# A Survey on Limit Theorems for Toeplitz Type Quadratic Functionals of
Stationary Processes and Applications

Mamikon S. Ginovyan and Murad S. Taqqu

###### Abstract

This is a survey of recent results on central and non-central limit theorems
for quadratic functionals of stationary processes. The underlying processes
are Gaussian, linear or Lévy-driven linear processes with memory, and are
defined either in discrete or continuous time. We focus on limit theorems for
Toeplitz and tapered Toeplitz type quadratic functionals of stationary
processes with applications in parametric and nonparametric statistical
estimation theory. We discuss questions concerning Toeplitz matrices and
operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-
type multiple stochastic integrals. These are the main tools for obtaining
limit theorems.

Keywords. Central and non-central limit theorems; Toeplitz type quadratic
functional; stationary processes; spectral density; Brownian motion;
parametric and nonparametric estimation. 2010 Mathematics Subject
Classification. Primary: 60F05, 60G10, 60G15; secondary: 62F12, 62G05.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation and conventions.
    2. 1.2 The functionals under consideration
    3. 1.3 A brief history
    4. 1.4 Frequency-domain conditions
    5. 1.5 Methods and tools.
    6. 1.6 The structure of the paper
  2. 2 The model: second-order stationary process
    1. 2.1 Key notions and some basic results
      1. 2.1.1 Second-order (wide-sense) stationary process
      2. 2.1.2 Spectral representations
      3. 2.1.3 Kolmogorov’s isometric isomorphism theorem
    2. 2.2 Linear processes. Existence of spectral density functions
    3. 2.3 Lévy-driven linear process
    4. 2.4 Dependence (memory) structure of the model
      1. 2.4.1 Short memory models
      2. 2.4.2 Discrete-time long-memory and anti-persistent models
      3. 2.4.3 Continuous-time long-memory and anti-persistent models
  3. 3 CLT for Toeplitz type quadratic functionals for Gaussian and linear processes
    1. 3.1 Frequency domain conditions
    2. 3.2 Time domain conditions
    3. 3.3 Operator conditions
  4. 4 Functional limit theorems for Gaussian and linear models
    1. 4.1 Functional CLT for $Q_{T}(\tau)$
    2. 4.2 Functional NCLT for $Q_{T}(\tau)$
      1. 4.2.1 Non-central limit theorems (discrete-time)
      2. 4.2.2 Non-central limit theorems (continuous-time)
  5. 5 Functional limit theorems for Lévy-driven linear models
    1. 5.1 Central limit theorems
    2. 5.2 Non-central limit theorems
  6. 6 CLT for Tapered Toeplitz Quadratic Functionals
    1. 6.1 The problem
    2. 6.2 Statistical motivation
    3. 6.3 Central limit theorems for tapered quadratic functional $Q_{T}^{h}$
      1. 6.3.1 CLT for Gaussian models
      2. 6.3.2 CLT for Lévy-driven stationary linear models
  7. 7 Applications
    1. 7.1 Nonparametric estimation of spectral functionals
    2. 7.2 Parametric estimation: The Whittle procedure
  8. 8 Methods and tools
    1. 8.1 The characteristic functions and cumulant criteria for the CLT
    2. 8.2 Approximation of traces of products of Toeplitz matrices and operators
    3. 8.3 Approximation method for the CLT
    4. 8.4 Fejér-type singular integrals
    5. 8.5 Lévy-Itô-type and Stratonovich-type multiple stochastic integrals
    6. 8.6 Power counting theorems

##  1 Introduction

A significant part of large-sample statistical inference relies on limit
theorems of probability theory, which involves sums and quadratic functionals
of stationary observations. Depending on the memory (or dependence) structure
of the underlying processes, these functionals, once normalized, can have
different limits, and the proofs of such limit theorems generally use
different methods. In this paper, we focus on the quadratic functionals. The
term ’central limit theorem’ (CLT) refers to a statement that a suitably
standardized quadratic functional converges in distribution to a Gaussian
random variable. Limit theorems where a suitably standardized quadratic
functional converges in distribution to a non-Gaussian random variable are
termed ’non-central limit theorems’ (NCLT).

We present results on central and non-central limit theorems for Toeplitz and
tapered Toeplitz type quadratic functionals of stationary processes with
applications in parametric and nonparametric statistical estimation theory.
The underlying processes are Gaussian, linear or Lévy-driven linear processes
with memory, and are defined either in discrete or continuous time. We also
discuss some questions concerning Toeplitz matrices and operators, Fejér-type
singular integrals, Lévy-Itô-type and Stratonovich-type multiple stochastic
integrals, and power counting theorems. These are the main tools for obtaining
limit theorems, but they are also of interest in themselves.

###  1.1 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; NCLT:= non-central limit theorem. The symbol
’$\stackrel{{\scriptstyle d}}{{=}}$’ stands for equality of the finite-
dimensional distributions. The symbol ’$\overset{d}{\rightarrow}$’ stands for
convergence in distribution. The symbol ’$\overset{f.d.d.}{\longrightarrow}$’
stands for convergence of finite-dimensional distributions. The symbol
’$\Longrightarrow$’ stands for weak convergence. 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. 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 $L^{p}$ and $||\cdot||_{p}$, respectively. By $l^{p}$ ($p\geq$1)
we denote the the space of $p$-summable sequences. 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.

###  1.2 The functionals under consideration

Let $\\{X(u),\ u\in\mathbb{U}\\}$ be a centered real-valued stationary process
with spectral density (s.d.) $f(\lambda)$, $\lambda\in\Lambda$ and covariance
function (c.f.) $r(t)$, $t\in\mathbb{U}$. 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 first survey the recent results concerning the asymptotic distribution (as
$T\to\infty$) of the following Toeplitz type quadratic functionals of the
process $X(u)$:

$Q_{T}:=\left\\{\begin{array}[]{ll}\sum_{t=1}^{T}\sum_{s=1}^{T}\widehat{g}(t-s)X(t)X(s)&\mbox{in
the d.t.\ case},\\\ \\\
\int_{0}^{T}\int_{0}^{T}\widehat{g}(t-s)X(t)X(s)\,dt\,ds&\mbox{in the c.t.\
case},\end{array}\right.$ (1.1)

where

$\widehat{g}(t):=\int_{\Lambda}e^{i\lambda t}\,g(\lambda)\,d\lambda,\quad
t\in\mathbb{U}$ (1.2)

is the Fourier transform of some real, even, integrable function $g(\lambda),$
$\lambda\in\Lambda$. We will refer to $g(\lambda)$ and to its Fourier
transform $\widehat{g}(t)$ as a generating function and generating kernel for
the functional $Q_{T}$, respectively. In the d.t. case the functions
$f(\lambda)$ and $g(\lambda)$ are assumed to be $2\pi$-periodic and
periodically extended to $\mathbb{R}$. In the c.t. case the process $X(u)$ is
assumed measurable and mean square continuous.

The limit distributions of the functionals in (1.1) are completely determined
by the spectral density $f(\lambda)$ (or covariance function $r(t)$) and the
generating function $g(\lambda)$ (or generating kernel $\widehat{g}(t)$), and
depending on their properties, the limits can be either Gaussian (that is,
$Q_{T}$ with an appropriate normalization obeys the central limit theorem), or
non-Gaussian.

The following two questions arise naturally:

  * (a)

Under what conditions on $f(\lambda)$ (resp. $r(t)$) and $g(\lambda)$ (resp.
$\widehat{g}(t)$) will the limits be Gaussian? (CLT-problem).

  * (b)

Describe the limit distributions, if they are non-Gaussian (NCLT-problem).

We discuss here these questions both for d.t. and c.t. stationary processes,
and survey the recent results.

We also survey recent results concerning functional central and non-central
limit theorems for the following processes, generated by quadratic functionals
in (1.1):

$Q_{T}(\tau):=\left\\{\begin{array}[]{ll}\sum_{t=1}^{[T\tau]}\sum_{s=1}^{[T\tau]}\widehat{g}(t-s)X(t)X(s)&\mbox{in
the d.t.\ case},\\\ \\\
\int_{0}^{T\tau}\int_{0}^{T\tau}\widehat{g}(t-s)X(t)X(s)\,dt\,ds&\mbox{in the
c.t.\ case},\end{array}\right.$ (1.3)

where $\tau\in[0,1]$ and $[\,\cdot\,]$ stands for the greatest integer.

We say that a functional central limit theorem (FCLT) for $Q_{T}(\tau)$ holds
if the process $Q_{T}(\tau)$ with an appropriate normalization converges
weakly in $C[0,1]$ in the c.t. case (and in $D[0,1]$ in the d.t. case) to
Brownian motion. We say that we have a functional non-central limit theorem
(FNCLT) for the process $Q_{T}(\tau)$ if the limit is non-Gaussian or, if
Gaussian, it is not Brownian motion.

The tapered case. In the spectral analysis of stationary processes the data
are frequently tapered before calculating the statistics of interest. Instead
of the original data $\\{X(t)$, $0\leq t\leq T\\}$ the tapered data
$\\{h(t)X(t)$, $0\leq t\leq T\\}$ with the data taper $h(t)$ are used for all
further calculations. The benefits of tapering the data have been widely
reported in the literature (see, e.g., Bloomfield [14], Brillinger [17],
Dahlhaus [24, 25, 26], Dahlhaus and Künsch [28], Guyon [69], 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 case, to establish asymptotic properties of the corresponding
estimators, we have to study the asymptotic distribution (as $T\to\infty$) of
the following Toeplitz type tapered quadratic functionals of the process
$X(u)$:

$Q_{T}^{h}:=\left\\{\begin{array}[]{ll}\sum_{t=1}^{T}\sum_{s=1}^{T}\widehat{g}(t-s)h_{T}(t)h_{T}(s)X(t)X(s)&\mbox{in
the d.t.\ case},\\\ \\\
\int_{0}^{T}\int_{0}^{T}\widehat{g}(t-s)h_{T}(t)h_{T}(s)X(t)X(s)\,dt\,ds&\mbox{in
the c.t.\ case},\end{array}\right.$ (1.4)

where $\widehat{g}(t)$, $t\in\mathbb{U}$ is the Fourier transform of some
integrable even function $g(\lambda)$, $\lambda\in\Lambda$, and
$h_{T}(t):=h(t/T)$ with a taper function $h(t)$, $t\in[0,1]$.

Quadratic functionals of the form (1.1) and (1.4) arise naturally in the
context of nonparametric and parametric statistical estimation of the spectrum
of stationary processes based on the non-tapered and tapered data,
respectively. And their limiting distributions are necessary to establish
asymptotic properties of the corresponding estimators. For instance, when we
are interested in nonparametric estimation of a linear integral functional
$J(f)$ in $L^{p}(\Lambda)$, $p>1$, then a natural statistical estimator for
$J(f)$ is the linear integral functional of the empirical spectral density
(periodogram) of the process $X(t)$, which is a quadratic functional of the
form (1.1) in the non-tapered case, and of the form (1.4) in the tapered case.
In the case of parametric estimation, for example, the Whittle estimation
procedure is based on the minimization of quadratic functionals of the form
(1.1) and (1.4).

###  1.3 A brief history

The problem of describing the asymptotic distribution of Toeplitz type
quadratic forms and functionals of stationary processes has a long history,
and goes back to the classical monograph by Grenander and Szegő [67], where
the CLT-problem for Gaussian processes was considered as an application of the
authors’ theory of the asymptotic behavior of the trace of products of
truncated Toeplitz matrices and operators.

Later the CLT-problem has been studied by a number of authors. Here we mention
only some significant contributions. For the d.t. short memory processes, the
problem was studied by Bentkus [10], Ibragimov [75] and Rosenblatt [87], in
connection with statistical estimation of the spectral and covariance
functions. Since 1986, there has been a renewed interest in this problem,
related to the statistical inferences for long memory (long-range dependence)
and intermediate memory (anti-persistent) processes (see, e.g., Avram [6], Fox
and Taqqu [39], Giraitis and Surgailis [60], Giraitis and Taqqu [62],
Has’minskii and Ibragimov [70], Ginovian and Sahakian [52], Terrin and Taqqu
[102], and references therein). In particular, Avram [6], Fox and Taqqu [39],
Ginovian and Sahakian [52], Giraitis and Surgailis [60], Giraitis and Taqqu
[62] have obtained sufficient conditions for quadratic form $Q_{T}$ to obey
the central limit theorem.

In the case of c.t. stationary Gaussian processes the CLT-problem for Toeplitz
type quadratic functionals was studied in a number of papers. We cite merely
the papers Avram et al. [7], Bai et al. [8, 9], Bryc and Dembo [19], Ginovyan
[43, 45, 48], Ginovyan and Sahakyan [53], Ibragimov [75], Leonenko and Sakhno
[80], where additional references can be found. The NCLT-problem have been
studied in Bai et al. [9], Giraitis and Taqqu [65], and Terrin and Taqqu
[101].

Central and non-central limit theorems for tapered quadratic forms of a d.t.
long memory Gaussian stationary fields have been proved in Doukhan et al.
[32]. A central limit theorem for tapered quadratic functionals $Q_{T}^{h}$,
in the case where the underlying model $X(t)$ is a Lévy-driven c.t. stationary
linear process has been proved in Ginovyan and Sahakyan [56].

The problem of nonparametric and parametric estimation of the spectrum of the
process $X(t)$ both for d.t. and c.t. cases based on the non-tapered data has
been considered by many authors (see, e.g., Avram et al. [7], Bentkus [10],
Dahlhaus [27], Dzhaparidze [27], Fox and Taqqu [38], Gao et al. [41], Ginovyan
[42, 43, 46, 47, 48, 49], Giraitis et al. [59], Giraitis and Surgailis [60],
Giraitis and Taqqu [64], Guyon [69], Hasminskii and Ibragimov [70], Ibragimov
[75, 76], Leonenko and Sakhno [80], Taniguchi [95], Taniguchi and Kakizawa
[96], Taqqu [98], and references therein). The problem in the tapered case was
studied in Alomari et al. [1], Dahlhaus [24, 26], Dahlhaus and Künsch [28],
Ginovyan [50], Ginovyan and Sahakyan [56], and Ludeña and Lavielle [82].

###  1.4 Frequency-domain conditions

Conditions that are expressed in terms of the spectral density $f(\lambda)$
and the generating function $g(\lambda)$ will be termed frequency-domain
conditions, while conditions that are in terms of the covariance function
$r(t)$ and the generating kernel $\widehat{g}(t)$ will be termed time-domain
conditions.

There are three sets of frequency-domain conditions for functionals of the
form (1.1) to obey the CLT, and these conditions separate the ranges of CLT
and NCLT:

  * (a)

the $(L^{p},L^{q})$ condition,

  * (b)

the $(\alpha,\beta)$ condition,

  * (c)

the trace condition.

All three are compensation conditions, meaning that the good behavior of one
function, say $g$ can compensate for the bad behavior of $f$ and vice versa.

(a) The $(L^{p},L^{q})$ condition. Let $f\in L^{p}$ $(p\geq 2)$ and $g\in
L^{q}$ $(q\geq 2)$. If $1/p+1/q\leq 1/2,$ then the functional $Q_{T}$ with an
appropriate normalization obeys the CLT (see Theorem 3.1(C)), while when
$1/p+1/q>1/2$, then, in general, $Q_{T}$ does not obey the CLT. This condition
goes back to the classical works of Rosenblatt [86, 87], where when estimating
the covariance function $r(t)$ of a d.t. Gaussian process, it was shown that
for a sample covariance $\widehat{r}(t)$ (which is a functional of the form
(1.1) with $g(\lambda)=\cos(t\lambda)$) to obey the CLT, the spectral density
$f$ should satisfy the condition $f\in L^{p}$ $(p\geq 2)$.

(b) The $(\alpha,\beta)$ condition. If both the spectral density $f$ and the
generating function $g$ are regularly varying at the origin of orders $\alpha$
and $\beta$, respectively, then it is the sum $\alpha+\beta$ that determines
the limiting distribution of $Q_{T}$. When $\alpha+\beta\leq 1/2,$ then the
limiting distribution of $Q_{T}$ is Gaussian, that is, $Q_{T}$ with an
appropriate normalization obeys the CLT (see Theorem 3.3), while when
$\alpha+\beta>1/2,$ then the limiting distribution of $Q_{T}$ is non-Gaussian
(see Theorem 4.6).

