# Statistical inference for stationary linear models with tapered data

Mamikon S. Ginovyan
Department of Mathematics and Statistics
Boston University, Boston, MA, USA
e-mail<EMAIL_ADDRESS>

Artur A. Sahakyan
Department of Mathematics and Mechanics
Yerevan State University, Yerevan, Armenia
e-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we survey some recent results on statistical inference
(parametric and nonparametric statistical estimation, hypotheses testing)
about the spectrum of stationary models with tapered data, as well as, a
question concerning robustness of inferences, carried out on a linear
stationary process contaminated by a small trend.We also discuss some question
concerning tapered Toeplitz matrices and operators, central limit theorems for
tapered Toeplitz type quadratic functionals, and tapered Fejér-type kernels
and singular integrals. These are the main tools for obtaining the
corresponding results, and also are of interest in themselves. The processes
considered will be discrete-time and continuous-time Gaussian, linear or Lévy-
driven linear processes with memory.

Keywords. Tapered data; stationary processes; spectral density; parametric and
nonparametric estimation; goodness-of-fit test; robustness. 2010 Mathematics
Subject Classification. 62F10, 62F12, 60G10, 62G05, 62G10, 60G15.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 The model
      1. 2.1.1 Second-order (wide-sense) stationary process
      2. 2.1.2 Linear processes. Existence of spectral density functions
      3. 2.1.3 Lévy-driven linear process
      4. 2.1.4 Dependence (memory) structure of the model.
    2. 2.2 Data tapers and tapered periodogram
  3. 3 Nonparametric Estimation Problem
    1. 3.1 Estimation of linear spectral functionals
    2. 3.2 Asymptotic unbiasedness
    3. 3.3 Consistency
    4. 3.4 Asymptotic normality
  4. 4 Parametric Estimation Problem
    1. 4.1 The Whittle estimation procedure
    2. 4.2 Asymptotic properties of the Whittle estimator
  5. 5 Goodness-of-fit tests
    1. 5.1 A Goodness-of-fit test for simple hypothesis
    2. 5.2 A Goodness-of-fit test for composite hypothesis
  6. 6 Robustness to small trends of estimation
    1. 6.1 A relation between stationary and contaminated periodograms
    2. 6.2 Robustness to small trends of nonparametric estimation
    3. 6.3 Robustness to small trends of parametric estimation
  7. 7 Methods and tools
    1. 7.1 Approximation of traces of products of Toeplitz matrices and operators.
    2. 7.2 Central limit theorems for tapered quadratic functionals
      1. 7.2.1 CLT for Gaussian models
      2. 7.2.2 CLT for Lévy-driven stationary linear models
    3. 7.3 Fejér-type kernels and singular integrals

##  1 Introduction

Let $\\{X(t),\ t\in\mathbb{U}\\}$ be a centered real-valued stationary process
with spectral density $f(\lambda)$, $\lambda\in\Lambda$, and covariance
function $r(t)$, $t\in\mathbb{U}$. We consider simultaneously the continuous-
time (c.t.) case, where $\mathbb{U}=\mathbb{R}:=(-\infty,\infty)$, and the
discrete-time (d.t.) case, where $\mathbb{U}=\mathbb{Z}:=\\{0,\pm 1,\pm
2,\ldots\\}$. The domain $\Lambda$ of the frequency variable $\lambda$ is
$\Lambda=\mathbb{R}$ in the c.t. case, and $\Lambda:=[-\pi.\pi]$ in the d.t.
case.

We want to make statistical inferences (parametric and nonparametric
estimation, and hypotheses testing) about the spectrum of $X(t)$. In the
classical setting, the inferences are based on an observed finite realization
$\mathbf{X}_{T}$ of the process $X(t)$: $\mathbf{X}_{T}:=\\{X(t),\,t\in
D_{T}\\},$ where $D_{T}:=[0,T]$ in the c.t. case and $D_{T}:=\\{1,\ldots,T\\}$
in the d.t. case.

A sufficiently developed inferential theory is now available for stationary
models based on the standard (non-tapered) data $\mathbf{X}_{T}$. We cite
merely the following references Anh et al. [4], Avram et al. [5], Casas and
Gao [11], Dahlhaus [21], Dahlhaus and Wefelmeyer [24], Dzhaparidze [26, 27],
Dzhaparidze and Yaglom [29], Fox and Taqqu [30], Gao [31], Gao et al. [32],
Ginovyan [33, 34, 37, 39, 38, 41, 43], Giraitis et al. [52], Giraitis and
Surgailis [53], Guyon [56], Hannan [57], Has’minskii and Ibragimov [58], Heyde
and Dai [59], Heyde and Gay [60], Ibragimov [61, 62], Ibragimov and
Khas’minskii [63], Leonenko and Sakhno [65], Millar [67], Osidze [68, 69],
Taniguchi [71], Taniguchi and Kakizawa [72], Tsai and Chan [74], Walker [75],
Whittle [76], where can also be found additional references.

In the statistical analysis of stationary processes, however, the data are
frequently tapered before calculating the statistic of interest, and the
statistical inference procedure, instead of the original data
$\mathbf{X}_{T}$, is based on the tapered data:
$\mathbf{X}^{h}_{T}:=\\{h_{T}(t)X(t),\,t\in D_{T}\\}\\},$ where $D_{T}:=[0,T]$
in the c.t. case and $D_{T}:=\\{1,\ldots,T\\}$ in the d.t. case, and
$h_{T}(t):=h(t/T)$ with $h(t)$, $t\in\mathbb{R}$ being a taper function.

The use of data tapers in nonparametric time series was suggested by Tukey
[73]. The benefits of tapering the data have been widely reported in the
literature (see, e.g., Brillinger [8], Dahlhaus [17]–[20], [22], Dahlhaus and
Künsch [23], Guyon [56], and references therein). For example, data-tapers are
introduced to reduce the so-called ’leakage effects’, that is, to obtain
better estimation of the spectrum of the model in the case where it contains
high peaks. Other application of data-tapers is in situations in which some of
the data values are missing. Also, the use of tapers leads to 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 survey some recent results on statistical inference
(parametric and nonparametric statistical estimation, and hypotheses testing)
about the spectrum of stationary models with tapered data, as well as, a
question concerning robustness of inferences, carried out on a linear
stationary process contaminated by a small trend. We also discuss some
questions concerning tapered Toeplitz matrices and operators, central limit
theorems for tapered Toeplitz type quadratic functionals, and tapered Fejér-
type kernels and singular integrals. These are the main tools for obtaining
the corresponding results, and also are of interest in themselves. The
processes considered will be discrete-time and continuous-time Gaussian,
linear or Lévy-driven linear processes with memory.

##### Some notation and conventions.

The following notation and conventions are used throughout the paper.
The symbol ’$:=$’ stands for ’by definition’; c.t.: = continuous-time; d.t.:=
discrete-time; s.d.:= spectral density; c.f.:= covariance function; CLT:=
central limit theorem. The symbols ’$\overset{P}{\rightarrow}$’ and
’$\overset{d}{\rightarrow}$’ stand for convergence in probability and in
distribution, respectively. The notation
$X_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2})$ as $T\to\infty$ will
mean that the distribution of the random variable $X_{T}$ tends (as
$T\to\infty$) to the centered normal distribution with variance $\sigma^{2}$.
$\mathbb{E}[\cdot]$: = expectation operator; $\hbox{tr}[A]$: = trace of an
operator (matrix) $A$; ${\mathbb{I}}_{A}(\cdot)$: = indicator of a set
$A\subset\Lambda$; WN$(0,1)$: = standard white-noise. The standard symbols
$\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ denote the sets of natural,
integer and real numbers, respectively; $\mathbb{N}_{0}:=\mathbb{N}\cup 0$. By
$\Lambda$ we denote the frequency domain, that is, $\Lambda:=\mathbb{R}$ in
the c.t. case, and $\Lambda:=[-\pi.\pi]$ in the d.t. case. By
$L^{p}(\mu):=L^{p}(\Lambda,\mu)$ ($p\geq$1) we denote the weighted Lebesgue
space with respect to the measure $\mu$, and by $||\cdot||_{p,\mu}$ we denote
the norm in $L^{p}(\mu)$. In the special case where $d\mu(\lambda)=d\lambda$,
we will use the notation $L^{p}$ and $||\cdot||_{p}$, respectively. The
letters $C$ and $c$ with or without indices are used to denote positive
constants, the values of which can vary from line to line. Also, in the d.t.
case all the considered functions are assumed to be $2\pi$-periodic and
periodically extended to $\mathbb{R}$.

##### The structure of the paper.

The rest of the paper is structured as follows. In Section 2 we specify the
model of interest - a stationary process, recall some key notions and results
from the theory of stationary processes, and introduce the data tapers and
tapered periodogram. In Section 3 we discuss the nonparametric estimation
problem. We analyze the asymptotic properties, involving asymptotic
unbiasedness, bias rate convergence, consistency, a central limit theorem and
asymptotic normality of the empirical spectral functionals. In Section 4 we
discuss the parametric estimation problem. We present sufficient conditions
for consistency and asymptotic normality of minimum contrast estimator based
on the Whittle contrast functional for stationary linear models with tapered
data. Section 5 is devoted to the construction of goodness-of-fit tests for
testing hypotheses that the hypothetical spectral density of a stationary
Gaussian model has the specified form, based on the tapered data. A question
concerning robustness of inferences, carried out on a linear stationary
process contaminated by a small trend is discussed in Section 6. In Section 7
we briefly discuss the methods and tools, used to prove the results stated in
Sections 3–6.

##  2 Preliminaries

In this section we specify the model of interest - a stationary process, and
introduce the data tapers and tapered periodogram.

###  2.1 The model

####  2.1.1 Second-order (wide-sense) stationary process

Let $\\{X(u),\ u\in\mathbb{U}\\}$ be a centered real-valued second-order
(wide-sense) stationary process defined on a probability space
$(\Omega,\mathcal{F},P)$ with covariance function $r(t)$, that is,
$\mathbb{E}[X(u)]=0$, $r(u)=\mathbb{E}[X(t+u)X(t)]$, $u,t\in\mathbb{U}$, where
$\mathbb{E}[\cdot]$ stands for the expectation operator with respect to
measure $P$. We consider simultaneously the c.t. case, where
$\mathbb{U}=\mathbb{R}:=(-\infty,\infty)$, and the d.t. case, where
$\mathbb{U}=\mathbb{Z}:=\\{0,\pm 1,\pm 2,\ldots\\}$. We assume that $X(u)$ is
a non-degenerate process, that is, ${\rm
Var}[X(u)]=\mathbb{E}|X(u)|^{2}=r(0)>0$. (Without loss of generality, we
assume that $r(0)=1$). In the c.t. case the process $X(u)$ is also assumed
mean-square continuous, that is, $\mathbb{E}[X(t)-X(s)]^{2}\to 0$ as $t\to s$.
This assumption is equivalent to that of the covariance function $r(u)$ be
continuous at $u=0$ (see, e.g., Cramér and Leadbetter [16], Section 5.2).

By the Herglotz theorem in the d.t. case, and the Bochner-Khintchine theorem
in the c.t. case (see, e.g., Cramér and Leadbetter [16]), there is a finite
measure $\mu$ on $(\Lambda,\mathfrak{B}(\Lambda))$, where $\Lambda=\mathbb{R}$
in the c.t. case, and $\Lambda=[-\pi.\pi]$ in the d.t. case, and
$\mathfrak{B}(\Lambda)$ is the Borel $\sigma$-algebra on $\Lambda$, such that
for any $u\in\mathbb{U}$ the covariance function $r(u)$ admits the following
spectral representation:

$r(u)=\int_{\Lambda}\exp\\{i\lambda u\\}d\mu(\lambda),\quad u\in\mathbb{U}.$
(2.1)

The measure $\mu$ in (2.1) is called the spectral measure of the process
$X(u)$. The function $F(\lambda):=\mu[-\pi,\lambda]$ in the d.t. case and
$F(\lambda):=\mu[-\infty,\lambda]$ in the c.t. case, is called the spectral
function of the process $X(t)$. If $F(\lambda)$ is absolutely continuous (with
respect to Lebesgue measure), then the function
$f(\lambda):=dF(\lambda)/d\lambda$ is called the spectral density of the
process $X(t)$. Notice that if the spectral density $f(\lambda)$ exists, then
$f(\lambda)\geq 0$, $f(\lambda)\in L^{1}(\Lambda)$, and (2.1) becomes

$r(u)=\int_{\Lambda}\exp\\{i\lambda u\\}f(\lambda)d\lambda,\quad
u\in\mathbb{U}.$ (2.2)

Thus, the covariance function $r(u)$ and the spectral function $F(\lambda)$
(resp. the spectral density function $f(\lambda)$) are equivalent
specifications of the second order properties for a stationary process
$\\{X(u),\ u\in\mathbb{U}\\}$.

####  2.1.2 Linear processes. Existence of spectral density functions

We will consider here stationary processes possessing spectral density
functions. For the following results we refer to Cramér and Leadbetter [16],
Doob [25], and Ibragimov and Linnik [64].

  * (a)

The spectral function $F(\lambda)$ of a d.t. stationary process
$\\{X(u),\,u\in\mathbb{Z}\\}$ is absolutely continuous (with respect to the
Lebesgue measure), $F(\lambda)=\int_{-\pi}^{\lambda}f(x)dx$, if and only if it
can be represented as an infinite moving average:

$X(u)=\sum_{k=-\infty}^{\infty}a(u-k)\xi(k),\qquad\sum_{k=-\infty}^{\infty}|a(k)|^{2}<\infty,$
(2.3)

where $\\{\xi(k),k\in\mathbb{Z}\\}\sim$ WN(0,1) is a standard white-noise,
that is, a sequence of orthonormal random variables.

