# Asymptotic behavior of the prediction error for stationary
sequences††thanks: The work is dedicated to our teacher Academician, Professor
Il’dar Abdullovich Ibragimov on the occasion of his 90th birthday.

Nikolay M. Babayan and Mamikon S. Ginovyan  Russian-Armenian University,
Yerevan, Armenia, e-mail<EMAIL_ADDRESS>University, Boston, USA,
e-mail<EMAIL_ADDRESS>
###### Abstract

One of the main problem in prediction theory of discrete-time second-order
stationary processes $X(t)$ is to describe the asymptotic behavior of the best
linear mean squared prediction error in predicting $X(0)$ given $X(t),$
$-n\leq t\leq-1$, as $n$ goes to infinity. This behavior depends on the
regularity (deterministic or nondeterministic) and on the dependence structure
of the underlying observed process $X(t)$. In this paper we consider this
problem both for deterministic and nondeterministic processes and survey some
recent results. We focus on the less investigated case - deterministic
processes. It turns out that for nondeterministic processes the asymptotic
behavior of the prediction error is determined by the dependence structure of
the observed process $X(t)$ and the differential properties of its spectral
density $f$, while for deterministic processes it is determined by the
geometric properties of the spectrum of $X(t)$ and singularities of its
spectral density $f$.

Key words and phrases. Prediction error, deterministic and nondeterministic
process, spectral density, Ibragimov’s theorems, Rosenblatt’s theorems,
Szegő’s condition, transfinite diameter, eigenvalues of truncated Toeplitz
matrices.

2010 Mathematics Subject Classification. Primary: 60G10, 60G25, 62M15, 62M20.
Secondary: 15A18, 30C10.

###### Contents

  1. 1 Introduction
    1. 1.1 The finite prediction problem
    2. 1.2 A brief history
    3. 1.3 Notation and conventions.
    4. 1.4 The structure of the paper
  2. 2 The model. Key notions and some basic results
    1. 2.1 Second-order (wide-sense) stationary processes
    2. 2.2 Linear processes. Existence of spectral density functions
    3. 2.3 Dependence (memory) structure of the model
      1. 2.3.1 Short memory models
      2. 2.3.2 Long-memory and anti-persistent models
    4. 2.4 Deterministic and nondeterministic processes.
      1. 2.4.1 Kolmogorov’s isometric isomorphism theorem
      2. 2.4.2 The infinite prediction problem
      3. 2.4.3 Deterministic (singular) and nondeterministic processes. Characterizations
  3. 3 Formulas for the finite prediction error $\sigma_{n}^{2}(f)$ and some properties
    1. 3.1 Formulas for the prediction error $\sigma_{n}^{2}(f)$
    2. 3.2 Properties of the prediction error $\sigma_{n}^{2}(f)$
  4. 4 Asymptotic behavior of the prediction error $\delta_{n}(f)$ for nondeterministic processes
    1. 4.1 Asymptotic behavior of $\delta_{n}(f)$ for short-memory processes
      1. 4.1.1 Exponential rate of decrease of $\delta_{n}(f)$
      2. 4.1.2 Hyperbolic rate of decrease of $\delta_{n}(f)$
    2. 4.2 Asymptotic behavior of $\delta_{n}(f)$ for anti-persistent and long-memory processes
      1. 4.2.1 An example
      2. 4.2.2 The ARFIMA$(p,d,q)$ Model
      3. 4.2.3 The Jacobian Model
  5. 5 Asymptotic behavior of the finite prediction error $\sigma_{n}^{2}(f)$ for deterministic (singular) processes
    1. 5.1 Background
    2. 5.2 Rosenblatt’s results about speed of convergence
    3. 5.3 Extensions of Rosenblatt’s first theorem
      1. 5.3.1 Extensions of Theorem 5.1
      2. 5.3.2 Examples. Calculation of transfinite diameters of some special sets
      3. 5.3.3 A consequence of Theorem 5.3
    4. 5.4 Davisson’s theorem and its extension
    5. 5.5 Extensions of Rosenblatt’s second theorem
      1. 5.5.1 Preliminaries
      2. 5.5.2 Extensions of Rosenblatt’s second theorem
      3. 5.5.3 Examples
      4. 5.5.4 Rosenblatt’s second theorem revisited
  6. 6 An Application. Asymptotic behavior of the extreme eigenvalues of truncated Toeplitz matrices
    1. 6.1 Extreme eigenvalues of truncated Toeplitz matrices
    2. 6.2 The relationship between the prediction error and the minimal eigenvalue.
      1. 6.2.1 A model with spectral density $f$ which is discontinuous, zero on an interval, and positive elsewhere.
      2. 6.2.2 A model with spectral density $f$ possessing exponential order zeros.
  7. 7 Tools
    1. 7.1 Some metric characteristics of bounded closed sets in the plane
      1. 7.1.1 The transfinite diameter, the Chebyshev constant and the capacity
      2. 7.1.2 Transfinite diameters of related sets
      3. 7.1.3 An extension of Robinson’s theorem
    2. 7.2 Weakly varying sequences
    3. 7.3 Some properties of the geometric mean and trigonometric polynomials.

##  1 Introduction

###  1.1 The finite prediction problem

Let $X(t),$ $t\in\mathbb{Z}:=\\{0,\pm 1,\ldots\\}$, be a centered discrete-
time second-order stationary process. The process is assumed to have an
absolutely continuous spectrum with spectral density function $f(\lambda),$
$\lambda\in[-\pi,\pi].$ The ’finite’ linear prediction problem is as follows.

Suppose we observe a finite realization of the process $X(t)$:

$\\{X(t),\,\,-n\leq t\leq-1\\},\quad n\in\mathbb{N}:=\\{1,2,\ldots\\}.$

We want to make an one-step ahead prediction, that is, to predict the
unobserved random variable $X(0)$, using the linear predictor

$Y=\sum_{k=1}^{n}c_{k}X(-k).$

The coefficients $c_{k}$, $k=1,2,\ldots,n$, are chosen so as to minimize the
mean-squared error: $\rm I\\!E\left|X(0)-Y\right|^{2},$ where $\rm
I\\!E[\cdot]$ stands for the expectation operator. If such minimizing
constants $\widehat{c}_{k}:=\widehat{c}_{k,n}$ can be found, then the random
variable

$\widehat{X}_{n}(0):=\sum_{k=1}^{n}\widehat{c}_{k}X(-k)$

is called the best linear one-step ahead predictor of $X(0)$ based on the
observed finite past:
$X(-n),\ldots,X(-1)$. The minimum mean-squared error:

$\sigma_{n}^{2}(f):=\rm I\\!E\left|X(0)-\widehat{X}_{n}(0)\right|^{2}\geq 0$

is called the best linear one-step ahead prediction error of $X(t)$ based on
the past of length $n$.

One of the main problem in prediction theory of second-order stationary
processes, called the ’direct’ prediction problem is to describe the
asymptotic behavior of the prediction error $\sigma_{n}^{2}(f)$ as
$n\to\infty$. This behavior depends on the regularity nature (deterministic or
nondeterministic) of the observed process $X(t)$.

Observe that $\sigma_{n+1}^{2}(f)\leq\sigma_{n}^{2}(f)$, $n\in\mathbb{N}$, and
hence the limit of $\sigma_{n}^{2}(f)$ as $n\to\infty$ exists. Denote by
$\sigma^{2}(f):=\sigma_{\infty}^{2}(f)$ the prediction error of $X(0)$ by the
entire infinite past: $\\{X(t)$, $t\leq-1\\}$.

From the prediction point of view it is natural to distinguish the class of
processes for which we have error-free prediction by the entire infinite past,
that is, $\sigma^{2}(f)=0$. Such processes are called deterministic or
singular. Processes for which $\sigma^{2}(f)>0$ are called nondeterministic.

Note. The term ”deterministic” here is not used in the usual sense of absence
of randomness. Instead determinism of a process means that there is an
extremely strong dependence between the successive random variables forming
the process, yielding error-free prediction when using the entire infinite
past (for more about this term see Section 2.4, and also Bingham [11], and
Grenander and Szegő [32], p.176.)

Define the ’relative’ prediction error

$\delta_{n}(f):=\sigma^{2}_{n}(f)-\sigma^{2}(f),$

and observe that $\delta_{n}(f)$ is non-negative and tends to zero as
$n\to\infty$. But what about the speed of convergence of $\delta_{n}(f)$ to
zero as $n\to\infty$? The paper deals with this question. Specifically, the
prediction problem we are interested in is to describe the rate of decrease of
$\delta_{n}(f)$ to zero as $n\to\infty,$ depending on the regularity nature of
the observed process $X(t)$.

We consider the problem both for deterministic and nondeterministic processes
and survey some recent results. We focus on the less investigated case -
deterministic processes. It turns out that for nondeterministic processes the
asymptotic behavior of the prediction error is determined by the dependence
structure of the observed process $X(t)$ and the differential properties of
its spectral density $f$, while for deterministic processes it is determined
by the geometric properties of the spectrum of $X(t)$ and singularities of its
spectral density $f$.

###  1.2 A brief history

The prediction problem stated above goes back to classical works of A. N.
Kolmogorov [45, 46], G. Szegő [68, 71] and N. Wiener [76]. It was then
considered by many authors for different classes of nondeterministic processes
(see, e.g., Baxter [2], Devinatz [18], Geronimus [23, 24], Golinski [27],
Golinski and Ibragimov [28], Grenander and Rosenblatt [30, 31], Grenander and
Szegő [32], Helson and Szegő [34], Hirshman [35], Ibragimov [37, 38],
Ibragimov and Rozanov [40], Ibragimov and Solev [41], Inoue [42], Pourahmadi
[54], Rozanov [61], and reference therein). More references can be found in
the survey papers Bingham [11] and Ginovyan [26]. In Section 4 of the paper we
state some important known results for nondeterministic processes.

We focus in this paper on deterministic processes, that is, on the class of
processes for which $\sigma^{2}(f)=0$. This case is not only of theoretical
interest, but is also important from the point of view of applications. For
example, as pointed out by Rosenblatt [60] (see also Pierson [51]), situations
of this type arise in Neumann’s theoretical model of storm-generated ocean
waves. Such models are also of interest in meteorology (see, e.g., Fortus
[21]).

Only few works are devoted to the study of the speed of convergence of
$\delta_{n}(f)=\sigma^{2}_{n}(f)$ to zero as $n\to\infty$, that is, the
asymptotic behavior of the prediction error for deterministic processes. One
needs to go back to the classical work of M. Rosenblatt [60]. Using the
technique of orthogonal polynomials on the unit circle (OPUC), M. Rosenblatt
investigated the asymptotic behavior of the prediction error
$\sigma^{2}_{n}(f)$ for deterministic processes in the following two cases:

  * (a)

the spectral density $f(\lambda)$ is continuous and positive on a segment of
$[-\pi,\pi]$ and is zero elsewhere,

  * (b)

the spectral density $f(\lambda)$ has a very high order of contact with zero
at points $\lambda=0,\pm\pi$, and is strictly positive otherwise.

Later the problems (a) and (b) were studied by Babayan [3, 4], Babayan and
Ginovyan [5, 6, 7], Babayan et al. [8] (see also Davisson [16] and Fortus
[21]), where some generalizations and extensions of Rosenblatt’s results have
been obtained.

###  1.3 Notation and conventions.

Throughout the paper we will use the following notation and conventions.
The standard symbols $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$ and $\mathbb{C}$
denote the sets of natural, integer, real and complex numbers, respectively.
Also, we denote $\mathbb{Z}_{+}:=\\{0,1,2,\ldots\\}$, $\Lambda:=[-\pi,\pi],$
$\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$,
$\mathbb{T}:=\\{z\in\mathbb{C}:\,|z|=1\\}$. For a point
$\lambda_{0}\in\Lambda$ and a number $\delta>0$ by $O_{\delta}(\lambda_{0})$
we denote a $\delta$-neighborhood of $\lambda_{0}$, that is,
$O_{\delta}(\lambda_{0}):=\\{\lambda\in\Lambda:\
|\lambda-\lambda_{0}|<\delta\\}$. By $L^{p}(\mu):=L^{p}(\mathbb{T},\mu)$
($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 $\mu$ is the Lebesgue measure, we will use the notation
$L^{p}$ and $||\cdot||_{p}$, respectively. For a function $h\geq 0$ by $G(h)$
we denote the geometric mean of $h$. For two functions $f(\lambda)\geq 0$ and
$g(\lambda)\geq 0$ we will write $f(\lambda){\sim}g(\lambda)$ as
${\lambda\to\lambda_{0}}$ if
$\lim_{\lambda\to\lambda_{0}}{f(\lambda)}/{g(\lambda)}=1$;
$f(\lambda){\simeq}g(\lambda)$ as ${\lambda\to\lambda_{0}}$ if
$\lim_{\lambda\to\lambda_{0}}{f(\lambda)}/{g(\lambda)}=c>0$, and
$f(\lambda){\asymp}g(\lambda)$ if there are constants $c_{1},c_{2}$
($0<c_{1}\leq c_{2}<\infty$) such that $0<c_{1}\leq f(\lambda)/g(\lambda)\leq
c_{2}<\infty$ for all $\lambda\in\Lambda$. We will use similar notation for
sequences: for two sequences $\\{a_{n}\geq 0,n\in\mathbb{N}\\}$ and
$\\{b_{n}>0,n\in\mathbb{N}\\}$, we will write $a_{n}\sim b_{n}$ if
$\lim_{n\to\infty}{a_{n}}/{b_{n}}=1$, $a_{n}{\simeq}b_{n}$ if
$\lim_{n\to\infty}{a_{n}}/{b_{n}}=c>0$, $a_{n}{\asymp}b_{n}$ if
$c_{1}\leq{a_{n}}/{b_{n}}\leq c_{2}$ for all $\lambda\in\mathbb{N}$,
$a_{n}=O(b_{n})$ if ${a_{n}}/{b_{n}}$ is bounded, and $a_{n}=o(b_{n})$ if
${a_{n}}/{b_{n}}\to 0$ as $n\to\infty$. For a set $E$ by $\overline{E}$ we
denote the closure of $E$. The letters $C$, $c$, $M$ and $m$ with or without
indices are used to denote positive constants, the values of which can vary
from line to line.

We will use the abbreviations: OPUC for ’orthogonal polynomials on the unit
circle’, PACF for ’partial autocorrelation function’, and ’a.e.’ for ’almost
everywhere’ (with respect to the Lebesgue measure). We will assume that all
the relevant objects are defined in terms of Lebesgue integrals, and so are
invariant under change of the integrand on a null set.

###  1.4 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 formulas for the
finite prediction error $\sigma_{n}^{2}(f)$, and state some preliminary
results. In Section 4 we state some well known results on asymptotic behavior
of the prediction error for nondeterministic processes. Asymptotic behavior of
the finite prediction error $\sigma_{n}^{2}(f)$ for deterministic processes is
discussed in Section 5. Here we state extensions of Rosenblatt’s and
Davisson’s results, and discuss a number of examples. In Section 6 we analyze
the relationship between the rate of convergence to zero of the prediction
error $\sigma_{n}^{2}(f)$ and the minimal eigenvalue of a truncated Toeplitz
matrix generated by the spectral density $f$. In Section 7 we briefly discuss
the tools, used to prove the theorems stated in the paper.