(c) The trace condition. This condition, which is more general and implies
both conditions (a) and (b), is an implicit condition. It is expressed in
terms of traces of products of truncated Toeplitz matrices (in the d.t. case)
and operators (in the c.t. case). The idea here is to approximate the traces
of products of Toeplitz matrices and operators (which are no longer Toeplitz)
by the traces of a Toeplitz matrix and a Toeplitz operator, respectively. Let
$A_{T}(f)$ denote either the $T\times T$ Toeplitz matrix $B_{T}(f)$ or the
$T$-truncated Toeplitz operator $W_{T}(f)$ generated by the spectral density
$f$, and let $A_{T}(g)$ denote either the $T\times T$ Toeplitz matrix
$B_{T}(g)$, or the $T$-truncated Toeplitz operator $W_{T}(g)$ generated by the
function $g$ (for definitions see Section 3.1). The trace condition is: if

$fg\in L^{2}$

and

$T^{-1}\bigl{[}\hbox{tr}\bigl{(}A_{T}(f)A_{T}(g)\bigr{)}^{2}-\hbox{tr}\bigl{(}A_{T}(f^{2}g^{2})\bigr{)}\bigr{]}\to
0\quad{\rm as}\quad T\to\infty,$

then the quadratic functional $Q_{T}$ in (1.1) with an appropriate
normalization obeys the CLT (see Theorems 3.1(A) and 3.4).

We will also discuss the time-domain counterparts of $(L^{p},L^{q})$ and
$(\alpha,\beta)$ conditions.

###  1.5 Methods and tools.

The most commonly used methods to prove central limit theorems for Toeplitz
type quadratic forms and functionals are:

  * (a)

the method of cumulants or moments,

  * (b)

the approximation method,

  * (c)

the method of characteristic functions.

To prove the non-central limit theorems for Toeplitz type quadratic
functionals, was used:

  * (a)

the spectral representation of the underlying process,

  * (b)

the properties of Lévy-Itô-type and Stratonovich-type multiple stochastic
integrals,

  * (c)

the power counting theorem.

Some details of the above methods are described in Section 8.

###  1.6 The structure of the paper

The paper is structured as follows. In Section 2 we describe the model of
interest - a stationary process, and recall some key notions and results from
the theory of stationary processes. In Section 3 we present sufficient
conditions for Toeplitz type quadratic forms and functionals of the form (1.1)
to obey the CLT in the case where the model is either a Gaussian or a linear
process. Section 4 contains functional central and noncentral limit theorems
for processes generated by Toeplitz type quadratic forms and functionals for
Gaussian and linear models. Section 5 is devoted to the functional central and
noncentral limit theorems for Lévy-driven linear models. In Section 6 we
discuss the case of tapered Toeplitz quadratic functionals, and state central
limit theorems. Section 7 contains some applications, involving nonparametric
estimation of spectral functionals and Whittle parametric estimation
procedure. In Section 8 we briefly discuss the methods and tools, used to
prove central and noncentral limit theorems for Toeplitz type quadratic forms
and functionals.

##  2 The model: second-order stationary process

In this section we introduce the model of interest - a second-order stationary
process, and recall some key notions and results from the theory of stationary
processes.

###  2.1 Key notions and some basic results

####  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,

$\mathbb{E}[X(u)]=0,\quad r(u)=\mathbb{E}[X(t+u)X(t)],\quad u,t\in\mathbb{U},$

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 [23], Section 5.2).

####  2.1.2 Spectral representations

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 [23], Doob [31], Ibragimov
and Linnik [77]), 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 $t\in\mathbb{U}$
the covariance function $r(t)$ 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
$\\{X(u),\ u\in\mathbb{U}\\}$.

By the well-known Cramér theorem (see, e.g., Cramér and Leadbetter [23]) for
any stationary process $\\{X(u),\,u\in\mathbb{U}\\}$ with spectral measure
$\mu$ there exists an orthogonal stochastic measure $Z=Z(B)$,
$B\in\mathfrak{B}(\Lambda)$, such that for every $u\in\mathbb{U}$ the process
$X(u)$ admits the following spectral representation:

$X(u)=\int_{\Lambda}\exp\\{i\lambda u\\}dZ(\lambda),\quad u\in\mathbb{U}.$
(2.3)

Moreover, $E\left[|Z(B)|^{2}\right]=\mu(B)$ for every
$B\in\mathfrak{B}(\Lambda)$. For definition and properties of orthogonal
stochastic measures and stochastic integral in (2.3) we refer, e.g., Cramér
and Leadbetter [23].

####  2.1.3 Kolmogorov’s isometric isomorphism theorem

Given a probability space $(\Omega,\mathcal{F},P)$, define the $L^{2}$-space
of random variables $\xi=\xi(\omega)$, $\mathbb{E}[\xi]=0$:

$L^{2}(\Omega):=\left\\{\xi:\,||\xi||^{2}:=\int_{\Omega}|\xi(\omega)|^{2}dP(\omega)<\infty\right\\}.$
(2.4)

Then $L^{2}(\Omega)$ becomes a Hilbert space with the following inner product:
for $\xi,\eta\in L^{2}(\Omega)$

$(\xi,\eta)=\mathbb{E}[\xi\eta]=\int_{\Omega}\xi(\omega){\eta(\omega)}dP(\omega).$
(2.5)

For $a,b\in\mathbb{U}$, $-\infty\leq a\leq b\leq\infty,$ we define the space
$H_{a}^{b}(X)$ to be the closed linear subspace of the space $L^{2}(\Omega)$
spanned by the random variables $X(u,\omega)$, $u\in[a,b]$:

$H_{a}^{b}(X):=\overline{sp}\\{X(u),\,\,a\leq u\leq b\\}_{L^{2}(\Omega)}.$
(2.6)

Observe that the subspace $H_{a}^{b}(X)$ consists of all finite linear
combinations $\sum_{k=1}^{n}c_{k}X(u_{k})$ ($a\leq u_{k}\leq b$), as well as,
their $L^{2}(\Omega)$-limits.
The space $H(X):=H_{-\infty}^{\infty}(X)$ is called the Hilbert space
generated by the process $X(u)$, or the time-domain of $X(u)$.

Let $\mu$ be the spectral measure of the process
$\\{X(u),\,u\in\mathbb{U}\\}$. Consider the weighted $L^{2}$-space
$L^{2}(\mu):=L^{2}(\mu,\Lambda)$ of complex-valued functions
$\varphi(\lambda),\,\lambda\in\Lambda$, defined by

$L^{2}(\mu):=\left\\{\varphi(\lambda):\,||\varphi||^{2}_{\mu}:=\int_{\Lambda}|\varphi(\lambda)|^{2}d\mu(\lambda)<\infty\right\\}.$
(2.7)

Then $L^{2}(\mu)$ becomes a Hilbert space with the following inner product:
for $\varphi,\psi\in L^{2}(\mu)$

$(\varphi,\psi)_{\mu}=\int_{\Lambda}\varphi(\lambda)\overline{\psi}(\lambda)d\mu(\lambda).$
(2.8)

The Hilbert space $L^{2}(\mu,\Lambda)$ is called the frequency-domain of the
process $X(u)$.

Kolmogorov’s isometric isomorphism theorem states that for any stationary
process $X(u)$, $u\in\mathbb{U}$, with spectral measure $\mu$ there exists a
unique isometric isomorphism $V$ between the time- and frequency-domains
$H(X)$ and $L^{2}(\mu)$, such that $V[X(u)]=e^{iu\lambda}$ for any
$u\in\mathbb{U}.$

Thus, any linear problem in the time-domain $H(X)$ can be translated into one
in the frequency-domain $L^{2}(\mu)$, and vice versa. This fact allows to
study stationary processes using analytic methods.

###  2.2 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 [23],
Doob [31], Ibragimov and Linnik [77].

  * (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:

$X(u)=\sum_{k=-\infty}^{\infty}a(u-k)\xi(k),\quad\sum_{k=-\infty}^{\infty}|a(k)|^{2}<\infty,$
(2.9)

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:

$r(u)=\mathbb{E}X(u)X(0)=\sum_{k=-\infty}^{\infty}a(u+k)a(k),$ (2.10)

and

$f(\lambda)=\frac{1}{2\pi}|\widehat{a}(\lambda)|^{2}=\frac{1}{2\pi}\left|\sum_{k=-\infty}^{\infty}a(k)e^{-ik\lambda}\right|^{2},\quad\lambda\in\Lambda.$
(2.11)

  * (c)

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

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

  * (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:

$X(u)=\int_{\mathbb{R}}a(u-t)d\xi(t),\quad\int_{\mathbb{R}}|a(t)|^{2}dt<\infty,$
(2.12)

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:

$r(u)=\mathbb{E}X(u)X(0)=\int_{\mathbb{R}}a(u+x)a(x)dx,$ (2.13)

and

$f(\lambda)=\frac{1}{2\pi}|\widehat{a}(\lambda)|^{2}=\frac{1}{2\pi}\left|\int_{\mathbb{R}}e^{-i\lambda
t}a(t)dt\right|^{2},\quad\lambda\in\mathbb{R}.$ (2.14)

  * (c)

In the case where $\\{\xi(t),t\in\mathbb{R}\\}$ is a Gaussian process, the
process $\\{X(u),\,u\in\mathbb{Z}\\}$ is Gaussian.

###  2.3 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 (2.12), 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.

The function $a(\cdot)$ in representations (2.9) and (2.12) plays the role of
a time-invariant filter, and the linear processes defined by (2.9) and (2.12)
can be viewed as the output of a linear filter $a(\cdot)$ applied to the
process $\\{\xi(u),\ t\in\mathbb{U}\\}$, called the innovation or driving
process of $X(u)$.

Processes of the form (2.12) appear in many fields of science (economics,
finance, physics, etc.), and cover a large class of popular models in c.t.
time series modeling. For instance, the so-called c.t. autoregressive moving
average (CARMA) models, which are the c.t. analogs of the classical
autoregressive moving average (ARMA) models in d.t. case, are of the form
(2.12) and play a central role in the representations of c.t. stationary time
series (see, e.g., Brockwell [18]).

###  2.4 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 $f(\lambda).$

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).

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 [11], Beran et al. [12], Giraitis et al. [59],
Guégan [68], Robinson [85], and references therein).

It is convenient to characterize the memory structure in terms of the spectral
density function.

####  2.4.1 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 [19], Section 3.1):

$f(\lambda)=\frac{\sigma^{2}}{2\pi}\cdot\frac{|\theta_{q}(e^{-i\lambda})|^{2}}{|\psi_{p}(e^{-i\lambda})|^{2}}.$
(2.15)

A typical c.t. short-memory model example is the stationary c.t. ARMA$(p,q)$
processes, denoted by CARMA$(p,q)$, which is defined to be the solution of a
$p$th order stochastic differential equation with a suitable initial condition
and driven by a standard Brownian motion and its derivatives up to and
including order $0\leq q<p$. The spectral density function $f(\lambda)$ of the
CARMA$(p,q)$ process is given by the following formula (see, e.g., Brockwell
[18]):

$f(\lambda)=\frac{\sigma^{2}}{2\pi}\cdot\frac{|\beta(i\lambda)|^{2}}{|\alpha(i\lambda)|^{2}},$
(2.16)

where $\alpha(z)$ and $\beta(z)$ 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

$f(\lambda)=\frac{\sigma^{2}}{\pi}\cdot\frac{\alpha^{2}}{\lambda^{2}+\alpha^{2}},\quad\alpha>0,\,\lambda\in\mathbb{R}.$
(2.17)

####  2.4.2 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 [11], Guégan [68], Palma
[83], Taqqu [97] 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 [19], 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

$f(\lambda)=|1-e^{-i\lambda}|^{-2d}=(2\sin(\lambda/2))^{-2d},\quad
0<\lambda\leq\pi,\quad 0<d<1/2.$ (2.18)

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 $f(\lambda)=|1-e^{-i\lambda}|^{-2d}$ 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 $X(t)$
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

$f_{X}(\lambda)=|1-e^{-i\lambda}|^{-2d}f(\lambda),\quad d<1/2,$ (2.19)

where $f(\lambda)$ is the spectral density of an ARMA$(p,q)$ process, given by
(2.15). Observe that for $0<d<1/2$ the model $X(t)$ specified by (2.19)
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 (2.19) 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., Bondon and Palma [16], and
Brockweel and Davis [19]).

The ARFIMA$(p,d,q)$ processes, first introduced by Granger and Joyeux [66],
and Hosking [73], became very popular due to their ability in providing a good
characterization of the long-run properties of many economic and financial
time series. They are also very useful for modeling multivariate time series,
since they are able to capture a larger number of long term equilibrium
relations among economic variables than the traditional multivariate ARIMA
models (see, e.g., Guégan [68], and Henry and Zaffaroni [71] for a survey on
this topic).

Another important long-memory model is the fractional Gaussian noise (fGn). To
define fGn we first consider 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, that
is, $B_{H}(t)$ satisfies the following conditions:

(a) $B_{H}(0)=0$, $\mathbb{E}[B_{H}(t)]=0$, $t\in\mathbb{R}$;

(b) $\\{B_{H}(at),t\in\mathbb{R}\\}\stackrel{{\scriptstyle
d}}{{=}}\\{a^{H}B_{H}(t),t\in\mathbb{R}\\}$ for any $a>0$;

(c) $\\{B_{H}(t+u)-B_{H}(u),t\in\mathbb{R}\\}\stackrel{{\scriptstyle
d}}{{=}}\\{B_{H}(t),t\in\mathbb{R}\\}$ for each fixed $u\in\mathbb{R}$;

(d) the covariance function is given by

${\rm
Cov}(B_{H}(s),B_{H}(t))=\frac{\sigma_{0}^{2}}{2}\left[|t|^{2H}-|s|^{2H}-|t-s|^{2H}\right],$

where the symbol $\stackrel{{\scriptstyle d}}{{=}}$ stands for equality of the
finite-dimensional distributions, and $\sigma_{0}^{2}={\rm Var}B_{H}(1)$. 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 covariance function

$r(k)=\frac{\sigma_{0}^{2}}{2}\left[|k+1|^{2H}-|k|^{2H}-|k-1|^{2H}\right],\quad
k\in\mathbb{Z},$ (2.20)

and spectral density function

$f(\lambda)=c\,|1-e^{-i\lambda}|^{2}\sum_{k=-\infty}^{\infty}|\lambda+2\pi
k|^{-(2H+1)},\quad-\pi\leq\lambda\leq\pi,$ (2.21)

where $c$ is a positive constant.

It follows from (2.21) 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 (2.18) and (2.21), 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
[89] and Taqqu [97, 99].

####  2.4.3 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”:

$f(\lambda)=c|\lambda|^{-(2H+1)},\quad c>0,\ \,\,0<H<1,\
\,\,\lambda\in\mathbb{R}.$ (2.22)

The form (2.22) can be understood in a generalized sense (see Yaglom [108],
Section 24, Flandrin [36], Solo [91]), 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

$f(\lambda)=c\,|\lambda|^{-2\alpha}(1+\lambda^{2})^{-\beta},\quad\lambda\in\mathbb{R},\,0<c<\infty,\,0<\alpha<1,\,\beta>0.$
(2.23)

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. [41]).

Notice that the process $X(t)$, specified by (2.23), is stationary if
$0<\alpha<1/2$ and is non-stationary with stationary increments if
$1/2\leq\alpha<1.$ Observe also that the spectral density (2.23) 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 (2.23) 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 (2.22) and (2.23), 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$(p,H,q)$ processes,
which is defined to be the solution of a $p$th order stochastic differential
equation with a suitable initial condition and driven by a fractional Brownian
motion with Hurst parameter $H$ and its derivatives up to and including order
$0\leq q<p$. The spectral density function $f(\lambda)$ of the
CARFIMA$(p,H,q)$ process is given by the following formula (see, e.g., Tsai
and Chan [104]):

$f(\lambda)=\frac{\sigma^{2}}{2\pi}\Gamma(2H+1)\sin(\pi
H)|\lambda|^{1-2H}\frac{|\beta(i\lambda)|^{2}}{|\alpha(i\lambda)|^{2}},$
(2.24)

where $\alpha(z)$ and $\beta(z)$ are polynomials of degrees $p$ and $q$,
respectively. Notice that for $H=1/2$, the spectral density function given by
(2.24) becomes that of the short-memory CARMA$(p,q)$ process, given by (2.16).