  * (b)

The covariance function $r(u)$ and the spectral density $f(\lambda)$ of $X(u)$
are given by formulas:

$r(u)=\mathbb{E}X(u)X(0)=\sum_{k=-\infty}^{\infty}a(u+k)a(k),$ (2.4)

and

$f(\lambda)=\frac{1}{2\pi}|\widehat{a}(\lambda)|^{2}=\frac{1}{2\pi}\left|\sum_{k=-\infty}^{\infty}a(k)e^{-ik\lambda}\right|^{2},\quad\lambda\in[-\pi,\pi].$
(2.5)

  * (c)

In the case where $\xi(k)$ is a sequence of Gaussian random variables, the
process $X(u)$ is Gaussian.

Similar results hold for c.t. processes. Indeed, the following holds.

  * (a)

The spectral function $F(\lambda)$ of a c.t. stationary process
$\\{X(u),\,u\in\mathbb{R}\\}$ is absolutely continuous (with respect to
Lebesgue measure), $F(\lambda)=\int_{-\infty}^{\lambda}f(x)dx$, if and only if
it can be represented as an infinite continuous moving average:

$X(u)=\int_{\mathbb{R}}a(u-t)d\xi(t),,\qquad\int_{\mathbb{R}}|a(t)|^{2}dt<\infty,$
(2.6)

where $\\{\xi(t),t\in\mathbb{R}\\}$ is a process with orthogonal increments
and $\mathbb{E}|d\,\xi(t)|^{2}=dt$.

  * (b)

The covariance function $r(u)$ and the spectral density $f(\lambda)$ of $X(u)$
are given by formulas:

$r(u)=\mathbb{E}X(u)X(0)=\int_{\mathbb{R}}a(u+x)a(x)dx,$ (2.7)

and

$f(\lambda)=\frac{1}{2\pi}|\widehat{a}(\lambda)|^{2}=\frac{1}{2\pi}\left|\int_{\mathbb{R}}e^{-i\lambda
t}a(t)dt\right|^{2},\quad\lambda\in\mathbb{R}.$ (2.8)

  * (c)

In the case where $\xi(t)$ is a Gaussian process, the process $X(u)$ is
Gaussian.

####  2.1.3 Lévy-driven linear process

We first recall that a Lévy process, $\\{\xi(t),\ t\in\mathbb{R}\\}$ is a
process with independent and stationary increments, continuous in probability,
with sample-paths which are right-continuous with left limits
(c$\grave{a}$dl$\grave{a}$g) and $\xi(0)=\xi(0-)=0$. The Wiener process
$\\{B(t),\ t\geq 0\\}$ and the centered Poisson process
$\\{N(t)-\mathbb{E}N(t),\ t\geq 0\\}$ are typical examples of centered Lévy
processes. A Lévy-driven linear process $\\{X(t),\ t\in\mathbb{R}\\}$ is a
real-valued c.t. stationary process defined by (2.6), where $\xi(t)$ is a Lévy
process satisfying the conditions: $\mathbb{E}\xi(t)=0$,
$\mathbb{E}\xi^{2}(1)=1$ and $\mathbb{E}\xi^{4}(1)<\infty$. In the case where
$\xi(t)=B(t)$, $X(t)$ is a Gaussian process (see, e.g., Bai et al. [6]):

The function $a(\cdot)$ in representations (2.3) and (2.6) plays the role of a
time-invariant filter, and the linear processes defined by (2.3) and (2.6) can
be viewed as the output of a linear filter $a(\cdot)$ applied to the process
$\xi(t)$, called the innovation or driving process of $X(t)$.

Processes of the form (2.3) and (2.6) appear in many fields of science
(economics, finance, physics, etc.), and cover large classes of popular models
in time series modeling. For instance, the classical autoregressive moving
average (ARMA) models and their continuous counterparts the c.t.
autoregressive moving average (CARMA) models are of the form (2.3) and (2.6),
respectively, and play a central role in the representations of stationary
time series (see, e.g., Brockwell [9], Brockwell and Davis [10]).

####  2.1.4 Dependence (memory) structure of the model.

In the frequency domain setting, the statistical and spectral analysis of
stationary processes requires two types of conditions on the spectral density
$f(\lambda).$ The first type controls the singularities of $f(\lambda)$, and
involves the dependence (or memory) structure  of the process, while the
second type – controls the smoothness of $f(\lambda).$ The memory structure of
a stationary process is essentially a measure of the dependence between all
the variables in the process, considering the effect of all correlations
simultaneously. Traditionally memory structure has been defined in the time
domain in terms of decay rates of the autocorrelations, or in the frequency
domain in terms of rates of explosion of low frequency spectra (see, e.g.,
Beran et al. [7], Giraitis et al. [52], Guégan [55]). It is convenient to
characterize the memory structure in terms of the spectral density function.

We will distinguish the following types of stationary models:

(a) short memory (or short-range dependent),

(b) long memory (or long-range dependent),

(c) intermediate memory (or anti-persistent).

Short-memory models. Much of statistical inference is concerned with short-
memory stationary models, where the spectral density $f(\lambda)$ of the model
is bounded away from zero and infinity, that is, there are constants $C_{1}$
and $C_{2}$ such that

$0<C_{1}\leq f(\lambda)\leq C_{2}<\infty.$

A typical d.t. short memory model example is the stationary Autoregressive
Moving Average (ARMA)$(p,q)$ process $X(t)$ defined to be a stationary
solution of the difference equation:

$\psi_{p}(B)X(t)=\theta_{q}(B)\varepsilon(t),\quad t\in\mathbb{Z},$

where $\psi_{p}$ and $\theta_{q}$ are polynomials of degrees $p$ and $q$,
respectively, $B$ is the backshift operator defined by $BX(t)=X(t-1)$, and
$\\{\varepsilon(t),t\in\mathbb{Z}\\}$ is a d.t. white noise, that is, a
sequence of zero-mean, uncorrelated random variables with variance
$\sigma^{2}$. The covariance $r(k)$ of (ARMA)$(p,q)$ process is exponentially
bounded:

$|r(k)|\leq Cr^{-k},\quad k=1,2,\ldots;\quad 0<C<\infty;\,\,0<r<1,$

and the spectral density $f(\lambda)$ is a rational function (see, e.g.,
Brockwell and Davis [10], Section 3.1):

$f(\lambda)=\frac{\sigma^{2}}{2\pi}\cdot\frac{|\theta_{q}(e^{-i\lambda})|^{2}}{|\psi_{p}(e^{-i\lambda})|^{2}}.$
(2.9)

A typical c.t. short-memory model example is the stationary c.t. ARMA$(p,q)$
processes, denoted by CARMA$(p,q)$, The spectral density function $f(\lambda)$
of a CARMA$(p,q)$ process $X(t)$ is given by the following formula (see, e.g.,
Brockwell [9]):

$f(\lambda)=\frac{\sigma^{2}}{2\pi}\cdot\frac{|\beta_{q}(i\lambda)|^{2}}{|\alpha_{p}(i\lambda)|^{2}},$
(2.10)

where $\alpha_{p}(z)=z^{p}-\alpha_{p}z^{(p-1)}-\cdots-\alpha_{1}$ and
$\beta_{q}(z)=1+\beta_{1}z+\cdots+\beta_{q}z^{q}$ are polynomials of degrees
$p$ and $q$, respectively.

Another important c.t. short-memory model is the Ornstein-Uhlenbeck process,
which is a Gaussian stationary process with covariance function
$r(t)=\sigma^{2}e^{-\alpha|t|}$ ($t\in\mathbb{R}$), and spectral density

$f(\lambda)=\frac{\sigma^{2}}{\pi}\cdot\frac{\alpha^{2}}{\lambda^{2}+\alpha^{2}},\quad\alpha>0,\,\lambda\in\mathbb{R}.$
(2.11)

Discrete-time long-memory and anti-persistent models. Data in many fields of
science (economics, finance, hydrology, etc.), however, is well modeled by
stationary processes whose spectral densities are unbounded or vanishing at
some fixed points (see, e.g., Beran et al. [7], Guégan [55], and references
therein).

A long-memory model is defined to be a stationary process with unbounded
spectral density, and an anti-persistent model – a stationary process with
vanishing (at some fixed points) spectral density.

In the discrete context, a basic long-memory model is the Autoregressive
Fractionally Integrated Moving Average (ARFIMA)$(0,d,0))$ process $X(t)$
defined to be a stationary solution of the difference equation (see, e.g.,
Brockwell and Davis [10], Section 13.2):

$(1-B)^{d}X(t)=\varepsilon(t),\quad 0<d<1/2,$

where $B$ is the backshift operator and $\varepsilon(t)$ is a d.t. white noise
defined above. The spectral density $f(\lambda)$ of $X(t)$ is given by

$f(\lambda)=|1-e^{-i\lambda}|^{-2d}=(2\sin(\lambda/2))^{-2d},\quad
0<\lambda\leq\pi,\quad 0<d<1/2.$ (2.12)

Notice that $f(\lambda)\thicksim c\,|\lambda|^{-2d}$ as $\lambda\to 0$, that
is, $f(\lambda)$ blows up at $\lambda=0$ like a power function, which is the
typical behavior of a long memory model.

A typical example of an anti-persistent model is the ARFIMA$(0,d,0)$ process
$X(t)$ with spectral density specified by (2.12) with $d<0$, which vanishes at
$\lambda=0$. Note that the condition $d<1/2$ ensures that
$\int_{-\pi}^{\pi}f(\lambda)d\lambda<\infty$, implying that the process $X(t)$
with spectral density (2.12) is well defined because
$E|X(t)|^{2}=\int_{-\pi}^{\pi}f(\lambda)d\lambda.$

Data can also occur in the form of a realization of a ’mixed’ short-long-
intermediate-memory stationary process $X(t)$. A well-known example of such a
process, which appears in many applied problems, is an ARFIMA$(p,d,q)$ process
$X(t)$ defined to be a stationary solution of the difference equation:

$\psi_{p}(B)(1-B)^{d}X(t)=\theta_{q}(B)\varepsilon(t),\quad d<1/2,$

where $B$ is the backshift operator, $\varepsilon(t)$ is a d.t. white noise,
and $\psi_{p}$ and $\theta_{q}$ are polynomials of degrees $p$ and $q$,
respectively. The spectral density $f_{X}(\lambda)$ of $X(t)$ is given by

$f_{X}(\lambda)=|1-e^{-i\lambda}|^{-2d}f(\lambda),\quad d<1/2,$ (2.13)

where $f(\lambda)$ is the spectral density of an ARMA$(p,q)$ process, given by
(2.9). Observe that for $0<d<1/2$ the model $X(t)$ specified by the spectral
density (2.13) displays long-memory, for $d<0$ – intermediate-memory, and for
$d=0$ – short-memory. For $d\geq 1/2$ the function $f_{X}(\lambda)$ in (2.13)
is not integrable, and thus it cannot represent a spectral density of a
stationary process. Also, if $d\leq-1$, then the series $X(t)$ is not
invertible in the sense that it cannot be used to recover a white noise
$\varepsilon(t)$ by passing $X(t)$ through a linear filter (see, e.g.,
Brockweel and Davis [10]).

Another important long-memory model is the fractional Gaussian noise (fGn). To
define the fGn we first introduce the fractional Brownian motion (fBm)
$\\{B_{H}(t),t\in\mathbb{R}\\}$ with Hurst index $H$, $0<H<1$, defined to be a
centered Gaussian $H$-self-similar process having stationary increments (see,
e.g., Samorodnisky and Taqqu [70]). Then the increment process
$\\{X(k):=B_{H}(k+1)-B_{H}(k),k\in\mathbb{Z}\\},$ called fractional Gaussian
noise (fGn), is a d.t. centered Gaussian stationary process with spectral
density function:

$f(\lambda)=c\,|1-e^{-i\lambda}|^{2}\sum_{k=-\infty}^{\infty}|\lambda+2\pi
k|^{-(2H+1)},\quad-\pi\leq\lambda\leq\pi,$ (2.14)

where $c$ is a positive constant.

It follows from (2.14) that $f(\lambda)\thicksim c\,|\lambda|^{1-2H}$ as
$\lambda\to 0$, that is, $f(\lambda)$ blows up if $H>1/2$ and tends to zero if
$H<1/2$. Also, comparing (2.12) and (2.14), we observe that, up to a constant,
the spectral density of fGn has the same behavior at the origin as
ARFIMA$(0,d,0)$ with $d=H-1/2.$

Thus, the fGn $\\{X(k),k\in\mathbb{Z}\\}$ has long-memory if $1/2<H<1$ and is
anti-percipient if $0<H<1/2$. The variables $X(k)$, $k\in\mathbb{Z}$, are
independent if $H=1/2$. For more details we refer to Samorodnisky and Taqqu
[70].

Continuous-time long-memory and anti-persistent models. In the continuous
context, a basic process which has commonly been used to model long-range
dependence is the fractional Brownian motion (fBm) $B_{H}$ with Hurst index
$H$, defined above, which can be regarded as a Gaussian process having a
’spectral density’:

$f(\lambda)=c|\lambda|^{-(2H+1)},\quad c>0,\ \,\,0<H<1,\
\,\,\lambda\in\mathbb{R}.$ (2.15)

The form (2.15) can be understood in a generalized sense (see, e.g., Yaglom
[77]), since the fBm $B_{H}$ is a nonstationary process.