##  2 The model. Key notions and some basic results

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 Second-order (wide-sense) stationary processes

Let $\\{X(t),\ t\in\mathbb{Z}\\}$ 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,

${\rm I\\!E}[X(t)]=0,\quad r(t)={\rm I\\!E}[X(t+s)X(s)],\quad
s,t\in\mathbb{Z},$

where ${\rm I\\!E}[\cdot]$ stands for the expectation operator with respect to
the measure $P$.

By the Herglotz theorem (see, e.g., Brockwell and Davis [13], p. 117-118),
there is a finite measure $\mu$ on $\Lambda$ such that the covariance function
$r(t)$ admits the following spectral representation:

$r(t)=\int_{-\pi}^{\pi}e^{-it\lambda}d\mu(\lambda),\quad t\in\mathbb{Z}.$
(2.1)

The measure $\mu$ in (2.1) is called the spectral measure of the process
$X(t)$. If $\mu$ is absolutely continuous (with respect to the Lebesgue
measure), then the function $f(\lambda):=d\mu(\lambda)/d\lambda$ is called the
spectral density of $X(t)$. We assume that $X(t)$ is a non-degenerate process,
that is, ${\rm Var}[X(0)]:={\rm I\\!E}|X(0)|^{2}=r(0)>0$ and, without loss of
generality, we may take $r(0)=1$. Also, to avoid the trivial cases, we assume
that the spectral measure $\mu$ is non-trivial, that is, $\mu$ has infinite
support.

Notice that if the spectral density $f(\lambda)$ exists, then $f(\lambda)\geq
0$, $f(\lambda)\in L^{1}(\Lambda)$, and (2.1) becomes

$r(t)=\int_{-\pi}^{\pi}e^{-it\lambda}f(\lambda)d\lambda,\quad t\in\mathbb{Z}.$
(2.2)

Thus, the covariance function $r(t)$ 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
$\\{X(t),\ t\in\mathbb{Z}\\}$.

######  Remark 2.1.

The parametrization of the unit circle $\mathbb{T}$ by the formula
$z=e^{i\lambda}$ establishes a bijection between $\mathbb{T}$ and the interval
$[-\pi,\pi)$. By means of this bijection the measure $\mu$ on $\Lambda$
generates the corresponding measure on the unit circle $\mathbb{T}$, which we
also denote by $\mu$. Thus, depending on the context, the measure $\mu$ will
be supported either on $\Lambda$ or on $\mathbb{T}$. We use the standard
Lebesgue decomposition of the measure $\mu$:

$d\mu(\lambda)=d\mu_{a}(\lambda)+d\mu_{s}(\lambda)=f(\lambda)d\lambda+d\mu_{s}(\lambda),$
(2.3)

where $\mu_{a}$ is the absolutely continuous part of $\mu$ (with respect to
the Lebesgue measure) and $\mu_{s}$ is the singular part of $\mu$, which is
the sum of the discrete and continuous singular components of $\mu$.

By the well-known Cramér theorem (see, e.g., Cramér and Leadbetter [15]) for
any stationary process $\\{X(t),\,t\in\mathbb{Z}\\}$ with spectral measure
$\mu$ there exists an orthogonal stochastic measure $Z=Z(B)$,
$B\in\mathfrak{B}(\Lambda)$, such that for every $t\in\mathbb{Z}$ the process
$X(t)$ admits the following spectral representation:

$X(t)=\int_{\Lambda}e^{-it\lambda}dZ(\lambda),\quad t\in\mathbb{Z}.$ (2.4)

Moreover, ${\rm I\\!E}\left[|Z(B)|^{2}\right]=\mu(B)$ for every
$B\in\mathfrak{B}(\Lambda)$. Here $\mathfrak{B}(\Lambda)$ stands for the Borel
$\sigma$-algebra of the sets of $\Lambda$. For definition and properties of
orthogonal stochastic measures and stochastic integral in (2.4) we refer,
e.g., Cramér and Leadbetter [15], Ibragimov and Linnik [39], and Shiryaev
[67].

###  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 [15],
Doob [19], Ibragimov and Linnik [39].

######  Theorem 2.1.

The following assertions hold.

  * (a)

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

$X(t)=\sum_{k=-\infty}^{\infty}a(t-k)\xi(k),\quad\sum_{k=-\infty}^{\infty}|a(k)|^{2}<\infty,$
(2.5)

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 $r(t)$ and the spectral density $f(\lambda)$ of $X(t)$
are given by formulas:

$r(t)={\rm I\\!E}X(t)X(0)=\sum_{k=-\infty}^{\infty}a(t+k)a(k),$ (2.6)

and

$f(\lambda)=\frac{1}{2\pi}\left|\sum_{k=-\infty}^{\infty}a(k)e^{-ik\lambda}\right|^{2}=\frac{1}{2\pi}|\widehat{a}(\lambda)|^{2},\quad\lambda\in\Lambda.$
(2.7)

  * (c)

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

###  2.3 Dependence (memory) structure of the model

Depending on the memory (dependence) structure, 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 et al. [10], and references therein).

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

####  2.3.1 Short memory models

A stationary process $\\{X(t),\,t\in\mathbb{Z}\\}$ with spectral density
function $f(\lambda)$ is said to be a short memory process if the spectral
density $f(\lambda)$ 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 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 respectively of degrees $p$
and $q$ having no zeros on the unit circle $\mathbb{T}$, $B$ is the backshift
operator defined by $BX(t)=X(t-1)$, and $\\{\varepsilon(t),t\in\mathbb{Z}\\}$
is a WN(0,$\sigma^{2}$) 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:

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

and the spectral density $f(\lambda)$ is a rational function (see, e.g.,
Brockwell and Davis [13], Sec. 3.1):

$f(\lambda)=\frac{\sigma^{2}}{2\pi}\cdot\frac{|\theta_{q}(e^{-i\lambda})|^{2}}{|\psi_{p}(e^{-i\lambda})|^{2}}.$
(2.8)

####  2.3.2 Long-memory and anti-persistent models

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 (see, e.g., Beran et al.
[10], Brockwell and Davis [13], and references therein).

A typical model example that displays long-memory and intermediate memory
(anti-persistent) is the Autoregressive Fractionally Integrated Moving Average
(ARFIMA)$(p,d,q)$ process $X(t)$ defined to be a stationary solution of the
difference equation (see, e.g., Brockwell and Davis [13], Section 13.2):

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

where $B$ is the backshift operator, $\varepsilon(t)$ is a WN(0,$\sigma^{2}$)
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

$f_{X}(\lambda)=|1-e^{-i\lambda}|^{-2d}f(\lambda)=(2\sin(\lambda/2))^{-2d}f(\lambda),\quad
d<1/2,$ (2.9)

where $f(\lambda)$ is the spectral density of an ARMA$(p,q)$ process, given by
(2.8). 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 ${\rm
I\\!E}|X(t)|^{2}=\int_{-\pi}^{\pi}f(\lambda)d\lambda.$

Observe that for $0<d<1/2$ the model $X(t)$ specified by the spectral density
(2.9) displays long-memory. In this case we have $f(\lambda)\thicksim
c\,|\lambda|^{-2d}$ 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. For $d<0$, the model $X(t)$ displays intermediate-memory, and in
this case, the spectral density in (2.9) vanishes at $\lambda=0$. For $d=0$
the model $X(t)$ displays short-memory. For $d\geq 1/2$ the function
$f_{X}(\lambda)$ in (2.9) is not integrable, and thus it cannot represent a
spectral density of a stationary process.

###  2.4 Deterministic and nondeterministic processes.

In this section we state Kolmogorov’s isometric isomorphism theorem and the
infinite prediction problem. We give time-domain (Wold’s theorem) and
frequency-domain (Kolmogorov-Szegő’s theorem) characterizations of
deterministic and nondeterministic processes.

####  2.4.1 Kolmogorov’s isometric isomorphism theorem

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

$L^{2}(\Omega):=\left\\{\xi:\,||\xi||^{2}:=\int_{\Omega}|\xi(\omega)|^{2}dP(\omega)<\infty\right\\}.$
(2.10)

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

$(\xi,\eta)=\rm
I\\!E[\xi\eta]=\int_{\Omega}\xi(\omega){\eta(\omega)}dP(\omega).$ (2.11)

For $a,b\in\mathbb{Z}$, $-\infty\leq a\leq b\leq\infty,$ 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 $X(t)=X(t,\omega)$, $t\in[a,b]$,
$\omega\in\Omega$:

$H_{a}^{b}(X):=\overline{sp}\\{X(t),\,\,a\leq t\leq b\\}_{L^{2}(\Omega)}.$
(2.12)

Observe that the space $H_{a}^{b}(X)$ consists of all finite linear
combinations of the form $\sum_{k=a}^{b}c_{k}X(k)$, 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 $X(t)$, or the time-domain of $X(t)$.

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

$L^{2}(\mu):=\left\\{\varphi(\lambda):\,||\varphi||^{2}_{\mu}:=\int_{\Lambda}|\varphi(\lambda)|^{2}d\mu(\lambda)<\infty\right\\}.$
(2.13)

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

$(\varphi,\psi)_{\mu}=\int_{\Lambda}\varphi(\lambda)\overline{\psi}(\lambda)d\mu(\lambda).$
(2.14)

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

######  Theorem 2.2 (Kolmogorov’s isometric isomorphism theorem).

For any stationary process $X(t)$, $t\in\mathbb{Z}$, with spectral measure
$\mu$ there exists a unique isometric isomorphism $V$ between the time-domain
$H(X)$ and the frequency-domain $L^{2}(\mu)$, such that
$V[X(t)]=e^{it\lambda}$ for any $t\in\mathbb{Z}.$

Thus, in view of Theorem 2.2, 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.

####  2.4.2 The infinite prediction problem

Observe first that since by assumption $X(t)$ is a non-degenerate process, the
time-domain $H(X)$ of $X(t)$ is non-trivial, that is, $H(X)$ contains elements
different from zero.

######  Definition 2.1.

The space $H_{-T}^{t}(X)$ is called the finite history, or past of length $T$
and present of the process $X(u)$ up to time $t$. The space
$H_{t}(X):=H_{-\infty}^{t}(X)$ is called the entire history, or infinite past
and present of the process $X(u)$ up to time $t$. The space

$H_{-\infty}(X):=\cap_{t}H_{-\infty}^{t}(X)$ (2.15)

is called the remote past of the process $X(u)$.

It is clear that

$H_{-\infty}(X)\subset\cdots\subset H_{-\infty}^{t}(X)\subset
H_{-\infty}^{t+\tau}(X)\subset\cdots\subset H(X),\quad\tau\in\mathbb{N}.$
(2.16)

The Hilbert space setting provides a natural framework for stating and solving
the problem of predicting future values of the process $X(u)$ from the
observed past values. Assume that a realization of the process $X(u)$ for
times $u\leq t$ is observed and we want to predict the value $X(t+\tau)$ for
some $\tau\geq 1$ from the observed values. Since we will never know what
particular realization is being observed, it is reasonable to consider as a
predictor $\widehat{X}(t,\tau)$ for $X(t+\tau)$ a function of the observed
values, $g(\\{X(u),u\leq t\\})$, which is good ”on the average”. So, as an
optimality criterion for our predictor we take the $L^{2}$-distance, that is,
the mean squared error, and consider only the linear predictors. With these
restrictions, the infinite linear prediction problem can be stated as follows.

The infinite linear prediction problem. Given a ’parameter’ of the process
$X(u)$ (e.g., the covariance function $r(t)$ or the spectral function
$F(\lambda)$), the entire history $H_{-\infty}^{t}(X)$ of $X(u)$, and a number
$\tau\in\mathbb{N}$, find a random variable $\widehat{X}(t,\tau)$ such that

  * a)

$\widehat{X}(t,\tau)$ is linear, that is, $\widehat{X}(t,\tau)\in
H_{-\infty}^{t}(X)$,

  * b)

$\widehat{X}(t,\tau)$ is mean-square optimal (best) among all elements $Y\in
H_{-\infty}^{t}(X)$, that is, $\widehat{X}(t,\tau)$ minimizes the mean-squared
error $||X(t+\tau)-Y||^{2}_{L^{2}(\Omega)}:$

$||X(t+\tau)-\widehat{X}(t,\tau)||^{2}_{L^{2}(\Omega)}=\min_{Y\in
H_{-\infty}^{t}(X)}||X(t+\tau)-Y||^{2}_{L^{2}(\Omega)}.$ (2.17)

The solution - the random variable $\widehat{X}(t,\tau)$ satisfying a) and b),
is called the best linear $\tau$-step ahead predictor for an element
$X(t+\tau)\in H(X)$. The quantity

$\sigma^{2}(\tau):=||X(t+\tau)-\widehat{X}(t,\tau)||^{2}_{L^{2}(\Omega)}=||X(t+\tau)||^{2}_{L^{2}(\Omega)}-||\widehat{X}(t,\tau)||^{2}_{L^{2}(\Omega)},$
(2.18)

which is independent of $t$, is called the prediction error (variance).

The advantage of the Hilbert space setting now becomes apparent. Namely, by
the projection theorem in Hilbert spaces (see, e.g., Pourahmadi [54], p. 312),
such a predictor $\widehat{X}(t,\tau)$ exists, is unique, and is given by

$\widehat{X}(t,\tau)=P_{t}X(t+\tau),$ (2.19)

where $P_{t}:=P_{-\infty}^{t}$ is the orthogonal projection operator in $H(X)$
onto $H_{-\infty}^{t}(X)$.

######  Remark 2.2.

The reason for restricting attention to linear predictors is that the best
linear predictor $\widehat{X}(t,\tau)$, in this case, depends only on
knowledge of the covariance function $r(t)$ or the spectral function
$F(\lambda)$. The prediction problem becomes much more difficult when
nonlinear predictors are allowed (see, e.g., Hannan [33], Koopmans [47]).

####  2.4.3 Deterministic (singular) and nondeterministic processes.
Characterizations

From prediction point of view it is natural to distinguish the class of
processes for which we have error-free prediction, that is, $\sigma(\tau)=0$
for all $\tau\geq 1$, or equivalently, $\widehat{X}(t,\tau)=X(t+\tau)$ for all
$t\in\mathbb{Z}$ and $\tau\geq 1$. In this case, the prediction is called
perfect. It is clear that a process $X(t)$ possessing perfect prediction
represents a singular case of extremely strong dependence between the random
variables forming the process. Such a process $X(t)$ is called deterministic
or singular. From the physical point of view, singular processes are
exceptional. From application point of view, of considerable interest is the
class of processes for which we have $\sigma(\tau)>0$ for all $\tau\geq 1$. In
this case the prediction is called imperfect, and the process $X(t)$ is called
nondeterministic.

Observe that the time-domain $H(X)$ of any non-degenerate stationary process
$\\{X(t)$, $t\in\mathbb{Z}\\}$ can be represented as the orthogonal sum
$H(X)=H_{1}(X)\oplus H_{-\infty}(X),$ where $H_{-\infty}(X)$ is the remote
past of $X(t)$ defined by (2.15), and $H_{1}(X)$ is the orthogonal complement
of $H_{-\infty}(X)$. So, we can give the following geometric definition of the
deterministic (singular), nondeterministic and purely nondeterministic
(regular) processes.