##  3 CLT for Toeplitz type quadratic functionals for Gaussian and linear
processes

In this section we present sufficient conditions for quadratic forms and
functionals of the form (1.1) to obey the CLT. The processes considered will
be d.t. and c.t. Gaussian or linear processes with memory. The matrix and the
operator that characterize the quadratic form and functional are Toeplitz.

As it was mentioned in the introduction, the limit distributions of the
functionals in (1.1) are completely determined by the spectral density
$f(\lambda)$ (or covariance function $r(t)$) and the generating function
$g(\lambda)$ (or generating kernel $\widehat{g}(t)$). Conditions that are in
terms of the spectral density $f(\lambda)$ and the generating function
$g(\lambda)$ will be called frequency-domain conditions, while conditions that
are in terms of the covariance function $r(t)$ and the generating kernel
$\widehat{g}(t)$ will be called time-domain conditions.

###  3.1 Frequency domain conditions

Let $\\{X(u),\ u\in\mathbb{U}\\}$ be a centered real-valued Gaussian
stationary process with spectral density $f(\lambda)$, $\lambda\in\Lambda$ and
covariance function $r(t):=\widehat{f}(t)$, $t\in\mathbb{U}$, where
$\mathbb{U}$ and $\Lambda$ are as in Section 2. We are interested in the
asymptotic distribution (as $T\to\infty$) of the following Toeplitz type
quadratic functionals of the process $X(u)$:

$Q_{T}:=Q_{T}(f,g)=\left\\{\begin{array}[]{ll}\sum_{t=1}^{T}\sum_{s=1}^{T}\widehat{g}(t-s)X(t)X(s)&\mbox{in
the d.t.\ case},\\\ \\\
\int_{0}^{T}\int_{0}^{T}\widehat{g}(t-s)X(t)X(s)\,dt\,ds&\mbox{in the c.t.\
case},\end{array}\right.$ (3.1)

where $\widehat{g}(t)$ is the Fourier transform of some real, even, integrable
function $g(\lambda),$ $\lambda\in\Lambda$. In the d.t. case the functions
$f(\lambda)$ and $g(\lambda)$ are assumed to be $2\pi$-periodic and
periodically extended to $\mathbb{R}$.

Note. We include the function $f$ in the notation $Q_{T}(f,g)$ to emphasize
that the distribution of the quadratic form depends also on the spectral
density $f$. Let $Q_{T}$ be as in (3.1). By $\widetilde{Q}_{T}$ we denote the
standard normalized quadratic functional:

$\widetilde{Q}_{T}:=T^{-1/2}\,\left(Q_{T}-\mathbb{E}[Q_{T}]\right).$ (3.2)

As before, the notation

$\widetilde{Q}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2})\quad{\rm
as}\quad T\to\infty$ (3.3)

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

Toeplitz matrices and operators arise naturally in the theory of stationary
processes, and serve as tools, to study many topics of the spectral and
statistical analysis of d.t. and c.t. stationary processes.

We first define the truncated Toeplitz matrices and operators, generated by
integrable real symmetric functions.

Let $\psi(\lambda)$ be an integrable real symmetric function defined on
$\Lambda=[-\pi,\pi]$. For $T=1,2,\ldots$, the $(T\times T)$-truncated Toeplitz
matrix generated by $\psi(\lambda)$, denoted by $B_{T}(\psi)$, is defined by
the following equation (see, e.g., Ginovyan and Sahakyan [52], and Grenander
and Szegő [67]):

$B_{T}(\psi):=\|\widehat{\psi}(t-s)\|_{t,s=1,2\ldots,T}=\left(\begin{array}[]{llll}\widehat{\psi}(0)&\widehat{\psi}(-1)&\cdots&\widehat{\psi}(1-T)\\\
\widehat{\psi}(1)&\widehat{\psi}(0)&\cdots&\widehat{\psi}(2-T)\\\
\cdots&\cdots&\cdots&\cdots\\\
\widehat{\psi}(T-1)&\widehat{\psi}(T-2)&\cdots&\widehat{\psi}(0)\end{array}\right),$
(3.4)

where $\widehat{\psi}(t)=\int_{\Lambda}e^{i\lambda
t}\,\psi(\lambda)\,d\lambda$ $(t\in\mathbb{Z})$ are the Fourier coefficients
of $\psi$.

Given a real number $T>0$ and an integrable real symmetric function
$\psi(\lambda)$ defined on $\mathbb{R}:=(-\infty,\infty)$, the $T$-truncated
Toeplitz operator (also called Wiener-Hopf operator) generated by
$\psi(\lambda)$, denoted by $W_{T}(\psi)$, is defined by the following
equation (see, e.g., Ginovyan [45], Ginovyan and Sahakyan [53], Grenander and
Szegő [67], Ibragimov [75], and Kac [79]):

$[W_{T}(\psi)u](t)=\int_{0}^{T}\hat{\psi}(t-s)u(s)ds,\quad u(s)\in
L^{2}[0,T],$ (3.5)

where $\widehat{\psi}(t)=\int_{\mathbb{R}}e^{i\lambda
t}\,\psi(\lambda)\,d\lambda$ $(t\in\mathbb{R})$ is the Fourier transform of
$\psi(\lambda)$.

Let $A_{T}(f)$ and $A_{T}(g)$ denote either the $T$-truncated Toeplitz
operators (in the c.t. case), or the $T\times T$ Toeplitz matrices (in the
d.t. case) generated by the functions $f$ and $g$, respectively. Observe that
$A_{T}(f)$ is the covariance matrix (or operator) of the process $\\{X(u),\
u\in\mathbb{U}\\}$.

We assume below that $f,g\in L^{1}(\Lambda)$, and with no loss of generality,
that $g\geq 0$. Also, we set

$\sigma^{2}_{0}:=16\pi^{3}\int_{\Lambda}f^{2}(\lambda)g^{2}(\lambda)\,d\lambda.$
(3.6)

As usual $\Lambda=[-\pi,\pi]$ in the d.t. case and $\Lambda=\mathbb{R}$ in the
c.t. case.

The theorems that follow contain sufficient conditions expressed in terms of
$f(\lambda)$ and $g(\lambda)$ to ensure central limit theorems for standard
normalized quadratic functionals $\widetilde{Q}_{T}$ both for d.t. and c.t.
Gaussian processes. Some of the assumptions imposed on $f$ allow for long-
range dependence ($f(0)=\infty$), others for discontinuities at other
frequencies. Sometimes the good behavior of one function, say $g$ can
compensate for the bad behavior of $f$ and vice versa.

######  Theorem 3.1.

Let $f$, $g$, $A_{T}(f)$, $A_{T}(g)$, and $\widetilde{Q}_{T}$ be as above.
Each of the following conditions is sufficient for functional $Q_{T}$ to obey
the CLT, that is,

$\widetilde{Q}_{T}\overset{d}{\rightarrow}\eta\sim
N(0,\sigma_{0}^{2})\quad{\rm as}\quad T\to\infty,$ (3.7)

with $\sigma^{2}_{0}$ given by (3.6).

  * (A)

$f\cdot g\in L^{1}(\Lambda)\cap L^{2}(\Lambda)$ and

$\chi_{2}(\widetilde{Q}_{T}):=\frac{2}{T}\hbox{tr}\bigl{[}A_{T}(f)A_{T}(g)\bigr{]}^{2}\longrightarrow\sigma_{0}^{2},$
(3.8)

where $\hbox{tr}[A]$ stands for the trace of the operator (or the matrix) $A$.

  * (B)

The function

$\varphi({\bf
u}):=\varphi(u_{1},u_{2},u_{3})=\int_{\Lambda}f(\lambda)g(\lambda-
u_{1})f(\lambda-u_{2})g(\lambda-u_{3})\,d\lambda$ (3.9)

belongs to $L^{2}(\Lambda^{3})$ and is continuous at ${\bf 0}=(0,0,0).$

  * (C)

$f\in L^{1}(\Lambda)\cap L^{p}(\Lambda)$ $(p\geq 2)$ and $g\in
L^{1}(\Lambda)\cap L^{q}(\Lambda)$ $(q\geq 2)$ with $1/p+1/q\leq 1/2.$

  * (D)

$f\in L^{1}(\Lambda)\cap L^{2}(\Lambda)$, $g\in L^{1}(\Lambda)\cap
L^{2}(\Lambda)$, $fg\in L^{2}(\Lambda)$ and

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

######  Remark 3.1.

Observe that assertion (A) implies assertions (B) – (D), and assertion (B)
implies assertions (C) and (D) (see Giraitis and Surgailis [60], Ginovyan and
Sahakyan [52, 53]). For the d.t. case: assertions (A) and (D) were proved in
Giraitis and Surgailis [60] (see also Giraitis et al. [59]); assertions (B)
was proved in Ginovyan and Sahakyan [52]; assertion (C) for $p=q=\infty$ was
first established by Grenander and Szegő ([67], Section 11.7), while the case
$p=2$, $q=\infty$ was proved by Ibragimov [75] and Rosenblatt [87], in the
general d.t. case assertion (C) was proved by Avram [6]. For the c.t. case,
assertions (A) – (D) were proved in Ginovyan [45] and in Ginovyan and Sahakyan
[53].

The following theorem provides conditions for the CLT to hold when either $f$
or $g$ is a sufficiently smooth function. Given a number $\alpha$
($0<\alpha<1$). Denote by $\rm{Lip}_{\alpha}(\Lambda)$ the class of Lipschitz
functions on $\Lambda$. By definition, $\phi\in\rm{Lip}_{\alpha}(\Lambda)$ if
there exists a constant $C<\infty$ such that

$|\phi(x)-\phi(y)|\leq C|x-y|^{\alpha}\quad\text{for all}\quad x,y\in\Lambda.$

######  Theorem 3.2.

Let either $f\in\rm{Lip}_{\alpha}(\Lambda)$ or
$g\in\rm{Lip}_{\alpha}(\Lambda)$ with $\alpha>1/2$, and let $fg\in
L^{2}(\Lambda)$. Then $\widetilde{Q}_{T}\overset{d}{\rightarrow}\eta\sim
N(0,\sigma^{2}_{0})$ as $T\to\infty$ with $\sigma^{2}_{0}$ given by (3.6).

Theorem 3.2 for the d.t. case was proved in Giraitis and Taqqu [61]. For the
c.t. case it can be proved similarly.

To state the next theorem, we need to introduce a class of slowly varying
functions at zero. Recall that a function $u(\lambda)$,
$\lambda\in\mathbb{R}$, is called slowly varying at zero if it is non-negative
and for any $t>0$

$\lim_{\lambda\rightarrow 0}\frac{u(t\lambda)}{u(\lambda)}\rightarrow 1.$

Denote by $SV_{0}(\Lambda)$ the class of slowly varying functions at zero
$u(\lambda)$, $\lambda\in\Lambda$, 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
$0<\lambda<\mu<a$. An example of a function belonging to $SV_{0}(\Lambda)$ is
$u(\lambda)=\left|\ln|\lambda|\right|^{-\gamma}$ with $\gamma>0$ and $a=1$.

######  Theorem 3.3.

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

$f(\lambda)\leq|\lambda|^{-\alpha}L_{1}(\lambda),\quad|g(\lambda)|\leq|\lambda|^{-\beta}L_{2}(\lambda),\quad\lambda\in[-a,a],$
(3.10)

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

$\displaystyle L_{i}\in SV_{0}(\mathbb{R}),\ \
\lambda^{-(\alpha+\beta)}L_{i}(\lambda)\in L^{2}[-a,a],\ i=1,2.$ (3.11)

Then $\widetilde{Q}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{0})$
as $T\to\infty$ with $\sigma^{2}_{0}$ given by (3.6).

######  Remark 3.2.

For the d.t. case the result of Theorem 3.3 under the condition
$\alpha+\beta<1/2$ was first obtained by Fox and Taqqu [39]. For the general
case, including the critical value $\alpha+\beta=1/2$ it was proved in
Ginovyan and Sahakyan [52]. For the c.t. case the result was proved in
Ginovyan and Sahakyan [53].

######  Remark 3.3.

The conditions $\alpha<1$ and $\beta<1$ in Theorem 3.3 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 $f\cdot g\in
L^{1}(\mathbb{R})$, $f,g\in L^{\infty}(\mathbb{R}\setminus[-a,a])$ and (3.11)
imply that $f\cdot g\in L^{2}(\mathbb{R})$, so that the variance
$\sigma^{2}_{0}$ in (3.6) is finite.

######  Remark 3.4.

In Theorem 3.3, the assumption that $L_{1}(x)$ and $L_{2}(x)$ belong to
$SV_{0}(\mathbb{R})$ instead of merely being slowly varying at zero is done in
order to deal with the critical case $\alpha+\beta=1/2$. Suppose that we are
away from this critical case, namely, $f(x)=|x|^{-\alpha}l_{1}(x)$ and
$g(x)=|x|^{-\beta}l_{2}(x)$, where $\alpha+\beta<1/2$, and $l_{1}(x)$ and
$l_{2}(x)$ are slowly varying at zero functions. Assume also that $f(x)$ and
$g(x)$ are integrable and bounded on $(-\infty,-a)\cup(a,+\infty)$ for any
$a>0$. We claim that Theorem 3.3 applies. Indeed, choose
$\alpha^{\prime}>\alpha$, $\beta^{\prime}>\beta$ with
$\alpha^{\prime}+\beta^{\prime}<1/2$. Write
$f(x)=|x|^{-\alpha^{\prime}}|x|^{\delta}l_{1}(x)$, where
$\delta=\alpha^{\prime}-\alpha>0$. Since $l_{1}(x)$ is slowly varying, when
$|x|$ is small enough, for some $\epsilon\in(0,\delta)$ we have
$|x|^{\delta}l_{1}(x)\leq|x|^{\delta-\epsilon}$. Then one can bound
$|x|^{\delta-\epsilon}$ by $c\left|\ln|x|\right|^{-1}\in SV_{0}(\mathbb{R})$
for small $|x|<1$. Hence one has when $|x|<1$ is small enough,

$f(x)\leq|x|^{-\alpha^{\prime}}\left(c\left|\ln|x|\right|^{-1}\right).$

Similarly, when $|x|<1$ is small enough, one has

$g(x)\leq|x|^{-\beta^{\prime}}\left(c\left|\ln|x|\right|^{-1}\right).$

All the assumptions in Theorem 3.3 are now readily checked with $\alpha$ and
$\beta$ replaced by $\alpha^{\prime}$ and $\beta^{\prime}$, respectively.

######  Remark 3.5.

The functions

$f(\lambda)=|\lambda|^{-\alpha}|\ln|\lambda||^{-\gamma}\quad\mbox{and}\quad
g(\lambda)=|\lambda|^{-\beta}|\ln|\lambda||^{-\gamma},$ (3.12)

where $\alpha<1,$ $\beta<1,$ $\alpha+\beta\leq 1/2$ and $\gamma>1/2$, provide
examples of spectral density $f(\lambda)$ and generating function $g(\lambda)$
satisfying the conditions of Theorem 3.3 (see Ginovyan and Sahakyan [52]).

######  Remark 3.6.

The slowly varying functions $L_{1}$ and $L_{2}$ in (3.10) are important
because they provide a great flexibility in the choice of spectral density $f$
and generating function $g$. Theorem 3.3 shows that in the critical case
$\alpha+\beta=1/2$ the limit distribution of the standard normalized quadratic
form $\widetilde{Q}_{T}$ given by (3.2) is Gaussian and depends on the slowly
varying factors $L_{1}$ and $L_{2}$ through $f$ and $g$.

######  Remark 3.7.

The functions $f(\lambda)$ and $g(\lambda)$ in Theorem 3.3 have singularities
at the point $\lambda=0$, and are bounded in any neighborhood of this point.
It can be shown that the choice of the point $\lambda=0$ is not essential, and
instead it can be taken to be any point $\lambda_{0}\in[-a,a]$. Using the
properties of the products of Toeplitz matrices and operators it can be shown
that Theorem 3.3 remains valid when $f(\lambda)$ and $g(\lambda)$ have
singularities of the form (3.12) at the same finite number of points of the
segment $[-a,a]$ (see Ginovyan and Sahakyan [52]).