A proper stationary model in lieu of fBm is the fractional Riesz-Bessel motion
(fRBm), introduced in Anh et al. [2], and defined as a c.t. Gaussian process
$X(t)$ with spectral density

$f(\lambda)=c\,|\lambda|^{-2\alpha}(1+\lambda^{2})^{-\beta},\quad\lambda\in\mathbb{R},\,0<c<\infty,\,0<\alpha<1,\,\beta>0.$
(2.16)

The exponent $\alpha$ determines the long-range dependence, while the exponent
$\beta$ indicates the second-order intermittency of the process (see, e.g.,
Anh et al. [3] and Gao et al. [32]).

Notice that the process $X(t)$, specified by the spectral density (2.16), is
stationary if $0<\alpha<1/2$ and is non-stationary with stationary increments
if $1/2\leq\alpha<1.$ Observe also that the function $f(\lambda)$ in (2.16)
behaves as $O(|\lambda|^{-2\alpha})$ as $|\lambda|\to 0$ and as
$O(|\lambda|^{-2(\alpha+\beta)})$ as $|\lambda|\to\infty$. Thus, under the
conditions $0<\alpha<1/2$, $\beta>0$ and $\alpha+\beta>1/2,$ the function
$f(\lambda)$ in (2.16) is well-defined for both $|\lambda|\to 0$ and
$|\lambda|\to\infty$ due to the presence of the component
$(1+\lambda^{2})^{-\beta}$, $\beta>0$, which is the Fourier transform of the
Bessel potential.

Comparing (2.15) and (2.16), we observe that the spectral density of fBm is
the limiting case as $\beta\to 0$ that of fRBm with Hurst index
$H=\alpha-1/2.$

Another important c.t. long-memory model is the CARFIMA$(p,H,q)$ process. The
spectral density function $f(\lambda)$ of a CARFIMA$(p,H,q)$ process $X(t)$ is
given by the following formula (see, e.g., Brockwell [9], and Tsai and Chan
[74]):

$f(\lambda)=\frac{\sigma^{2}}{2\pi}\Gamma(2H+1)\sin(\pi
H)|\lambda|^{1-2H}\frac{|\beta_{q}(i\lambda)|^{2}}{|\alpha_{p}(i\lambda)|^{2}},$
(2.17)

where $\alpha_{p}(z)=z^{p}-\alpha_{p}z^{(p-1)}-\cdots-\alpha_{1}$ and
$\beta_{q}(z)=1+\beta_{1}z+\cdots+\beta_{q}z^{q}$ are polynomials of degrees
$p$ and $q$, respectively. Notice that for $H=1/2$, the spectral density given
by (2.17) becomes that of the short-memory CARMA$(p,q)$ process, given by
(2.10).

###  2.2 Data tapers and tapered periodogram

Our inference procedures will be based on the tapered data
$\mathbf{X}_{T}^{h}$:

$\mathbf{X}_{T}^{h}:=\left\\{\begin{array}[]{ll}\\{h_{T}(t)X(t),\,t=1,\ldots,T\\}&\mbox{in
the d.t.\ case},\\\ \\{h_{T}(t)X(t),\,0\leq t\leq T\\}&\mbox{in the c.t.\
case},\end{array}\right.$ (2.18)

where

$h_{T}(t):=h(t/T)$ (2.19)

with $h(t)$, $t\in\mathbb{R}$ being a taper function.

Throughout the paper, we will assume that the taper function $h(\cdot)$
satisfies the following assumption.

######  Assumption 2.1.

The taper $h:\mathbb{R}\to\mathbb{R}$ is a continuous nonnegative function of
bounded variation and of bounded support $[0,1]$, such that $H_{k}\neq 0$,
where

$H_{k}:=\int_{0}^{1}h^{k}(t)dt,\quad k\in\mathbb{N}:=\\{1,2,\ldots\\}.$ (2.20)

Note. The case $h(t)={\mathbb{I}}_{[0,1]}(t)$, where
${\mathbb{I}}_{[0,1]}(\cdot)$ denotes the indicator of the segment $[0,1]$,
will be referred to as the non-tapered case.

######  Remark 2.1.

For the d.t. case, an example of a taper function $h(t)$ satisfying Assumption
2.1 is the Tukey-Hanning taper function $h(t)=0.5(1-\cos(\pi t))$ for
$t\in[0,1]$. For the c.t. case, a simple example of a taper function $h(t)$
satisfying Assumption 2.1 is the function $h(t)=1-t$ for $t\in[0,1]$. More
examples of taper functions satisfying Assumption 2.1 can be found in Dahlhaus
[22] and in Guyon [56].

Denote by $H_{k,T}(\lambda)$ the tapered Dirichlet type kernel, defined by

$H_{k,T}(\lambda):=\left\\{\begin{array}[]{ll}\sum_{t=1}^{T}h_{T}^{k}(t)e^{-i\lambda
t}&\mbox{in the d.t.\ case},\\\ \\\\[-2.84526pt]
\int_{0}^{T}h_{T}^{k}(t)e^{-i\lambda t}dt&\mbox{in the c.t.\
case}.\end{array}\right.$ (2.21)

Define the finite Fourier transform of the tapered data (2.18):

$d^{h}_{T}(\lambda):=\left\\{\begin{array}[]{ll}\sum_{t=0}^{T}h_{T}(t)X(t)e^{-i\lambda
t}&\mbox{in the d.t.\ case},\\\ \\\\[-2.84526pt]
\int_{0}^{T}h_{T}(t)X(t)e^{-i\lambda t}dt&\mbox{in the c.t.\
case}.\end{array}\right.$ (2.22)

and the tapered periodogram $I^{h}_{T}(\lambda)$ of the process $X(t)$:

$\displaystyle
I^{h}_{T}(\lambda):=\frac{1}{C_{T}}\,d^{h}_{T}(\lambda)d^{h}_{T}(-\lambda)=\left\\{\begin{array}[]{ll}\frac{1}{C_{T}}\left|\sum_{t=0}^{T}h_{T}(t)X(t)e^{-i\lambda
t}\right|^{2}&\mbox{in the d.t.\ case},\\\ \\\\[-2.84526pt]
\frac{1}{C_{T}}\left|\int_{0}^{T}h_{T}(t)X(t)e^{-i\lambda
t}dt\right|^{2}&\mbox{in the c.t.\ case}.\end{array}\right.$ (2.26)

where

$C_{T}:=2\pi H_{2,T}(0)=2\pi\int_{0}^{T}h_{T}^{2}(t)dt=2\pi H_{2}\,T\neq 0.$
(2.27)

Notice that for non-tapered case ($h(t)={\mathbb{I}}_{[0,1]}(t)$), we have
$C_{T}=2\pi T$.

##  3 Nonparametric Estimation Problem

Suppose we observe a finite realization $\mathbf{X}_{T}:=\\{X(u)$, $0\leq
u\leq T$ (or $u={1,\ldots,T}$ in the d.t. case)} of a centered stationary
process $X(u)$ with an unknown spectral density function $f(\lambda)$,
$\lambda\in\Lambda$. We assume that $f(\lambda)$ belongs to a given (infinite-
dimensional) class $\mathcal{F}\subset L^{p}:=L^{p}(\Lambda)$ $(p\geq 1)$ of
spectral densities possessing some specified smoothness properties. The
problem is to estimate the value $J(f)$ of a given functional $J(\cdot)$ at an
unknown ’point’ $f\in\mathcal{F}$ on the basis of an observation
$\mathbf{X}_{T},$ and investigate the asymptotic (as $T\to\infty$) properties
of the suggested estimators, depending on the dependence structure of the
model $X(u)$ and the smoothness structure of the ’parametric’ set
$\mathcal{F}\subset L^{p}(\Lambda)$ $(p\geq 1)$.

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 functionals (see,
e.g., Anh et al. [4], Dzhaparidze [27], Guyon [56], Leonenko and Sakhno [65],
Taniguchi and Kakizawa [72], and references therein). In this section, we
review the asymptotic properties, involving asymptotic unbiasedness, bias rate
convergence, consistency, a central limit theorem and asymptotic normality of
the empirical spectral functionals based on the tapered data. Some of these
properties were discussed and proved in Ginovyan and Sahakyan [49, 50]. For
non-tapered case, these properties were established in the papers Ginovyan
[35, 39]. The results stated in this section are used to prove consistency and
asymptotic normality of the minimum contrast estimator based on the Whittle
contrast functional for stationary linear models with tapered data (see
Section 4). Here we follow the papers Ginovyan [36, 39, 40], and Ginovyan and
Sahakyan [49, 50].

###  3.1 Estimation of linear spectral functionals

We are interested in the nonparametric estimation problem, based on the
tapered data (2.18), of the following linear spectral functional:

$J=J(f,g):=\int_{\Lambda}f(\lambda)g(\lambda)d\lambda,$ (3.1)

where $g(\lambda)\in L^{q}(\Lambda)$, $1/p+1/q=1$.

As an estimator $J_{T}^{h}$ for functional $J(f)$, given by (3.1), based on
the tapered data (2.18), we consider the averaged tapered periodogram (or a
simple ’plug-in’ statistic), defined by

$\displaystyle J_{T}^{h}$ $\displaystyle=$ $\displaystyle
J(I^{h}_{T}):=\int_{\Lambda}I^{h}_{T}(\lambda)g(\lambda)d\lambda,$ (3.2)

where $I^{h}_{T}(\lambda)$ is the tapered periodogram of the process $X(t)$
given by (2.26). Denote

$Q_{T}^{h}:=\left\\{\begin{array}[]{ll}\sum_{t=1}^{T}\sum_{s=1}^{T}\widehat{g}(t-s)h_{T}(t)h_{T}(s)X(t)X(s)&\mbox{in
the d.t.\ case},\\\ \\\
\int_{0}^{T}\int_{0}^{T}\widehat{g}(t-s)h_{T}(t)h_{T}(s)X(t)X(s)\,dt\,ds&\mbox{in
the c.t.\ case},\end{array}\right.$ (3.3)

where $\widehat{g}(t)$ is the Fourier transform of function $g(\lambda)$:

$\widehat{g}(t):=\int_{\Lambda}e^{i\lambda t}g(\lambda)d\lambda,\quad
t\in\Lambda.$ (3.4)

In view of (2.26) and (3.2) – (3.4) we have

$\displaystyle J_{T}^{h}=C_{T}^{-1}Q_{T}^{h},$ (3.5)

where $C_{T}$ is as in (2.27). We will refer to $g(\lambda)$ and to its
Fourier transform $\widehat{g}(t)$ as a generating function and generating
kernel for the functional $J_{T}^{h}$, respectively.

Thus, to study the asymptotic properties of the estimator $J_{T}^{h}$, we have
to study the asymptotic distribution (as $T\to\infty$) of the tapered Toeplitz
type quadratic functional $Q_{T}^{h}$ given by (3.3) (for details see Section
7.2).

###  3.2 Asymptotic unbiasedness

We begin with the following assumption.

######  Assumption 3.1.

The function

$\displaystyle\Psi(u)=\int_{\Lambda}f(v)g(u+v)\,dv$ (3.6)

belongs to $L^{1}(\Lambda)\cap L^{2}(\Lambda)$ and is continuous at $u=0$.

######  Theorem 3.1.

Let the functionals $J:=J(f,g)$ and $J_{T}^{h}:=J(I^{h}_{T},g)$ be defined by
(3.1) and (3.2), respectively. Then under Assumptions 2.1 and 3.1 the
statistic $J_{T}^{h}$ is an asymptotically unbiased estimator for $J(f)$, that
is, the following relation holds:

$\displaystyle\lim_{T\to\infty}[E(J_{T}^{h})-J]=0.$ (3.7)

######  Remark 3.1.

Using Hölder inequality, it can easily be shown that if $f\in
L^{1}(\Lambda)\cap L^{p_{1}}(\Lambda)$ and $g\in L^{1}(\Lambda)\cap
L^{p_{2}}(\Lambda)$ with $1\leq p_{1},p_{2}\leq\infty$, $1/p_{1}+1/p_{2}\leq
1$, then the relation (3.7) is satisfied.

Under additional smoothness conditions on functions $f(\lambda)$ and
$g(\lambda)$ we can estimate the rate of convergence in (3.7). To state the
corresponding result, we first introduce some notation and assumptions.

Given numbers $p\geq 1$, $0<\alpha<1$,
$r\in\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$, where $\mathbb{N}$ is the set of
natural numbers, we set $\beta=\alpha+r$ and denote by $H_{p}(\beta)$ the
$L^{p}$-Hölder class, that is, the class of those functions $\psi(\lambda)\in
L^{p}(\Lambda)$, which have $r$-th derivatives in $L^{p}(\Lambda)$ and with
some positive constant $C$ satisfy

$||\psi^{(r)}(\cdot+h)-\psi^{(r)}(\cdot)||_{p}\leq C|h|^{\alpha}.$

######  Assumption 3.2.

We say that a pair of integrable functions $(f(\lambda),g(\lambda))$,
$\lambda\in\Lambda$, satisfies condition $(\mathcal{H})$, and write
$(f,g)\in(\mathcal{H})$, if $f\in H_{p}(\beta_{1})$ for $\beta_{1}>0$, $p>1$
and $g\in H_{q}(\beta_{2})$ for $\beta_{2}>0$, $q>1$ with $1/p+1/q=1$, and one
of the conditions a) – d) is satisfied:

a) $\beta_{1}>1/p$, $\beta_{2}>1/q$,

b) $\beta_{1}\leq 1/p$, $\beta_{2}\leq 1/q$ and $\beta_{1}+\beta_{2}>1/2$,

c) $\beta_{1}>1/p$, $1/q-1/2<\beta_{2}\leq 1/q$,

d) $\beta_{2}>1/q$, $1/p-1/2<\beta_{1}\leq 1/p$.

