < 1, r ∈ N 0 := N ∪ {0}, we set β = α + r and denote by H p (β) the L p -Hölder class, that is, the class of those functions ψ(λ) ∈ L p (R), which have r-th derivatives in L p (R) and with some positive Throughout the paper the letters C, c and M are used to denote positive constants, the values of which can vary from line to line. Also, by I A (·) we denote the indicator of a set A ⊂ R.
The paper is structured as follows. In Section 2 we state the main results of the paper (Theorems 2.1 -2.3). In Section 3 we give a number of preliminary results that are used in the proofs of main results, and also represent independent interest. In Section 4 we analyze the bias of the estimator J h T , and prove Theorem 2.1. In Section 5 we study the asymptotic distribution of a stochastic process generated by a tapered Toeplitz type quadratic functional of a Lévy-driven continuous-time linear process, and prove Theorems 2.2.

Main results
In this section we state the main results of this paper, involving bias rate convergence theorem, a central limit theorem and asymptotic normality of the estimator J h T . To this end, we first formulate conditions on model, generating function and taper function needed to state the results.
(A1) The filter a(·) and the generating kernel b(·) are such that The spectral density f and the generating function g satisfy one of the following conditions.
The next theorem contains sufficient conditions for functional J h T to obey the central limit theorem (CLT), and is proved in Section 5.
where the symbol d → stands for convergence in distribution, and η is a normally distributed random variable with mean zero and variance σ 2 h (J) given by Here κ 4 is the fourth cumulant of ξ (1), and Taking into account the equality as an immediate consequence of Theorems 2.1 and 2.2, we obtain the next result that contains sufficient conditions for a simple "plug-in" statistic J(I h T ) to be an asymptotically normal estimator for a linear spectral functional J(f ).
where η is as in Theorem 2.2, that is, η is a normally distributed random variable with mean zero and variance σ 2 h (J) given by (2.5) and (2.6). Remark 2.1. Notice that if the underlying process X(t) is Gaussian, then in formula (2.5) we have only the first term. Using the results from Ginovyan [17] and Ginovyan and Sahakyan [22], it can be shown that in this case the conditions (A2 ) and (T) are sufficient for Theorem 2.3 to be true.

Remark 2.2.
The result of Theorem 2.3 under different more restrictive conditions were stated in Avram et al. [2] (see also Sakhno [31]). For non-tapered case (h(t) = I (0,1) (t)), Theorems 2.1-2.3 were proved in Ginovyan [20,21]. Remark 2.3. One of the common used approaches to derive central limit theorems for random quadratic functionals is the method-of-moments (see, e.g., R. Dahlhaus [9,10], Avram et al. [2], and references therein). Taking into account the complexity of computing the moments of multiple integrals with respect to non-Gaussian Lévy noise (see Peccati and Taqqu [30], Chapter 7), it is not clear how this method can be carried out for our model. In this paper, similar to Bai et al. [3] and Ginovyan and Sahakyan [22], our proofs of the central limit theorems are based on a new approximation approach which reduces the quadratic integral form to a single integral form. This method can also be adapted to the discrete-time case.

Remark 2.4.
Notice that linear and non-linear functionals of the periodogram play a key role in the parametric estimation of the spectrum of stationary processes, when using the minimum contrast estimation method with various contrast functions (see, e.g., Anh et al. [1], Ginovyan and Sahakyan [23], Leonenko and Sakhno [29], Sakhno [31], Taniguchi [32], and references therein). So, the results obtained in the present paper can be applied to prove consistency and asymptotic normality of minimum contrast estimators based on the Whittle and Ibragimov's contrast functionals for Lévy-driven continuous-time stationary linear models with tapered data. The details will be reported elsewhere.

Preliminaries
In this section we prove a number of auxiliary lemmas involving properties of continuous-time tapered Dirichlet and Fejér type kernels. Some of these properties for discrete-time tapered case were proved in Dahlhaus [9], and for continuous-time non-tapered case were established in Ginovyan and Sahakyan [22].
An important role in our analysis of the above mentioned properties is played by the function L T (·) : R → R, T ∈ R + , defined by The function L T (t) possesses a number of interesting properties, allowing to estimate the cumulants of functional J h T defined (1.12) (see Dahlhaus [9] for discrete-time case, and Eichler [13] for continuous-time case). In this paper we will use the following property of function L T (t).
Then for u ∈ R we have where M is a constant depending on m.
Proof. For |u| ≤ 2 we have In the case when |u| > 2, we can write Now we estimate the functions Q i (u), i = 1, 2, 3. Assuming, without loss of generality, that u > 2, for Q 1 (u) we have Similarly, for Q 2 (u) we get the estimate For Q 3 (u) we have We have Similarly, Finally, Combining the inequalities (3.4)-(3.11) we obtain (3.3). Lemma 3.1 is proved.
The next two lemmas contain some properties of the Dirichlet type kernel H k,T (λ) defined by (1.7). The next assertion is the continuous analog of formula (4) in Dahlhaus [9].