######  Definition 2.2.

A stationary process $\\{X(t)$, $t\in\mathbb{Z}\\}$ is called

  * •

deterministic or singular if $H_{-\infty}(X)=H(X),$ that is,
$H_{-\infty}^{t}(X)=H_{-\infty}^{s}(X)$ for all $t,s\in\mathbb{Z}$,

  * •

nondeterministic if $H_{-\infty}(X)$ is a proper subspace of $H(X)$, that is,
$H_{-\infty}(X)\subset H(X)$,

  * •

purely nondeterministic (PND) or regular if $H_{-\infty}(X)=\\{0\\}$, that is,
the remote past $H_{-\infty}(X)$ of $X(t)$ is the trivial subspace, consisting
of the singleton zero.

The next theorem contains a characterization of deterministic and purely
nondeterministic processes in terms of prediction error.

######  Theorem 2.3.

A stationary process $\\{X(t)$, $t\in\mathbb{Z}\\}$ is

  * (a)

deterministic if and only if $\sigma(\tau_{0})=0$ for some $\tau_{0}\geq 1$,
$\tau_{0}\in\mathbb{N}$ (then $\sigma(\tau)=0$ for all $\tau\in\mathbb{N}$).

  * (b)

purely nondeterministic if and only if
$\lim_{\tau\to\infty}\sigma^{2}(\tau)=E|X(t)|^{2}=r(0)(=1).$

######  Remark 2.3.

Every purely nondeterministic process $X(t)$ is nondeterministic, but the
converse is generally not true. An example of such process provides the
process $X(t)=\varepsilon(t)+\xi$, where
$\\{\varepsilon(t),\,t\in\mathbb{Z}\\}$ is a ${\rm WN}(0,\sigma^{2})$ (a
centered white noise with variance $\sigma^{2}$), and $\xi$ is a random
variable such that $Var(\xi)=\sigma^{2}_{0}$ and $(\varepsilon(t),\xi)=0$ for
all $t\in\mathbb{Z}$ (see Pourahmadi [54], p. 163). Observe also that the only
process $X(t)$ that is both deterministic and purely nondeterministic is the
degenerate process. Assuming that $X(t)$ is a non-degenerate process, we
exclude this trivial case.

The next result, known as Wold’s decomposition theorem, provides a key step
for solution of the infinite prediction problem in the time-domain setting,
and essentially says that any stationary process can be represented in the
form of a sum of two orthogonal stationary components, one of which is
perfectly predictable (singular component), while for the other (regular
component) an explicit formula for the predictor can be obtained.

######  Theorem 2.4 (Wold’s decomposition).

Every centered non-degenerate discrete-time stationary process $X(t)$ admits a
unique decomposition:

$X(t)=X_{S}(t)+X_{R}(t),$

where

  * (a)

the processes $X_{R}(t)$ and $X_{S}(t)$ are stationary, centered, mutually
uncorrelated (orthogonal), and completely subordinated to $X(t)$, that is,
$H_{-\infty}^{t}(X_{R})\subseteq H_{-\infty}^{t}(X)$ and
$H_{-\infty}^{t}(X_{S})\subseteq H_{-\infty}^{t}(X)$ for all $t\in\mathbb{Z}.$

  * (b)

the process $X_{S}(t)$ is deterministic (singular),

  * (c)

the process $X_{R}(t)$ is purely nondeterministic (regular) and has the
infinite moving-average representation:

$X_{R}(t)=\sum_{k=0}^{\infty}a_{k}\varepsilon_{0}(t-k),\quad\sum_{k=0}^{\infty}|a_{k}|^{2}<\infty,$
(2.20)

where $\varepsilon_{0}(t)$ is an innovation of $X_{R}(t),$ that is,
$\varepsilon_{0}(t)$ is a standard white-noise process, such that
$H_{-\infty}^{t}(X_{R})=H_{-\infty}^{t}(\varepsilon_{0})$ for all
$t\in\mathbb{Z}$.

The next result describes the asymptotic behavior of the prediction error
$\sigma_{n}^{2}(\mu)$ for a stationary process $X(t)$ with spectral measure
$\mu$ of the form (2.3) and gives spectral characterizations of deterministic,
nondeterministic and purely nondeterministic processes (see, e.g., Grenander
and Szegő [32], p. 44, and Ibragimov and Rozanov [40], p. 35-36).

######  Theorem 2.5.

Let $X(t)$ be a non-degenerate stationary process with spectral measure $\mu$
of the form (2.3). The following assertions hold.

  * (a)

(Kolmogorov-Szegő’s Theorem). The following relations hold.

$\displaystyle\lim_{n\to\infty}\sigma_{n}^{2}(\mu)=\lim_{n\to\infty}\sigma_{n}^{2}(f)=\sigma^{2}(f)=2\pi
G(f),$ (2.21)

where $G(f)$ is the geometric mean of $f$, namely

$G(f):=\left\\{\begin{array}[]{ll}\exp\left\\{\frac{1}{2\pi}\int_{-\pi}^{\pi}\ln
f(\lambda)\,d\lambda\right\\}&\mbox{if \, $\ln f\in{L}^{1}(\Lambda)$}\\\
0,&\mbox{ otherwise,}\end{array}\right.$ (2.22)

  * (b)

$H_{-\infty}^{0}(\mu_{s})=H(\mu_{s})\Leftrightarrow\sigma^{2}(\mu)=0\Leftrightarrow
X(t)\,\,is\,\,deterministic,$

  * (c)

(Kolmogorov-Szegő’s alternative). Either

$H_{-\infty}^{0}(\mu_{a})=H(\mu_{a})\Leftrightarrow\int_{-\pi}^{\pi}\ln
f(\lambda)\,d\lambda=-\infty\Leftrightarrow\sigma^{2}(f)=0\Leftrightarrow
X(t)\,\,is\,\,deterministic,$

or else

$H_{-\infty}^{0}(\mu_{a})\neq H(\mu_{a})\Leftrightarrow\int_{-\pi}^{\pi}\ln
f(\lambda)\,d\lambda>-\infty\Leftrightarrow\sigma^{2}(f)>0\Leftrightarrow
X(t)\,\,is\,\,nondeterministic.$

  * (d)

The process $X(t)$ is regular (PND) if and only if it is nondeterministic and
$\mu_{s}\equiv 0$.

######  Remark 2.4.

The second equality in (2.21) was proved by Szegő in 1920, while the first
equality was proved by Kolmogorov in 1941 (see, e.g., Hoffman [36], p. 49). It
is remarkable that (2.21) is independent of the singular part $\mu_{s}$.

The condition $\ln f\in{L}^{1}(\Lambda)$ in (2.22) is equivalent to the Szegő
condition:

$\int_{-\pi}^{\pi}\ln f(\lambda)\,d\lambda>-\infty$ (2.23)

(this equivalence follows because $\ln f(\lambda)\leq f(\lambda)$). The Szegő
condition (2.23) is also called the non-determinism condition.

In this paper we consider the class of deterministic processes with absolutely
continuous spectra.

We will say that the spectral density $f(\lambda)$ has a very high order of
contact with zero at a point $\lambda_{0}$ if $f(\lambda)$ is positive
everywhere except for the point $\lambda_{0}$, due to which the Szegő
condition (2.23) is violated. Observe that the Szegő condition is related to
the character of the singularities (zeroes and poles) of the spectral density
$f$, and does not depend on the differential properties of $f$. For example,
for any $a>0$, the function
$\widehat{f}_{a}(\lambda)=\exp\\{-|\lambda|^{-a}\\}$ is infinitely
differentiable. In addition, for $a<1$ Szegő’s condition is satisfied, and
hence the corresponding process $X(t)$ is nondeterministic, while for $a\geq
1$ Szegő’s condition is violated, and $X(t)$ is deterministic (see, e.g.,
Pourahmadi [54], p.68, Rakhmanov [56]). Thus, according to the above
definition, for $a\geq 1$ this function has a very high order of contact with
zero at the point $\lambda=0$.

##  3 Formulas for the finite prediction error $\sigma_{n}^{2}(f)$ and some
properties

In this section, we provide various formulas for the prediction error
$\sigma_{n}^{2}(f)$ in terms of orthogonal polynomials and their parameters
(Verblunsky’s coefficients), Toeplitz determinants, state Szegő’s,
Verblunsky’s and Rakhmanov’s theorems, and list a number of properties of
$\sigma_{n}^{2}(f)$.

###  3.1 Formulas for the prediction error $\sigma_{n}^{2}(f)$

We present here formulas for the finite prediction error $\sigma_{n}^{2}(f)$
and state some preliminary results, which will be used in the sequel.

Suppose we have observed the values $X(-n),\ldots,X(-1)$ of a centered, real-
valued stationary process $X(t)$ with spectral measure $\mu$ of the form
(2.3). The one-step ahead linear prediction problem in predicting a random
variable $X(0)$ based on the observed values $X(-n),\ldots,X(-1)$ involves
finding constants $\widehat{c}_{k}:=\widehat{c}_{k,n}$, $k=1,2,\ldots,n$, that
minimize the one-step ahead prediction error:

$\displaystyle\sigma_{n}^{2}(\mu):=\min_{\\{c_{k}\\}}\rm
I\\!E\left|X(0)-\sum_{k=1}^{n}c_{k}X(-k)\right|^{2}=\rm
I\\!E\left|X(0)-\sum_{k=1}^{n}\widehat{c}_{k}X(-k)\right|^{2}.$ (3.1)

Using Kolmogorov’s isometric isomorphism $V:\,X(t)\leftrightarrow
e^{it\lambda}$, in view of (3.1), for the prediction error
$\sigma_{n}^{2}(\mu)$ we can write

$\displaystyle\sigma_{n}^{2}(\mu)=\min_{\\{c_{k}\\}}\left\|1-\sum_{k=1}^{n}c_{k}e^{-ik\lambda}\right\|^{2}_{2,\mu}=\min_{\\{q_{n}\in\mathcal{Q}_{n}\\}}\left\|q_{n}\right\|^{2}_{2,\mu},$
(3.2)

where $||\cdot||_{2,\mu}$ is the norm in $L^{2}(\mathbb{T},\mu)$, and

$\mathcal{Q}_{n}:=\left\\{q_{n}:q_{n}(z)=z^{n}+c_{1}z^{n-1}+\cdots
c_{n}\right\\}$ (3.3)

is the class of monic polynomials (i.e. with $c_{0}=1$) of degree $n$. Thus,
the problem of finding $\sigma_{n}^{2}(\mu)$ becomes to the problem of finding
the solution of the minimum problem (3.2)-(3.3).

The polynomial $p_{n}(z):=p_{n}(z,\mu)$ which solves the minimum problem
(3.2)-(3.3) is called the optimal polynomial for $\mu$ in the class
$\mathcal{Q}_{n}$. This minimum problem was solved by G. Szegő by showing that
the optimal polynomial $p_{n}(z)$ exists, is unique and can be expressed in
terms of orthogonal polynomials on the unit circle with respect to the measure
$\mu$ (see Theorem 3.2 below).

To state Szegő’s solution of the minimum problem (3.2)-(3.3), we first recall
some facts from the theory of orthogonal polynomials on the unit circle
(OPUC).

The system of orthogonal polynomials on the unit circle associated with the
measure $\mu$:

$\\{\varphi_{n}(z)=\varphi_{n}(z;f),\quad z=e^{i\lambda},\quad
n\in\mathbb{Z}_{+}\\}$

is uniquely determined by the following two conditions:

  * (i)

$\varphi_{n}(z)=\kappa_{n}z^{n}+\cdots+l_{n}$ is a polynomial of degree $n$,
in which the leading coefficient $\kappa_{n}$ is positive;

  * (ii)

$(\varphi_{k},\varphi_{j})_{\mu}=\delta_{kj}$ for arbitrary
$k,j\in\mathbb{Z}_{+}$, where $\delta_{kj}$ is the Kronecker delta.

Define the monic ($p_{n}(z)$) and the reciprocal ($p^{*}_{n}(z)$) polynomials
(see, e.g., Simon [66], p. 2):

$\displaystyle
p_{n}(z):=p_{n}(z,\mu)=\kappa^{-1}_{n}\varphi_{n}(z)=z^{n}+\cdots+l_{n}\kappa_{n}^{-1},$
(3.4) $\displaystyle
p^{*}_{n}(z):=p^{*}_{n}(z,\mu)=z^{n}\overline{p_{n}(1/\overline{z})}=\overline{l}_{n}\kappa_{n}^{-1}z^{n}+\cdots+1.$
(3.5)

We have

$||p_{n}||_{2,\mu}=||p^{*}_{n}||_{2,\mu}=\kappa_{n}^{-2}.$ (3.6)

The polynomials $p_{n}(z)$ and $p^{*}_{n}(z)$ satisfy Szegő’s recursion
relation (see Simon [66], p. 56):

$p_{n+1}(z)=zp_{n}(z)-\overline{v}_{n+1}p^{*}_{n}(z),\quad n\in\mathbb{Z}_{+}$
(3.7)

where

$v_{n+1}=-\overline{p_{n+1}(0)}=\overline{l}_{n+1}\kappa_{n+1}^{-1},\quad|v_{n+1}|<1.$
(3.8)

In view of (3.7) we have (see Simon [66], p. 56)

$||p_{n}||_{2,\mu}^{2}=(1-|v_{n}|^{2})||p_{n-1}||_{2,\mu}^{2}=\prod_{j=1}^{n}(1-|v_{j}|^{2}),\quad
n\in\mathbb{N}.$ (3.9)

From (3.6) and (3.9) we obtain

$\kappa_{n}^{2}\kappa_{n+1}^{-2}=1-|v_{n+1}|^{2}.$ (3.10)

The parameters $v_{n}:=v_{n}(\mu)$ ($n\in\mathbb{N}$), which play an important
role in the theory of OPUC, are called Verblunsky’s coefficients (also known
as the Szegő, Schur, and canonical moments; see Simon [66], Sect. 1.1, and
Dette and Studden [17], Sect. 9.4).

Note. The term ”Verblunsky coefficient” is from Simon [66]. Observe that we
write $v_{n+1}$ for Simon’s $\alpha_{n}$, and so one has $n\in\mathbb{N}$ for
Simon’s $n\in\mathbb{Z}_{+}$. Our notational convention is already established
in the time-series literature and is more convenient in our context of the
PACF (defined below), where $n\in\mathbb{N}$ (see Bingham [11], Brockwell and
Davis [13], Sec. 5.2, Inoue [42], Pourahmadi [54], Sec. 7.3).

The following result shows that Verblunsky’s coefficients provide a convenient
way for the parametrization of probability measures on the unit circle
$\mathbb{T}$ (see, e.g., Verblunsky [73, 74], Ramsey [57], Simon [66], p. 2).

######  Theorem 3.1 (Verblunsky [73]).

Let $\mathbb{D}^{\infty}:=\times_{k=0}^{\infty}\mathbb{D}$ be the set of
complex sequences $v:=(v_{n},n\in\mathbb{N})$ with $v_{n}\in\mathbb{D}$. The
map $\mathcal{S}:\mu\longmapsto v$ is a bijection between the set of
nontrivial probability measures $\\{\mu\\}$ on $\mathbb{T}$ and
$\mathbb{D}^{\infty}$.

This result was established by Verblunsky [73] in 1935, in connection with
OPUC. It was re-discovered by Ramsey [57] in 1974, in connection with
parametrization of time-series models.