######  Remark 3.2.

In Ginovian [35] it was proved that if $(f,g)\in(\mathcal{H})$, then there
exist numbers $p_{1}$ $(p_{1}>p)$ and $q_{1}$ $(q_{1}>q)$, such that
$H_{p}(\beta_{1})\subset L_{p_{1}}$, $H_{q}(\beta_{2})\subset L_{q_{1}}$ and
$1/{p_{1}}+1/{q_{1}}\leq 1/2$.

######  Assumption 3.3.

The spectral density $f$ and the generating function $g$ are such that $f,g\in
L^{1}(\Lambda)\cap L^{2}(\Lambda)$ and $g$ is of bounded variation.

The following theorem controls the bias $E(J_{T}^{h})-J$ and provides
sufficient conditions assuring the proper rate of convergence of bias to zero,
necessary for asymptotic normality of the estimator $J_{T}^{h}$. Specifically,
we have the following result.

######  Theorem 3.2.

Let the functionals $J:=J(f,g)$ and $J_{T}^{h}:=J(I^{h}_{T},g)$ be defined by
(3.1) and (3.2), respectively. Then under Assumptions 2.1 and 3.2 (or 3.3),
the following asymptotic relation holds:

$\displaystyle T^{1/2}\left[\mathbb{E}(J_{T}^{h})-J\right]\to 0\quad{\rm
as}\quad T\to\infty.$ (3.8)

######  Remark 3.3.

We call an estimator $J_{T}^{h}$ of $J$ asymptotically unbiased of the order
of $T^{\beta}$, $\beta>0$ if $\lim_{T\to\infty}T^{\beta}[E(J_{T}^{h})-J]=0.$
Thus, Theorem 3.2 states that the statistic $J_{T}^{h}$ is an asymptotically
unbiased estimator for $J$ of the order of $T^{1/2}$.

###  3.3 Consistency

Recall that an estimator $J_{T}^{h}$ of $J$ is said to be (a) consistent if
$J_{T}^{h}\to J$ in probability as $T\to\infty$, (b) mean square consistent if
$\mathbb{E}(J_{T}^{h}-J)^{2}\to 0$ as $T\to\infty$, (c) $\sqrt{T}$-consistent
in the mean square sense if
$\mathbb{E}\left([\sqrt{T}(J_{T}^{h}-J)]^{2}\right)=O(1)$ as $T\to\infty$,

To state the corresponding results we first introduce the following
assumption.

######  Assumption 3.4.

The filter $a(\cdot)$ and the generating kernel $\widehat{g}(\cdot)$ are such
that

$a(\cdot)\in L^{p}(\Lambda)\cap L^{2}(\Lambda),\quad\widehat{g}(\cdot)\in
L^{q}(\Lambda)\quad{\rm with}\quad 1\leq p,q\leq
2,\quad\frac{2}{p}+\frac{1}{q}\geq\frac{5}{2}.$

We begin with results on the asymptotic behavior of the variance
$\mathrm{Var}(J_{T}^{h})=\mathbb{E}(J_{T}^{h}-\mathbb{E}(J_{T}^{h}))^{2}$. The
proof of the next theorem can be found in Ginovyan and Sahakyan [49].

######  Theorem 3.3.

Let the functionals $J:=J(f,g)$ and $J_{T}^{h}:=J(I^{h}_{T},g)$ be defined by
(3.1) and (3.2), respectively. Then under Assumptions 2.1 and 3.4 the
following asymptotic relation holds:

$\displaystyle\lim_{T\to\infty}T\mathrm{Var}(J_{T}^{h})=\sigma^{2}_{h}(J),$
(3.9)

where

$\sigma^{2}_{h}(J):=4\pi
e(h)\int_{\Lambda}f^{2}(\lambda)g^{2}(\lambda)d\lambda+\kappa_{4}e(h)\left[\int_{\Lambda}f(\lambda)g(\lambda)d\lambda\right]^{2}.$
(3.10)

Here $\kappa_{4}$ is the fourth cumulant of $\xi(1)$, and

$e(h):=\frac{H_{4}}{H_{2}^{2}}=\int_{0}^{1}h^{4}(t)dt\left(\int_{0}^{1}h^{2}(t)dt\right)^{-2}.$
(3.11)

From Theorems 3.1–3.3 we infer the following result.

######  Theorem 3.4.

The following assertions hold.

  * (a)

Under Assumptions 2.1, 3.1 and 3.4 the statistic $J_{T}^{h}$ is a mean square
consistent estimator for $J$.

  * (b)

Under Assumptions 2.1, 3.2 (or 3.3) and 3.4 the statistic $J_{T}^{h}$ is a
$\sqrt{T}$-consistent in the mean square sense estimator for $J$.

###  3.4 Asymptotic normality

The next result contains sufficient conditions for functional $J_{T}^{h}$ to
obey the central limit theorem (CLT), and was proved in Ginovyan and Sahakyan
[49].

######  Theorem 3.5 (CLT).

Let $J:=J(f,g)$ and $J_{T}^{h}:=J(I^{h}_{T},g)$ be defined by (3.1) and (3.2),
respectively. Then under Assumptions 2.1 and 3.4 the functional $J_{T}^{h}$
obeys the central limit theorem. More precisely, we have

$\displaystyle
T^{1/2}\left[J_{T}^{h}-\mathbb{E}(J_{T}^{h})\right]\overset{d}{\rightarrow}\eta\quad{\rm
as}\quad T\to\infty,$ (3.12)

where the symbol $\overset{d}{\rightarrow}$ stands for convergence in
distribution, and $\eta$ is a normally distributed random variable with mean
zero and variance $\sigma^{2}_{h}(J)$ given by (3.10) and (3.11).

Taking into account the equality

$\displaystyle
T^{1/2}\left[J_{T}^{h}-J\right]=T^{1/2}\left[\mathbb{E}(J_{T}^{h})-J\right]+T^{1/2}\left[J_{T}^{h}-\mathbb{E}(J_{T}^{h})\right],$
(3.13)

as an immediate consequence of Theorems 3.2 and 3.5, 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 $J$.

######  Theorem 3.6.

Let the functionals $J:=J(f,g)$ and $J_{T}^{h}:=J(I^{h}_{T},g)$ be defined by
(3.1) and (3.2), respectively. Then under Assumptions 2.1, 3.2 (or 3.3) and
3.4 the statistic $J_{T}^{h}$ is an asymptotically normal estimator for
functional $J$. More precisely, we have

$\displaystyle
T^{1/2}\left[J_{T}^{h}-J\right]\overset{d}{\rightarrow}\eta\quad{\rm as}\quad
T\to\infty,$ (3.14)

where $\eta$ is as in Theorem 3.5, that is, $\eta$ is a normally distributed
random variable with mean zero and variance $\sigma^{2}_{h}(J)$ given by
(3.10) and (3.11).

######  Remark 3.4.

Notice that if the underlying process $X(u)$ is Gaussian, then in formula
(3.10) we have only the first term. Using the results from Ginovyan [35] and
Ginovyan and Sahakyan [44, 45], it can be shown that in this case Theorem 3.6
is true under Assumptions 2.1 and 3.4.

######  Example 3.1 (Estimation of covariance function).

Assume that $X(t)$ is a c.t. process, and let $g(\lambda)=e^{iu\lambda}$, then

$J(f)=\int_{-\infty}^{\infty}e^{iu\lambda}f(\lambda)\,d\lambda:=r(u).$

Thus, in this special case our problem becomes to the estimation of the the
covariance function $r(u)=\mathbb{E}[X(t+u)X(t)]$ of the process $X(t)$. By
Theorem 3.6 the simple ”plug-in” statistic

$\widehat{J}_{T}^{h}=\widehat{r}_{T}(u)=\int_{-\infty}^{\infty}e^{iu\lambda}I_{T}^{h}(\lambda)\,d\lambda$

is asymptotically normal estimator for $r(u)$ with asymptotic variance

$\sigma_{u}^{2}\,=\,4\pi
e(h)\int_{-\infty}^{\infty}\cos^{2}(u\lambda)f^{2}(\lambda)\,d\lambda,$

where $e(h)$ is given by (3.11).

######  Example 3.2 (Estimation of spectral function).

Assume that $X(t)$ is a d.t. process, and let
$g(\lambda)=\chi_{[0,\mu]}(\lambda)$ be the indicator of an interval
$[0,\mu]$, then

$J(f)=\int_{-\pi}^{\pi}\chi_{[0,\mu]}(\lambda)f(\lambda)\,d\lambda=\int_{0}^{\mu}f(\lambda)\,d\lambda:=F(\mu).$

Thus, in this case the estimand functional is the spectral function $F(\mu)$
of the process $X(u),$ and by Theorem 3.6 the simple ”plug-in” statistic

$\widehat{J}_{T}^{h}=\widehat{F}_{T}(\mu)=\int_{0}^{\mu}I_{T}^{h}(\lambda)\,d\lambda$
(3.15)

is asymptotically normal estimator for $F(\mu)$ with asymptotic variance

$\sigma^{2}(\mu)=4\pi e(h)\int_{0}^{\mu}f^{2}(s)\,ds,$

where $e(h)$ is given by (3.11).

##  4 Parametric Estimation Problem

We assume here that the spectral density $f(\lambda)$ belongs to a given
parametric family of spectral densities
$\mathcal{F}:=\\{f(\lambda,\theta):\,\theta\in\Theta\\}$, where
$\theta:=(\theta_{1},\ldots,\theta_{p})$ is an unknown parameter and $\Theta$
is a subset in the Euclidean space $\mathbb{R}^{p}$. The problem of interest
is to estimate the unknown parameter $\theta$ on the basis of the tapered data
(2.18), and investigate the asymptotic (as $T\to\infty$) properties of the
suggested estimators, depending on the dependence (memory) structure of the
model $X(t)$ and the smoothness of its spectral density $f$.

There are different methods of estimation: maximum likelihood, Whittle,
minimum contrast, etc. Here we focus on the Whittle method.

###  4.1 The Whittle estimation procedure

The Whittle estimation procedure, originally devised for d.t. short memory
stationary processes, is based on the smoothed periodogram analysis on a
frequency domain, involving approximation of the likelihood function and
asymptotic properties of empirical spectral functionals (see Whittle [76]).
The Whittle estimation method since its discovery has played a major role in
the asymptotic theory of parametric estimation in the frequency domain, and
was the focus of interest of many statisticians. Their aim was to weaken the
conditions needed to guarantee the validity of the Whittle approximation for
d.t. short memory models, to find analogues for long and intermediate memory
models, to find conditions under which the Whittle estimator is asymptotically
equivalent to the exact maximum likelihood estimator, and to extend the
procedure to the c.t. models and random fields.

For the d.t. case, it was shown that for Gaussian and linear stationary models
the Whittle approach leads to consistent and asymptotically normal estimators
under short, intermediate and long memory assumptions. Moreover, it was shown
that in the Gaussian case the Whittle estimator is also asymptotically
efficient in the sense of Fisher (see, e. g., Dahlhaus [21], Dzhaparidze [26],
Fox and Taqqu [30], Giraitis and Surgailis [53], Guyon [56], Heyde and Gay
[60], Taniguchi and Kakizawa [72], Walker [75], and references therein).

For c.t. models, the Whittle estimation procedure has been considered, for
example, in Anh et al. [4], Avram et al. [5], Casas and Gao [11], Dzhaparidze
[26], Dzhaparidze and Yaglom [29], Gao [31], Gao et al. [32], Leonenko and
Sakhno [65], Tsai and Chan [74], where can also be found additional
references. In this case, it was proved that the Whittle estimator is
consistent and asymptotically normal.

The Whittle estimation procedure based on the d.t. tapered data has been
studied in Alomari et al. [1], Dahlhaus [17], Dahlhaus and Künsch [23], Guyon
[56], Ludeña and Lavielle [66]. In the case where the underlying model is a
Lévy-driven c.t. linear process with possibly unbounded or vanishing spectral
density function, consistency and asymptotic normality of the Whittle
estimator was established in Ginovyan [42].

To explain the idea behind the Whittle estimation procedure, assume for
simplicity that the underlying process $X(t)$ is a d.t. Gaussian process, and
we want to estimate the parameter $\theta$ based on the sample
$X_{T}:=\\{X(t),\,t=1,\ldots,T\\}$. A natural approach is to find the maximum
likelihood estimator (MLE) $\widehat{\theta}_{T,MLE}$ of $\theta$, that is, to
maximize the likelihood function, or to minimize the $-1/T\times$log-
likelihood function $L_{T}(\theta)$, which in this case takes the form:

$L_{T}(\theta):=\frac{1}{2}\ln 2\pi+\frac{1}{2T}\ln\det
B_{T}(f_{\theta})+\frac{1}{2T}X^{\prime}_{T}[B_{T}(f_{\theta})]^{-1}X_{T},$

where $B_{T}(f_{\theta})$ is the Toeplitz matrix generated by $f_{\theta}$.
Unfortunately, the above function is difficult to handle, and no explicit
expression for the estimator $\widehat{\theta}_{T,MLE}$ is known (even in the
case of simple models). An approach, suggested by P. Whittle, called the
Whittle estimation procedure, is to approximate the term $\ln\det
B_{T}(f_{\theta})$ by $\frac{T}{2}\int_{-\pi}^{\pi}\ln
f_{\theta}(\lambda)d\lambda$ and the inverse matrix $[B_{T}(f_{\theta})]^{-1}$
by the Toeplitz matrix $B_{T}(1/f_{\theta})$. This leads to the following
approximation of the log-likelihood function $L_{T}(\theta)$, introduced by
Whittle [76], and called Whittle functional:

$L_{T,W}(\theta)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\left[\ln
f_{\theta}(\lambda)+\frac{I_{T}(\lambda)}{f_{\theta}(\lambda)}\right]\,d\lambda,$

where $I_{T}(\lambda)$ is the ordinary periodogram of the process $X(t)$.