######  Remark 3.8.

Assertion (A) of Theorem 3.1 implies Theorem 3.3. On the other hand, for
functions $f(\lambda)=\lambda^{-3/4}$ and $g(\lambda)=\lambda^{3/4}$
satisfying the conditions of Theorem 3.3, the function
$\varphi(u_{1},u_{2},u_{3})$ in (3.9) is not defined for $u_{2}=0$, $u_{1}\neq
0$, $u_{3}\neq 0$, showing that assertion (B) of Theorem 3.1 generally does
not imply Theorem 3.3 (see Ginovyan and Sahakyan [52]).

Giraitis and Surgailis [60] proved that assertion (A) of Theorem 3.1 remains
valid for d.t. linear processes. More precisely, they proved the following
theorem.

######  Theorem 3.4.

Let $\\{X(u),\,u\in\mathbb{Z}\\}$ be a d.t. stationary linear process of the
form (2.9) with spectral density $f$. Let $Q_{T}$ be a quadratic form
generated by a function $g$ given by (3.1), and let $B_{T}(f)$ and $B_{T}(g)$
be the $T\times T$ Toeplitz matrices generated by the functions $f$ and $g$,
respectively (see (3.4)). Assume that

$\frac{1}{T}\hbox{tr}\bigl{[}B_{T}(f)B_{T}(g)\bigr{]}^{2}\longrightarrow
8\pi^{3}\int_{-\pi}^{\pi}f^{2}(\lambda)g^{2}(\lambda)d\lambda<\infty,$ (3.13)

Then the CLT holds for $Q_{T}$, that is,

$\widetilde{Q}_{T}:=T^{-1/2}\,\left(Q_{T}-\mathbb{E}[Q_{T}]\right)\overset{d}{\rightarrow}\eta\sim
N(0,\sigma^{2})\quad{\rm as}\quad T\to\infty,$ (3.14)

where

$\sigma^{2}=16\pi^{3}\int_{-\pi}^{\pi}f^{2}(\lambda)g^{2}(\lambda)d\lambda+\kappa_{4}\left[2\pi\int_{-\pi}^{\pi}f(\lambda)g(\lambda)d\lambda\right]^{2},$
(3.15)

and $\kappa_{4}$ is the fourth cumulant of $\xi(0)$.

The next result, which was proved in Giraitis and Taqqu [61], extends Theorem
3.2 to the case of d.t. linear processes.

######  Theorem 3.5.

Let $\\{X(u),\,u\in\mathbb{Z}\\}$ be a d.t. stationary linear process of the
form (2.9) with spectral density $f$ and $\mathbb{E}[\xi^{2}(0)]=1$, and let
$Q_{T}$ be a quadratic form generated by a function $g$ given by (3.1).
Suppose either $f\in\rm{Lip}_{\alpha}(\Lambda)$ or
$g\in\rm{Lip}_{\alpha}(\Lambda)$ with $\alpha>1/2$, and $fg\in
L^{2}(\Lambda)$. Then $\widetilde{Q}_{T}\overset{d}{\rightarrow}\eta\sim
N(0,\sigma^{2})$ as $T\to\infty$ with $\sigma^{2}$ given by (3.15).

Giraitis and Surgailis [60] conjectured that the relation (3.13), and hence,
the CLT for $Q_{T}$, holds under the single condition that the integral on the
right hand side of (3.13) is finite. Ginovyan [44] showed that the finiteness
of this integral does not guarantee convergence in (3.13), and conjectured
that positiveness and finiteness of the integral in (3.13) must be sufficient
for $Q_{T}$ to obey the CLT.

The next proposition shows that the condition of positiveness and finiteness
of the integral in (3.13) also is not sufficient for $Q_{T}$ to obey the CLT
(see Ginovyan and Sahakyan [52]).

######  Proposition 3.1.

There exist a spectral density $f(\lambda)$ and a generating function
$g(\lambda)$ such that

$0<\int_{-\pi}^{\pi}f^{2}(\lambda)\,g^{2}(\lambda)\,d\lambda<\infty$ (3.16)

and

$\lim_{T\to\infty}\sup\chi_{2}(\widetilde{Q}_{T})=\lim_{n\to\infty}\sup\frac{2}{T}\hbox{tr}\left(B_{T}(f)B_{T}(g)\right)^{2}=\infty,$
(3.17)

that is, the condition (3.16) does not guarantee convergence in (3.13), and
hence is not sufficient for $Q_{T}$ to obey the CLT.

To construct functions $f(\lambda)$ and $g(\lambda)$ satisfying (3.16) and
(3.17), for a fixed $p\geq 2$ we choose a number $q>1$ to satisfy $1/p+1/q>1$,
and for such $p$ and $q$ we consider the functions $f_{0}(\lambda)$ and
$g_{0}(\lambda)$ defined by

$f_{0}(\lambda)=\left\\{\begin{array}[]{ll}\left({2^{s}}/{s^{2}}\right)^{1/p},&\mbox{if
$2^{-s-1}\leq\lambda\leq 2^{-s},\,s=2m$}\\\ 0,&\mbox{if
$2^{-s-1}\leq\lambda\leq 2^{-s},\,s=2m+1$},\end{array}\right.$ (3.18)
$g_{0}(\lambda)=\left\\{\begin{array}[]{ll}\left({2^{s}}/{s^{2}}\right)^{1/q},&\mbox{if
$2^{-s-1}\leq\lambda\leq 2^{-s},\,s=2m+1$}\\\ 0,&\mbox{if
$2^{-s-1}\leq\lambda\leq 2^{-s},\,s=2m$},\end{array}\right.$ (3.19)

where $m$ is a positive integer. For an arbitrary finite positive constant $C$
we set $g_{\pm}(\lambda)=g_{0}(\lambda)\pm C$. Then the functions $f=f_{0}$
and $g=g_{+}$ or $g=g_{-}$ satisfy (3.16) and (3.17) (for details we refer to
Ginovyan [44], and Ginovyan and Sahakyan [52]). Consequently, for these
functions the quadratic form $Q_{T}$ does not obey the CLT, and it is of
interest to describe the limiting non-Gaussian distribution of $Q_{T}$ in this
special case.

###  3.2 Time domain conditions

In this subsection we present time-domain sufficient conditions for a
quadratic form $Q_{T}$ of the form (3.1) to obey the CLT. That is, the
conditions are in terms of the covariance function $r(t)$ and the generating
kernel $\widehat{g}(t)$. The processes considered here will be d.t. Gaussian
or linear processes with memory. Observe that the time-domain conditions
stated below are, in general, not equivalent to the frequency-domain
conditions stated in Theorems 3.1 and 3.3. The method of proof for
establishing the CLT is also different under the time-domain conditions. In
this case, diagrams are used and the method of moments is applied. This is
because, in the frequency-domain, one can use an approximation technique,
essentially replacing the possibly unbounded spectral density by a bounded
one, which allowed us to approximate the bivariate quadratic forms by
univariate sums of $m$-dependent random variables. Such an approximation
technique, however, does not work with time-domain conditions, and one has to
deal directly with the covariances, which decrease slowly.

The next results contain sufficient conditions in terms of the covariance
function $r(t)$ and the generating kernel $\widehat{g}(t)$ ensuring central
limit theorems for standard normalized quadratic form $\widetilde{Q}_{T}$ for
the d.t. Gaussian processes (see Fox and Taqqu [37], and Giraitis and Taqqu
[62]).

######  Theorem 3.6.

If the covariance function $r(t)$ and the generating kernel $\widehat{g}(t)$
satisfy the following condition:

$r\in l^{p}\,(p\geq 1)\quad\text{and}\quad\widehat{g}\in l^{q}\,(q\geq
1)\quad\text{with}\quad 1/p+1/q\geq 3/2,$

then the CLT holds for the quadratic form $Q_{T}$ with limiting variance
$\sigma^{2}_{0}$ given by (3.6).

######  Remark 3.9.

In fact, in Giraitis and Taqqu [62] (see also Giraitis and Taqqu [63]) was
proved more general result stating that Theorem 3.6 is true for quadratic
forms of the form:

$Q_{T,m,n}=\sum_{t=1}^{T}\sum_{s=1}^{T}\widehat{g}(t-s)P_{m,n}(X(t),X(s)),\quad
T\in\mathbb{N},$ (3.20)

where $P_{m,n}(X(t),X(s))$ is a bivariate Appell polynomial (Wick power) of
the linear variables $X(t)$ and $X(s)$, $m,n\geq 0$, $m+n\geq 1$, and $X(t)$
is a d.t. linear process of the form (2.9). Also, observe that
$Q_{T,1,1}=Q_{T}$.

The next theorem, which was proved in in Giraitis and Taqqu [62], shows that
under some rather restrictive conditions on the covariance function $r(t)$ and
the generating kernel $\widehat{g}(t)$, the long-range dependence of the
process $X(t)$ can be compensated by the fast decay of the generating kernel
$\widehat{g}(t)$ in such a way that the CLT for $Q_{T}$ holds. These
conditions ensure, in fact, that the sufficient conditions in the frequency
domain, provided in Theorem 3.3, are satisfied. The theorem involves quasi-
monotone sequences: a sequence $\\{b(t),t\in\mathbb{Z}\\}$ is said to be
quasi-monotonically convergent to $0$ if $b(t)\to 0$ and $b(t+1)\leq
b(t)(1+c/t)$ as $t\to 0$ for some $c>0$. The sequence $b(t)$ has bounded
variation if $\sum_{t=1}^{\infty}|b(t+1)-b(t)|<\infty$.

######  Theorem 3.7.

Assume that $r(t)=|t|^{-\gamma_{1}}L_{1}(|t|),$
$\widehat{g}(t)=|t|^{-\gamma_{2}}L_{2}(|t|)$ with $0<\gamma_{1},\gamma_{2}<3$
and $\gamma_{1}+\gamma_{2}>3/2$. Suppose in addition that both sequences
$\\{r(t)\\}$ and $\\{{\widehat{g}(t)}\\}$ have bounded variation and are
quasi-monotonically convergent to $0$; if $1<\gamma_{1}<3$, $r(t)$ has the
same sign for large $t$ and satisfies $\sum_{t\in\mathbb{Z}}r(t)=0$; if
$1<\gamma_{2}<3$, $\widehat{g}(t)$ has the same sign for large $t$ and
satisfies $\sum_{t\in\mathbb{Z}}\widehat{g}(t)=0$. Then the CLT holds for the
quadratic form $Q_{T}$. The limiting variance is expressed by (3.6) provided
that $0<\gamma_{1},\gamma_{2}<1$.

###  3.3 Operator conditions

In this subsection we assume that the model $X(t)$ is a d.t. Gaussian process
defined on a probability space $(\Omega,\mathcal{F},P)$, and we are interested
in asymptotic normality of the quadratic form $Q_{T}:=Q_{T}(f,g)$ given by
(3.1) (d.t. case). Recall the notation (see formula (3.2)

$\widetilde{Q}_{T}(f,g):=T^{-1/2}\,\left(Q_{T}(f,g)-\mathbb{E}[Q_{T}(f,g)]\right).$
(3.21)

and (see formula (3.6)

$\sigma^{2}(f,g):=16\pi^{3}\int_{-\pi}^{\pi}f^{2}(\lambda)g^{2}(\lambda)\,d\lambda.$
(3.22)

We denote by $L^{2}(dP)$ the $L^{2}$-space constructed from a probability
measure $P$ such that $X(t)\in L^{2}(dP)$, $t\in\mathbb{Z}$.

Solev and Gerville-Reache [90] observed that for a fixed spectral density $f$
and a number $T\in\mathbb{N}$ the quadratic form $\widetilde{Q}_{T}(f,g)$ in
(3.21) can be regarded as the value of a linear operator
$\mathbf{\widetilde{Q}}_{T}:g\mapsto\widetilde{Q}_{T}(f,g)$. It turns out that
in order to study the asymptotic normality of the quadratic form
$\widetilde{Q}_{T}(f,g)$, it is enough to understand for what sets
$B_{\varepsilon}$ (possibly, depending on $f$) we have

$\limsup_{T\to\infty}\sup_{g\in
B_{\varepsilon}}||\widetilde{Q}_{T}(f,g)||_{L^{2}(dP)}\leq\varepsilon.$ (3.23)

It can be shown that the function $||\widetilde{Q}_{T}(f,g)||_{L^{2}(dP)}$
possesses the following symmetry property regarded as a function of $(f,g)$:
if $g$ is a nonnegative function, then

$||\widetilde{Q}_{T}(f,g)||^{2}_{L^{2}(dP)}=||\widetilde{Q}_{T}(g,f)||^{2}_{L^{2}(dP)}.$
(3.24)

The next result plays a key role in the proof of the asymptotic normality of
$\widetilde{Q}_{T}(f,g)$ (see Solev and Gerville-Reache [90]).

######  Theorem 3.8.

Let $\mathcal{G}_{0}$ and $\mathcal{G}$ ($\mathcal{G}_{0}\subset\mathcal{G}$)
be linear subsets in the space $L^{1}[-\pi,\pi]$. Assume that the following
conditions hold:

  * (a)

for every $g_{0}\in\mathcal{G}_{0}$ the limit

$\lim_{T\to\infty}||\widetilde{Q}_{T}(f,g_{0})||^{2}_{L^{2}(dP)}=\sigma^{2}(f,g_{0})<\infty.$

exists, and

$\widetilde{Q}_{T}(f,g_{0})\overset{d}{\rightarrow}\eta_{0}\sim
N(0,\sigma^{2}(f,g_{0}))\quad{\rm as}\quad T\to\infty;$

  * (b)

for every $g\in\mathcal{G}$ and a number $\varepsilon>0$ there exists a
function $g_{0}\in\mathcal{G}_{0}$ such that

$\limsup_{T\to\infty}||\widetilde{Q}_{T}(f,g-g_{0}||_{L^{2}(dP)}\leq\varepsilon,\quad||g-g_{0})||_{f^{2}}\leq\varepsilon,$

where $||\cdot||_{L^{2}(dP)}$ and $||\cdot||_{f^{2}}$ are the norms in the
spaces $L^{2}(dP)$ and $L^{2}(f^{2},[-\pi,\pi])$, respectively. Then
$\sigma^{2}(f,g)<\infty$, the limit

$\lim_{T\to\infty}||\widetilde{Q}_{T}(f,g)||^{2}_{L^{2}(dP)}=\sigma^{2}(f,g)$

exists, and

$\widetilde{Q}_{T}(f,g)\overset{d}{\rightarrow}\eta\sim
N(0,\sigma^{2}(f,g))\quad{\rm as}\quad T\to\infty.$ (3.25)

Thus, if the results holds for every $g_{0}\in\mathcal{G}_{0}$ (part (a)),
then it holds for every $g\in\mathcal{G}$ close to $g_{0}$ (part (b)).

Given a function $\phi\in L^{1}[-\pi,\pi]$ and an interval
$I\subset[-\pi,\pi]$ of length $|I|$, we set

$\\{\phi\\}_{I}:=\frac{1}{|I|}\int_{I}\phi(x)dx,$

and define

$\lambda(\phi):=\sup_{I}\\{\phi\\}_{I}\times\\{1/\phi\\}_{I},$ (3.26)

where the supremum is taken over all intervals $I\subset[-\pi,\pi]$. The
condition $\lambda(\phi)<\infty$ implies a type of smoothness of the function
$\psi=\log|\phi|$ in the neighborhoods of points where it turns to infinity
(for details, see the paper Hunt et al. [74]).

The following result was obtained by Solev and Gerville-Reache [90].

######  Theorem 3.9.

Assume that the following conditions hold:

  * (a)

$f\leq f_{*}$, $|g|\leq g_{*}$ and $\lambda(f_{*}),\lambda(g_{*})<\infty$,

  * (b)

$f_{*}g_{*}\in L^{2}[-\pi,\pi]$.

Then the quadratic form $Q_{T}(f,g)$ obeys the CLT, that is, the relation
(3.25) is satisfied.