Lemma 3.2.
The function H k,T (λ) defined by (1.7) satisfies the following identity: (3.12) Proof. Without loss of generality we assume that μ = 0, and observe that H k,T ∈ L 2 (R). Hence by (1.7) we have H k,T (ζ) = 2πh k T (−ζ) and H k,T (λ − ·)(ζ) = 2πh k T (ζ)e −iλζ , whereĝ is the Fourier transform of g. Hence using Plansherel's identity, we get and the result follows. Lemma 3.2 is proved. [13]). Let the functions H k,T (·) and L T (·) be defined by (1.7) and (3.1), respectively. Then for all k ∈ N and a constant C k independent of T , the following inequality holds:

Lemma 3.3 (see Eichler
Proof. Since the taper function h is assumed to be of bounded variation, then denoting by V (h) the total variation of h, we can write (3.14) Therefore Taking into account that |H k,T (λ)| ≤ ||h|| k ∞ T , we obtain (3.13). Lemma 3.3 is proved.
For a number k (k = 2, 3, . . .) and a taper function h satisfying assumption (T) consider the following Fejér type kernel function: where 17) and the function H k,T is defined by (1.7) with H k,T (0) = T · H k = 0 (see (1.8)). The next lemma shows that the kernel Φ h k,T is an approximation identity.
To prove assertion c) we write Using the same arguments as in (3.22), for s = 1, 2 . . . , k − 1 we get and the result follows.
In the case where k ≥ 4 we have We estimate I 1 , the integrals I 2 , . . . , I k−1 can be estimated similarly. We have According to (3.16) and (3.23), we have Note that in the integral I 1k , we have |u 1 + · · · + u k−1 | > δ − δ(k − 2)/k > δ/k, and hence by (3.23) we obtain Next, for any ε > 0 we can find δ > 0 to satisfy where C 1 is the constant from assertion a).
and hence, by assertions a) and c) of the lemma and (3.29), for sufficiently large T we obtain If k > 2 we consider the decomposition Ψ = Ψ 1 + Ψ 2 such that where M δ is as in assertion d). Applying assertions a) -d) of the lemma and formulas (3.29) -(3.31) for sufficiently large T we obtain This combined with (3.29) yields (3.19) for v = 0. Lemma 3.4 is proved.

The Bias. Proof of Theorem 2.1
In this section we prove Theorem 2.1, that is, we show that under the conditions (A2) (or (A2 ) and (T) the bias E(J h T ) − J of estimator J h T satisfies the asymptotic relation (2.3).
We first prove two lemmas.

Lemma 4.1. Assume that the conditions (A2) and (T) are satisfied. Then the following asymptotic relation holds as T → ∞:
where Φ h 2,T (μ) is given by (3.16). Proof. We have According to (3.16) and (3.20) we can write (4.4) where H 2 = Observe that for function q(λ) : and hence, by (4.2) and (4.3), we obtain Therefore, we can write Since by assumption g and h are of bounded variation, we have where the constant K does not depend on ζ. Hence, we have as T → ∞. Combining (4.5) -(4.7), we obtain (4.1). Lemma 4.1 is proved.

Lemma 4.2. Assume that the conditions (A2 ) and (T) are satisfied. Then the following inequality holds:
where C h is a constant depending on h.
Proof. According to (3.16) and (3.17) we have where H 2 = 1 0 h 2 (t)dt > 0. The following properties of the kernel Φ h 2,T follow from Lemma 3.4, and formulas (3.13) and (4.10): if α > 1. (4.11) Since the function Φ h 2,T (μ) is even, we have and hence, using the equality Using Hölder's inequality from (4.12) we get (4.13) and hence by the conditions of the lemma, From (4.11) and (4.14) follows (4.9). Lemma 4.2 is proved. Proof of Theorem 2.1. We first show that the expected value of the estimator J h T is given by formula: where Φ h 2,T (·) is defined by (3.16). Indeed, using (1.4), (1.12) and (3.16), we can write and (4.15) follows. Thus, for the bias Theorem 2.1 is proved.