Now minimizing the Whittle functional $L_{T,W}(\theta)$ with respect to
$\theta$, we get the Whittle estimator $\widehat{\theta}_{T}$ for $\theta$. It
can be shown that if

$T^{1/2}(L_{T}(\theta)-L_{T,W}(\theta)\to 0\quad{\rm as}\quad
n\to\infty\quad{\rm in\,\,probability,}$

then the MLE $\widehat{\theta}_{T,MLE}$ and the Whittle estimator
$\widehat{\theta}_{T}$ are asymptotically equivalent in the sense that
$\widehat{\theta}_{T}$ also is consistent, asymptotically normal and
asymptotically Fisher-efficient (see, e.g., Coursol and Dacunha-Castelle [14],
Dzhaparidze [26], and Dzhaparidze and Yaglom [29]).

In the continuous context, the Whittle procedure of estimation of a spectral
parameter $\theta$ based on the sample $X_{T}:=\\{X(t),\,0\leq t\leq T\\}$ is
to choose the estimator $\widehat{\theta}_{T}$ to minimize the weighted
Whittle functional:

$U_{T}(\theta):=\frac{1}{4\pi}\int_{\mathbb{R}}\left[\ln
f(\lambda,\theta)+\frac{I_{T}(\lambda)}{f(\lambda,\theta)}\right]\cdot
w(\lambda)\,d\lambda,$ (4.1)

where $I_{T}(\lambda)$ is the continuous periodogram of $X(t)$, and
$w(\lambda)$ is a weight function ($w(-\lambda)=w(\lambda)$, $w(\lambda)\geq
0$, $w(\lambda)\in L^{1}(\mathbb{R})$) for which the integral in (4.1) is well
defined. An example of common used weight function is
$w(\lambda)=1/(1+\lambda^{2})$.

The Whittle procedure of estimation of a spectral parameter $\theta$ based on
the tapered sample (2.18) is to choose the estimator $\widehat{\theta}_{T,h}$
to minimize the weighted tapered Whittle functional:

$U_{T,h}(\theta):=\frac{1}{4\pi}\int_{\Lambda}\left[\log
f(\lambda,\theta)+\frac{I^{h}_{T}(\lambda)}{f(\lambda,\theta)}\right]\cdot
w(\lambda)\,d\lambda,$ (4.2)

where $I^{h}_{T}(\lambda)$ is the tapered periodogram of $X(t)$, given by
(2.26), and $w(\lambda)$ is a weight function for which the integral in (4.2)
is well defined. Thus,

$\widehat{\theta}_{T,h}:=\underset{\theta\in\Theta}{\rm
Arg\,min}\,U_{T,h}(\theta).$ (4.3)

###  4.2 Asymptotic properties of the Whittle estimator

To state results involving properties of the Whittle estimator, we first
introduce the following set of assumptions.

######  Assumption 4.1.

The true value $\theta_{0}$ of the parameter $\theta$ belongs to a compact set
$\Theta$, which is contained in an open set $S$ in the $p$-dimensional
Euclidean space $\mathbb{R}^{p}$, and $f(\lambda,\theta_{1})\neq
f(\lambda,\theta_{2})$ whenever $\theta_{1}\neq\theta_{2}$ almost everywhere
in $\Lambda$ with respect to the Lebesgue measure.

######  Assumption 4.2.

The functions $f(\lambda,\theta)$, $f^{-1}(\lambda,\theta)$ and
$\frac{\partial}{\partial\theta_{k}}f^{-1}(\lambda,\theta)$, $k=1,\ldots,p$,
are continuous in $(\lambda,\theta)$.

######  Assumption 4.3.

The functions $f:=f(\lambda,\theta)$ and
$g:=w(\lambda)\frac{\partial}{\partial\theta_{k}}f^{-1}(\lambda,\theta)$
satisfy Assumptions 3.3 or 3.4 for all $k=1,\ldots,p$ and $\theta\in\Theta$.

######  Assumption 4.4.

The functions $a:=a(\lambda,\theta)$ and $b:=\widehat{g}$, where $g$ is as in
Assumption 4.3, satisfy Assumption 3.1.

######  Assumption 4.5.

The functions
$\frac{\partial^{2}}{\partial\theta_{k}\partial\theta_{j}}f^{-1}(\lambda,\theta)$
and
$\frac{\partial^{3}}{\partial\theta_{k}\partial\theta_{j}\partial\theta_{j}}f^{-1}(\lambda,\theta)$,
$k,j,l=1,\ldots,p$, are continuous in $(\lambda,\theta)$ for
$\lambda\in\Lambda$, $\theta\in N_{\delta}(\theta_{0})$, where
$N_{\delta}(\theta_{0}):=\\{\theta:\,|\theta-\theta_{0}|<\delta\\}$ is some
neighborhood of $\theta_{0}$.

######  Assumption 4.6.

The matrices

$\displaystyle W(\theta):=\|w_{ij}(\theta)\|_{i,j=1,\ldots,p},\quad
A(\theta):=\|a_{ij}(\theta)\|_{i,j=1,\ldots,p},\quad
B(\theta):=\|b_{ij}(\theta)\|_{i,j=1,\ldots,p}$ (4.4)

are positive definite, where

$\displaystyle w_{ij}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int_{\Lambda}\frac{\partial}{\partial\theta_{i}}\ln
f(\lambda,\theta)\frac{\partial}{\partial\theta_{j}}\ln
f(\lambda,\theta)w(\lambda)d\lambda,$ (4.5) $\displaystyle a_{ij}(\theta)$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int_{\Lambda}\frac{\partial}{\partial\theta_{i}}\ln
f(\lambda,\theta)\frac{\partial}{\partial\theta_{j}}\ln
f(\lambda,\theta)w^{2}(\lambda)d\lambda,$ (4.6) $\displaystyle b_{ij}(\theta)$
$\displaystyle=$
$\displaystyle\frac{\kappa_{4}}{16\pi^{2}}\int_{\Lambda}\frac{\partial}{\partial\theta_{i}}\ln
f(\lambda,\theta)w(\lambda)d\lambda\int_{\mathbb{R}}\frac{\partial}{\partial\theta_{j}}\ln
f(\lambda,\theta)w(\lambda)d\lambda,$ (4.7)

and $\kappa_{4}$ is the fourth cumulant of $\xi(1)$.

The next theorem contains sufficient conditions for Whittle estimator to be
consistent (see Ginovyan [42]).

######  Theorem 4.1.

Let $\widehat{\theta}_{T,h}$ be the Whittle estimator defined by (4.3) and let
$\theta_{0}$ be the true value of parameter $\theta$. Then, under Assumptions
4.1–4.4 and 2.1, the statistic $\widehat{\theta}_{T,h}$ is a consistent
estimator for $\theta$, that is, $\widehat{\theta}_{T,h}\to\theta_{0}$ in
probability as $T\to\infty$.

Having established the consistency of the Whittle estimator
$\widehat{\theta}_{T,h}$, we can go on to obtain the limiting distribution of
$T^{1/2}\left(\widehat{\theta}_{T,h}-\theta_{0}\right)$ in the usual way by
applying the Taylor’s formula, the mean value theorem, and Slutsky’s
arguments. Specifically we have the following result, showing that under the
above assumptions, the Whittle estimator $\widehat{\theta}_{T,h}$ is
asymptotically normal (see Ginovyan [42]).

######  Theorem 4.2.

Suppose that Assumptions 4.1–4.6 and 2.1 are satisfied. Then the Whittle
estimator $\widehat{\theta}_{T,h}$ of an unknown spectral parameter $\theta$
based on the tapered data (2.18) is asymptotically normal. More precisely, we
have

$\displaystyle
T^{1/2}\left(\widehat{\theta}_{T,h}-\theta_{0}\right)\overset{d}{\rightarrow}N_{p}\left(0,e(h)\Gamma(\theta_{0})\right)\quad{\rm
as}\quad T\to\infty,$ (4.8)

where $N_{p}(\cdot,\cdot)$ denotes the $p$-dimensional normal law,
$\overset{d}{\rightarrow}$ stands for convergence in distribution,

$\displaystyle\Gamma(\theta_{0})=W^{-1}(\theta_{0})\left(A(\theta_{0})+B(\theta_{0})\right)W^{-1}(\theta_{0}),$
(4.9)

where the matrices $W$, $A$ and $B$ are defined in (4.4)-(4.7), and the
tapering factor $e(h)$ is given by formula (3.11).

######  Remark 4.1.

In the d.t. case as a weight function we take $w(\lambda)\equiv 1$, and the
matrices $A(\theta_{0})$ and $W(\theta_{0})$ coincide (see (4.4) – (4.6)). So,
in this case, formula (4.9) becomes
$\Gamma(\theta_{0})=W^{-1}(\theta_{0})\left(W(\theta_{0})+B(\theta_{0})\right)W^{-1}(\theta_{0})$.
If, in addition, the underlying process is Gaussian ($\kappa_{4}=0$, and hence
$B(\theta_{0})=0$), and the taper $h$ is chosen so that the tapering factor
$e(h)$ is equal to one, then we have $\Gamma(\theta_{0})=W^{-1}(\theta_{0})$,
that is, the Whittle estimator $\widehat{\theta}_{T,h}$ is Fisher-efficient.

##  5 Goodness-of-fit tests

In this section we consider the following problem of hypotheses testing.

Based on the tapered sample $\mathbf{X}_{T}^{h}$ given by (2.18), we want to
construct goodness-of-fit tests for testing a hypothesis $H_{0}$ that the
spectral density of the process $X(t)$ has the specified form $f(\lambda)$. We
will distinguish the following two cases.

  * a)

The hypothesis $H_{0}$ is simple, that is, the hypothetical spectral density
$f(\lambda)$ of $X(t)$ does not depend on unknown parameters.

  * b)

The hypothesis $H_{0}$ is composite, that is, the hypothetical spectral
density $f(\lambda)$ of $X(t)$ depends on an unknown $p$–dimensional vector
parameter ${\theta}=(\theta_{1},\ldots,\theta_{p})$, that is,
$f(\lambda)=f(\lambda,{\theta})$, $\lambda\in\Lambda$,
${\theta}\in\Theta\subset\mathbb{R}^{p}$.

The above stated problem in the non-tapered case has been considered by many
authors for different models. For instance, for independent observations, the
problem was considered in Chernov and Lehman [12], Chibisov [13], Cramer [15],
and Dzhaparidze and Nikulin [28]. For observations generated by d.t. Gaussian
stationary processes it was considered in Dzhaparidze [27], Ginovyan [38],
Hannan [57], and Osidze [68, 69]. For c.t. Gaussian stationary observations,
the problem was discussed in Ginovyan [41] and Osidze [68, 69]. For tapered
case the problem has been considered in Ginovyan [43].

To test the hypothesis $H_{0}$, similar to the non-tapered case, it is natural
to introduce a measure of divergence (disparity) of the hypothetical and
empirical spectral densities, and construct a goodness-of-fit test based on
the distribution of the chosen measure (see, e.g., Dzhaparidze [27], Ginovyan
[38, 41, 43], Hannan [57], and Osidze [68, 69]).

###  5.1 A Goodness-of-fit test for simple hypothesis

We first consider the relatively easy case a) of a simple hypothesis $H_{0}$.
As a measure of divergence of the hypothetical spectral density $f(\lambda)$
and the tapered empirical spectral density $I^{h}_{T}(\lambda)$, we consider
the $m$–dimensional random vector

${\Phi}^{h}_{T}:=\bigl{(}\Phi^{h}_{1T},\ldots,\Phi^{h}_{mT}\bigr{)}$ (5.1)

with elements

$\Phi^{h}_{jT}:\,=\,\Phi_{jT}(\mathbf{X}_{T}^{h})\,=\,\frac{\sqrt{T}}{\sqrt{4\pi
e(h)}}\int_{\Lambda}\biggl{[}\frac{I^{h}_{T}(\lambda)}{f(\lambda)}-1\biggr{]}\,\varphi_{j}(\lambda)d\lambda,\quad
j=1,2,\ldots,m,$ (5.2)

where $e(h)$ is as in (3.11) and $\\{\varphi_{j}(\lambda),\,j=1,2,\ldots,m\\}$
is some orthonormal system on $\Lambda$:

$\int_{\Lambda}\varphi_{k}(\lambda)\varphi_{j}(\lambda)\,d\lambda=\delta_{kj}=\left\\{\begin{array}[]{ll}1&\mbox{for
$k=j$},\\\ \\\\[-11.38109pt] 0&\mbox{for $k\neq j$}.\end{array}\right.$ (5.3)

In Ginovyan [43] it was shown that under wide conditions on $f(\lambda)$ and
$\varphi_{j}(\lambda)$, the random vector ${\Phi}^{h}_{T}$ in (5.1) – (5.2)
has asymptotically $N(0,I_{m})$–normal distribution as $T\to\infty$, where
$I_{m}$ is the $m\times m$ identity matrix. Therefore in the case of simple
hypothesis $H_{0}$, we can use the statistic

$S_{T}^{h}=S_{T}(\mathbf{X}_{T}^{h}):={\Phi}^{\prime}_{T}(\mathbf{X}_{T}^{h}){\Phi}_{T}(\mathbf{X}_{T}^{h})=\sum_{j=1}^{m}\Phi^{2}_{jT}(\mathbf{X}_{T}^{h}),$
(5.4)

which for $T\to\infty$ will have a $\chi^{2}$–distribution with $m$ degrees of
freedom.