##  4 Functional limit theorems for Gaussian and linear models

In this section, we establish weak convergence in $C[0,1]$ for c.t. case and
in $D[0,1]$ for d.t. case of normalized stochastic processes, generated by
Toeplitz type quadratic functionals of a Gaussian stationary process,
exhibiting long-range dependence. Specifically, we are interested in
describing the limiting behavior (as $T\to\infty$) of the following process
$Q_{T}(\tau)$, generated by Toeplitz type quadratic functionals of a Gaussian
stationary process $\\{X(u),\ u\in\mathbb{U}\\}$ with spectral density $f$:

$Q_{T}(\tau)=\left\\{\begin{array}[]{ll}\sum_{t=1}^{[T\tau]}\sum_{s=1}^{[T\tau]}\widehat{g}(t-s)X(t)X(s)&\mbox{in
the d.t.\ case},\\\ \\\
\int_{0}^{T\tau}\int_{0}^{T\tau}\widehat{g}(t-s)X(t)X(s)\,dt\,ds&\mbox{in the
c.t.\ case},\end{array}\right.$ (4.1)

where $\tau\in[0,1]$, $\widehat{g}(t)$, $t\in\mathbb{U}$ is the Fourier
transform of some integrable even function $g(\lambda)$, $\lambda\in\Lambda$,
and $[\,\cdot\,]$ stands for the greatest integer.

The limit of the process (4.1) again is completely determined by the spectral
density $f(\lambda)$ (or covariance function $r(t)$) and the generating
function $g(\lambda)$ (or generating kernel $\widehat{g}(t)$), and depending
on their properties, the limiting process can be Gaussian or not. We say that
we have a functional non-central limit theorem (FNCLT) for the process
$Q_{T}(\tau)$ if the limit is non-Gaussian or, if Gaussian, it is not Brownian
motion. We say that a functional central limit theorem (FCLT) for
$Q_{T}(\tau)$ holds if we have the following weak convergence in $C[0,1]$ in
the c.t. case (and in $D[0,1]$ in the d.t. case):

$\widetilde{Q}_{T}(\tau)\Longrightarrow\sigma B(\tau),\quad\tau\in[0,1],$
(4.2)

where $\sigma>0$, $B(\tau)$ is a standard Brownian motion, and
$\widetilde{Q}_{T}(\tau)$ is the standard normalized version of the process
$Q_{T}(\tau)$ in (4.1):

$\widetilde{Q}_{T}(\tau):=T^{-1/2}\left(Q_{T}(\tau)-\mathbb{E}[Q_{T}(\tau)]\right),\quad\tau\in[0,1].$
(4.3)

###  4.1 Functional CLT for $Q_{T}(\tau)$

We first state functional central limit theorems, when the limit of the
normalized process (4.3) is Brownian motion.

In the d.t. case the next result was obtained in Giraitis and Taqqu [65] (see
also Fox and Taqqu [38] and Giraitis and Taqqu [61]).

######  Theorem 4.1.

If the covariance function $r(t)$ and the generating kernel $\widehat{g}(t)$
satisfy the following condition: $r\in l^{p}$ $(p\geq 1)$ and $\widehat{g}\in
l^{q}$ $(q\geq 1)$ with $1/p+1/q\geq 3/2,$ then for the process $Q_{T}(\tau)$
the FCLT holds, that is, the convergence (4.2) holds in $D[0,1]$ with limiting
variance $\sigma^{2}$ given by

$\sigma^{2}:=\sum_{t,s,v\in\mathbb{Z}}\widehat{g}(t)\widehat{g}(s)\mathrm{Cov}\left(P(X(v),X(v+t)),P(X(0),X(s)\right),$
(4.4)

where $P(X(t),X(s)):=X(t)X(s)-\mathbb{E}[X(t)X(s)]$.

Now we consider the c.t. case. The next result, which is an extension of
Theorem 3.1(A), involves the convergence of finite-dimensional distributions
of the process $\widetilde{Q}_{T}(\tau)$ to those of a standard Brownian
motion (see Bai et al. [8]).

######  Theorem 4.2.

Assume that the spectral density $f(\lambda)$ and the generating function
$g(\lambda)$ satisfy the following conditions:

$f\cdot g\in L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$ (4.5)

and

$\mathbb{E}[\widetilde{Q}_{T}^{2}(1)]\rightarrow
16\pi^{3}\int_{\mathbb{R}}f^{2}(x)g^{2}(x)dx~{}\text{ as }T\rightarrow\infty.$
(4.6)

Then we have the following convergence of the finite-dimensional distributions

$\widetilde{Q}_{T}(\tau)\overset{f.d.d.}{\longrightarrow}\sigma_{0}B(\tau),$

where $\widetilde{Q}_{T}(\tau)$ is as in (4.3), $B(\tau)$ is a standard
Brownian motion, and (see (3.6)):

$\sigma^{2}_{0}:=16\pi^{3}\int_{\mathbb{R}}f^{2}(x)g^{2}(x)dx.$ (4.7)

To extend the convergence of finite-dimensional distributions in Theorem 4.2
to the weak convergence in the space $C[0,1]$, we impose an additional
condition on the underlying Gaussian process $X(t)$ and on the generating
function $g$. It is convenient to impose this condition in the time domain,
that is, on the covariance function $r:=\hat{f}$ and the generating kernel
$a:=\hat{g}$. The following condition is an analog of the assumption in
Theorem 4.1

$r(\cdot)\in L^{p}(\mathbb{R}),\quad a(\cdot)\in
L^{q}(\mathbb{R})\quad\text{for some}\quad p,q\geq
1,~{}\frac{1}{p}+\frac{1}{q}\geq\frac{3}{2}.$ (4.8)

######  Remark 4.1.

In fact under (4.5), the condition (4.8) is sufficient for the convergence in
(4.6). Indeed, let $\bar{p}=p/(p-1)$ and $\bar{q}=q/(q-1)$ be the Hölder
conjugates of $p$ and $q$, respectively. Since $1\leq p,q\leq 2$, one has by
the Hausdorff-Young inequality that $\|f\|_{\bar{p}}\leq
c_{p}\|r\|_{p},~{}\|g\|_{\bar{q}}\leq c_{q}\|a\|_{q},$ and hence, in view if
(4.8) we have

$f(\cdot)\in L^{\bar{p}},\quad g(\cdot)\in
L^{\bar{q}},\quad\frac{1}{\bar{p}}+\frac{1}{\bar{q}}=2-\frac{1}{p}-\frac{1}{q}\leq
1/2.$ (4.9)

Then the convergence in (4.6) follows from Theorem 3.1(C) (see the proof of
Theorem 3 of Ginovyan and Sahakyan [53] in the c.t. case, and Giraitis and
Surgailis [60] in the d.t. case).

######  Remark 4.2.

Observe that condition (4.8) is satisfied if the functions $r(t)$ and $a(t)$
satisfy the following: there exist constants $C>0$, $\alpha^{*}$ and
$\beta^{*}$, such that

$|r(t)|\leq C(1\wedge|t|^{\alpha^{*}-1}),\qquad|a(t)|\leq
C(1\wedge|t|^{\beta^{*}-1}),$ (4.10)

where $0<\alpha^{*},\beta^{*}<1/2$ and $\alpha^{*}+\beta^{*}<1/2$. Indeed, to
see this, note first that $r(\cdot),\ a(\cdot)\in L^{\infty}(\mathbb{R})$.
Then one can choose $p,q\geq 1$ such that $p(\alpha^{*}-1)<-1$ and
$q(\beta^{*}-1)<-1$, which entails that $r(\cdot)\in L^{p}(\mathbb{R})$ and
$a(\cdot)\in L^{q}(\mathbb{R})$. Since $1/p+1/q<2-\alpha^{*}-\beta^{*}$ and
$2-\alpha^{*}-\beta^{*}>3/2$, one can further choose $p,q$ to satisfy
$1/p+1/q\geq 3/2$.

The next results, two functional central limit theorems, extend Theorems
3.1(A) and 3.3 to weak convergence in the space $C[0,1]$ of the stochastic
process $\widetilde{Q}_{T}(\tau)$ to a standard Brownian motion.

######  Theorem 4.3.

Let the spectral density $f$ and the generating function $g$ satisfy condition
(4.5). Let the covariance function $r(t)$ and the generating kernel $a(t)$
satisfy condition (4.8). Then we have the following weak convergence in
$C[0,1]$:

$\widetilde{Q}_{T}(\tau)\Longrightarrow\sigma_{0}B(\tau),$

where $\widetilde{Q}_{T}(\tau)$ is as in (4.3), $\sigma_{0}$ is as in (4.7),
and $B(\tau)$ is a standard Brownian motion.

Recall the class $SV_{0}(\mathbb{R})$ of slowly varying at zero functions
$u(x)$, $x\in\mathbb{R}$, satisfying the following conditions: for some $a>0$,
$u(x)$ is bounded on $[-a,a]$, $\lim_{x\to 0}u(x)=0,$ $u(x)=u(-x)$ and
$0<u(x)<u(y)$ for $0<x<y<a$.

######  Theorem 4.4.

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

$f(x)\leq|x|^{-\alpha}L_{1}(x),\quad|g(x)|\leq|x|^{-\beta}L_{2}(x),\quad
x\in[-a,a]$ (4.11)

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

$L_{i}\in SV_{0}(\mathbb{R}),\quad x^{-(\alpha+\beta)}L_{i}(x)\in
L^{2}[-a,a],\quad i=1,2.$ (4.12)

Let, in addition, the covariance function $r(t)$ and the generating kernel
$a(t)$ satisfy condition (4.8). Then we have the following weak convergence in
$C[0,1]$:

$\widetilde{Q}_{T}(\tau)\Longrightarrow\sigma_{0}B(\tau),$

where $\widetilde{Q}_{T}(\tau)$ is as in (4.3), $\sigma_{0}$ is as in (4.7),
and $B(\tau)$ is a standard Brownian motion.

######  Remark 4.3.

The proofs of Theorems 4.2 \- 4.4, given in Bai et al. [8] uses the method
developed in Ginovyan and Sahakyan [53], which itself is based on the
approximations of traces of the products of truncated Toeplitz operators (see
Section 8.1).

###  4.2 Functional NCLT for $Q_{T}(\tau)$

####  4.2.1 Non-central limit theorems (discrete-time)

For d.t. Gaussian processes the problem of description of the limit
distribution of the quadratic form:

$Q_{T}:=\sum_{t=1}^{T}\sum_{s=1}^{T}\widehat{g}(t-s)X(t)X(s),\quad
T\in\mathbb{N},$ (4.13)

if it is non-Gaussian, goes back to the classical papers of Rosenblatt
[86]–[88]. Later this problem was studied in a series of papers by Fox and
Taqqu, Taqqu, and Terrin and Taqqu (see, e.g., [37], [97], [100], [101],
[102], and references therein). Suppose that the spectral density $f(\lambda)$
and the generating function $g(\lambda)$ are regularly varying functions at
the origin:

$f(\lambda)=|\lambda|^{-\alpha}L_{1}(\lambda)\quad{\rm and}\quad
g(\lambda)=|\lambda|^{-\beta}L_{2}(\lambda),\quad\alpha<1,\beta<1,$ (4.14)

where $L_{1}(\lambda)$ and $L_{2}(\lambda)$ are slowly varying functions at
zero, which are bounded on bounded intervals. The conditions $\alpha<1$ and
$\beta<1$ ensure that the Fourier coefficients of $f$ and $g$ are well
defined. When $\alpha>0$ the model $\\{X(t),t\in\mathbb{Z}\\}$ exhibits long
memory.

It is the sum $\alpha+\beta$ that determines the asymptotic behavior of the
quadratic form $Q_{T}$. If $\alpha+\beta\leq 1/2$, then by Theorem 3.3 the
standard normalized quadratic form

$T^{-1/2}\left(Q_{T}-\mathbb{E}[Q_{T}]\right)$

converges in distribution to a Gaussian random variable. If
$\alpha+\beta>1/2$, the convergence to a Gaussian distribution fails.

Consider the embedding of the discrete sequence $\\{Q_{T},\,T\in\mathbb{N}\\}$
into a continuous-time process
$\\{Q_{T}(\tau),\,T\in\mathbb{N},\,\tau\in[0,1]\\}$ defined by

$Q_{T}(\tau):=\sum_{t=1}^{[T\tau]}\sum_{s=1}^{[T\tau]}\widehat{g}(t-s)X(t)X(s),$
(4.15)

where $[\,\cdot\,]$ stands for the greatest integer. The problem of interest
is to describe the limit distribution of the following normalized process:

$\widetilde{Q}_{T}(\tau):=d_{T}^{-1}\left(Q_{T}(\tau)-\mathbb{E}[Q_{T}(\tau)]\right)$
(4.16)

where $d_{T}$ is a suitably chosen normalization factor.

In [86] (see also [88]) Rosenblatt showed that if a d.t. centered Gaussian
process $X(t)$ has covariance function $r(t)=(1+t^{2})^{\alpha/2-1/2}$ with
$1/2<\alpha<1$, then the random variable

$\widetilde{Q}_{T}:=T^{-\alpha}\sum_{t=1}^{T}\left[X^{2}(t)-1\right]$

has a non-Gaussian limiting distribution, and described this distribution in
terms of characteristic functions. This is a special case of (4.16) with
$\tau=1$, $1/2<\alpha<1$, $\beta=0$ and $d_{T}=T^{\alpha}$. In [97] (see also
[100]) Taqqu extended Rosenblatt’s result by showing that the stochastic
process

$\widetilde{Q}_{T}(\tau):=T^{-\alpha}\sum_{t=1}^{[T\tau]}\left[X^{2}(t)-1\right]$

converges (as $T\to\infty$) weakly in $D[0,1]$ to a process (called the
Rosenblatt process) which has the double Wiener-Itô integral representation

$Q(\tau):=C_{\alpha}\int_{\mathbb{R}^{2}}^{{}^{\prime\prime}}\frac{e^{i\tau(x+y)}-1}{i(x+y)}|x|^{-\alpha/2}|y|^{-\alpha/2}dZ(x)dZ(y),$
(4.17)

where $Z(\cdot)$ is a complex-valued Gaussian random measure defined on the
Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$, and satisfying $EZ(B)=0$,
$E|Z(B)|^{2}=|B|$, and $\overline{Z(-B)}=Z(B)$ for any
$B\in\mathcal{B}(\mathbb{R})$. The double prime in the integral (4.17)
indicates that the integration excludes the diagonals $x=\pm y$.

Notice that the distribution of the random variable $Q(\tau)$ in (4.17) for
$\tau=1$ is described in Veillette and Taqqu [105].

The next result, proved in Terrin and Taqqu [101] using power counting
theorems (see Section 8.6), describes the non-Gaussian limit distribution of
the suitable normalized process $Q_{T}(\tau)$.

######  Theorem 4.5.

Let $f(\lambda)$ and $g(\lambda)$ be as in (4.14) with $\alpha<1$, $\beta<1$
and slowly varying at zero and bounded on bounded intervals factors
$L_{1}(\lambda)$ and $L_{2}(\lambda)$. Let the process $Q_{T}(\tau)$ be as in
(4.15). Then for $\alpha+\beta>1/2$ the process:

$Z_{T}(\tau):=\frac{1}{T^{\alpha+\beta}L_{1}(1/T)L_{2}(1/T)}\,\left(Q_{T}(\tau)-\mathbb{E}[Q_{T}(\tau)]\right)$
(4.18)

converges (as $T\to\infty$) weakly in $D[0,1]$ to

$Z(\tau):=\int_{\mathbb{R}^{2}}^{{}^{\prime\prime}}K_{\tau}(x,y)dZ(x)dZ(y),$
(4.19)

where

$K_{\tau}(x,y)=|xy|^{-\alpha/2}\int_{\mathbb{R}}\frac{e^{i\tau(x+u)}-1}{i(x+u)}\cdot\frac{e^{i\tau(y-u)}-1}{i(y-u)}|u|^{-\beta}du,$
(4.20)

The double prime in the integral (4.19) indicates that the integration
excludes the diagonals $x=\pm y$.

######  Remark 4.4.

The limiting process in (4.19) is real-valued, non-Gaussian, and satisfies
$\mathbb{E}[Z(\tau)]=0$ and
$\mathbb{E}[Z^{2}(\tau)]=\int_{\mathbb{R}^{2}}|K_{\tau}(x,y)|^{2}dxdy$. It is
self-similar with parameter $H=\alpha+\beta\in(1/2,2)$, that is, the processes
$\\{Z(a\tau),\tau\in[0,1]\\}$ and $\\{a^{H}Z(\tau),\tau\in[0,1]\\}$ have the
same finite dimensional distributions for all $a>0$.