Partial autocorrelation function(PACF). For a stationary process $X(t)$ with a
non-trivial spectral measure $\mu$ the partial autocorrelation function (PACF)
of $X(t)$, denoted by $\pi_{n}=\pi_{n}(\mu)$ ($n\in\mathbb{N}$), is defined to
be the correlation coefficient between the forward and backward residuals in
the linear prediction of the variables $X(n)$ and $X(0)$ on the basis of the
intermediate observations $X(1),\ldots,X(n-1)$, that is,

$\pi_{n}:={\rm corr}(X(n)-\widehat{X}(n),X(0)-\widehat{X}(0)).$

It turns out that the Verblunsky coefficients $v_{n}$ and the PACF $\pi_{n}$
coincide, that is, $v_{n}=\pi_{n}$ for all $n\in\mathbb{N}$ (see Dette and
Studden [17], Sect. 9.6). Thus, the Verblunsky sequence
$v:=(v_{n},n\in\mathbb{N})$ provides a link between OPUC and time-series
analysis, and, in view of the equality $v_{n}=\pi_{n}$, the Verblunsky
bijection gives a simple and unconstrained parametrization of stationary
processes, in contrast to using the covariance function, which has to be
positive-definite.

The next result by Szegő solves the minimum problem (3.2)-(3.3) (see, e.g.,
Grenander and Szegő [32], p. 38).

######  Theorem 3.2 (Szegő).

The unique solution of the minimum problem (3.2)-(3.3) is the monic polynomial
$p_{n}(\mu):=p_{n}(z,\mu)$ given by formula (3.4), and the minimum in (3.2) is
equal to $||p_{n}||_{2,\mu}=\kappa^{-2}_{n}$ (see (3.6)).

Thus, for the prediction error $\sigma_{n}^{2}(\mu)$ we have the following
formula:

$\displaystyle\sigma^{2}_{n}(\mu)=\min_{\\{q_{n}\in\mathcal{Q}_{n}\\}}\left\|q_{n}\right\|^{2}_{2,\mu}=\left\|p_{n}(\mu)\right\|^{2}_{2,\mu}=\kappa^{-2}_{n}.$
(3.11)

######  Remark 3.1.

Define

$\mathcal{Q}^{*}_{n}:=\left\\{q_{n}:q_{n}(z)=c_{0}z^{n}+c_{1}z^{n-1}+\cdots
c_{n},\,c_{n}=1\right\\},$ (3.12)

and observe that the classes of polynomials $\mathcal{Q}_{n}$ and
$\mathcal{Q}^{*}_{n}$ defined in (3.3) and (3.12), respectively, differ by
normalization: in (3.12) we have $c_{n}=1$, while in (3.3) we have $c_{0}=1$.
Also, the optimal polynomial for $\mu$ in the class $\mathcal{Q}^{*}_{n}$ is
the reciprocal polynomial $p^{*}_{n}(z)$ (see (3.5)). Taking into account
(3.6), we have the following formula for the prediction error
$\sigma^{2}_{n}(\mu)$ in terms of the optimal polynomial $p^{*}_{n}(z)$:

$\displaystyle\sigma^{2}_{n}(\mu)=\min_{\\{q_{n}\in\mathcal{Q}^{*}_{n}\\}}\left\|q_{n}\right\|^{2}_{2,\mu}=\left\|p^{*}_{n}(\mu)\right\|^{2}_{2,\mu}.$
(3.13)

######  Remark 3.2.

Denote by $D_{n}=D_{n}(\mu):=\det[r(t-s),\,t,s=0,1,\ldots n]$ the $n^{th}$
Toeplitz determinant generated by the measure $\mu$, where $r(t)$ is the
covariance function given by (2.1). Taking into account that
$\kappa^{2}_{n}={D_{n-1}/D_{n}}$ (see, e.g., Grenander and Szegő [32], p. 38),
in view of (3.11) we obtain the following formula for the prediction error
$\sigma_{n}^{2}(\mu)$ in terms of $D_{n}(\mu)$:

$\displaystyle\sigma^{2}_{n}(\mu)=\frac{D_{n}(\mu)}{D_{n-1}(\mu)}.$ (3.14)

######  Remark 3.3.

In view of (3.11), the formulas (3.9) and (3.10) can be written as follows

$\sigma^{2}_{n}(\mu)=\prod_{j=1}^{n}(1-|v_{j}|^{2})\quad{\rm
and}\quad\frac{\sigma^{2}_{n+1}(\mu)}{\sigma^{2}_{n}(\mu)}=1-|v_{n}|^{2}.$
(3.15)

From the second formula in (3.15), it follows that the convergence of the
sequences $|v_{n}|$ and ${\sigma_{n+1}(\mu)}/{\sigma_{n}(\mu)}$ are
equivalent, and, the greater the limiting value of $|v_{n}|$, the faster the
rate of decrease of $\sigma_{n}(\mu)$.

For a general measure $\mu$ of the form (2.3) the asymptotic relation

$\lim_{n\to\infty}v_{n}(\mu)=0$ (3.16)

is of special interest in the theory of OPUC. In this respect the following
question arises naturally: what is the ’minimal’ sufficient condition on the
measure $\mu$ ensuring the relation (3.16)? The next result of Rakhmanov [55,
56] shows that for (3.16) (or equivalently, for
$\lim_{n\to\infty}{\sigma_{n+1}(\mu)}/{\sigma_{n}(\mu)}=1$) it is enough only
to have a.e. positiveness on $\mathbb{T}$ of the spectral density $f$ (see
also Babayan et al. [8] and Simon [66], p. 5).

######  Theorem 3.3 (Rakhmanov [55]).

Let the measure $\mu$ have the form (2.3) with $f>0$ a.e. on $\mathbb{T}$.
Then the asymptotic relation (3.16) is satisfied.

Note that the converse of Rakhmanov’s theorem, in general, is not true. A
partial converse of Rakhmanov’s theorem is stated in Theorem 5.7.

Bello and López [9] proved the following extension of Rakhmanov’s theorem: Let
$\Gamma_{\delta}$ be a closed arc of the unit circle of length $2\delta$
$(0<\delta\leq\pi)$, and let $\mu$ and $(v_{n})$ be as in Rakhmanov’s Theorem.
Assume that the measure $\mu$ is supported on the arc $\Gamma_{\delta}$ with
$f>0$ a.e. on $\Gamma_{\delta}$. Then
$\lim_{n\to\infty}|v_{n}|=\cos(\delta/2)$. The case $\delta=\pi$ corresponds
to Rakhmanov’s theorem.

###  3.2 Properties of the prediction error $\sigma_{n}^{2}(f)$

In what follows we assume that the spectral measure $\mu$ is absolutely
continuous with spectral density $f$, and instead of $\sigma^{2}_{n}(\mu)$,
$p_{n}(\mu)$ and $D_{n}(\mu)$ we use the notation $\sigma^{2}_{n}(f)$,
$p_{n}(f)$ and $D_{n}(f)$, respectively.

In the next proposition we list a number of properties of the prediction error
$\sigma_{n}^{2}(f)$. The proof can be found in Babayan and Ginovyan [6].

######  Proposition 3.1.

The prediction error $\sigma_{n}^{2}(f)$ possesses the following properties.

  * (a)

$\sigma_{n}^{2}(f)$ is a non-decreasing functional of $f$:
$\sigma_{n}^{2}(f_{1})\leq\sigma_{n}^{2}(f_{2})$ when $f_{1}(\lambda)\leq
f_{2}(\lambda)$, $\lambda\in[-\pi,\pi]$.

  * (b)

If $f(\lambda)=g(\lambda)$ almost everywhere on $[-\pi,\pi]$, then
$\sigma_{n}^{2}(f)=\sigma_{n}^{2}(g).$

  * (c)

For any positive constant $c$ we have $\sigma_{n}^{2}(cf)=c\sigma_{n}^{2}(f)$.

  * (d)

If $\bar{f}(\lambda)=f(\lambda-\lambda_{0})$, $\lambda_{0}\in[-\pi,\pi]$, then
$\sigma_{n}^{2}(\bar{f})=\sigma_{n}^{2}(f).$

##  4 Asymptotic behavior of the prediction error $\delta_{n}(f)$ for
nondeterministic processes

In this section we study the asymptotic behavior of the the relative
prediction error $\delta_{n}(f)=\sigma^{2}_{n}(f)-\sigma^{2}(f)$ for
nondeterministic processes, and review some important known results.

###  4.1 Asymptotic behavior of $\delta_{n}(f)$ for short-memory processes

Recall that a short memory processes is a second order stationary processes
possessing a spectral density $f$ which is bounded away from zero and
infinity.

####  4.1.1 Exponential rate of decrease of $\delta_{n}(f)$

The first result of this type goes back to the Grenander and Rosenblatt [30].
The next theorem was proved by Ibragimov [37].

######  Theorem 4.1 (Ibragimov [37]).

A necessary and sufficient condition for

$\delta_{n}(f)=O(q^{n}),\quad q=e^{-c},\quad c>0,\quad n\rightarrow\infty$
(4.1)

is that $f(\lambda)$ is a spectral density of a short-memory process, and
$1/{f(\lambda)}\in A_{c}$, where $A_{c}$ is the class of $2\pi$–periodic
continuous functions $\varphi(\lambda),\,\lambda\in\mathbb{R}$, admitting an
analytic continuation into the strip $z=\lambda+i\mu$,
$-\infty<\lambda<\infty$, $|\mu|\leq c$.

Observe that (4.1) will be true for all $c>0$ if and only if the analytic
continuation of $f(\lambda)$ is an entire function of $z=\lambda+i\mu$.

Thus, to have exponential rate of decrease to zero for $\delta_{n}(f)$ the
underlying model should be short-memory process with sufficiently smooth
spectral density.

####  4.1.2 Hyperbolic rate of decrease of $\delta_{n}(f)$

Here we are interested in estimates for $\delta_{n}(f)$ of type

$\displaystyle\delta_{n}(f)$ $\displaystyle=$ $\displaystyle
O(n^{-\gamma}),\quad\gamma>0,\quad n\rightarrow\infty.$ (4.2)
$\displaystyle\delta_{n}(f)$ $\displaystyle=$ $\displaystyle
o(n^{-\gamma}),\quad\gamma>0,\quad n\rightarrow\infty.$ (4.3)

Bounds of type (4.2) with $\gamma>1$ for different classes of spectral
densities were obtained by Baxter [2], Devinatz [18], Geronimus [24, 25],
Grenander and Rosenblatt [30], Grenander and Szegő [32], and others (see,
e.g., Ginovyan [26], and references therein). The most general result in this
direction has been obtained by Ibragimov [37]. To state Ibragimov’s theorem,
we first introduce Hölder classes of functions.

For a function $\varphi(\lambda)\in L^{p}(\mathbb{T})$, we define its
$L^{p}$-modulus of continuity by

$\displaystyle\omega_{p}(\varphi;\delta)=\sup_{0<|t|\leq\delta}||\varphi(\cdot+t)-\varphi(\cdot)||_{p},\quad\delta>0.$
(4.4)

Given numbers $0<\alpha<1$, $r\in\mathbb{Z}_{0}:=\\{0,1,2\ldots\\}$, and
$p\geq 1$, we put $\gamma:=r+\alpha$. A Hölder class of functions, denoted by
$H_{p}(\gamma)$, is defined to be the set of those functions
$\varphi(\lambda)\in L^{p}(\mathbb{T})$ that have $r$-th derivative
$\varphi^{(r)}(\lambda)$, such that $\varphi^{(r)}(\lambda)\in
L^{p}(\mathbb{T})$ and $\omega_{p}(\varphi^{(r)};\delta)=O(\delta^{\alpha})$
as $\delta\to 0$.

######  Theorem 4.2 (Ibragimov [37]).

A necessary and sufficient condition for

$\displaystyle\delta_{n}(f)=O(n^{-\gamma}),\quad\gamma=2(r+\alpha)>1;\,\,0<\alpha<1,\,r\in\mathbb{Z}_{0},\quad{\rm
as}\quad n\rightarrow\infty$ (4.5)

is that $f(\lambda)$ is a spectral density of a short-memory process belonging
to $H_{2}(\gamma)$.

Bounds of type (4.3) for short memory models have been obtained by Baxter [2],
Devinatz [18], Hirshman [35] and Golinskii [27]. For ’large’ values of
$\gamma$ ($\gamma>1$), Hirshman has obtained the following necessary and
sufficient condition for (4.3) (see Hirshman [35], p. 314).

######  Theorem 4.3 (Hirshman [35]).

If $X(t)$ is a short-memory process, then $\delta_{n}(f)=o(n^{-\gamma})$ with
$\gamma>1$ as $n\rightarrow\infty$ if and only if $\sum_{|t|\geq
n}{|r(t)|^{2}}=o(n^{-\gamma})$ as $n\rightarrow\infty$, where $r(t)$ is the
covariance function of $X(t)$.

The next theorem was proved by G. Baxter (see Baxter [2], Theorem 3.1).

######  Theorem 4.4 (Baxter [2]).

If $X(t)$ is a short-memory process with covariance function $r(t)$ satisfying
$\sum_{t=1}^{\infty}{t^{\gamma}|r(t)|}<\infty$, $\gamma>0$, then
$\delta_{n}(f)=o(n^{-2\gamma})$ as $n\rightarrow\infty$.

######  Remark 4.1.

It follows from Theorem 4.2 that if $\delta_{n}(f)=O(n^{-\gamma})$ with
$\gamma>1$, then the underlying model $X(t)$ is necessarily a short-memory
process. Moreover, as it was pointed out by Grenander and Rosenblatt [30]
(see, also, Devinatz [18], p. 118), if the model is not a short-memory
process, that is, the spectral density $f$ has zeros or is unbounded, then, in
general, we cannot expect $\delta_{n}(f)$ to go to zero faster than $1/n$ as
$n\to\infty$. This question we discuss in the next section.

###  4.2 Asymptotic behavior of $\delta_{n}(f)$ for anti-persistent and long-
memory processes

In this section we describe the rate of decrease of the relative prediction
error $\delta_{n}(f)$ to zero as $n\to\infty,$ in the case where the
underlying process $X(t)$ is nondeterministic and is anti-persistent or has
long-memory, that is, the spectral density $f(\lambda)$ of $X(t)$ has zeros or
is unbounded at a finite number of points, such that $\ln f(\lambda)\in
L^{1}(\mathbb{T})$. This case is of great interest because in many
applications the model is described by such type processes.

####  4.2.1 An example

We start with an example, which shows that the asymptotic behavior of the
prediction error $\delta_{n}(f)$ essentially depends on the dependence
(memory) structure of the underlying model $X(t)$ (see, e.g., Grenander and
Szegő [32], p. 191).