Thus, fixing an asymptotic level of significance $\alpha$ we can consider the
class of goodness-of-fit tests for testing the simple hypothesis $H_{0}$ about
the form of the spectral density $f$ with asymptotic level of significance
$\alpha$ determined by critical regions of the form:

$\\{\mathbf{X}_{T}^{h}:\,\,S_{T}(\mathbf{X}_{T}^{h})>d_{\alpha}\\},$

where $S_{T}(\mathbf{X}_{T}^{h})$ is given by (5.4), and $d_{\alpha}$ is the
$\alpha$-quantile of $\chi^{2}$–distribution with $m$ degrees of freedom, that
is, $d_{\alpha}$ is determined from the condition:

$P(\chi^{2}>d_{\alpha})=\int_{d_{\alpha}}^{\infty}k_{m}(x)\,dx=\alpha,$

where $k_{m}(x)$ is the density of $\chi^{2}$–distribution with $m$ degrees of
freedom.

The next theorem contains sufficient conditions for statistic $S_{T}^{h}$,
given by (5.4), to have a limiting (as $T\to\infty$) $\chi^{2}$–distribution
with $m$ degrees of freedom (see Ginovyan [43]).

######  Theorem 5.1.

Let the spectral density $f(\lambda)$ and the orthonormal functions
$\\{\varphi_{j}(\lambda),\,j=1,2,\ldots,m\\}$ be such that
$(f,g_{j})\in(\mathcal{H})$ for all $j=1,2,\ldots,m$, where
$g_{j}=\varphi_{j}/f$ (see Assumption 3.2). Then under Assumption 2.1 the
limiting (as $T\to\infty$) distribution of the statistic
$S_{T}^{h}=S_{T}(\mathbf{X}_{T}^{h})$ given by (5.4) is a
$\chi^{2}$–distribution with $m$ degrees of freedom.

######  Remark 5.1.

For the non-tapered case, for observations generated by d.t. short-memory
Gaussian stationary processes the result of Theorem 5.1 was first proved in
Hannan [57] (p. 94) (see, also, Dzhaparidze [27] and Osidze [68, 69]). In the
case where the spectral density has singularities (zeros and/or poles), the
result for d.t. processes was proved in Ginovyan [38]. The non-tapered
counterpart of Theorem 5.1 for c.t. processes was proved in Ginovyan [41].

###  5.2 A Goodness-of-fit test for composite hypothesis

Now we consider the case of composite hypothesis $H_{0}$, and assume that the
hypothetical spectral density $f=f(\lambda,\theta)$ is known with the
exception of a vector parameter
$\theta:=(\theta_{1},\ldots,\theta_{p})\in\Theta\subset\mathbb{R}^{p}$. In
this case, the problem of construction of goodness-of-fit tests becomes more
complex, because we first have to choose an appropriate statistical estimator
$\widehat{\theta}_{T}$ for the unknown parameter $\theta$, constructed on the
basis of the tapered sample (2.18). It is important to remark that in this
case the limiting distribution of the test statistic will change in accordance
with properties of an estimator of ${\theta}$, and generally will not be a
$\chi^{2}$–distribution.

For testing a composite hypothesis $H_{0}$, we again can use a statistic of
type (5.4), but with a statistical estimator $\widehat{{\theta}}_{T}$ instead
of unknown ${\theta}$. The corresponding statistic can be written as follows:

$S_{T}^{h}(\widehat{\theta}_{T})=S_{T}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T}):={\Phi}^{\prime}_{T}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T}){\Phi}_{T}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T})=\sum_{j=1}^{m}\Phi^{2}_{jT}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T}),$
(5.5)

where now

${\Phi}^{h}_{T}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T}):=\bigl{(}\Phi_{1T}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T}),\ldots,\Phi_{mT}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T})\bigr{)}$

with elements

$\Phi_{jT}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T}):\,=\,\frac{\sqrt{T}}{\sqrt{4\pi
e(h)}}\int_{\Lambda}\biggl{[}\frac{I^{h}_{T}(\lambda)}{f(\lambda,\widehat{\theta}_{T})}-1\biggr{]}\,\varphi_{j}(\lambda)d\lambda,\quad
j=1,2,\ldots,m.$ (5.6)

So, we must choose an appropriate statistical estimator $\widehat{\theta}_{T}$
for unknown ${\theta}$, and determine the limiting distribution of the
statistic (5.5). Then, having the limiting distribution of the statistic
(5.5), for given level of significance $\alpha$ we can consider the class of
goodness-of-fit tests for testing the composite hypothesis $H_{0}$ about the
form of the spectral density $f$ with asymptotic level of significance
$\alpha$ determined by critical regions of the form:

$\\{\mathbf{X}_{T}^{h}:\,\,S_{T}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T})>d_{\alpha}\\},$

where $d_{\alpha}$ is the $\alpha$-quantile of the limiting distribution of
the statistic (5.5), that is, $d_{\alpha}$ is determined from the condition:

$\int_{d_{\alpha}}^{\infty}\widehat{k}_{m}(x)\,dx=\alpha,$

where $\widehat{k}_{m}(x)$ is the density of the limiting distribution of
$S^{h}_{T}(\widehat{\theta}_{T})$ defined by (5.5).

To state the corresponding result we first introduce the following set of
assumptions:

######  Assumption 5.1.

For $\theta\in\Theta$, $(f,g_{j})\in(\mathcal{H})$ for all $j=1,2,\ldots,m$,
where $f:=f(\lambda,\theta)$ and
$g_{j}:=\varphi_{j}(\lambda)/f(\lambda,\theta)$.

######  Assumption 5.2.

For $\theta\in\Theta$, $(f,h_{kj})\in(\mathcal{H})$ for all $k=1,2,\ldots,p$
and $j=1,2,\ldots,m$, where $f:=f(\lambda,\theta)$ and
$h_{kj}:=\dfrac{\varphi_{j}(\lambda)}{f(\lambda,\theta)}\dfrac{\partial}{\partial\theta_{k}}\ln
f(\lambda,\theta)$.

######  Assumption 5.3.

The $(p\times p)$–matrix
$\Gamma(\theta_{0})=||\gamma_{kj}(\theta_{0})||_{k,j=\overline{1,p}}\,$ with
elements

$\gamma_{kj}(\theta_{0}):=\frac{1}{4\pi}\int_{\Lambda}\left[\frac{\partial}{\partial\theta_{k}}\ln
f(\lambda,\theta)\right]_{\theta=\theta_{0}}\left[\frac{\partial}{\partial\theta_{j}}\ln
f(\lambda,\theta)\right]_{\theta=\theta_{0}}\,d\lambda$ (5.7)

is non-singular.

######  Assumption 5.4.

There exists a $\sqrt{T}$–consistent estimator $\widehat{{\theta}}_{T}$ for
the parameter $\theta$ such that the following asymptotic relation holds:

$\sqrt{T}\,(\widehat{{\theta}}_{T}-\theta_{0})-\Gamma^{-1}(\theta_{0})\Delta_{T}(\theta_{0})=o_{P}(1),$
(5.8)

where $\Gamma^{-1}(\theta_{0})$ is the inverse of the matrix
$\Gamma(\theta_{0})$ defined in Assumption 5.3, and

$\Delta_{T}(\theta):=\Delta^{h}_{T}(\theta)=\bigl{(}\Delta_{1T}(\theta),\ldots,\Delta_{pT}(\theta)\bigr{)}$
(5.9)

is a $p$-dimensional random vector with components

$\Delta_{kT}(\theta):\,=\,\frac{\sqrt{T}}{\sqrt{4\pi
e(h)}}\int\limits_{\Lambda}\biggl{[}\frac{I^{h}_{T}(\lambda)}{f(\lambda,\theta)}-1\biggr{]}\,\frac{\partial}{\partial\theta_{k}}\ln
f(\lambda,\theta)\,d\lambda,\quad k=1,\ldots,p.$ (5.10)

The term $o_{P}(1)$ in (5.8) tends to zero in probability as $T\to\infty$.
(Recall that an estimator $\widehat{{\theta}}_{T}$ for $\theta$ is called
$\sqrt{T}$–consistent if $\sqrt{T}(\widehat{{\theta}}_{T}-\theta)$ is bounded
in probability).

######  Remark 5.2.

As an estimator $\widehat{{\theta}}_{T}$ for $\theta$ satisfying (5.8) can be
considered minimum contrast estimators (in particular, the Whittle estimator)
based on the tapered data. Minimum contrast estimators based on the tapered
data for d.t. processes have been studied in Dahlhaus [17, 19, 18], for
Gaussian c.t. processes in Ginovyan [42], and for some classes of c.t. non-
Gaussian processes in Alomari et al [1].

Let
$B(\theta):=B^{h}(\theta)=||b_{jk}(\theta)||_{j=\overline{1,m},\,k=\overline{1,p}},\,$
be a $\,(m\times p)$–matrix with elements

$b_{jk}(\theta):=\frac{1}{\sqrt{4\pi
e(h)}}\int_{\Lambda}\varphi_{j}(\lambda)\frac{\partial}{\partial\theta_{k}}\ln
f(\lambda,\theta)\,d\lambda,$ (5.11)

where $\varphi_{j}(\lambda)$ $(j=1,2,\ldots,m)$ are the functions from (5.2),
and $e(h)$ is as in (3.11).

The following theorem was proved in Ginovyan [43].

######  Theorem 5.2.

Under Assumptions 2.1, 4.1 and 5.1–5.4 the limiting distribution (as
$T\to\infty$) of the statistic
$S_{T}(\mathbf{X}_{T}^{h},\widehat{\theta}_{T})$ given by (5.5), coincides
with the distribution of the random variable

$\sum_{j=1}^{m-p}\xi^{2}_{j}+\sum_{j=1}^{p}\nu_{j}\,\xi^{2}_{m-p+j},$

where $\xi_{j}$, $j=1,2,\ldots,m,$ are iid $N(0,1)$ random variables, while
the numbers $\nu_{k}$ $(0\leq\nu_{k}<1)$, $k=1,2,\ldots,p$, are the roots
relative to $\nu$ of the following equation:

$\det\left[(1-\nu)\Gamma({\theta}_{0})-B^{\prime}({\theta}_{0})\,B({\theta}_{0})\right]=0.$
(5.12)

######  Remark 5.3.

For the non-tapered case, for independent observations the result of Theorem
5.2 was first obtained by Chernov and Lehman [12] (see, also, Chibisov [13]).
For observations generated by d.t. short-memory Gaussian stationary processes
the result was stated in Osidze [68] (see, also, Dzhaparidze [27]). In the
case where the spectral density has singularities, the result for d.t.
processes was proved in Ginovyan [38]. The non-tapered counterpart of Theorem
5.2 for c.t. processes was proved in Ginovyan [41].

######  Example 5.1.

Let $X(t)$ be a d.t. Autoregressive Process of order $p$ (AR$(p)$), that is,
$X(t)$ is a stationary process with the spectral density
$f(\lambda)=\frac{1}{2\pi}\cdot\bigl{|}\alpha_{p}(e^{-i\lambda})\bigr{|}^{-2},$
where $\alpha_{p}(z)=1-\theta_{1}z-\cdots-\theta_{p}z^{p}$. Consider the
functions $\varphi_{j}(\lambda)$ ($j=1,\ldots,m$) defined by
$\varphi_{j}(\lambda):=ce^{ij\lambda}\alpha_{p}(e^{-i\lambda})\bigl{|}\alpha_{p}(e^{-i\lambda})\bigr{|}^{-1}$
for $j>p$ and $\varphi_{j}(\lambda):=0$ for $j\leq p$, where $c$ is a
normalizing constant. As an estimator of
$\theta:=(\theta_{1},\ldots,\theta_{p})$ consider the Whittle estimator, and
as a taper the Tukey-Hanning taper function $h(t)$ (see Remark 2.1). Then it
is easy to check that the conditions of Theorem 5.2 are satisfied and
$B(\theta)=0$. Therefore, the limiting (as $T\to\infty$) distribution of the
statistic in (5.5) is a $\chi^{2}$–distribution with $m-p$ degrees of freedom.

##  6 Robustness to small trends of estimation

In time series analysis, much of statistical inferences about unknown spectral
parameters or spectral functionals are concerned with the stationary models,
in which case it is assumed that the models are centered, or have constant
means. In this section, we are concerned with the robustness of inferences,
carried out on a stationary models, possibly exhibiting long memory,
contaminated by a small trend. Specifically, let $\\{X(t),\,t\in\mathbb{U}\\}$
be a centered stationary process possessing a spectral density
$f_{X}(\lambda)$, $\lambda\in\Lambda$. Assuming that either $f_{X}$ is known
with the exception of a vector parameter
$\theta\in\Theta\subset\mathbb{R}^{p}$, or $f_{X}$ is completely unknown and
belongs to a given class $\mathcal{F}$, we want to make inferences about
$\theta$ or the value $J(f_{X})$ of a given functional $J(\cdot)$ at an
unknown point $f_{X}\in\mathcal{F}$ in the case where the actual observed data
are in the contaminated form:

$Y(t)=X(t)+M(t),\quad 0\leq t\leq T,$ (6.1)

where $M(t)$ is a deterministic trend.