####  4.2.2 Non-central limit theorems (continuous-time)

Now we state a _non-central limit theorem_ for the process $Q_{T}(\tau)$ in
the c.t. case, that is, when $Q_{T}(\tau)$ in (4.1) is defined through
integrals. Let the spectral density $f$ and the generating function $g$
satisfy

$f(x)=|x|^{-\alpha}L_{1}(x)\quad{\rm and}\quad g(x)=|x|^{-\beta}L_{2}(x),\quad
x\in\mathbb{R},\quad\alpha<1,\,\beta<1,$ (4.21)

with slowly varying at zero functions $L_{1}(x)$ and $L_{2}(x)$ such that

$\int_{\mathbb{R}}|x|^{-\alpha}L_{1}(x)dx<\infty\quad{\rm
and}\quad\int_{\mathbb{R}}|x|^{-\beta}L_{2}(x)dx<\infty.$

We assume in addition that the functions $L_{1}(x)$ and $L_{2}(x)$ satisfy the
following condition, called Potter’s bound (see Giraitis et al. [59], formula
(2.3.5)): for any $\epsilon>0$ there exists a constant $C>0$ so that if $T$ is
large enough, then

$\frac{L_{i}(u/T)}{L_{i}(1/T)}\leq C(|u|^{\epsilon}+|u|^{-\epsilon}),\quad
i=1,2.$ (4.22)

Note that a sufficient condition for (4.22) to hold is that $L_{1}(x)$ and
$L_{2}(x)$ are bounded on intervals $[a,\infty)$ for any $a>0$, which is the
case for the slowly varying functions in Theorem 4.4.

Now we are interested in the limit process of the following normalized version
of the process $Q_{T}(\tau)$ given by (4.1), with $f$ and $g$ as in (4.21):

$Z_{T}(\tau):=\frac{1}{T^{\alpha+\beta}L_{1}(1/T)L_{2}(1/T)}\left(Q_{T}(\tau)-\mathbb{E}[Q_{T}(\tau)]\right).$
(4.23)

######  Theorem 4.6.

Let $f$ and $g$ be as in (4.21) with $\alpha<1$, $\beta<1$ and slowly varying
at zero functions $L_{1}(x)$ and $L_{2}(x)$ satisfying (4.22), and let
$Z_{T}(\tau)$ be as in (4.23). Then if $\alpha+\beta>1/2$, we have the
following weak convergence in the space $C[0,1]$:

$Z_{T}(\tau)\Longrightarrow Z(\tau),$

where the limit process $Z(\tau)$ is given by

$Z(\tau)=\int_{\mathbb{R}^{2}}^{\prime\prime}H_{\tau}(x_{1},x_{2})W(dx_{1})W(dx_{2}),$
(4.24)

with

$H_{\tau}(x_{1},x_{2})=|x_{1}x_{2}|^{-\alpha/2}\int_{\mathbb{R}}\left[\frac{e^{i\tau(x_{1}+u)}-1}{i(x_{1}+u)}\right]\cdot\left[\frac{e^{i\tau(x_{2}-u)}-1}{i(x_{2}-u)}\right]|u|^{-\beta}du~{},$
(4.25)

where $W(\cdot)$ is a complex Gaussian random measure with Lebesgue control
measure, and the double prime in the integral (4.24) indicates that the
integration excludes the diagonals $x_{1}=\pm x_{2}$.

######  Remark 4.5.

Let $P_{T}$ and $P$ denote the measures generated in $C[0,1]$ by the processes
$Z_{T}(\tau)$ and $Z(\tau)$ given by (4.23) and (4.24), respectively. Then
Theorem 4.6 can be restated as follows: under the conditions of Theorem 4.6,
the measure $P_{T}$ converges weakly in $C[0,1]$ to the measure $P$ as
$T\rightarrow\infty$. Similar assertions can be stated for Theorems 4.3 and
4.4.

######  Remark 4.6.

Comparing Theorems 4.5 and 4.6, we see that the limit process $Z(\tau)$ is the
same both for continuous and discrete time models. Also, it is worth noting
that although the statements of Theorems 4.5 and 4.6 are similar, the proofs
are different. The proof of Theorem 4.5, given in Terrin and Taqqu [101], uses
hard technical analysis based on power counting theorems, while the proof of
Theorem 4.6, given in Bai et al. [9], is simple and uses the spectral
representation of the underlying process and properties of Wiener-Itô
integrals.

##  5 Functional limit theorems for Lévy-driven linear models

In this section, we survey results involving functional central and non-
central limit theorems for a suitably normalized stochastic process:

$Q_{T}(\tau):=\int_{0}^{T\tau}\int_{0}^{T\tau}\widehat{g}(t-s)X(t)X(s)dtds,\quad\tau\in[0,1]$
(5.1)

in the general case where the underlying model $\\{X(t),\ t\in\mathbb{R}\\}$
is a c.t. linear process driven from Lévy noise $\xi(t)$ with time invariant
filter $a(\cdot)$, that is,

$X(t)=\int_{\mathbb{R}}a(t-s)d\xi(s),\quad\int_{\mathbb{R}}|a(s)|^{2}ds<\infty.$
(5.2)

These theorems, which were stated and proved in Bai et al. [9], extend the
results stated in Section 4 for Wiener-driven processes and show that under
some $(L^{p},L^{q})$-type conditions imposed on the filter $a(\cdot)$ and the
generating kernel $\widehat{g}(\cdot)$ of the quadratic functional, the
process $Q_{T}(\tau)$ obeys a central limit theorem, that is, the finite-
dimensional distributions of the standard $\sqrt{T}$ normalized process
$Q_{T}(\tau)$ in (5.1) tend to those of a normalized standard Brownian motion.
In contrast, when the functions $a(\cdot)$ and $\widehat{g}(\cdot)$ have slow
power decay, then we have a non-central limit theorem for $Q_{T}(\tau)$, that
is, the finite-dimensional distributions of the process $Q_{T}(\tau)$,
normalized by $T^{\gamma}$ for some $\gamma>1/2$, tend to those of a non-
Gaussian non-stationary-increment self-similar process which can be
represented by a double stochastic Wiener-Itô integral on $\mathbb{R}^{2}$.

We point out that the proofs of the central limit theorems given in Bai et al.
[9] were based on an approximation approach, which reduces the quadratic
integral form to a single integral form, while the proofs of the non-central
limit theorems, used the spectral representation of the underlying process,
the properties of Wiener-Itô integrals, and a continuous analog of a method to
establish convergence in distribution of quadratic functionals to double
Wiener-Itô integrals, developed by Surgailis [94] (see also Giraitis et al.
[59]).

It is worth noting that if the underlying process $X(t)$ is not necessarily
Gaussian, additional complications arise due to the contribution of the random
diagonal term in the double stochastic integral with respect to Lévy noise,
which is not present in the case of Gaussian noise (see Remark 5.2 below). For
this model, in Avram et al. [7], a central limit theorem for the quadratic
functional $Q_{T}(1)$ was stated (without proof) under some
$(L^{p},L^{q})$-type conditions imposed on the spectral density $f(\lambda)$
and the generating function $g(\lambda)$ (see Remark 5.5 below). For a related
study of the sample covariances of Lévy-driven moving average processes we
refer to the papers by Cohen and Lindner [22], and Spangenberg [93].

In this section, we follow the paper Bai et al. [9], and will use the
following notation.
The symbol $\ast$ will stand for the convolution:

$(\phi\ast\psi)(u)=\int_{\mathbb{R}}\phi(u-x)\psi(x)dx,$

while the symbol $\bar{\ast}$ is used to denote the reversed convolution:

$(\phi^{\bar{\ast}2})(u)=(\phi\bar{\ast}\phi)(u)=\int_{\mathbb{R}}\phi(u+x)\phi(x)dx.$

By $\mathcal{F}$ and $\mathcal{F}^{-1}$ we denote the Fourier and the inverse
Fourier transforms:

$(\mathcal{F}\phi)(u)=\widehat{\phi}(u)=\int_{\mathbb{R}}e^{ixu}\phi(x)dx,\quad(\mathcal{F}^{-1}\phi)(u)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-ixu}\phi(x)dx.$

We will use the following well-known identities:

$\mathcal{F}(\phi\ast\psi)=\mathcal{F}(\phi)\cdot\mathcal{F}(\psi)\quad\text{and}\quad\mathcal{F}(\phi\bar{\ast}\psi)=\mathcal{F}(\phi)\cdot\overline{\mathcal{F}(\psi)}.$
(5.3)

###  5.1 Central limit theorems

Let $\\{X(t),\ t\in\mathbb{R}\\}$ be a centered real-valued linear process
given by (5.2) with filter $a(\cdot)\in L^{2}(\mathbb{R})$. Recall that the
covariance function $r(t)$ and the spectral density $f(\lambda)$ of $X(t)$ are
given by formulas (2.13) and (2.14), respectively.

The theorem that follows contains $(L^{p},L^{q})$-type time-domain sufficient
conditions for the process $Q_{T}(\tau)$ to obey the central limit theorem
(see Bai et al. [9]).

######  Theorem 5.1.

Let $X(t)$ be as in (5.2), and let $Q_{T}(\tau)$ be as in (5.1). Assume that

$a(\cdot)\in L^{p}(\mathbb{R})\cap
L^{2}(\mathbb{R}),\quad\widehat{g}(\cdot)\in L^{q}(\mathbb{R})$ (5.4)

with

$1\leq p,q\leq 2,\quad\frac{2}{p}+\frac{1}{q}\geq\frac{5}{2}.$ (5.5)

Then

$\widetilde{Q}_{T}(\tau):=T^{-1/2}\left(Q_{T}(\tau)-\mathbb{E}[Q_{T}(\tau)]\right)\overset{f.d.d.}{\longrightarrow}\sigma
B(\tau),$ (5.6)

where the symbol $\overset{f.d.d.}{\longrightarrow}$ stands for convergence of
finite-dimensional distributions, $B(\tau)$ is a standard Brownian motion, and

$\sigma^{2}=\int_{\mathbb{R}}[2K_{A}(v)+\kappa_{4}K_{B}(v)]~{}dv,$ (5.7)

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

$K_{A}(v)=\Big{(}(a\ast b)^{\bar{\ast}2}\cdot a^{\bar{\ast}2}\Big{)}(v),\quad
K_{B}(v)=\Big{(}(a{\ast}b)\cdot a\Big{)}^{\bar{\ast}2}(v).$ (5.8)

######  Remark 5.1.

Young’s convolution inequality (see, e.g., Bogachev [15], Theorem 3.9.4)
states that for any numbers $p,p_{1},q$ satisfying $1\leq p,q\leq
p_{1}\leq\infty$ and $\frac{1}{p_{1}}=\frac{1}{p}+\frac{1}{q}-1$, and for any
functions $f\in L^{p}(\mathbb{R})$, $g\in L^{q}(\mathbb{R})$, the function
$f\ast g$ is defined almost everywhere, $f\ast g\in L^{p_{1}}(\mathbb{R})$,
and

$\|f\ast g\|_{p_{1}}\leq\|f\|_{p}\|g\|_{q}.$ (5.9)

Applying this inequality to the convolution in (2.13), we get
$\|r\|_{p_{1}}\leq\|a\|_{p}^{2}<\infty$, where $1+1/p_{1}=2/p$. Hence the
relations (5.4) and (5.5) imply that

$r(\cdot)\in L^{p_{1}}(\mathbb{R}),\quad b(\cdot)\in
L^{q}(\mathbb{R}),\quad\frac{1}{p_{1}}+\frac{1}{q}=\frac{2}{p}-1+\frac{1}{q}\geq\frac{5}{2}-1=\frac{3}{2}.$
(5.10)

The condition (5.10) is sufficient for the convergence in Theorem 5.1 to hold
in the case where $\xi(t)$ is Brownian motion (see Theorem 4.3). In fact, in
this case, the convergence in Theorem 5.1 holds under even a weaker condition
imposed on the generating function $g(\lambda)$ and the spectral density
$f(\lambda)$ of $X(t)$ (see Theorem 4.2).

######  Remark 5.2.

In contrast to the cases where the model is either a d.t. linear process (see
Giraitis and Surgailis [60]), or a c.t. Gaussian process (see Bai et al. [8]),
it is convenient to impose the time-domain conditions (5.4) and (5.5) on the
functions $a(\cdot)$ and $\widehat{g}(\cdot)$, instead of on the spectral
density $f(\lambda)$ and the generating function $g(\lambda)$. This allows the
analysis of the random diagonal term which arises from the double stochastic
integral with respect to a non-Gaussian Lévy process. In the d.t. case the
random diagonal term is estimated by the full double sum (see Giraitis and
Surgailis [60], relation (2.3)), while in the c.t. Gaussian case, there is no
such random diagonal term (see Ginovyan and Sahakyan [53]). In the c.t. non-
Gaussian case, there is a random diagonal term in the form of a single
stochastic integral that cannot be controlled by the double integral, and
hence it should be treated separately (see the proof of Theorem 2.1 of Bai et
al. [9]).

######  Remark 5.3.

Observe that the long-run variance $\sigma^{2}$ given by (5.7) can be
expressed in terms of the spectral density $f(\lambda)$ and the generating
function $g(\lambda)$, provided that these functions satisfy some regularity
conditions. Indeed, using the equalities in (5.3) and the Parseval-Plancherel
theorem, under suitable integrability conditions on $a(\cdot)$ and $b(\cdot)$,
we can write

$\displaystyle\int_{\mathbb{R}}K_{A}(v)dv$
$\displaystyle=\int_{\mathbb{R}}(a\ast
b)^{\bar{\ast}2}(v)a^{\bar{\ast}2}(v)dv=\frac{1}{2\pi}\int_{\mathbb{R}}\mathcal{F}\left((a\ast
b)^{\bar{\ast}2}\right)(\lambda)\overline{\mathcal{F}\left(a^{\bar{\ast}2}\right)(\lambda)}d\lambda=$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}}|\mathcal{F}(a\ast
b)(\lambda)|^{2}|\mathcal{F}(a)(\lambda)|^{2}d\lambda=\frac{1}{2\pi}\int_{\mathbb{R}}|\widehat{a}(\lambda)\widehat{b}(\lambda)|^{2}|\widehat{a}(\lambda)|^{2}d\lambda$
$\displaystyle=8\pi^{3}\int_{\mathbb{R}}f(\lambda)^{2}g(\lambda)^{2}d\lambda,$

where in the last equality we used the fact $|\widehat{a}|^{2}=2\pi f$ and
$\widehat{b}=2\pi g$ (because $b(\cdot)$ is an even function). Similarly, we
have

$\displaystyle\int_{\mathbb{R}}K_{B}(v)dv$
$\displaystyle=\int_{\mathbb{R}}dv\int_{\mathbb{R}}dx\Big{(}(a{\ast}b)\cdot
a\Big{)}(x)\Big{(}(a{\ast}b)\cdot
a\Big{)}(x+v)=\left(\int_{\mathbb{R}}(a{\ast}b)(x)a(x)dx\right)^{2}$
$\displaystyle=\frac{1}{4\pi^{2}}\left(\int_{\mathbb{R}}\widehat{a}(x)\widehat{b}(x)\overline{\widehat{a}(x)}d\lambda\right)^{2}=4\pi^{2}\left[\int_{\mathbb{R}}f(\lambda)g(\lambda)d\lambda\right]^{2}.$

So an alternative expression for $\sigma^{2}$ in (5.7) is

$\sigma^{2}=16\pi^{3}\int_{\mathbb{R}}f(\lambda)^{2}g(\lambda)^{2}d\lambda+\kappa_{4}\left[2\pi\int_{\mathbb{R}}f(\lambda)g(\lambda)d\lambda\right]^{2},$
(5.11)

which should be compared with Avram et al. [7] (Theorem 4.1), and formula
(3.15) for an analogous expression in the d.t. case.

######  Remark 5.4.

The d.t. analog of Theorem 5.1 with $t=1$ and $\xi$ being Gaussian was
established in Giraitis and Surgailis [60]. A special case of Theorem 5.1 with
$t=1$ and $\xi$ being Gaussian was established in Ginovian [45], and in
Ginovian and Sahakyan [53]. Theorem 5.1 for Wiener-driven model
($\kappa_{4}=0$) was proved in Bai et al. [8].