######  Example 4.1.

Consider a first-order moving-average $MA(1)$ process $X(t)$:

$X(t)=\varepsilon(t)-b\cdot\varepsilon(t-1),\quad\varepsilon(t)\sim
WN(0,\sigma^{2}_{\varepsilon}),\quad 0\leq b\leq 1.$

The spectral density is
$f(\lambda)=\frac{\sigma^{2}_{\varepsilon}}{2\pi}\cdot|1-be^{i\lambda}|^{2}$
(see formula (2.8)).

a) First assume that $X(t)$ has short-memory, that is, $0\leq b<1$. It is easy
to check that

$\delta_{n}(f)=\sigma_{n}^{2}(f)-\sigma^{2}(f)=\frac{b^{2n}(b^{2}-b^{4})}{1-b^{2n+2}}\sim
b^{2n}\quad{\rm as}\quad n\to\infty,$

showing that in this case $\delta_{n}(f)$ goes to zero with exponential rate.

b) Now let $b=1$. We have
$f(\lambda)=\frac{\sigma^{2}_{\varepsilon}}{2\pi}\cdot|1-e^{i\lambda}|^{2},$
that is, the process $X(t)$ is anti-persistent. In this case we have

$\delta_{n}(f)=\sigma_{n}^{2}(f)-\sigma^{2}(f)=\frac{1}{n+2}\sim\frac{1}{n}\quad{\rm
as}\quad n\to\infty,$

showing that $\delta_{n}(f)$ goes to zero fat precisely the rate $1/n$. The
slow rate is due to the presence of a zero of $f(\lambda)$ at $\lambda=0$ (see
Grenander and Szegő [32], p. 191).

It can be shown that for models with spectral densities of the form
$f(\lambda)=\frac{\sigma^{2}_{\varepsilon}}{2\pi}\cdot|q(e^{i\lambda})|^{2},$
where $q(e^{i\lambda})$ is a polynomial with at least one root on the unit
circle $\mathbb{T}$, we have $\delta_{n}(f)\sim\frac{1}{n}$ as $n\to\infty.$

####  4.2.2 The ARFIMA$(p,d,q)$ Model

As it was mentioned in Section 2.3 a well-known example of processes that
displays long memory or is anti-persistent is an ARFIMA$(p,d,q)$ process
$X(t)$ with spectral density $f_{X}(\lambda)$ given by (see (2.9)):

$f_{X}(\lambda)=|1-e^{-i\lambda}|^{-2d}f(\lambda),\quad d<1/2,$ (4.6)

where $f(\lambda)$ is the spectral density of an ARMA$(p,q)$ process, given by
(2.8). Recall that for $0<d<1/2$ the model $X(t)$ specified by spectral
density (4.6) displays long-memory, for $d<0$ it is anti-persistent, and for
$d=0$ it displays short-memory.

The following theorem was proved by A. Inoue (see Inoue [42], Theorem 4.3).

######  Theorem 4.5 (Inoue [42]).

Let $f_{X}(\lambda)$ have the form (4.6) with $0<d<1/2$, where $f(\lambda)$ is
the spectral density of an ARMA$(p,q)$ process. Then

$\delta_{n}(f)\sim\frac{d^{2}}{n}\quad{\rm as}\quad n\to\infty.$ (4.7)

######  Remark 4.2.

Note that for ARFIMA$(0,d,0)$ model the asymptotic relation (4.7) remains
valid for all $d<1/2$ ($d\neq 0$). In this case, for the Verblunsky
coefficients (parameters) $v_{n}$ we have $v_{n}=\frac{d}{n-d+1}$ (see
Golinskii [27], p. 703).

####  4.2.3 The Jacobian Model

Another well-known example of processes that displays long memory or is anti-
persistent is the Jacobian model. We say that a stationary process $X(t)$ is a
Jacobian process, and the corresponding model is a Jacobian model, if its
spectral density $f(\lambda)$ has the following form:

$f(\lambda)=f_{1}(\lambda)\prod_{k=1}^{m}|e^{i\lambda}-e^{i\lambda_{k}}|^{-2d_{k}},$
(4.8)

where $f_{1}(\lambda)$ is the spectral density of a short-memory process, the
points $\lambda_{k}\in[-\pi,\pi]$ are distinct, and $d_{k}\leq 1/2$,
$k=1,\ldots,m$.

The asymptotic behavior of $\delta_{n}(f)$ as $n\to\infty$ for Jacobian model
(4.8) has been considered in a number of papers (see, e.g., Golinskii [27],
Grenander and Rosenblatt [30], Ibragimov [37], Ibragimov and Solev [41].)

The following theorem was proved in Ibragimov and Solev [41].

######  Theorem 4.6 (Ibragimov and Solev [41]).

Let $f(\lambda)$ have the form (4.8), where $f_{1}(\lambda)$ is the spectral
density of a short-memory process, the points $\lambda_{k}\in[-\pi,\pi]$ are
distinct, and $d_{k}\leq 1/2$, $k=1,\ldots,m$. If the function
$f_{1}(\lambda)$ satisfies a Lipschitz condition with exponent $\alpha\geq
1/2$, then

$\delta_{n}(f)\sim{1}/n\quad{\rm as}\quad n\to\infty.$ (4.9)

##  5 Asymptotic behavior of the finite prediction error $\sigma_{n}^{2}(f)$
for deterministic (singular) processes

###  5.1 Background

The linear prediction problem has been studied most intensively for
nondeterministic processes, that is, in the case where the prediction error is
known to be positive $(\sigma^{2}(f)>0)$ (see Section 4).

In this section we focus on the less investigated case - deterministic
processes, that is, when $\sigma^{2}(f)=0$. This case is not only of
theoretical interest, but is also important from the point of view of
applications. For example, as pointed out by Rosenblatt [60] (see also Pierson
[51]), situations of this type arise in Neumann’s theoretical model of storm-
generated ocean waves. Such models are also of interest in meteorology (see,
e.g., Fortus [21]).

Only few works are devoted to the study of the speed of convergence of
$\delta_{n}(f)=\sigma^{2}_{n}(f)$ to zero as $n\to\infty$, that is, the
asymptotic behavior of the prediction error for deterministic processes. One
needs to go back to the classical work of M. Rosenblatt [60]. Using the
technique of orthogonal polynomials on the unit circle, M. Rosenblatt
investigated the asymptotic behavior of the prediction error
$\sigma^{2}_{n}(f)$ for deterministic processes in the following two cases:

  * (a)

the spectral density $f(\lambda)$ is continuous and positive on a segment of
$[-\pi,\pi]$ and zero elsewhere.

  * (b)

the spectral density $f(\lambda)$ has a very high order of contact with zero
at points $\lambda=0,\pm\pi$, and is strictly positive otherwise.

Later the problems (a) and (b) were studied by Babayan [3, 4], Babayan and
Ginovyan [5, 6, 7], and Babayan et al. [8] (see also Davisson [16] and Fortus
[21]), where some generalizations and extensions of Rosenblatt’s results have
been obtained.

We start by describing Rosenblatt’s results concerning the asymptotic behavior
of the prediction error $\sigma^{2}_{n}(f)$, obtained in Rosenblatt [60] for
the above stated cases (a) and (b).

###  5.2 Rosenblatt’s results about speed of convergence

For the case (a) above, M. Rosenblatt proved in [60] that the prediction error
$\sigma^{2}_{n}(f)$ decreases to zero exponentially as $n\to\infty$. More
precisely, M. Rosenblatt proved the following theorem.

######  Theorem 5.1 (Rosenblatt’s first theorem).

Let the spectral density $f$ of a discrete-time stationary process $X(t)$ be
positive and continuous on the segment $[\pi/2-\alpha,\pi/2+\alpha],$
$0<\alpha<\pi,$ and zero elsewhere. Then the prediction error
$\sigma^{2}_{n}(f)$ approaches zero exponentially as $n\to\infty$. More
precisely, the following asymptotic relation holds:

$\sigma^{2}_{n}(f)\simeq\left(\sin(\alpha/2)\right)^{2n+1}\quad{\rm as}\quad
n\to\infty.$ (5.1)

Thus, when the spectral density $f$ is continuous and vanishes on an entire
segment, then the prediction error $\sigma^{2}_{n}(f)$ approaches zero with a
sufficiently high speed, namely as a geometric progression with common ratio
$\sin^{2}(\alpha/2)<1$. Notice that (5.1) implies that

$\lim_{n\to\infty}\sqrt[n]{\sigma_{n}(f)}=\sin(\alpha/2).$ (5.2)

Concerning the case (b) above, for a specific deterministic process $X(t)$,
Rosenblatt proved in [60] that the prediction error $\sigma^{2}_{n}(f)$
decreases to zero like a power as $n\to\infty$. More precisely, the
deterministic process $X(t)$ considered in Rosenblatt [60] has the spectral
density

$f_{a}(\lambda):=\frac{e^{(2\lambda-\pi)\varphi(\lambda)}}{\cosh\left(\pi\varphi(\lambda)\right)},\quad
f_{a}(-\lambda)=f_{a}(\lambda),\quad 0\leq\lambda\leq\pi,$ (5.3)

where $\varphi(\lambda)=(a/2)\cot\lambda$ and $a$ is a positive parameter.

Using the technique of orthogonal polynomials on the unit circle and Szegő’s
results, Rosenblatt [60] proved the following theorem.

######  Theorem 5.2 (Rosenblatt’s second theorem).

Suppose that the process $X(t)$ has spectral density $f_{a}$ given by (5.3).
Then the following asymptotic relation for the prediction error
$\sigma^{2}_{n}(f_{a})$ holds:

$\sigma^{2}_{n}(f_{a})\sim\frac{\Gamma^{2}\left(({a+1)}/2\right)}{\pi
2^{2-a}}\ n^{-a}\quad{\rm as}\quad n\to\infty.$ (5.4)

Note that the function in (5.3) was first considered by Pollaczek [52], and
then by Szegő [70], as a weight-function of a class of orthogonal polynomials
that serve as illustrations for certain ’irregular’ phenomena in the theory of
orthogonal polynomials. For the function $f_{a}$ in (5.3), we have the
following asymptotic relation (for details see Szegő [70], Babayan and
Ginovyan [6], and Section 5.5.4):

$f_{a}(\lambda)\sim\left\\{\begin{array}[]{ll}2e^{a}\exp\left\\{-{a\pi}/{|\lambda|}\right\\}&\mbox{as
$\lambda\to 0$},\\\ 2\exp\left\\{-{a\pi}/{(\pi-|\lambda|)}\right\\}&\mbox{as
$\lambda\to\pm\pi$}.\end{array}\right.$ (5.5)

Thus, the function $f_{a}$ in (5.3) has a very high order of contact with zero
at points $\lambda=0,\pm\pi$, due to which the process with spectral density
$f_{a}$ is deterministic and the prediction error $\sigma^{2}_{n}(f_{a})$ in
(5.4) decreases to zero like a power as $n\to\infty$.

######  Remark 5.1.

In view of formulas in (3.15), under the conditions of Theorem 5.1, we have

$\lim_{n\to\infty}{\sigma^{2}_{n+1}(f)}/{\sigma^{2}_{n}(f)}=\sin^{2}(\alpha/2)\quad{\rm
and}\quad\lim_{n\to\infty}|v_{n}(f)|=\cos(\alpha/2).$

Similarly, under the conditions of Theorem 5.2, we have

$\lim_{n\to\infty}{\sigma_{n+1}(f_{a})}/{\sigma_{n}(f_{a})}=1\quad{\rm
and}\quad\lim_{n\to\infty}v_{n}(f_{a})=0,$

where $v_{n}(f)$ and $v_{n}(f_{a})$ are the Verblunsky coefficients
corresponding to functions $f$ and $f_{a}$, respectively.

In the papers Babayan [3, 4], Babayan and Ginovyan [5, 6, 7], and Babayan et
al. [8], the above stated Theorems 5.1 and 5.2 were extended to broader
classes of spectral densities.

Concerning Theorem 5.1, was described an extension of the asymptotic relation
(5.2) to the case of several arcs, without having to stipulate continuity of
the spectral density $f$.

As for the extension of Theorem 5.2, it was proved that if the spectral
density $f$ is such that the sequence $\sigma_{n}(f)$ is weakly varying (a
term defined in Section 7.2) and if, in addition, $g$ is a nonnegative
function that can have arbitrary power type singularities, then the sequences
$\\{\sigma_{n}(fg)\\}$ and $\\{\sigma_{n}(f)\\}$ have the same asymptotic
behavior as $n\to\infty$, up to some positive multiplicative factor. This
allows to derive the asymptotic behavior of $\sigma_{n}(fg)$ from that of
$\sigma_{n}(f)$.

Using this result, Rosenblatt’s Theorem 5.2 was extended in Babayan and
Ginovyan [6] and in Babayan et al. [8] to a class of spectral densities of the
form $f=f_{a}g$, where $f_{a}$ is as in (5.3) and $g$ is a nonnegative
function that can have arbitrary power type singularities (see Corollary 5.5
in Section 5.5.2).

###  5.3 Extensions of Rosenblatt’s first theorem

In this section we extend Rosenblatt’s first theorem (Theorem 5.1) to a
broader class of deterministic processes, possessing spectral densities. More
precisely, we extend the asymptotic relation (5.2) to the case of several
arcs, without having to stipulate continuity of the spectral density $f$.
Besides, we state necessary as well as sufficient conditions for the
exponential decay of the prediction error $\sigma_{n}(f)$. Also, we calculate
the transfinite diameters of some subsets of the unit circle, and thus, obtain
explicit asymptotic relations for $\sigma_{n}(f)$ similar to Rosenblatt’s
relation (5.2).

####  5.3.1 Extensions of Theorem 5.1

In what follows, by $E_{f}$ we denote the spectrum of the process $X(t)$, that
is,

$E_{f}:=\\{e^{i\lambda}:f(\lambda)>0\\}.$ (5.6)

Thus, the closure $\overline{E}_{f}$ of $E_{f}$ is the support of the spectral
density $f$.

The next theorem extends Rosenblatt’s first theorem (Theorem 5.1). More
precisely, the result that follows extends the asymptotic relation (5.2) to
the case of several arcs, without having to stipulate continuity of the
spectral density $f$.

######  Theorem 5.3 (Babayan et al. [8]).

Let the support $\overline{E}_{f}$ of the spectral density $f$ of the process
$X(t)$ consists of a finite number of closed arcs of the unit circle
$\mathbb{T}$, and let $f>0$ a.e. on $\overline{E}_{f}$. Then the sequence
$\sqrt[n]{\sigma_{n}(f)}$ converges, and

$\lim_{n\to\infty}\sqrt[n]{\sigma_{n}(f)}=\tau_{f},$ (5.7)

where $\tau_{f}:=\tau(\overline{E}_{f})$ is the transfinite diameter of
$\overline{E}_{f}$.

######  Remark 5.2.

In Theorem 5.1,
$\overline{E}_{f}=\\{e^{i\lambda}:\lambda\in[\pi/2-\alpha,\pi/2+\alpha]\\},$
which represents a closed arc of length $2\alpha$, and, according to
Proposition 7.2(d), we have
$\tau(\overline{E}_{f})=\sin(2\alpha/4)=\sin(\alpha/2)$. Thus, the asymptotic
relation (5.2) is a special case of (5.7).

We will need the following definition, which characterizes the rate of
variation of a sequence of non-negative numbers compared with a geometric
progression (see also Simon [66], p. 91).