The process $X(t)$ is what we believe is being observed but in reality the
data are in the contaminated form $Y(t)$. In this case standard inferences can
be carried on the basis of the stationary model $X(t)$, and we are interested
in question whether the conclusions are robust against this kind of departure
from the stationarity.

This problem for d.t. models was considered in Heyde and Dai [59] (see also
Taniguchi and Kakizawa [72], Theorems 6.4.1 and 6.4.2). For c.t. models it was
studied in Ginovyan and Sahakyan [48].

The results stated below show that if the trend $M(t)$ is ’small’, then the
asymptotic properties of estimators of the parameter $\theta$ and the
functional $J(f)$, stated in Sections 3 and 4 for a stationary model $X(t)$,
remain valid for the contaminated model $Y(t)$, that is, both the parametric
and nonparametric estimating procedures are robust against replacing the
stationary model $X(t)$ by the non-stationary $Y(t)$. To this end, we first
establish an asymptotic relation between stationary and contaminated
periodograms.

###  6.1 A relation between stationary and contaminated periodograms

The next result shows that a small trend of the form $|M(t)|\leq
C|t|^{-\beta}$, $\beta>1/4$, does not effect the asymptotic properties of the
empirical spectral linear functionals of a periodogram. Note that this result
is of general nature, and do not require from the model $X(t)$ to be linear.

######  Theorem 6.1.

Let $\\{X(t),\,t\in\mathbb{U}\\}$ be a stationary mean zero process,
$\\{M(t),\,t\in\mathbb{U}\\}$ be a deterministic trend, $Y(t)=X(t)+M(t)$, and
let $I^{h}_{TX}(\lambda)$ and $I^{h}_{TY}(\lambda)$ be the periodograms of
$X(t)$ and $Y(t)$, respectively. Let $g(\lambda),$ $\lambda\in\Lambda$ be an
even integrable function. If the trend $M(t)$ and the Fourier transform
$a(t):=\widehat{g}(t)$ of $g(\lambda)$ are such that $M(t)$ is locally
integrable on ${\mathbb{R}}$ and

$|M(t)|\leq C|t|^{-\beta},\qquad|a(t)|\leq C|t|^{-\gamma},\quad
t\in\Lambda,\quad 2\beta+\gamma>\frac{3}{2},$ (6.2)

with some constants $C>0$, $\gamma>0$ and $\beta>1/4$, then

$\displaystyle
T^{1/2}\int_{\Lambda}g(\lambda)\left[I^{h}_{TY}(\lambda)-I^{h}_{TX}(\lambda)\right]d\lambda\overset{P}{\rightarrow}0\quad{\rm
as}\quad T\to\infty,$ (6.3)

where $\overset{P}{\rightarrow}$ stands for convergence in probability,
provided that one of the following conditions holds:

  * (i)

the process $X(t)$ has short or intermediate memory, that is, the covariance
function $r(t):=r_{X}(t)$ of $X(t)$ satisfies $\ r\in L^{1}(\Lambda)$, and
$\beta+\gamma>1$,

  * (ii)

the process $X(t)$ has long memory with covariance function $r(t)$ satisfying

$|r(t)|\leq C|t|^{-\alpha},\quad
t\in\Lambda,\quad\alpha+\gamma\geq\frac{3}{2}$ (6.4)

with some constants $C>0$, $0<\alpha\leq 1$, and $\alpha+2\beta>1$ if
$\beta<1<\gamma$.

######  Remark 6.1.

It is easy to check that the statement of Theorem 6.1 holds, in particular, if
the parameters $\alpha$, $\beta$ and $\gamma$ satisfy the following
conditions:

in the case (i): $\beta>1/2,\ \gamma\geq 1/2$,

in the case (ii): $\alpha\geq 3/4,\ \beta>3/8,\ \gamma\geq 3/4$.

######  Remark 6.2.

In the d.t. case, Theorem 6.1 (with additional conditions $\gamma=1$ in the
case (i), and $\gamma>1,\ \alpha<1/2$ in the case (ii)), was proved by Heyde
and Dai [59] (see also Taniguchi and Kakizawa [72], Theorems 6.4.1 and 6.4.2).

###  6.2 Robustness to small trends of nonparametric estimation

The next result shows that a small trend of the form $|M(t)|\leq
C|t|^{-\beta}$ does not effect the asymptotic properties of the estimator of a
linear spectral functional $J(f)$, that is, the nonparametric estimation
procedure is robust to the presence of a small trend in the model.

######  Theorem 6.2.

Suppose that the assumptions of Theorems 3.6 and 6.1 are fulfilled. Then the
statistic $J(I^{h}_{TY})$ is consistent and asymptotically normal estimator
for functional $J(f)$ with asymptotic variance $\sigma^{2}_{h}(J)$ given by
(3.10) and (3.11), that is, the asymptotic relation (3.14) is satisfied with
$I^{h}_{TX}(\lambda)$ replaced by the contaminated periodogram
$I^{h}_{TY}(\lambda)$:

$\displaystyle
T^{1/2}\left[J(I^{h}_{TY})-J(f)\right]\overset{d}{\rightarrow}\eta\quad{\rm
as}\quad T\to\infty,$ (6.5)

where $\eta$ is $N(0,\sigma^{2}_{h}(J))$ with $\sigma^{2}_{h}(J)$ given by
(3.10) and (3.11).

###  6.3 Robustness to small trends of parametric estimation

The next result shows that a small trend of the form $|M(t)|\leq
C|t|^{-\beta}$, $\beta>1/4$, does not effect the asymptotic properties of the
Whittle estimator of an unknown spectral parameter $\theta$, that is, the
Whittle parametric estimation procedure is robust to the presence of a small
trend in the model.

######  Theorem 6.3.

Suppose that the assumptions of Theorem 6.1 with
$g=f^{-1}(\lambda,\theta)\cdot w(\lambda)$ are satisfied. Then under the
conditions of Theorems 4.2 the Whittle estimator $\widehat{\theta}_{TY}$,
constructed on the basis of the contaminated periodogram
$I_{T,Y}^{h}(\lambda)$, is consistent and asymptotically normal estimator for
an unknown spectral parameter $\theta$, that is, the asymptotic relation (4.8)
is satisfied with $I^{h}_{TX}(\lambda)$ replaced by the contaminated
periodogram $I^{h}_{TY}(\lambda)$:

$\displaystyle
T^{1/2}\left(\widehat{\theta}_{TY}-\theta_{0}\right)\overset{d}{\rightarrow}N_{p}\left(0,R(\theta_{0})\right)\quad{\rm
as}\quad T\to\infty,$ (6.6)

where the matrix $R(\theta_{0})$ is defined in (4.9).

######  Remark 6.3.

In the non-tapered case, Theorems 6.1 – 6.3 were proved in Ginovyan and
Sahakyan [48]. In the tapered case, the theorems can be proved similarly by
using the tapered tools stated in Section 7.

##  7 Methods and tools

In this section we briefly discuss the methods and tools, used to prove the
results stated in Sections 3–6.

###  7.1 Approximation of traces of products of Toeplitz matrices and
operators.

The trace approximation problem for truncated Toeplitz operators and matrices
has been discussed in detail in the survey paper Ginovyan et al. [51] in the
non-tapered case. Here we present some important results in the tapered case,
which were used to prove the results stated in Sections 3–6.

Let $\psi(\lambda)$ be an integrable real symmetric function defined on
$[-\pi,\pi]$, and let $h(t)$, $t\in[0,1]$ be a taper function. For
$T=1,2,\ldots$, the $(T\times T)$-truncated tapered Toeplitz matrix generated
by $\psi$ and $h$, denoted by $B_{T}^{h}(\psi)$, is defined by the following
equation:

$B_{T}^{h}(\psi):=\|\widehat{\psi}(t-s)h_{T}(t)h_{T}(s)\|_{t,s=1,2\ldots,T},$
(7.1)

where $\widehat{\psi}(t)$ $(t\in\mathbb{Z})$ are the Fourier coefficients of
$\psi$.

Given a real number $T>0$ and an integrable real symmetric function
$\psi(\lambda)$ defined on $\mathbb{R}$, the $T$-truncated tapered Toeplitz
operator (also called tapered Wiener-Hopf operator) generated by $\psi$ and a
taper function $h$, denoted by $W_{T}^{h}(\psi)$ is defined as follows:

$[{W}^{h}_{T}(\psi)u](t)=\int_{0}^{T}\hat{\psi}(t-s)u(s)h_{T}(s)ds,\quad
u(s)\in L^{2}([0,T];h_{T}),$ (7.2)

where $\hat{\psi}(\cdot)$ is the Fourier transform of $\psi(\cdot)$, and
$L^{2}([0,T];h_{T})$ denotes the weighted $L^{2}$-space with respect to the
measure $h_{T}(t)dt$.

Let $h$ be a taper function satisfying Assumption 2.1, and let
$A^{h}_{T}(\psi)$ be either the $T\times T$ tapered Toeplitz matrix
$B^{h}_{T}(\psi)$, or the $T$-truncated tapered Toeplitz operator
$W^{h}_{T}(\psi)$ generated by a function $\psi$ (see (7.1) and (7.2)).

Observe that, in view of (2.20), (2.27), (7.1) and (7.2), we have

$\frac{1}{T}\hbox{tr}\left[A^{h}_{T}(\psi)\right]=\frac{1}{T}\cdot\widehat{\psi}(0)\cdot\int_{0}^{T}h_{T}^{2}(t)dt=2\pi
H_{2}\int_{\Lambda}\psi(\lambda)d\lambda.$ (7.3)

What happens to the relation (7.3) when $A^{h}_{T}(\psi)$ is replaced by a
product of Toeplitz matrices (or operators)? Observe that the product of
Toeplitz matrices (resp. operators) is not a Toeplitz matrix (resp. operator).

The idea is to approximate the trace of the product of Toeplitz matrices
(resp. operators) by the trace of a Toeplitz matrix (resp. operator) generated
by the product of the generating functions. More precisely, let
$\\{\psi_{1},\psi_{2},\ldots,\psi_{m}\\}$ be a collection of integrable real
symmetric functions defined on $\Lambda$. Let $A_{T}^{h}(\psi_{i})$ be either
the $T\times T$ tapered Toeplitz matrix $B_{T}^{h}(\psi_{i})$, or the
$T$-truncated tapered Toeplitz operator $W_{T}^{h}(\psi_{i})$ generated by a
function $\psi_{i}$ and a taper function $h$. Define

$\displaystyle
S_{A,\mathcal{H},h}(T):=\frac{1}{T}\hbox{tr}\left[\prod_{i=1}^{m}A_{T}^{h}(\psi_{i})\right],\quad
M_{\Lambda,\mathcal{H},h}:=(2\pi)^{m-1}H_{m}\int_{\Lambda}\left[\prod_{i=1}^{m}\psi_{i}(\lambda)\right]\,d\lambda,$

and let

$\displaystyle\Delta(T):=\Delta_{A,{\Lambda},\mathcal{H},h}(T)=|S_{A,\mathcal{H},h}(T)-M_{{\Lambda},\mathcal{H},h}|.$
(7.4)

######  Proposition 7.1.

Let $\Delta(T)$ be as in (7.4). Each of the following conditions is sufficient
for

$\Delta(T)=o(1)\quad{\rm as}\quad T\to\infty.$ (7.5)

  * (C1)

$\psi_{i}\in L^{1}(\Lambda)\cap L^{p_{i}}(\Lambda)$, $p_{i}>1$,
$i=1,2,\ldots,m$, with $1/p_{1}+\cdots+1/p_{m}\leq 1$.

  * (C2)

The function $\varphi({\bf u})$ defined by

$\varphi({\bf u}):\,=\,\int_{\Lambda}\psi_{1}(\lambda)\psi_{2}(\lambda-
u_{1})\psi_{3}(\lambda-u_{2})\cdots\psi_{m}(\lambda-u_{m-1})\,d\lambda,$ (7.6)

where ${\bf u}=(u_{1},u_{2},\ldots,u_{m-1})\in\Lambda^{m-1}$, belongs to
$L^{m-2}(\Lambda^{m-1})$ and is continuous at ${\bf
0}=(0,0,\ldots,0)\in\Lambda^{m-1}$.

######  Remark 7.1.

In the non-tapered case, Proposition 7.1 was proved in Ginovyan et al. [51],
while in the tapered case, it was proved in Ginovyan [43]. Proposition 7.1 was
used to prove Theorems 3.5, 3.6, 4.2 and 5.2.

######  Remark 7.2.

More results concerning the trace approximation problem for truncated Toeplitz
operators and matrices can be found in Ginovyan and Sahakyan [46, 47], and in
Ginovyan et al. [51].

###  7.2 Central limit theorems for tapered quadratic functionals

In this subsection we state central limit theorems for tapered quadratic
functional $Q^{h}_{T}$ given by (3.3), which were used to prove the results
stated in Sections 3–6.

Let $A_{T}^{h}(f)$ be either the $T\times T$ tapered Toeplitz matrix
$B_{T}^{h}(f)$, or the $T$-truncated tapered Toeplitz operator $W_{T}^{h}(f)$
generated by the spectral density $f$ and taper $h$, and let $A_{T}^{h}(g)$
denote either the $T\times T$ tapered Toeplitz matrix, or the $T$-truncated
tapered Toeplitz operator generated by the functions $g$ and $h$ (for
definitions see formulas (7.1) and (7.2)). Similar to the non-tapered case, we
have the following results (cf. Ginovyan et al. [51], Grenander and Szegő
[54], Ibragimov [61]).

  * 1.