######  Remark 5.5.

For Lévy-driven model with $t=1$ and $\sigma^{2}$ given by (5.11), a version
of Theorem 5.1 was stated in Avram et al. [7] (Theorem 4.1). They impose
$(L^{p},L^{q})$-type conditions on the spectral density $f(\cdot)$ and the
generating function $g(\cdot)$, and assume the existence of all moments of the
driving Lévy process $\xi(t)$. The details of the proof of Theorem 4.1 in
Avram et al. [7] is unfortunately omitted, and it is not clear how the omitted
details of the method-of-moment proof can be carried out given the complexity
of computing the moments of multiple integrals with respect to non-Gaussian
Lévy noise (see Peccati and Taqqu [84], Chapter 7).

The following corollary contains sufficient conditions for the assumptions in
Theorem 5.1 to hold. These conditions involve bounds on the tails of functions
$a(\cdot)$ and $b(\cdot)$ by suitable power functions (see Bai et al. [9]).

######  Corollary 5.1.

The convergence in (5.6) holds if the functions $a(\cdot)$ and $b(\cdot)$
satisfy the following conditions:

$a(\cdot),\,\,b(\cdot)\in L^{\infty}(\mathbb{R}),\quad|a(x)|\leq
c|x|^{\alpha/2-1},\quad|b(x)|\leq c|x|^{\beta-1}$ (5.12)

with

$0<\alpha,\beta<1,\quad\alpha+\beta<1/2.$

###  5.2 Non-central limit theorems

We now state the non-central limit theorems. We make the following assumptions
on the functions $a(\cdot)$ and $b(\cdot)$, and on their Fourier transforms
$\widehat{a}(\cdot)$ and $\widehat{b}(\cdot)$.

######  Assumption 5.1.

The Fourier transform $\widehat{a}(\cdot)$ of $a(\cdot)\in L^{2}(\mathbb{R})$
satisfies

$\widehat{a}(x)=A(x)|x|^{-\alpha/2}L_{1}^{1/2}(x),$

where $L_{1}(x)$ is an even non-negative function slowly varying at zero and
bounded on intervals $[c,\infty)$ for any $c>0$, and $A(x)$ is a complex-
valued function satisfying $|A(x)|=1$, and $\lim_{x\rightarrow
0^{+}}A(x)=A_{0}$ for some $A_{0}$ on the complex unit circle (since
$\widehat{a}(-x)=\overline{\widehat{a}(x)}$, we also have $\lim_{x\rightarrow
0^{-}}A(x)=\overline{A_{0}}$).

######  Assumption 5.2.

The generating function $\widehat{b}(\cdot)\in L^{1}(\mathbb{R})$ and
satisfies

$\widehat{b}(x)=|x|^{-\beta}L_{2}(x),$

where $L_{2}(x)$ is an even non-negative function slowly varying at zero and
bounded on intervals $[c,\infty)$ for any $c>0$.

######  Assumption 5.3.

The parameters $\alpha$ and $\beta$ above satisfy

$-1/2<\alpha<1,\quad-1/2<\beta<1,\quad\alpha+\beta>1/2.$ (5.13)

######  Assumption 5.4.

There exist numbers $\alpha^{*}$ and $\beta^{*}$ satisfying

$0<\alpha^{*},\beta^{*}<1\quad 1<\alpha^{*}+\beta^{*}<\alpha+\beta+1/2,$

such that

$|a(x)|\leq C|x|^{\alpha^{*}/2-1},\quad|b(x)|\leq C|x|^{\beta^{*}-1}.$

The proof of the following theorem can be found in Bai et al. [9].

######  Theorem 5.2.

Suppose that Assumptions 5.1 \- 5.4 hold. Then as $T\rightarrow\infty$ we have

$Z_{T}(\tau):=\frac{1}{T^{\alpha+\beta}L_{1}(1/T)L_{2}(1/T)}\left(Q_{T}(\tau)-\mathbb{E}Q_{T}(\tau)\right)\overset{f.d.d.}{\longrightarrow}Z_{\alpha,\beta}(\tau),$
(5.14)

where

$Z_{\alpha,\beta}(\tau)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}^{\prime\prime}|x_{1}x_{2}|^{-\alpha/2}\int_{\mathbb{R}}\frac{e^{i\tau(x_{1}+u)}-1}{i(x_{1}+u)}\frac{e^{i\tau(x_{2}-u)}-1}{i(x_{2}-u)}|u|^{-\beta}du~{}W(dx_{1})W(dx_{2}),$
(5.15)

where the double prime ′′ indicates the exclusion of the hyper-diagonals
$u_{p}=\pm u_{q}$, $p\neq q$ and $W(\cdot)$ is a complex-valued Brownian
motion (see Section 8.5).

######  Remark 5.6.

The regular variation conditions on $\widehat{a}(\cdot)$ and
$\widehat{b}(\cdot)$ in Assumptions 5.1 \- 5.3 generally do not follow from
the corresponding regular variation conditions imposed on the inverse Fourier
transforms $a(\cdot)$ and $b(\cdot)$. This implication only holds under some
additional assumptions on the slowly varying factors $L_{1}(\cdot)$ and
$L_{2}(\cdot)$ of $a(\cdot)$ and $b(\cdot)$. For instance, it will hold if we
have (see Bingham et al. [13], formula (4.3.7))

$a(x)=x^{\alpha/2-1}\ell_{1}(x)1_{[0,\infty)}(x),\quad
b(x)=|x|^{\beta-1}\ell_{2}(x),$ (5.16)

where $0<\alpha<1$, $0<\beta<1$, $\alpha+\beta>1/2$, and $\ell_{1}(x)$ and
$\ell_{2}(x)$ are even non-negative functions which are locally bounded,
slowly varying at infinity and _quasi-monotone_. Recall that a slowly varying
function $l(\cdot)$ is said to be quasi-monotone if it has locally bounded
variation, and for all $\delta>0$, one has (see Bingham et al. [13], Section
2.7):

$\int_{0}^{x}t^{\delta}|d\ell(t)|=O(x^{\delta}l(x))\quad{\rm as}\quad
x\to\infty.$

A sufficient condition for a slowly varying $\ell(x)$ with locally bounded
variation to be quasi-monotone is that $x^{\delta}\ell(x)$ is increasing and
$x^{-\delta}\ell(x)$ is decreasing when $x$ is large enough, for any
$\delta>0$ (see Theorem 1.5.5 and Corollary 2.7.4 in Bingham et al. [13]).

Notice also that Assumption 5.4 will be satisfied if (5.16) holds.

######  Remark 5.7.

Let the functions $a(\cdot)$ and $b(\cdot)$ be as in (5.16) with $\alpha<0$ or
$\beta<0$ (by (5.13) only one of $\alpha$ and $\beta$ can be negative). Assume
that $\alpha<0$ and $\beta>0$. Then for the corresponding regular variation of
$\widehat{a}(\cdot)$ to hold, one needs to impose in addition that
$\int_{0}^{\infty}a(x)dx=0$. In this case, one does not need to assume quasi-
monotonicity for $\ell_{1}$ (see Corollary 1.40 of Soulier [92]). Similar
considerations hold if $\beta<0$ and $\alpha>0$ instead.

######  Remark 5.8.

Note that Assumption 5.1 holds with $\alpha=0$ if $a(\cdot)\in
L^{1}(\mathbb{R})$ and $\int_{0}^{\infty}a(x)\neq 0$, and Assumption 5.2 holds
with $\beta=0$ if $b(\cdot)\in L^{1}(\mathbb{R})$ and
$\int_{0}^{\infty}b(x)\neq 0$.

The next theorem contains time-domain representations for the limiting process
$Z_{\alpha,\beta}(\tau)$ in (5.15) in the case $\alpha,\beta\geq 0$ (see Bai
et al. [9], Theorem 2.3).

######  Theorem 5.3.

The limiting process $Z_{\alpha,\beta}(\tau)$ in (5.15) admits the following
time-domain representations:

  1. (a)

when $\alpha>0$, $\beta>0$:

$Z_{\alpha,\beta}(\tau)\overset{f.d.d.}{=}c_{\alpha,\beta}\int_{\mathbb{R}^{2}}^{\prime}\int_{0}^{\tau}\int_{0}^{\tau}|u-v|^{\beta-1}(u-x_{1})_{+}^{\alpha/2-1}(v-x_{2})_{+}^{\alpha/2-1}dudv~{}B(dx_{1})B(dx_{2}),$
(5.17)

where

$c_{\alpha,\beta}=\frac{\Gamma(1-\beta)\sin(\beta\pi/2)}{\pi\Gamma(\alpha/2)^{2}};$

  2. (b)

when $\alpha>1/2$, $\beta=0$:

$Z_{\alpha,\beta}(\tau)\overset{f.d.d.}{=}c_{\alpha}\int_{\mathbb{R}^{2}}^{\prime}\int_{0}^{\tau}(u-x_{1})_{+}^{\alpha/2-1}(u-x_{2})_{+}^{\alpha/2-1}du~{}B(dx_{1})B(dx_{2}),$
(5.18)

where

$c_{\alpha}=\frac{\sin(\alpha\pi/2)\Gamma(1-\alpha/2)}{\pi\Gamma(\alpha/2)};$

  3. (c)

when $\alpha=0$, $\beta>1/2$:

$Z_{\alpha,\beta}(\tau)\overset{f.d.d.}{=}c_{\beta}\int_{[0,\tau]^{2}}^{\prime}|x_{1}-x_{2}|^{\beta-1}~{}B(dx_{1})B(dx_{2}),$
(5.19)

where

$c_{\beta}=\frac{\Gamma(1-\beta)\sin(\beta\pi/2)}{\pi}.$

Here $B(\cdot)$ is the real Brownian random measure and the prime ′ in the
integrals indicates the exclusion of the diagonals.

######  Remark 5.9.

In view of (5.1) and (5.16), the representation (5.17) gives an explicit
insight of the convergence in Theorem 5.2 (see Theorem 5.4 below). The process
in (5.18) is known as Rosenblatt process (see Taqqu [97]), and the
corresponding convergence in Theorem 5.2 is the c.t. analog of the d.t. case
considered in Fox and Taqqu [37]. The representation (5.19) is obtained
because for $\alpha=0$, the underlying process $X(t)$ has short memory and in
this case, one expects that in the limit $X(t)dt$ in (5.1) can be replaced by
the white noise $B(dt)$.

######  Remark 5.10.

It is of interest to obtain appropriate elementary expressions for the time-
domain representation of the limiting process $Z_{\alpha,\beta}(\tau)$,
similar to (5.17) - (5.19), in the cases where either $\alpha$ or $\beta$
satisfying (5.13) is negative.

Using the time-domain representation (5.17), one can state a non-central limit
theorem in the case where $\alpha,\beta>0$ without going to the spectral
domain. This simplifies the assumptions imposed on the functions $a(\cdot)$
and $b(\cdot)$ (see Bai et al. [9]).

######  Theorem 5.4.

Suppose that the functions $a(\cdot)$ and $b(\cdot)$ are given by (5.16),
where $0<\alpha<1$, $0<\beta<1$, $\alpha+\beta>1/2$, and $\ell_{1}(x)$ and
$\ell_{2}(x)$ are even functions slowly varying at infinity and bounded on
bounded intervals. Then as $T\rightarrow\infty$ we have

$\displaystyle\frac{1}{T^{\alpha+\beta}\ell_{1}(T)\ell_{2}(T)}\left(Q_{T}(\tau)-\mathbb{E}[Q_{T}(\tau)]\right)\overset{f.d.d.}{\longrightarrow}$
$\displaystyle\int_{\mathbb{R}^{2}}^{\prime}\int_{0}^{\tau}\int_{0}^{\tau}|u-v|^{\beta-1}(u-x_{1})_{+}^{\alpha/2-1}(v-x_{2})_{+}^{\alpha/2-1}dudv~{}B(dx_{1})B(dx_{2}).$

##  6 CLT for Tapered Toeplitz Quadratic Functionals

###  6.1 The problem

In this section we consider a question concerning asymptotic distribution (as
$T\to\infty$) of the following tapered Toeplitz type quadratic functionals of
the centered stationary process $X(u)$, $u\in\mathbb{U}$, with spectral
density $f(\lambda)$, $\lambda\in\Lambda$:

$Q_{T}^{h}:=\left\\{\begin{array}[]{ll}\sum_{t=1}^{T}\sum_{s=1}^{T}\widehat{g}(t-s)h_{T}(t)h_{T}(s)X(t)X(s)&\mbox{in
the d.t.\ case},\\\ \\\
\int_{0}^{T}\int_{0}^{T}\widehat{g}(t-s)h_{T}(t)h_{T}(s)X(t)X(s)\,dt\,ds&\mbox{in
the c.t.\ case},\end{array}\right.$ (6.1)

where $\widehat{g}(t)$ is the Fourier transform of some integrable even
function $g(\lambda)$ and

$h_{T}(t):=h(t/T)$ (6.2)

with a taper function $h(\cdot)$ to be specified below.

The limit distribution of the functional (6.1) is completely determined by the
functions $f$, $g$ and $h$, and depending on their properties it can be either
Gaussian (that is, $Q^{h}_{T}$ with an appropriate normalization obey central
limit theorem), or non-Gaussian.

We discuss here the case where the limit distribution is Gaussian, and present
sufficient conditions in terms of functions $f$, $g$ (and $h$) ensuring
central limit theorems for a standard normalized tapered quadratic functional
$Q_{T}^{h}$.

We will assume that the taper function $h(\cdot)$ satisfies the following
assumption.

######  Assumption 6.1.

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

$H_{k}:=\int_{0}^{1}h^{k}(t)dt,\quad k\in\mathbb{N}:=\\{1,2,\ldots\\}.$ (6.3)

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

######  Remark 6.1.

In the d.t. case, an example of a taper function $h(t)$ satisfying Assumption
6.1 is the Tukey-Hanning taper function $h(t)=0.5(1-\cos(\pi t))$ for
$t\in[0,1]$. In the c.t. case, a simple example of a taper function $h(t)$
satisfying Assumption 6.1 is the function $h(t)=1-t$ for $t\in[0,1]$ (see,
e.g., Anh et al. [5]).

###  6.2 Statistical motivation

Much of the statistical inferences (parametric and nonparametric estimation,
hypotheses testing) about the spectrum or the covariance of a stationary
process $\\{X(u),\ u\in\mathbb{U}\\}$ is based on the sample:

$\mathbf{X}_{T}:=\left\\{\begin{array}[]{ll}\\{X(t),\,t=1,\ldots,T\\}&\mbox{in
the d.t.\ case},\\\ \\{X(t),\,0\leq t\leq T\\}&\mbox{in the c.t.\
case}\end{array}\right.$ (6.4)

In the statistical analysis of stationary processes, however, the data are
frequently tapered before calculating the statistics of interest, and the
statistical inference procedure, instead of the original data $\mathbf{X}_{T}$
given by (6.4), is based on the tapered data $\mathbf{X}_{T}^{h}$:

$\mathbf{X}_{T}^{h}:=\left\\{\begin{array}[]{ll}\\{h_{T}(t)X(t),\,t=1,\ldots,T\\}&\mbox{in
the d.t.\ case},\\\ \\{h_{T}(t)X(t),\,0\leq t\leq T\\}&\mbox{in the c.t.\
case},\end{array}\right.$ (6.5)

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

The benefits of tapering the data have been widely reported in the literature
(see, e.g., Bloomfield [14], Brillinger [17], Dahlhaus [24, 25, 26], Dahlhaus
and Künsch [28], Guyon [69], 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’.

Quadratic functionals of the form (6.1) appear both in nonparametric and
parametric estimation of the spectrum of the process $X(t)$ based on the
tapered data (6.5). For instance, when we are interested in nonparametric
estimation of a linear integral functional in $L^{p}(\Lambda)$, $p>1$ of the
form:

$J=J(f):=\int_{\Lambda}f(\lambda)g(\lambda)d\lambda,$ (6.6)

where $g(\lambda)\in L^{q}(\Lambda)$, $1/p+1/q=1$, then a natural statistical
estimator for $J(f)$ is the linear integral functional of the empirical
spectral density (periodogram) of the process $X(t)$. To define this
estimator, we first introduce some notation.