######  Definition 5.1.

(a) A sequence $\\{a_{n}\geq 0,\,n\in\mathbb{N}\\}$ is called exponentially
neutral if

$\lim_{n\rightarrow\infty}\sqrt[n]{a_{n}}=1.$

(b) A sequence $\\{b_{n}\geq 0,\,n\in\mathbb{N}\\}$ is called exponentially
decreasing if

$\limsup_{n\rightarrow\infty}\sqrt[n]{b_{n}}<1.$

For instance, the sequence
$\\{a_{n}=n^{\alpha},\,\alpha\in\mathbb{R},\,n\in\mathbb{N}\\}$ is
exponentially neutral because
$\ln\sqrt[n]{n^{\alpha}}=({\alpha}/{n})\ln{n}\rightarrow 0$ as $n\to\infty$.
The geometric progression $\\{b_{n}=q^{n},\,0<q<1,\,n\in\mathbb{N}\\}$ is
exponentially decreasing because $\sqrt[n]{b_{n}}=q^{n/n}=q<1$. The sequence
$\\{b_{n}=n^{\alpha}q^{n},\,\alpha\in\mathbb{R},\,0<q<1,\,n\in\mathbb{N}\\}$
is also exponentially decreasing because
$\sqrt[n]{b_{n}}=n^{\alpha/n}q\rightarrow q<1.$ In fact, it can easily be
shown that a sequence $\\{c_{n}\geq 0,\,n\in\mathbb{N}\\}$ is exponentially
decreasing if and only if there exists a number $q$ ($0<q<1$) such that
$c_{n}=O(q^{n})$ as $n\to\infty$.

######  Remark 5.3.

It follows from relation (5.7) that the question of exponential decay of the
prediction error $\sigma_{n}(f)$ as $n\to\infty$ is determined solely by the
value of the transfinite diameter of the support $\overline{E}_{f}$ of the
spectral density $f$, and does not depend on the values of $f$ on
$\overline{E}_{f}$. Denoting $\gamma_{n}:=\sigma_{n}(f)/\tau_{f}^{n}$, from
(5.7) we infer that $\lim_{n\to\infty}\sqrt[n]{\gamma_{n}}=1$ and

$\sigma_{n}(f)=\tau_{f}^{n}\cdot\gamma_{n}.$ (5.8)

Thus, in the case where $\tau_{f}<1$, the prediction error $\sigma_{n}(f)$ is
decomposed into a product of two factors, one of which ($\tau_{f}^{n}$) is a
geometric progression, and the second ($\gamma_{n}$) is an exponentially
neutral sequence. Also, if $g$ is another spectral density satisfying the
conditions of Theorem 5.3, then in view of (5.8), we have

$\frac{\sigma_{n}(g)}{\sigma_{n}(f)}=\left(\frac{\tau_{g}}{\tau_{f}}\right)^{n}\cdot\gamma^{\prime}_{n},$

where $\gamma^{\prime}_{n}$ is an exponentially neutral sequence. It is worth
noting that the last relation does not depend on the structures of the
supports $\overline{E}_{f}$ and $\overline{E}_{g}$ (viz., the number and the
lengths of arcs constituting these sets, as well as, their location on the
unit circle $\mathbb{T}$).

The following result provides a sufficient condition for the exponential decay
of the prediction error $\sigma_{n}(f)$.

######  Theorem 5.4 (Babayan et al. [8]).

If the spectral density $f$ of the process $X(t)$ vanishes on an arc, then the
prediction error $\sigma_{n}(f)$ decreases to zero exponentially. More
precisely, if $f$ vanishes on an arc $\Gamma_{\delta}\subset\mathbb{T}$ of
length $2\delta$ $(0<\delta<\pi)$, then

$\limsup_{n\rightarrow\infty}\sqrt[n]{\sigma_{n}(f)}\leq\cos({\delta}/{2})<1.$
(5.9)

The next result gives a necessary condition for the exponential decay of
$\sigma_{n}(f)$.

######  Theorem 5.5 (Babayan et al. [8]).

A necessary condition for the prediction error $\sigma_{n}(f)$ to tend to zero
exponentially is that the spectral density $f$ should vanish on a set of
positive Lebesgue measure.

######  Remark 5.4.

Theorem 5.5 shows that if the spectral density $f$ is almost everywhere
positive, then it is impossible to obtain exponential decay of the prediction
error $\sigma_{n}(f)$, no matter how high the orders of the zeros of $f$.

In view of (3.15), as a consequence of Theorem 5.3, we obtain the following
result.

######  Theorem 5.6 (Babayan et al. [8]).

Let the support $\overline{E}_{f}$ and the spectral density $f$ satisfy the
conditions of Theorem 5.3. If the sequence of Verblunsky coefficients
$v_{n}(f)$ converges in modulus, then

$\lim_{n\to\infty}|v_{n}(f)|=\sqrt{1-\tau_{f}^{2}}.$ (5.10)

######  Remark 5.5.

It is well-known that for an arbitrary sequence of positive numbers $a_{n}$
the convergence $a_{n+1}/a_{n}\to a$ implies the convergence
$\sqrt[n]{a_{n}}\to a$. The converse, in general, is not true, that is, the
sequence $\sqrt[n]{a_{n}}$ can be convergent, while $a_{n+1}/a_{n}$ divergent.
Indeed, for the sequence $a_{n}$:

$a_{n}:=\left\\{\begin{array}[]{ll}2^{-3k}&\mbox{if \, $n=2k-1$}\\\
2^{-(3k+1)}&\mbox{if \, $n=2k$},\end{array}\quad k\in\mathbb{N},\right.$

we have $\lim_{n\to\infty}\sqrt[n]{a_{n}}=2^{-3/2}$, while the limit
$\lim_{n\to\infty}a_{n+1}/a_{n}$ does not exists.

In the context of the considered sequences, $|v_{n}(f)|$ and
$\sqrt[n]{\sigma_{n}(f)}$, in view of (3.15), we can assert that the
convergence of $|v_{n}(f)|$ (or equivalently the convergence of
${\sigma_{n+1}(f)}/{\sigma_{n}(f)}$) implies the convergence of
$\sqrt[n]{\sigma_{n}(f)}$, but not the converse. Hence, the condition of
convergence (in modulus) of Verblunsky’s sequence in Theorem 5.6 is essential.

As a consequence of Theorem 5.3 we obtain the following result (see Geronimus
[22]), which is a partial converse of Rakhmanov’s theorem:

######  Theorem 5.7.

If the sequence $\sigma_{n}(f)$ satisfies the following condition:

$\limsup_{n\rightarrow\infty}\sqrt[n]{\sigma_{n}(f)}=1$ (5.11)

(in particular, if $\lim_{n\to\infty}v_{n}(f)=0$), then
$\overline{E}_{f}=\mathbb{T}$, i.e. the spectrum of the process is dense in
$\mathbb{T}$.

The next theorem extends Theorem 5.3.

######  Theorem 5.8 (Babayan [3]).

Let $E_{f}$ be the spectrum of a stationary process $X(t)$ possessing a
spectral density $f(\lambda)$, that is, $E_{f}=\\{\lambda:f(\lambda)>0\\}$,
and let $\tau(E_{f})$, $\tau_{*}(E_{f})$ and $\tau^{*}(E_{f})$ be the
transfinite diameter and the inner and the outer transfinite diameters of
$E_{f}$, respectively (for definition of $\tau_{*}(E_{f})$ and
$\tau^{*}(E_{f})$ see formula (7.8)). Then the following assertions hold.

  * (a)

The following inequalities hold:

$\displaystyle\limsup_{n\to\infty}(\sigma_{n}(f))^{1/n}\leq\tau^{*}(E_{f}),$
(5.12)
$\displaystyle\liminf_{n\to\infty}(\sigma_{n}(f))^{1/n}\geq\tau_{*}(E_{f}).$
(5.13)

  * (b)

If the spectrum $E_{f}$ consists of a countable number of open arcs of the
unit circle $\mathbb{T}$ and is $\tau$-measurable, that is,
$\tau_{*}(E_{f})=\tau^{*}(E_{f})=\tau(E_{f})$, then

$\lim_{n\rightarrow\infty}\sqrt[n]{\sigma_{n}(f)}=\tau(E_{f}).$ (5.14)

As an immediate corollary of Theorem 5.8 we have the following result.

######  Corollary 5.1.

A sufficient condition for the prediction error $\sigma_{n}(f)$ of a
deterministic stationary sequence to decrease to zero at least exponentially
as $n\to\infty$, that is, $\sigma_{n}(f)=O(e^{-bn})$ for some $b>0$, is that
the outer transfinite diameter of the spectrum $E_{f}$ should be less than
unity.

####  5.3.2 Examples. Calculation of transfinite diameters of some special
sets

Motivated by Theorems 5.1 and 5.3 and Remark 5.2, the following question
arises naturally: calculate the transfinite diameter $\tau(\overline{E}_{f})$
of the set $\overline{E}_{f}$ consisting of several closed arcs of the unit
circle $\mathbb{T}$, and thus, obtain an explicit asymptotic relation for the
prediction error $\sigma_{n}(f)$ similar to Rosenblatt’s relation (5.2). The
calculation of the transfinite diameter (and hence, the capacity and the
Chebyshev constant) is a challenging problem (for details see Section 7.1),
and in only very few cases has the transfinite diameter been exactly
calculated (see, e.g., Landkof [49], p. 172-173, Ransford [58], p.135, and
Proposition 7.2). One such example provides Theorem 5.1, in which case the
transfinite diameter of the set
$\overline{E}_{f}:=\\{e^{i\lambda}:\lambda\in[\pi/2-\alpha,\pi/2+\alpha]\\}$
is $\sin(\alpha/2)$. Below we give some other examples, where we can
explicitly calculate the Chebyshev constant (and hence the transfinite
diameter and the capacity) by using some properties of the transfinite
diameter, stated in Proposition 7.2, and results due to Fekete [20] and
Robinson [59] concerning the relation between the transfinite diameters of
related sets (see Propositions 7.3 and 7.5).

The examples given below show that Fekete’s formula (7.9) and Robinson’s
formula (7.12) give a simple way to calculate the transfinite diameters of
some subsets of the unit circle, based only on the formula of the transfinite
diameter of a line segment (see Proposition 7.2(e)).

We will use the following notation: given $0<\beta<2\pi$ and
$z_{0}=e^{i\theta_{0}}$, $\theta_{0}\in[-\pi,\pi)$, we denote by
$\Gamma_{\beta}(\theta_{0})$ an arc of the unit circle of length $\beta$ which
is symmetric with respect to the point $z_{0}=e^{i\theta_{0}}$:

$\Gamma_{\beta}(\theta_{0}):=\\{e^{i\theta}:\,|\theta-\theta_{0}|\leq\beta/2\\}=\\{e^{i\theta}:\,\theta\in[\theta_{0}-\beta/2,\theta_{0}+\beta/2]\\}.$
(5.15)

######  Example 5.1.

Let $\Gamma_{2\alpha}:=\Gamma_{2\alpha}(0)$. Then the projection
$\Gamma_{2\alpha}^{x}$ of $\Gamma_{2\alpha}$ onto the real axis is the segment
$[\cos\alpha,1]$ (see Figure 1a)), and by Proposition 7.2(e) for the
transfinite diameter $\tau(\Gamma_{2\alpha}^{x})$ we have

$\tau(\Gamma_{2\alpha}^{x})=\frac{1-\cos\alpha}{4}=\frac{\sin^{2}(\alpha/2)}{2}.$

Hence, according to Robinson’s formula (7.12), we obtain

$\tau(\Gamma_{2\alpha})=[2\tau(\Gamma_{2\alpha}^{x})]^{1/2}=\left[2\frac{\sin^{2}(\alpha/2)}{2}\right]^{1/2}=\sin(\alpha/2).$
(5.16)

Taking into account that the transfinite diameter is invariant with respect to
rotation (see Proposition 7.2(b)), from (5.16) for any
$\theta_{0}\in[-\pi,\pi)$ we have

$\tau(\Gamma_{2\alpha}(\theta_{0}))=\sin(\alpha/2).$ (5.17)

Figure 1: a) The sets $\Gamma_{2\alpha}$ and $\Gamma^{x}_{2\alpha}$. b) The
set $\Gamma(k,\alpha)$ with $k=2$.

######  Remark 5.6.

Notice that the expression $\sin(\alpha/2)$ in (5.16) was first obtained by
Szegő [69], where he calculated it as the Chebyshev constant of the arc
$\Gamma_{2\alpha}(\pi/2)$, then it was deduced by Rosenblatt [60], as the
capacity of $\Gamma_{2\alpha}(\pi/2)$.

######  Example 5.2.

Let $\Gamma_{2\alpha}(\alpha)$ be an arc of length $2\alpha$, defined by
(5.15): $\Gamma_{2\alpha}(\alpha)=\\{e^{i\theta}:\,\theta\in[0,2\alpha]\\},$
and let $\Gamma(2,\alpha)$ be the preimage of the arc
$\Gamma_{2\alpha}(\alpha)$ under the mapping $p(z)=z^{2}$. It can be shown
(see Babayan et al. [8] for details) that the set $\Gamma(2,\alpha)$ is the
union of two closed arcs of equal lengths $\alpha$, symmetrically located with
respect to the center of the unit circle (see Figure 1b):

$\Gamma(2,\alpha)=\\{e^{i\omega}:\,\omega\in[-\pi,-\pi+\alpha]\cup[0,\alpha]\\}.$
(5.18)

Then, by the Fekete theorem (see Proposition 7.3) and formula (5.17), for the
transfinite diameter $\tau(\Gamma(2,\alpha))$ we have

$\tau(\Gamma(2,\alpha))=[\tau(\Gamma_{2\alpha}(\alpha))]^{1/2}=\left(\sin(\alpha/2)\right)^{1/2}.$

The above result can easily be extended to the case of $k$ ($k>2$) arcs. Let
$\Gamma(k,\alpha)$ be the union of $k$ ($k\in\mathbb{N},\,k\geq 2$) closed
arcs of equal lengths $\alpha$, which are symmetrically located on the unit
circle (the arcs are assumed to be equidistant). It can be shown that the set
$\Gamma(k,\alpha)$ is the preimage (to within rotation) under the mapping
$p(z)=z^{k}$ of the arc $\Gamma_{k\alpha}(k\alpha/2)$ of length $k\alpha$
defined by (5.15). Therefore, by Fekete’s formula (7.9) and the invariance
property of the transfinite diameter with respect to rotation (see Proposition
7.2(b)), for the transfinite diameter $\tau(\Gamma(k,\alpha))$, we have

$\tau(\Gamma(k,\alpha))=\left(\sin(k\alpha/4)\right)^{1/k}.$ (5.19)

######  Example 5.3.

Let $\alpha>0,$ $\delta\geq 0$ and $\alpha+\delta\leq\pi$. Consider the set

$\Gamma_{\alpha,\delta}(\theta_{0}):=\Gamma_{\alpha+\delta}(\theta_{0})\setminus\Gamma_{\delta}(\theta_{0})$
(5.20)

consisting of the union of two arcs of the unit circle of lengths $\alpha$,
the distance of which (over the circle) is equal to $2\delta$. Define (see
Fig. 2a)):

$\Gamma_{\alpha,\delta}:=\Gamma_{\alpha,\delta}(0)=\\{e^{i\theta}:\,\theta\in[-(\delta+\alpha),-\delta]\cup[\delta,\delta+\alpha]\\}.$
(5.21)

Then the projection $\Gamma_{\alpha,\delta}^{x}$ of $\Gamma_{\alpha,\delta}$
onto the real axis is the segment
$\Gamma_{\alpha,\delta}^{x}=[\cos(\alpha+\delta),\cos\delta]$, and by
Proposition 7.2(e) for the transfinite diameter
$\tau(\Gamma_{\alpha,\delta}^{x})$ we have

$\tau(\Gamma_{\alpha,\delta}^{x})=\frac{\cos\delta-\cos(\alpha+\delta)}{4}=\frac{\sin(\alpha/2)\sin(\alpha/2+\delta)}{2}.$

Hence, according to Robinson’s formula (7.12), for the transfinite diameter
$\tau(\Gamma_{\alpha,\delta})$, we obtain

$\tau(\Gamma_{\alpha,\delta})=[2\tau(\Gamma_{\alpha,\delta}^{x})]^{1/2}=\left(\sin(\alpha/2)\sin(\alpha/2+\delta)\right)^{1/2}.$
(5.22)

In view of Proposition 7.2(b), from (5.22) for any $\theta_{0}\in[-\pi,\pi)$
we have

$\tau(\Gamma_{\alpha,\delta}(\theta_{0}))=\left(\sin(\alpha/2)\sin(\alpha/2+\delta)\right)^{1/2}.$
(5.23)

Observe that for $\delta=0$ we have
$\Gamma_{\alpha,\delta}(\theta_{0})=\Gamma_{2\alpha}(\theta_{0})$ (see (5.15)
and (5.20)), and formula (5.23) becomes (5.17).