The quadratic functional $Q^{h}_{T}$ in (3.3) has the same distribution as the
sum $\sum_{j=1}^{\infty}\lambda_{j,T}^{2}\xi_{j}^{2}$, where $\\{\xi_{j},j\geq
1\\}$ are independent $N(0,1)$ Gaussian random variables and
$\\{\lambda_{j,T},j\geq 1\\}$ are the eigenvalues of the operator
$A_{T}^{h}(f)\,A_{T}^{h}(g)$.

  * 2.

The characteristic function $\varphi(t)$ of $Q^{h}_{T}$ is given by formula:

$\varphi(t)=\prod_{j=1}^{\infty}|1-2it\lambda_{j,T}|^{-1/2}.$ (7.7)

  * 3.

The $k$–th order cumulant $\chi_{k}(Q^{h}_{T})$ of $Q^{h}_{T}$ is given by
formula:

$\displaystyle\chi_{k}(Q_{T})=2^{k-1}(k-1)!\sum_{j=1}^{\infty}\lambda_{j,T}^{k}=2^{k-1}(k-1)!\,\hbox{tr}\,[A_{T}^{h}(f)\,A_{T}^{h}(g)]^{k}.$
(7.8)

Thus, to describe the asymptotic distribution of the quadratic functional
$Q^{h}_{T}$, we have to control the traces and eigenvalues of the products of
truncated tapered Toeplitz operators and matrices.

####  7.2.1 CLT for Gaussian models

We assume that the model process $X(t)$ is Gaussian, and with no loss of
generality, that $g\geq 0$. We will use the following notation. By
$\widetilde{Q}^{h}_{T}$ we denote the standard normalized quadratic
functional:

$\widetilde{Q}^{h}_{T}=T^{-1/2}\,\left(Q^{h}_{T}-\mathbb{E}[Q^{h}_{T}]\right).$
(7.9)

Also, we set

$\sigma^{2}_{h}:=16\pi^{3}H_{4}\int_{\Lambda}f^{2}(\lambda)g^{2}(\lambda)\,d\lambda,$
(7.10)

where $H_{4}$ is as in (2.20). The notation

$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim
N(0,\sigma^{2}_{h})\quad{\rm as}\quad T\to\infty$ (7.11)

will mean that the distribution of the random variable $\widetilde{Q}^{h}_{T}$
tends (as $T\to\infty$) to the centered normal distribution with variance
$\sigma^{2}_{h}$.

The following theorems were proved in Ginovyan and Sahakyan [50].

######  Theorem 7.1.

Assume that $f\cdot g\in L^{1}(\Lambda)\cap L^{2}(\Lambda)$, the taper
function $h$ satisfies Assumption 2.1, and for $T\to\infty$

$\chi_{2}(\widetilde{Q}^{h}_{T})=\frac{2}{T}\hbox{tr}\bigl{[}A^{h}_{T}(f)A^{h}_{T}(g)\bigr{]}^{2}\longrightarrow\sigma^{2}_{h},$
(7.12)

where $\sigma^{2}_{h}$ is as in (7.10). Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{h})$ as
$T\to\infty$.

######  Theorem 7.2.

Assume that the function

$\varphi(x_{1},x_{2},x_{3})=\int_{\Lambda}f(u)g(u-x_{1})f(u-x_{2})g(u-x_{3})\,du$
(7.13)

belongs to $L^{2}(\Lambda^{3})$ and is continuous at $(0,0,0)$, and the taper
function $h$ satisfies Assumption 2.1. Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{h})$ as
$T\to\infty$.

######  Theorem 7.3.

Assume that $f(\lambda)\in L^{p}(\Lambda)$ $(p\geq 1)$ and $g(\lambda)\in
L^{q}(\Lambda)$ $(q\geq 1)$ with $1/p+1/q\leq 1/2$, and the taper function $h$
satisfies Assumption 2.1. Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{h})$ as
$T\to\infty$.

######  Theorem 7.4.

Let $f\in L^{2}(\Lambda)$, $g\in L^{2}(\Lambda)$, $fg\in L^{2}(\Lambda)$,

$\int_{\Lambda}f^{2}(\lambda)g^{2}(\lambda-\mu)\,d\lambda\longrightarrow\int_{\Lambda}f^{2}(\lambda)g^{2}(\lambda)\,d\lambda\quad{\rm
as}\quad\mu\to 0,$ (7.14)

and let the taper function $h$ satisfy Assumption 2.1. Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{h})$ as
$T\to\infty$.

To state the next theorem, we recall the class $SV_{0}(\mathbb{R})$ of slowly
varying functions at zero $u(\lambda)$, $\lambda\in\mathbb{R}$, satisfying the
following conditions: for some $a>0$, $u(\lambda)$ is bounded on $[-a,a]$,
$\lim_{\lambda\to 0}u(\lambda)=0,$ $u(\lambda)=u(-\lambda)$ and
$0<u(\lambda)<u(\mu)$ for $0<\lambda<\mu<a$.

######  Theorem 7.5.

Assume that the functions $f$ and $g$ are integrable on $\mathbb{R}$ and
bounded outside any neighborhood of the origin, and satisfy for some $a>0$

$f(\lambda)\leq|\lambda|^{-\alpha}L_{1}(\lambda),\quad|g(\lambda)|\leq|\lambda|^{-\beta}L_{2}(\lambda),\quad\lambda\in[-a,a],$
(7.15)

for some $\alpha<1,\ \beta<1$ with $\alpha+\beta\leq 1/2$, where $L_{1}(x)$
and $L_{2}(x)$ are slowly varying functions at zero satisfying

$\displaystyle L_{i}\in SV_{0}(\mathbb{R}),\ \
\lambda^{-(\alpha+\beta)}L_{i}(\lambda)\in L^{2}[-a,a],\ i=1,2.$ (7.16)

Also, let the taper function $h$ satisfy Assumption 2.1. Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{h})$ as
$T\to\infty$.

The conditions $\alpha<1$ and $\beta<1$ in Theorem 7.5 ensure that the Fourier
transforms of $f$ and $g$ are well defined. Observe that when $\alpha>0$ the
process $X(t)$ may exhibit long-range dependence. We also allow here
$\alpha+\beta$ to assume the critical value 1/2. The assumptions $f\cdot g\in
L^{1}(\Lambda)$, $f,g\in L^{\infty}(\Lambda\setminus[-a,a])$ and (7.16) imply
that $f\cdot g\in L^{2}(\Lambda)$, so that the variance $\sigma^{2}_{h}$ in
(7.10) is finite.

####  7.2.2 CLT for Lévy-driven stationary linear models

Now we assume that the underlying model $X(t)$ is a Lévy-driven stationary
linear process defined by (2.6), where $a(\cdot)$ is a filter from
$L^{2}(\mathbb{R})$, and $\xi(t)$ is a Lévy process satisfying the conditions:
$\mathbb{E}\xi(t)=0$, $\mathbb{E}\xi^{2}(1)=1$ and
$\mathbb{E}\xi^{4}(1)<\infty$.

The central limit theorem that follows was proved in Ginovyan and Sahakyan
[49].

######  Theorem 7.6.

Assume that the filter $a(\cdot)$ and the generating kernel
$\widehat{g}(\cdot)$ are such that

$a(\cdot)\in L^{p}(\mathbb{R})\cap
L^{2}(\mathbb{R}),\quad\widehat{g}(\cdot)\in L^{q}(\mathbb{R}),\quad 1\leq
p,q\leq 2,\quad\frac{2}{p}+\frac{1}{q}\geq\frac{5}{2},$ (7.17)

and the taper $h$ satisfies Assumption 2.1. Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{L,h})$
as $T\to\infty$, where

$\sigma^{2}_{L,h}=16\pi^{3}H_{4}\int_{\mathbb{R}}f^{2}(\lambda)g^{2}(\lambda)d\lambda+\kappa_{4}4\pi^{2}H_{4}\left[\int_{\mathbb{R}}f(\lambda)g(\lambda)d\lambda\right]^{2},$
(7.18)

where $H_{4}$ is as in (2.20).

######  Remark 7.3.

Notice that if the underlying process $X(t)$ is Gaussian, then in formula
(7.18) we have only the first term and so $\sigma^{2}_{L,h}=\sigma^{2}_{h}$
(see (7.10)), because in this case $\kappa_{4}=0$. On the other hand, the
condition (7.17) is more restrictive than the conditions in Theorems 7.1 \-
7.5. Thus, for Gaussian processes Theorems 7.1 \- 7.5 improve Theorem 7.6. For
non-tapered case Theorem 7.6 was proved in Bai et al. [6].

###  7.3 Fejér-type kernels and singular integrals

We define Fejér-type tapered kernels and singular integrals, and state some of
their properties.

For a number $k$ ($k=2,3,\ldots$) and a taper function $h$ satisfying
Assumption 2.1 consider the following Fejér-type tapered kernel function:

$F^{h}_{k,T}({\bf u}):=\frac{H_{T}({\bf u})}{(2\pi)^{k-1}H_{k,T}(0)},\quad{\bf
u}=(u_{1},\ldots,u_{k-1})\in\mathbb{R}^{k-1},$ (7.19)

where

$H_{T}({\bf u}):={H_{1,T}(u_{1})\cdots
H_{1,T}(u_{k-1})H_{1,T}\left(-\sum_{j=1}^{k-1}u_{j}\right)},$ (7.20)

and the function $H_{k,T}(\cdot)$ is defined by (2.21) with $H_{k,T}(0)=T\cdot
H_{k}\neq 0$ (see (2.20)).

The next result shows that, similar to the classical Fejér kernel, the tapered
kernel $F^{h}_{k,T}({\bf u})$ is an approximation identity (see Ginovyan and
Sahakyan [49], Lemma 3.4).

######  Proposition 7.2.

For any $k=2,3,\ldots$ and a taper function $h$ satisfying Assumption 2.1 the
kernel $F^{h}_{k,T}({\bf u})$, ${\bf
u}=(u_{1},\ldots,u_{k-1})\in{\mathbb{R}}^{k-1}$, possesses the following
properties:

  * a)

$\sup_{T>0}\int_{{\mathbb{R}}^{k-1}}\left|F^{h}_{k,T}({\bf u})\right|\,d{\bf
u}=C_{1}<\infty;$

  * b)

$\int_{{\mathbb{R}}^{k-1}}F^{h}_{k,T}({\bf u})\,d{\bf u}=1;$

  * c)

$\lim_{T\to\infty}\int_{{\mathbb{E}}^{c}_{\delta}}\left|F^{h}_{k,T}({\bf
u})\right|\,d{\bf u}=0$ for any $\delta>0;$

  * d)

If $k>2$ for any $\delta>0$ there exists a constant $M_{\delta}>0$ such that
for $T>0$

$\left\|F^{h}_{k,T}\right\|_{L^{p_{k}}({\mathbb{E}^{c}_{\delta}})}\leq
M_{\delta},$ (7.21)

where $p_{k}=\frac{k-2}{k-3}$ for $k>3$, $p_{3}=\infty$ and

${\mathbb{E}}_{\delta}^{c}={\mathbb{R}}^{k-1}\setminus{\mathbb{E}}_{\delta},\quad{\mathbb{E}}_{\delta}=\\{{\bf
u}=(u_{1},\ldots,u_{k-1})\in{\mathbb{R}}^{k-1}:\,|u_{i}|\leq\delta,\,i=1,\ldots,k-1\\}.$

  * e)

If the function $Q\in L^{1}(\mathbb{R}^{k-1})\bigcap
L^{k-2}(\mathbb{R}^{k-1})$ and is continuous at ${\bf
v}=(v_{1},\ldots,v_{k-1})$ ($L^{0}$ is the space of measurable functions),
then

$\lim_{T\to\infty}\int_{\mathbb{R}^{k-1}}Q({\bf u}+{\bf v})F^{h}_{k,T}({\bf
u})d{\bf u}=Q({\bf v}).$ (7.22)

Denote

$\Delta_{2,T}^{h}:=\int_{\mathbb{R}^{2}}f(\lambda)g(\lambda+\mu)F^{h}_{2,T}(\mu)d\lambda
d\mu-\int_{\mathbb{R}}f(\lambda)g(\lambda)d\lambda,$ (7.23)

where $F^{h}_{2,T}(\mu)$ is given by (7.19) and (7.20).

The next two propositions give information on the rate of convergence to zero
of $\Delta_{2,T}^{h}$ as $T\to\infty$ (see Ginovyan and Sahakyan [49], Lemmas
4.1 and 4.2).

######  Proposition 7.3.

Assume that Assumptions 2.1 and 3.3 are satisfied. Then the following
asymptotic relation holds:

$\Delta_{2,T}^{h}=o\left(T^{-1/2}\right)\quad{\rm as}\quad T\to\infty.$ (7.24)

######  Proposition 7.4.

Assume that Assumptions 2.1 and 3.2 are satisfied. Then the following
inequality holds:

$|\Delta_{2,T}^{h}|\leq
C_{h}\begin{cases}T^{-(\beta_{1}+\beta_{2})},&\text{if}\ \
\beta_{1}+\beta_{2}<1\\\ T^{-1}\ln T,&\text{if}\ \ \beta_{1}+\beta_{2}=1\\\
T^{-1},&\text{if}\ \ \beta_{1}+\beta_{2}>1,\end{cases}\qquad T>0,$ (7.25)

where $C_{h}$ is a constant depending on $h$.

Notice that for non-tapered case ($h(t)={\mathbb{I}}_{[0,1]}(t)$), the above
stated results were proved in Ginovyan and Sahakyan [45] (see also Ginovyan
and Sahakyan [46, 47]). In the d.t. tapered case, Proposition 7.3 under
different conditions was proved in Dahlhaus [17].

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