A central limit theorem for tapered Toeplitz type quadratic functionals. Proof of Theorem 2.2
In this section we study the asymptotic distribution of a suitable normalized stochastic process {Q h T (t), t ∈ [0, 1]}, generated by a tapered Toeplitz type quadratic functional of a Lévy-driven continuous-time linear process X(t) given by (1.1). We show that under conditions (A1) and (T) the process Q T (t) obeys a central limit theorem, that is, the finite-dimensional distributions of the standard √ T normalized process Q T (t) tend to those of a normalized standard Brownian motion. Then, using this result we prove Theorem 2.2.
We will use the following notation. The symbol * will stand for the convolution: ( while the symbol * will be used to denote the reversed convolution: Also, we will use the following well-known identities: where F(h) := h denotes the Fourier transform of a function h. Denote by Q h T the tapered Toeplitz type quadratic functional of the process X(t) from formula (1.12), that is, We are interested in the asymptotic distribution (as T → ∞) of the stochastic process {Q h T (t), t ∈ [0, 1]}, generated by the functional Q h T : The next theorem, which is the tapered version of Theorem 2.1 of Bai et al. [3], contains sufficient conditions for the process {Q h T (t), t ∈ [0, 1]} to obey central limit theorem.

Theorem 5.1. Under the conditions (A1) and (T) the process Q h T (t) defined in (5.3) obeys central limit theorem. More precisely, we have
where the symbol f.d.d.

−→ stands for convergence of finite-dimensional distributions, B(t) is a standard Brownian motion, and
where H 4 is as in (1.8), κ 4 is the fourth cumulant of ξ(1), and We first introduce the notions of multiple off-diagonal (Itô-type) and withdiagonal (Stratonovich-type) stochastic integrals with respect to Lévy noise, and briefly discuss their properties (see, e.g., Bai et al. [3], Farré et al. [14], Peccati and Taqqu [30]). Let f be a function in L 2 (R k ), then the following off-diagonal multiple stochastic integral, called Itô-Lévy integral, is well-defined: where ξ(t) is a Lévy process with Eξ(t) = 0 and Var[ξ(t)] = σ 2 ξ t, and the prime indicates that we do not integrate on the diagonals The multiple integral I ξ k (·) satisfies the following inequality: 8) and the inequality in (5.8) becomes equality if f is symmetric: We will need a stochastic Fubini's theorem (see Bai et al. [3], Lemma 3.1, or Peccati and Taqqu [30], Theorem 5.13.1). Lemma 5.1. Let (S, μ) be a measure space with μ(S) < ∞, and let f (s, x 1 , . . . , x k ) be a function on S × R k such that then we can change the order of the multiple stochastic integration I ξ k (·) and the deterministic integration S f (s, ·)μ(ds): The with-diagonal counterpart of the integral I ξ k (f ), called a Stratonovichtype stochastic integral, is defined bẙ which includes all the diagonals. We refer to Farré et al. [14] for a comprehensive treatment of Stratonovich-type integralsI ξ k (f ). Observe that for the with-diagonal integralI ξ k (f ) to be well-defined, the integrand f needs also to be square-integrable on all the diagonals of R k (see Bai et al. [3], Farré et al. [14]).
The with-diagonal integralI ξ k (f ) can be expressed by off-diagonal integrals of lower orders using the Hu-Meyer formula (see Farré et al. [14], Theorem 5.9). We will only use the special case when k = 2, in which case we have and ξ (2) (t) is the quadratic variation of ξ(t), which is non-deterministic if ξ(t) is non-Gaussian (see Farré et al. [14], equation (10)). The centered process ξ (2) c (t) is called a second order Teugels martingale, which is a Lévy process with the same filtration as ξ(t), whose quadratic variation is deterministic: where κ 4 is the fourth cumulant of ξ (1). For any f, g ∈ L 2 (R), one has (see Farré et al. [14], page 2153), The decomposition (5.11) implies that Consider now the following integrals, the first of which is an off-diagonal double integral and the second is a single integral with respect to Teugels martingale ξ (2) c (t): (2) c (dx). (5.14) Notice that for any f ∈ L 2 (R 2 ) and g ∈ L 2 (R) the integrals in (5.14) are uncorrelated (see Bai et al. [3]). To prove Theorem 5.1, we first establish two lemmas. We set where K A (·) is as in (5.6). Proof. In the proof we use Young's inequality for convolution (see, e.g., Bogachev [6], Theorem 3.9.4), stating that for any numbers p, q, r satisfying 1 ≤ p, q, r ≤ ∞ and 1 r = 1 p + 1 q − 1, and for any functions f ∈ L p (R), g ∈ L q (R) the function f * g is defined almost everywhere, f * g ∈ L r (R), and one has f * g r ≤ f p g q . (5.20) In view of Riesz-Thorin theorem, without loss of generality, we can assume that a(·) ∈ L p (R), b(·) ∈ L q (R), 2 p + 1 q = 5 2 . (5.21) Let p and q be as in (5.21). Define the numbers q 1 , q * 1 , q 2 to satisfy the following equations: (Going from the last to the first equality in (5.22), one can solve successively for q 2 , q * 1 , q 1 and then verify using (5.21) that the first equality in (5.22) holds.) To prove (5.17) observe first that Since h is a bounded and continuous function, we have by the dominated convergence theorem, that . (5.27)