Figure 2: a) The set $\Gamma_{\alpha,\delta}$. b) The set
$\Delta_{\alpha,\delta}$.

######  Example 5.4.

Let the arc $\Gamma_{\alpha,\delta}$ be as in Example 5.3 (see (5.21)) with
$\alpha,\delta$ satisfying $\alpha+\delta\leq\pi/2$, that is,
$\Gamma_{\alpha,\delta}$ is a subset of the right semicircle $\mathbb{T}$.
Denote by $\Gamma^{\prime}_{\alpha,\delta}$ the symmetric to
$\Gamma_{\alpha,\delta}$ set with respect to $y$-axis, that is,

$\Gamma^{\prime}_{\alpha,\delta}:=\\{e^{i\theta}:\,\theta\in[-\pi+\delta,-\pi+(\delta+\alpha)]\cup[\pi-(\delta+\alpha),\pi-\delta]\\}.$

Define
$\Delta_{\alpha,\delta}:=\Gamma_{\alpha,\delta}\cup\Gamma^{\prime}_{\alpha,\delta}$,
and observe that the set $\Delta_{\alpha,\delta}$ consists of four arcs of
equal lengths $\alpha$, which are symmetrically located with respect to both
axes (see Figure 2b)). It is easy to see that the set $\Delta_{\alpha,\delta}$
is the preimage (to within rotation) of the set $\Gamma_{2\alpha,2\delta}$
under the mapping $p(z)=z^{2}$. Hence, according to Fekete’s formula (7.9) and
(5.22), for the transfinite diameter $\tau(\Delta_{\alpha,\delta})$, we obtain

$\tau(\Delta_{\alpha,\delta})=\left(\tau(\Gamma_{2\alpha,2\delta})\right)^{1/2}=\left(\sin\alpha\sin(\alpha+2\delta)\right)^{1/4}.$
(5.24)

Denote by $\Delta_{\alpha,\delta}(\theta_{0})$ the image of the set
$\Delta_{\alpha,\delta}$ under mapping $q(z)=e^{i\theta_{0}}z$, that is, under
the rotation by the central angle $\theta_{0}$ around the origin. Then, in
view of Proposition 7.2(b)), from (5.24) for any $\theta_{0}\in[-\pi,\pi)$ we
have

$\tau(\Delta_{\alpha,\delta}(\theta_{0}))=\left(\sin\alpha\sin(\alpha+2\delta)\right)^{1/4}.$
(5.25)

####  5.3.3 A consequence of Theorem 5.3

Now we apply Theorem 5.3 to obtain the asymptotic behavior of the prediction
error $\sigma_{n}(f)$ in the cases where the spectrum of a stationary process
$X(t)$ is as in Examples 5.1-5.4. Specifically, putting together Theorem 5.3
and Examples 5.1-5.4, we obtain the following result.

######  Theorem 5.9 (Babayan et al. [8]).

Let $\overline{E}_{f}$ be the support of the spectral density $f$ of a
stationary process $X(t)$, and let $f>0$ a.e. on $\overline{E}_{f}$. Then for
the prediction error $\sigma_{n}(f)$ the following assertions hold.

  * (a)

If $\overline{E}_{f}=\Gamma_{2\alpha}(\theta_{0})$, where
$\Gamma_{2\alpha}(\theta_{0})$ is as in Example 5.1, then

$\lim_{n\to\infty}\sqrt[n]{\sigma_{n}(f)}=\sin(\alpha/2).$

  * (b)

If $\overline{E}_{f}=\Gamma(k,\alpha)$, where $\Gamma(k,\alpha)$ is as in
Example 5.2, then

$\lim_{n\to\infty}\sqrt[n]{\sigma_{n}(f)}=\left(\sin(k\alpha/4)\right)^{1/k}.$

  * (c)

If $\overline{E}_{f}=\Gamma_{\alpha,\delta}(\theta_{0})$, where
$\Gamma_{\alpha,\delta}(\theta_{0})$ is as in Example 5.3, then

$\lim_{n\to\infty}\sqrt[n]{\sigma_{n}(f)}=\left(\sin(\alpha/2)\sin(\alpha/2+\delta)\right)^{1/2}.$

  * (d)

If $\overline{E}_{f}=\Delta_{\alpha,\delta}(\theta_{0})$, where
$\Delta_{\alpha,\delta}(\theta_{0})$, is as in Example 5.4, then

$\lim_{n\to\infty}\sqrt[n]{\sigma_{n}(f)}=\left(\sin\alpha\sin(\alpha+2\delta)\right)^{1/4}.$

######  Remark 5.7.

The assertion (a) is a slight extension of the Rosenblatt relation (5.2). The
assertion (c) is an extension of assertion (a), which reduces to assertion (a)
if $\delta=0$.

###  5.4 Davisson’s theorem and its extension

In this section, we consider a question of bounding the prediction error
$\sigma^{2}_{n}(f)$. Using constructive methods, Davisson [16] obtained an
upper bound (rather than an asymptote) for the prediction error
$\sigma^{2}_{n}(f)$ without imposing continuity requirement on the spectral
density $f(\lambda)$. Specifically, in Davisson [16] was proved the following
result:

######  Theorem 5.10 (Davisson [16]).

Let the spectral density $f(\lambda)$, $\lambda\in[-\pi,\pi]$ of the process
$X(t)$ be identically zero on a closed interval of length $2\pi-2\alpha$,
$0<\alpha<\pi$. Then for the prediction error $\sigma^{2}_{n}(f)$ the
following inequality holds:

$\sigma^{2}_{n}(f)\leq 4c\left(\sin(\alpha/2)\right)^{2n-2},$ (5.26)

where $c=r(0)$ and $r(\cdot)$ is the covariance function of $X(t)$ (see
formula (2.2)).

The theorem that follows, proved in Babayan and Ginovyan [7], extends
Davisson’s theorem to the case where the spectrum of the process $X(t)$
consists of a union of two equal arcs.

Let $\alpha>0,$ $\delta\geq 0$ and $\alpha+\delta\leq\pi$, and let
$\Gamma_{\alpha,\delta}$ be the set defined by (5.21). Recall that
$\Gamma_{\alpha,\delta}$ is the union of two arcs of the unit circle of
lengths $\alpha$, the distance between which (over the circle) is equal to
$2\delta$ (see Example 5.3 and Figure 2a)).

######  Theorem 5.11 (Babayan and Ginovyan [7]).

Let the spectral density $f(\lambda)$, $\lambda\in[-\pi,\pi]$ of the process
$X(t)$ vanish outside the set $\Gamma_{\alpha,\delta}$. Then for the
prediction error $\sigma^{2}_{n}(f)$ the following inequality holds:

$\sigma^{2}_{n}(f)\leq
4c\left(\sin(\alpha/2)\right)^{n-1}\left(\sin(\alpha/2+\delta)\right)^{n-1},$
(5.27)

where $c$ is as in Theorem 5.10.

######  Remark 5.8.

For $\delta=0$ the set $\Gamma_{\alpha,\delta}$ defined by (5.21) is an arc of
length $2\alpha$, and, in this case, the inequality (5.27) becomes Davisson’s
inequality (5.26).

###  5.5 Extensions of Rosenblatt’s second theorem

In this section, we analyze the asymptotic behavior of the prediction error in
the case where the spectral density $f(\lambda)$, $\lambda\in[-\pi,\pi]$ of
the model is strictly positive except one or several points at which it has a
very high order contact with zero so that the Szegő condition (2.23) is
violated.

Based on Rosenblatt’s result for this case, namely Theorem 5.2, we can expect
that for any deterministic process with spectral density possessing a
singularity of the type (5.5), the rate of the prediction error
$\sigma^{2}_{n}(f)$ should be the same as in (5.4). However, the method
applied in Rosenblatt [60] does not allow to prove this assertion. In Babayan
and Ginovyan [5, 6] and in Babayan et al. [8], using a different approach,
Rosenblatt’s second theorem was extended to broader classes of spectral
densities. To state the corresponding results, we first examine the asymptotic
behavior as $n\to\infty$ of the ratio:

$\frac{\sigma_{n}^{2}(fg)}{\sigma_{n}^{2}(f)},$

where $g$ is a non-negative function.

To clarify the approach, we first assume that $f$ is the spectral density of a
nondeterministic process, in which case the geometric mean $G(f)$ is positive
(see (2.21) and (2.22)). We can then write

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fg)}{\sigma_{n}^{2}(f)}=\frac{\sigma_{\infty}^{2}(fg)}{\sigma_{\infty}^{2}(f)}=\frac{2\pi
G(fg)}{2\pi G(f)}=\frac{G(f)G(g)}{G(f)}=G(g).$ (5.28)

It turns out that under some additional assumptions imposed on functions $f$
and $g$, the asymptotic relation (5.28) remains also valid in the case of
deterministic processes, that is, when $G(f)=0$.

####  5.5.1 Preliminaries

In what follows we consider the class of deterministic processes possessing
spectral densities $f$ for which the sequence of prediction errors
$\\{\sigma_{n}(f)\\}$ is weakly varying (see Definition 7.1), and denote by
$\mathcal{F}$ the class of the corresponding spectral densities:

$\mathcal{F}:=\left\\{f\in L^{1}(\Lambda):\,\,f\geq
0,\,\,G(f)=0,\,\,\lim_{n\to\infty}\frac{\sigma_{n+1}(f)}{\sigma_{n}(f)}=1\right\\}.$
(5.29)

######  Remark 5.9.

According to Rakhmanov’s theorem (Theorem 3.3), a sufficient condition for
$f\in\mathcal{F}$ is that $f>0$ almost everywhere on $\Lambda$ and $G(f)=0$.
Thus, the considered class $\mathcal{F}$ includes all deterministic processes
($G(f)=0$) with almost everywhere positive spectral densities ($f>0$ a.e.). On
the other hand, according to Theorem 3.2 and Remark 5.5, the class
$\mathcal{F}$ does not contain spectral densities, which vanish on an entire
segment of $\Lambda$ (or on an arc of the unit circle $\mathbb{T}$). Also,
from Theorem 5.7 and Remark 5.5 we infer that a necessary condition for
$f\in\mathcal{F}$ is that the spectrum $E_{f}$ is dense in $\Lambda$.

######  Definition 5.2.

Let $\mathcal{F}$ be the class of spectral densities defined by (5.29). For
$f\in\mathcal{F}$ denote by $\mathcal{M}_{f}$ the class of nonnegative
functions $g(\lambda)$ $(\lambda\in\Lambda)$ satisfying the conditions:
$G(g)>0$, $fg\in L^{1}(\Lambda)$, and

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g),$ (5.30)

that is,

$\mathcal{M}_{f}:=\left\\{g\geq 0,\,\,G(g)>0,\,\,fg\in
L^{1}(\Lambda),\,\,\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)\right\\}.$
(5.31)

The next proposition shows that the class $\mathcal{F}$ is close under
multiplication by functions from the class $\mathcal{M}_{f}$.

######  Proposition 5.1 (Babayan and Ginovyan [6]).

If $f\in\mathcal{F}$ and $g\in\mathcal{M}_{f}$, then $fg\in\mathcal{F}$.

The next result shows that the class $\mathcal{M}_{f}$ in a certain sense is
close under multiplication.

######  Proposition 5.2 (Babayan and Ginovyan [6]).

Let $f\in\mathcal{F}$. If $g_{1}\in\mathcal{M}_{f}$ and
$g_{2}\in\mathcal{M}_{fg_{1}}$, then $g:=g_{1}g_{2}\in\mathcal{M}_{f}$ and
$fg\in\mathcal{F}$. In particular, if
$g\in\mathcal{M}_{f}\cap\mathcal{M}_{fg}$, then $g^{2}\in\mathcal{M}_{f}$.

In the next definition we introduce certain classes of bounded functions.

######  Definition 5.3.

We define the class $B$ to be the set of all nonnegative, Riemann integrable
on $\Lambda=[-\pi,\pi]$ functions $h(\lambda)$. Also, we define the following
subclasses:

$B_{+}:=\\{h\in B:\,h(\lambda)\geqslant m\\},\quad B^{-}:=\\{h\in
B:\,h(\lambda)\leqslant M\\},\quad B_{+}^{-}:=B_{+}\cap B^{-},$ (5.32)

where $m$ and $M$ are some positive constants.

In the next proposition we list some obvious properties of the classes
$B_{+}$, $B^{-}$ and $B_{+}^{-}$.

######  Proposition 5.3.

The following assertions hold.

  * a)

If $h\in B_{+}(B^{-})$, then $1/h\in B^{-}(B_{+})$.

  * b)

If $h_{1},h_{2}\in B_{+}(B^{-})$, then $h_{1}+h_{2}\in B_{+}(B^{-})$ and
$h_{1}h_{2}\in B_{+}(B^{-})$.

  * c)

If $h_{1},h_{2}\in B^{-}$ and $h_{1}/h_{2}$ is bounded, then $h_{1}/h_{2}\in
B^{-}$.

  * d)

If $h_{1},h_{2}\in B_{+}^{-}$, then $h_{1}+h_{2}\in B_{+}^{-}$, $h_{1}h_{2}\in
B_{+}^{-}$ and $h_{1}/h_{2}\in B_{+}^{-}$.

In the next proposition we list some properties of weakly varying sequences
for functions from the above defined classes $B_{+}$, $B^{-}$ and $B_{+}^{-}$
(see Babayan et al. [8]).

######  Proposition 5.4.

Let the spectral density $f$ be such that the sequence $\sigma_{n}(f)$ is
weakly varying. The following assertions hold.

  * a)

If $g\in B_{+}^{-}$, then the sequence $\sigma_{n}(fg)$ is also weakly
varying.

  * b)

If $g\in B^{-}$ with $G(g)=0$, then $\sigma_{n}(fg)=o(\sigma_{n}(f))$ as
$n\to\infty$. Thus, multiplying singular spectral densities we obtain a
spectral density with higher ’order of singularity’.