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

$H_{k,T}(\lambda):=\left\\{\begin{array}[]{ll}\sum_{t=1}^{T}h_{T}^{k}(t)e^{-i\lambda
t}&\mbox{in the d.t.\ case},\\\ \\\\[-2.84526pt]
\int_{0}^{T}h_{T}^{k}(t)e^{-i\lambda t}dt&\mbox{in the c.t.\
case}.\end{array}\right.$ (6.7)

Define the finite Fourier transform of the tapered data (6.5):

$d^{h}_{T}(\lambda):=\left\\{\begin{array}[]{ll}\sum_{t=0}^{T}h_{T}(t)X(t)e^{-i\lambda
t}&\mbox{in the d.t.\ case},\\\ \\\\[-2.84526pt]
\int_{0}^{T}h_{T}(t)X(t)e^{-i\lambda t}dt&\mbox{in the c.t.\
case}.\end{array}\right.$ (6.8)

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

$\displaystyle
I^{h}_{T}(\lambda):=\frac{1}{C_{T}}\,d^{h}_{T}(\lambda)d^{h}_{T}(-\lambda)=\left\\{\begin{array}[]{ll}\frac{1}{C_{T}}\left|\sum_{t=0}^{T}h_{T}(t)X(t)e^{-i\lambda
t}\right|^{2}&\mbox{in the d.t.\ case},\\\ \\\\[-2.84526pt]
\frac{1}{C_{T}}\left|\int_{0}^{T}h_{T}(t)X(t)e^{-i\lambda
t}dt\right|^{2}&\mbox{in the c.t.\ case}.\end{array}\right.$ (6.12)

where

$C_{T}:=2\pi H_{2,T}(0)=2\pi\int_{0}^{T}h_{T}^{2}(t)dt=2\pi H_{2}\,T\neq 0.$
(6.13)

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

As an estimator $J_{T}^{h}$ for functional $J(f)$, given by (6.6), based on
the tapered data (6.5), we consider the averaged tapered periodogram (or a
simple ”plug-in” statistic), defined by

$\displaystyle J_{T}^{h}$ $\displaystyle=$ $\displaystyle
J(I^{h}_{T}):=\int_{\Lambda}I^{h}_{T}(\lambda)g(\lambda)d\lambda.$ (6.14)

In view of (6.1), (6.12) and (6.14) we have

$\displaystyle
J_{T}^{h}=C_{T}^{-1}Q_{T}^{h}=\left\\{\begin{array}[]{ll}\frac{1}{C_{T}}\sum_{t=1}^{T}\sum_{s=1}^{T}\widehat{g}(t-s)h_{T}(t)h_{T}(s)X(t)X(s)&\mbox{in
the d.t.\ case},\\\ \\\\[-2.84526pt]
\frac{1}{C_{T}}\int_{0}^{T}\int_{0}^{T}\widehat{g}(t-s)h_{T}(t)h_{T}(s)X(t)X(s)\,dt\,ds&\mbox{in
the c.t.\ case},\end{array}\right.$ (6.18)

where $C_{T}$ is as in (6.13), and $\widehat{g}(t)$ is the Fourier transform
of function $g(\lambda)$.

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 (6.1) (for details see Section
7).

###  6.3 Central limit theorems for tapered quadratic functional $Q_{T}^{h}$

We will use the following notation. By $\widetilde{Q}^{h}_{T}$ we denote the
standard normalized quadratic functional:

$\widetilde{Q}^{h}_{T}=T^{-1/2}\,\left(Q^{h}_{T}-\mathbb{E}[Q^{h}_{T}]\right).$
(6.19)

Also, we set

$\sigma^{2}_{h}:=16\pi^{3}H_{4}\int_{\Lambda}f^{2}(\lambda)g^{2}(\lambda)\,d\lambda,$
(6.20)

where $H_{4}$ is as in (6.3). The notation

$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim
N(0,\sigma^{2}_{h})\quad{\rm as}\quad T\to\infty$ (6.21)

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
$\sigma^{2}_{h}$.

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 (see (3.4) for non-tapered case):

$B_{T}^{h}(\psi):=\|\widehat{\psi}(t-s)h_{T}(t)h_{T}(s)\|_{t,s=1,2\ldots,T},$
(6.22)

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 (see
(3.5) for non-tapered case):

$[{W}^{h}_{T}(\psi)u](t)=\int_{0}^{T}\hat{\psi}(t-s)u(s)h_{T}(s)ds,\quad
u(s)\in L^{2}([0,T];h_{T}),$ (6.23)

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 $A_{T}^{h}(f)$ 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 $h$.

####  6.3.1 CLT for Gaussian models

The theorems that follow extend the results of Theorems 3.1 and 3.3 to the
tapered case. We assume that the model process $X(t)$ is Gaussian, and with no
loss of generality, that $g\geq 0$. The following theorems were proved in
Ginovyan and Sahakyan [57].

######  Theorem 6.1.

Assume that $f\cdot g\in L^{1}(\Lambda)\cap L^{2}(\Lambda)$, the taper
function $h$ satisfies Assumption 6.1, and for $T\to\infty$

$\chi_{2}(\widetilde{Q}^{h}_{T})=\frac{2}{T}\hbox{tr}\bigl{[}A^{h}_{T}(f)A^{h}_{T}(g)\bigr{]}^{2}\longrightarrow\sigma^{2}_{h},$
(6.24)

where $\sigma^{2}_{h}$ is as in (6.20). Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{h})$ as
$T\to\infty$.

######  Theorem 6.2.

Assume that the function

$\varphi(x_{1},x_{2},x_{3})=\int_{\Lambda}f(u)g(u-x_{1})f(u-x_{2})g(u-x_{3})\,du$
(6.25)

belongs to $L^{2}(\Lambda^{3})$ and is continuous at $(0,0,0)$, and the taper
function $h$ satisfies Assumption 6.1. Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{h})$ as
$T\to\infty$.

######  Theorem 6.3.

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

######  Theorem 6.4.

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

$\int_{\Lambda}f^{2}(\lambda)g^{2}(\lambda-\mu)\,d\lambda\longrightarrow\int_{\Lambda}f^{2}(\lambda)g^{2}(\lambda)\,d\lambda\quad{\rm
as}\quad\mu\to 0,$ (6.26)

and let the taper function $h$ satisfy Assumption 6.1. 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 6.5.

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

$f(\lambda)\leq|\lambda|^{-\alpha}L_{1}(\lambda),\quad|g(\lambda)|\leq|\lambda|^{-\beta}L_{2}(\lambda),\quad\lambda\in[-a,a],$
(6.27)

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

$\displaystyle L_{i}\in SV_{0}(\mathbb{R}),\ \
\lambda^{-(\alpha+\beta)}L_{i}(\lambda)\in L^{2}[-a,a],\ i=1,2.$ (6.28)

Also, let the taper function $h$ satisfy Assumption 6.1. Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{h})$ as
$T\to\infty$.

As in Remark 3.3, the conditions $\alpha<1$ and $\beta<1$ in Theorem 6.5
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 (6.28) imply that $f\cdot g\in
L^{2}(\Lambda)$, so that the variance $\sigma^{2}_{h}$ in (6.20) is finite.

####  6.3.2 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 (2.12), 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 follow is a tapered counterpart of Theorem 5.1
and was proved in Ginovyan and Sahakyan [56].

######  Theorem 6.6.

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

$a(\cdot)\in L^{p}(\mathbb{R})\cap
L^{2}(\mathbb{R}),\quad\widehat{g}(\cdot)\in L^{q}(\mathbb{R}),\quad 1\leq
p,q\leq 2,\quad\frac{2}{p}+\frac{1}{q}\geq\frac{5}{2},$ (6.29)

and the taper $h$ satisfies Assumption 6.1. Then
$\widetilde{Q}^{h}_{T}\overset{d}{\rightarrow}\eta\sim N(0,\sigma^{2}_{L,h})$
as $T\to\infty$, where

$\sigma^{2}_{L,h}=16\pi^{3}H_{4}\int_{\mathbb{R}}f^{2}(\lambda)g^{2}(\lambda)d\lambda+\kappa_{4}4\pi^{2}H_{4}\left[\int_{\mathbb{R}}f(\lambda)g(\lambda)d\lambda\right]^{2},$
(6.30)

and $H_{4}$ is as in (6.3).

######  Remark 6.2.

Notice that if the underlying process $X(t)$ is Gaussian, then in formula
(6.30) we have only the first term and so $\sigma^{2}_{L,h}=\sigma^{2}_{h}$
(see (6.20)), because in this case $\kappa_{4}=0$. On the other hand, the
condition (6.29) is more restrictive than the conditions in Theorems 6.1 \-
6.5. Thus, for Gaussian processes Theorems 6.1 \- 6.5 improve Theorem 6.6.

######  Remark 6.3.

Central and non-central limit theorems for tapered quadratic forms of a d.t.
long memory Gaussian stationary fields have been proved in Doukhan et al.
[32].

##  7 Applications

###  7.1 Nonparametric estimation of spectral functionals

Suppose we observe a 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
$X(t)$ with an unknown spectral density function $f(\lambda)$,
$\lambda\in\Lambda$. We assume that $f(\lambda)$ belongs to a given (infinite-
dimensional) class $\mathbf{\Theta}\subset L^{p}(\Lambda)$ $(p\geq 1)$ of
spectral densities possessing some smoothness properties. Let $\Phi(\cdot)$ be
some known functional, the domain of definition of which contains
$\mathbf{\Theta}$. The problem is to estimate the value $\Phi(f)$ of the
functional $\Phi(\cdot)$ at an unknown point $f\in\mathbf{\Theta}$ on the
basis of observation $\mathbf{X}_{T},$ and investigate the asymptotic (as
$T\to\infty$) properties of the suggested estimators.

This problem for linear and some nonlinear smooth functionals for d.t. and
c.t. stationary models has been extensively discussed in the literature (see,
e.g., Dahlhaus and Wefelmeyer [29], Ginovyan [42, 43, 46, 48, 49], Ginovyan
and Sahakyan [56, 57], Has’minskii and Ibragimov [70], Taniguchi [95],
Taniguchi and Kakizawa [96], and references therein).

In this section we apply the results of Section 6.3 to show that the statistic
$J_{T}^{h}$ given by (6.14) is a consistent and asymptotically normal
estimator for the linear functional $J(f)$ given by (6.6). We follow the
papers Ginovyan and Sahakyan [56, 57]. To state the corresponding results, we
first introduce the $L^{p}$-Hölder class and set up a set of assumptions.

Given numbers $p\geq 1$, $0<\alpha<1$,
$r\in\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$, 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

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

Now we list the assumptions.

######  Assumption 7.1.

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

$a(\cdot)\in L^{p}(\Lambda)\cap L^{2}(\Lambda),\quad\widehat{g}(\cdot)\in
L^{q}(\Lambda)$ (7.1)

with

$1\leq p,q\leq 2,\quad\frac{2}{p}+\frac{1}{q}\geq\frac{5}{2}.$ (7.2)

######  Assumption 7.2.

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.

######  Assumption 7.3.

The spectral density $f$ and the generating function $g$ are such that $f\in
H_{p}(\beta_{1})$, $\beta_{1}>0$, $p\geq 1$ and $g(\lambda)\in
H_{q}(\beta_{2})$, $\beta_{2}>0$, $q\geq 1$ 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 7.1.

In Ginovian [45] it was proved that if Assumption 7.3 is satisfied, 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
$1/{p_{1}}+1/{q_{1}}\leq 1/2$.

The next theorem controls the bias $E(J_{T}^{h})-J$ and provides sufficient
conditions assuring the proper rate of convergence of the bias to zero, which
is necessary to obtain the asymptotic normality of the estimator $J_{T}^{h}$.
Specifically, we have the following result, which was proved in Ginovyan and
Sahakyan [56].

######  Theorem 7.1.

Let the functionals $J:=J(f)$ and $J_{T}^{h}:=J(I^{h}_{T})$ be defined by
(6.6) and (6.14), respectively. Then under Assumptions 7.2 (or 7.3) and 6.1
the following asymptotic relation holds:

$\displaystyle T^{1/2}\left[\mathbb{E}(J_{T}^{h})-J\right]\to 0\quad{\rm
as}\quad T\to\infty.$ (7.3)

The next theorem, which is an immediate consequence of Theorem 6.6, contains
sufficient conditions for functional $J_{T}^{h}$ to obey the central limit
theorem.

######  Theorem 7.2.

Let $J:=J(f)$ and $J_{T}^{h}:=J(I^{h}_{T})$ be defined by (6.6) and (6.14),
respectively. Then under Assumptions 7.1 and 6.1 the functional $J_{T}^{h}$
obeys the central limit theorem. More precisely, we have

$\displaystyle
T^{1/2}\left[J_{T}^{h}-\mathbb{E}(J_{T}^{h})\right]\overset{d}{\rightarrow}\eta\quad{\rm
as}\quad T\to\infty,$ (7.4)

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

$\sigma^{2}_{h}(J)=4\pi
e(h)\int_{\mathbb{R}}f^{2}(\lambda)g^{2}(\lambda)d\lambda+\kappa_{4}e(h)\left[\int_{\mathbb{R}}f(\lambda)g(\lambda)d\lambda\right]^{2}.$
(7.5)

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

$e(h):=\frac{H_{4}}{H_{2}^{2}}=\int_{0}^{1}h^{4}(t)dt\left(\int_{0}^{1}h^{2}(t)dt\right)^{-2}.$
(7.6)

Taking into account the equality

$\displaystyle
T^{1/2}\left[J_{T}^{h}-J\right]=T^{1/2}\left[\mathbb{E}(J_{T}^{h})-J\right]+T^{1/2}\left[J_{T}^{h}-\mathbb{E}(J_{T}^{h})\right],$
(7.7)

as an immediate consequence of Theorems 7.1 and 7.2, 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 $J(f)$.

######  Theorem 7.3.

Let the functionals $J:=J(f)$ and $J_{T}^{h}:=J(I^{h}_{T})$ be defined by
(6.6) and (6.14), respectively. Then under Assumptions 7.1, 7.2 (or 7.3) and
6.1 the statistic $J_{T}^{h}$ is an asymptotically normal estimator for
functional $J$. More precisely, we have

$\displaystyle
T^{1/2}\left[J_{T}^{h}-J\right]\overset{d}{\rightarrow}\eta\quad{\rm as}\quad
T\to\infty,$ (7.8)

where $\eta$ is as in Theorem 7.2, that is, $\eta$ is a normally distributed
random variable with mean zero and variance $\sigma^{2}_{h}(J)$ given by (7.5)
and (7.6).

In the Gaussian case we have more accurate result. The next theorem, which was
proved in Ginovyan and Sahakyan [57], states that for Gaussian models
Assumptions 7.3 and 6.1 are sufficient for the statistic $J_{T}^{h}$ to be an
asymptotically normal estimator for functional $J$.

######  Theorem 7.4.

Let the functionals $J:=J(f)$ and $J_{T}^{h}:=J(I^{h}_{T})$ be defined by
(6.6) and (6.14), respectively. Then under Assumptions 7.3 and 6.1 the
statistic $J_{T}^{h}$ is an asymptotically normal estimator for functional
$J$. More precisely, we have

$\displaystyle
T^{1/2}\left[J_{T}^{h}-J\right]\overset{d}{\rightarrow}\eta\quad{\rm as}\quad
T\to\infty,$ (7.9)

where $\eta$ is a normally distributed random variable with mean zero and
variance $\sigma^{2}_{h}(J)$ given by

$\sigma^{2}_{h}(J)=4\pi
e(h)\int_{\mathbb{R}}f^{2}(\lambda)g^{2}(\lambda)d\lambda,\quad
e(h):=H_{4}H_{2}^{-2},$ (7.10)

and $H_{k}$ is as in (6.3).

###  7.2 Parametric estimation: The Whittle procedure

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 an open 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 (6.5), 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.

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 [107]).
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 [27], Dzhaparidze [33],
Fox and Taqqu [39], Giraitis and Surgailis [60], Guyon [69], Heyde and Gay
[72], Taniguchi and Kakizawa [96], Walker [106], and references therein).

For c.t. models, the Whittle estimation procedure has been considered, for
example, in Anh et al. [4], Avram et al. [7], Casas and Gao [21], Dzhaparidze
[33], Gao [40], Gao et al. [41], Leonenko and Sakhno [80], Tsai and Chan
[104], 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 [24], Dahlhaus and Künsch [28], Guyon
[69], Ludeña and Lavielle [82]. 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 [50].

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 log-likelihood function $L_{T}(\theta)$, which in this case takes
the form:

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

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 P.
Whittle [107], 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 maximizing 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., Dzhaparidze [33]).