  * c)

If $g\in B_{+}$ with $G(g)=\infty$, and $fg\in B$, then
$\sigma_{n}(f)=o(\sigma_{n}(fg))$ as $n\to\infty$.

####  5.5.2 Extensions of Rosenblatt’s second theorem

The following theorem, proved in Babayan et al. [8], describes the asymptotic
behavior of the ratio $\sigma_{n}^{2}(fg)/\sigma_{n}^{2}(f)$ as $n\to\infty$,
and essentially states that if the spectral density $f$ is from the class
$\mathcal{F}$ (see (5.29)), and $g$ is a nonnegative function, which can have
polynomial type singularities, then the sequences $\\{\sigma_{n}(fg)\\}$ and
$\\{\sigma_{n}(f)\\}$ have the same asymptotic behavior as $n\to\infty$ up to
a positive numerical factor.

######  Theorem 5.12 (Babayan et al. [8]).

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $g$ be
a function of the form:

$g(\lambda)=h(\lambda)\cdot\frac{t_{1}(\lambda)}{t_{2}(\lambda)},\quad\lambda\in\Lambda,$
(5.33)

where $h\in B_{+}^{-}$, $t_{1}$ and $t_{2}$ are nonnegative trigonometric
polynomials, such that $fg\in L^{1}(\Lambda)$. Then $g\in\mathcal{M}_{f}$ and
$fg\in\mathcal{F}$, that is, $fg$ is the spectral density of a deterministic
process with weakly varying prediction error, and the relation (5.30) holds.

In view of Remark 5.9, as a consequence of Theorem 5.12 we obtain the
following result.

######  Corollary 5.2.

Let the spectral density $f$ of a deterministic process $X(t)$ be a.e.
positive, and let $g$ be as in Theorem 5.12. Then $g$ is the spectral density
of a nondeterministic process and the relation (5.30) holds.

As an immediate consequence of Theorem 5.12 and Proposition 7.7(d), we have
the following result.

######  Corollary 5.3.

Let the functions $f$ and $g$ be as in Theorem 5.12. Then the sequence
$\sigma_{n}(fg)$ is also weakly varying.

The theorems that follow extend the above stated Theorem 5.12 to a broader
class of spectral densities, for which the function $g$ can have arbitrary
power type singularities.

######  Theorem 5.13 (Babayan and Ginovyan [6]).

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $g$ be
a function of the form:

$g(\lambda)=h(\lambda)\cdot|t(\lambda)|^{\alpha},\quad\alpha>0,\,\,\lambda\in\Lambda,$
(5.34)

where $h\in B_{+}^{-}$ and $t$ is an arbitrary trigonometric polynomial. Then
$g\in\mathcal{M}_{f}$ and $fg\in\mathcal{F}$, that is, $fg$ is the spectral
density of a deterministic process with weakly varying prediction error, and
the relation (5.30) holds.

Using inductive arguments and Theorem 5.13 we can state the following result.

######  Corollary 5.4.

The conclusion of Theorem 5.13 remains valid if the function $g$ has the
following form:

$g(\lambda)=h(\lambda)\cdot|t_{1}(\lambda)|^{\alpha_{1}}\cdot|t_{2}(\lambda)|^{\alpha_{2}}\cdot\cdots\cdot|t_{m}(\lambda)|^{\alpha_{m}},\quad\lambda\in\Lambda,$

where $h\in B_{+}^{-}$, $t_{1},t_{2},\ldots,t_{m}$ are arbitrary trigonometric
polynomials, $\alpha_{1},\alpha_{2},\ldots,\alpha_{m}$ are arbitrary positive
numbers, and $m\in\mathbb{N}$.

######  Theorem 5.14 (Babayan and Ginovyan [6]).

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $g$ be
a function of the form:

$g(\lambda)=h(\lambda)\cdot
t^{-\alpha}(\lambda),\quad\alpha>0,\,\,\lambda\in\Lambda,$ (5.35)

where $h\in B_{+}^{-}$ and $t$ is a nonnegative trigonometric polynomial. Then
the following assertions hold.

  * (a)

$g\in\mathcal{M}_{f}$ and $fg\in\mathcal{F}$ provided that
$\alpha\in\mathbb{Z}$ and $ft^{-\alpha}\in L^{1}(\Lambda)$.

  * (b)

$g\in\mathcal{M}_{f}$ and $fg\in\mathcal{F}$ provided that
$\alpha\notin\mathbb{Z}$ and $ft^{-(k+1)}\in L^{1}(\Lambda)$, where
$k:=[\alpha]$ is the integer part of $\alpha$.

To state the next result we need the following definition.

######  Definition 5.4.

Let $E_{1}$ and $E_{2}$ be two numerical sets such that for any $x\in E_{1}$
and $y\in E_{2}$ we have $x<y$. We say that the sets $E_{1}$ and $E_{2}$ are
separated from each other if $\sup E_{1}<\inf E_{2}.$ Also, we say that a
numerical set $E$ is separated from infinity if it is bounded from above.

######  Theorem 5.15 (Babayan and Ginovyan [6]).

Let $f(\lambda)$ and $\hat{f}(\lambda)$ $(\lambda\in\Lambda)$ be spectral
densities of stationary processes satisfying the following conditions:

  * 1)

$f,\hat{f}\in B^{-}$;

  * 2)

the functions $f(\lambda)$ and $\hat{f}(\lambda)$ have $k$ common essential
zeros

$\lambda_{1},\lambda_{2},\ldots,\lambda_{k}\in\Lambda\quad(-\pi<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{k}\leq\pi,\,\,k\in\mathbb{N}),$

that is,

$\lim_{\lambda\to\lambda_{j}}f(\lambda)=\lim_{\lambda\to\lambda_{j}}\hat{f}(\lambda)=0,\quad
j=1,2,\ldots,k;$ (5.36)

  * 3)

the functions $f(\lambda)$ and $\hat{f}(\lambda)$ are infinitesimal of the
same order in a neighborhood of each point $\lambda_{j}$ $(j=1,2,\ldots,k)$,
that is,

$\lim_{\lambda\to\lambda_{j}}\frac{\hat{f}(\lambda)}{f(\lambda)}=c_{j}>0,\quad
j=1,2,\ldots,k;$ (5.37)

  * 4)

the functions $f(\lambda)$ and $\hat{f}(\lambda)$ are bounded away from zero
outside any neighborhood $O_{\delta}(\lambda_{j})$ $(j=1,2,\ldots,k)$, which
is separated from the neighboring zeros $\lambda_{j-1}$ and $\lambda_{j+1}$ of
$\lambda_{j}$, that is, there is a number $m:=m_{\delta}>0$ such that
$f(\lambda)\geq m$ and $\hat{f}(\lambda)\geq m$ for almost all
$\lambda\notin\cup_{j=1}^{k}O_{\delta}(\lambda_{j})$. Then the following
assertions hold:

  * a)

$h(\lambda):=\frac{\hat{f}(\lambda)}{f(\lambda)}\in B_{+}^{-};$

  * b)

the processes with spectral densities $f$ and $\hat{f}$ either both are
deterministic or both are nondeterministic;

  * c)

if one of the functions $f$ and $\hat{f}$ is from the class $\mathcal{F}$,
then so is the other, and the following relation holds:

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f})}{\sigma_{n}^{2}(f)}=G(h)>0.$
(5.38)

######  Remark 5.10.

The conditions of Theorem 5.15 mean that the points $\lambda_{j}$
$(j=1,2,\ldots,k)$ are the only common zeros of functions $f(\lambda)$ and
$\hat{f}(\lambda)$. Besides, in the case of deterministic processes, at least
one of these zeros should be of sufficiently high order. Also, notice that the
conditions 1) and 4) of Theorem 5.15 will be satisfied if the functions
$f(\lambda)$ and $\hat{f}(\lambda)$ are continuous on $\Lambda$.

######  Theorem 5.16 (Babayan and Ginovyan [6]).

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $g$ be
a function of the form:

$g(\lambda)=h(\lambda)\cdot|q(\lambda)|^{\alpha},\quad\alpha\in\mathbb{R},\,\,\lambda\in\Lambda,$
(5.39)

where $h\in B_{+}^{-}$, $q$ is an arbitrary algebraic polynomial with real
coefficients, and $fg\in L^{1}(\Lambda)$. Then $fg\in\mathcal{F}$ and
$g\in\mathcal{M}_{f}$.

Taking into account that the sequence
$\\{n^{-\alpha},\,\,n\in\mathbb{N},\,\alpha>0\\}$ is weakly varying, as an
immediate consequence of Theorems 5.13, 5.14, 5.16 or Corollary 5.4, we have
the following result (see Babayan and Ginovyan [6], and Babayan et al. [8]).

######  Corollary 5.5.

Let the functions $f$ and $g$ satisfy the conditions of one of Theorems 5.13,
5.14, 5.16 or Corollary 5.4, and let $\sigma_{n}(f)\sim cn^{-\alpha}$
($c>0,\alpha>0$) as $n\to\infty$. Then

$\sigma_{n}(fg)\sim cG(g)n^{-\alpha}\quad{\rm as}\quad n\to\infty,$

where $G(g)$ is the geometric mean of $g$.

The next result, which immediately follows from Theorem 5.2 and Corollary 5.5,
extends Rosenblatt’s second theorem (Theorem 5.2) (see Babayan and Ginovyan
[6], and Babayan et al. [8]).

######  Theorem 5.17 (Babayan and Ginovyan [6]).

Let $f=f_{a}g$, where $f_{a}$ is defined by (5.3), and let $g$ be a function
satisfying the conditions of one of Theorems 5.13, 5.14, 5.16 or Corollary
5.4. Then

$\delta_{n}(f)=\sigma^{2}_{n}(f)\sim\frac{\Gamma^{2}\left(\frac{a+1}{2}\right)G(g)}{\pi
2^{2-a}}\ n^{-a}\quad{\rm as}\quad n\to\infty,$

where $G(g)$ is the geometric mean of $g$.

We thus have the same limiting behavior for $\sigma^{2}_{n}(f)$ as in the
Rosenblatt’s relation (5.4) up to an additional positive factor $G(g)$.

######  Remark 5.11.

In view of Remark 5.9 it follows that all the above stated results remain true
if the condition $f\in\mathcal{F}$ is replaced by the following slightly
strong but more constructive condition: ’the spectral density $f$ is positive
($f>0$) almost everywhere on $\Lambda$ and $G(f)=0$’.

####  5.5.3 Examples

In this section we discuss examples demonstrating the result stated in Section
5.5.2. In these examples we assume that $\\{X(t),$ $t\in\mathbb{Z}\\}$ is a
stationary deterministic process with a spectral density $f$ satisfying the
conditions of Theorem 5.12, and the function $g$ is given by formula (5.33).
To compute the geometric means we use the properties stated in Proposition
7.8(a).

######  Example 5.5.

Let the function $g(\lambda)$ be as in (5.33) with $h(\lambda)=c>0$ and
$t_{1}(\lambda)=t_{2}(\lambda)=1$, that is, $g(\lambda)=c>0$. Then for the
geometric mean $G(g)$ we have

$G(g)=G(c)=c,$ (5.40)

and in view of (5.30), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)=c.$

Thus, multiplying the spectral density $f$ by a constant $c>0$ multiplies the
prediction error by $c$.

######  Example 5.6.

Let the function $g$ be as in (5.33) with $h(\lambda)=e^{\varphi(\lambda)}$,
where $\varphi(\lambda)$ is an arbitrary odd function, and let
$t_{1}(\lambda)=t_{2}(\lambda)=1$, that is, $g(\lambda)=e^{\varphi(\lambda)}$.
Then for the geometric mean $G(g)$ we have

$G(g)=G(e^{\varphi(\lambda)})=\exp\left\\{\frac{1}{2\pi}\int_{-\pi}^{\pi}\ln
g(\lambda)\,d\lambda\right\\}=\exp\left\\{\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(\lambda)\,d\lambda\right\\}=e^{0}=1,$
(5.41)

and in view of (5.30), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)=1.$

Thus, multiplying the spectral density $f$ by the function
$e^{\varphi(\lambda)}$ with odd $\varphi(\lambda)$ does not change the
asymptotic behavior of the prediction error.

######  Example 5.7.

Let the function $g$ be as in (5.33) with $h(\lambda)=\lambda^{2}+1$ and
$t_{1}(\lambda)=t_{2}(\lambda)=1$, that is, $g(\lambda)=\lambda^{2}+1$. Then
for the geometric mean $G(g)$ by direct calculation we obtain

$G(g)=\exp\left\\{\frac{1}{2\pi}\int_{-\pi}^{\pi}\ln(\lambda^{2}+1)\,d\lambda\right\\}=\exp\\{\ln(1+\pi^{2})-2+\frac{2}{\pi}\arctan\pi\\}\approx
3.3,$ (5.42)

and in view of (5.30), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)=\exp\\{\ln(1+\pi^{2})-2+\frac{2}{\pi}\arctan\pi\\}\approx
3.3.$

Thus, multiplying the spectral density $f$ by the function $\lambda^{2}+1$
multiplies the prediction error approximately by 3.3.

######  Example 5.8.

Let the function $g$ be as in (5.33) with $h(\lambda)=t_{2}(\lambda)=1$, and
$t_{1}(\lambda)=\sin^{2k}(\lambda-\lambda_{0})$, where $k\in\mathbb{N}$ and
$\lambda_{0}$ is an arbitrary point from $[-\pi,\pi]$, that is,
$g(\lambda)=\sin^{2k}(\lambda-\lambda_{0})$. To compute the geometric mean
$G(g)$, we first find the algebraic polynomial $s_{2}(z)$ in the Fejér-Riesz
representation (7.15) of the non-negative trigonometric polynomial
$\sin^{2}(\lambda-\lambda_{0})$ of degree 2. For any
$\lambda_{0}\in[-\pi,\pi]$ we have

$\displaystyle\sin^{2}(\lambda-\lambda_{0})=|\sin(\lambda-\lambda_{0})|^{2}=\left|\frac{1}{2}(e^{2i(\lambda-\lambda_{0})}-1)\right|^{2}=\left|s_{2}(e^{i\lambda})\right|^{2},$

where

$s_{2}(z)=\frac{1}{2}(e^{-2i\lambda_{0}}z^{2}-1).$ (5.43)

Therefore, by Proposition 7.8(d) and (5.43), we have

$G(\sin^{2}(\lambda-\lambda_{0}))=|s_{2}(0)|^{2}=\left(\frac{1}{2}\right)^{2}=\frac{1}{4}.$
(5.44)

Now, in view of Proposition 7.8(a) and (5.44), for the geometric mean of
$g(\lambda)=t_{1}(\lambda)=\sin^{2k}(\lambda-\lambda_{0})$ ($k\in\mathbb{N}$),
we obtain

$G(g)=G(\sin^{2k}(\lambda-\lambda_{0}))=G^{k}(\sin^{2}(\lambda-\lambda_{0}))={4^{-k}},$
(5.45)

and in view of (5.30), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)=\frac{1}{4^{k}}.$

Thus, multiplying the spectral density $f$ by the non-negative trigonometric
polynomial $\sin^{2k}(\lambda-\lambda_{0})$ of degree $2k$ ($k\in\mathbb{N}$),
yields a $4^{k}$-fold asymptotic reduction of the prediction error.

######  Example 5.9.

Let the function $g$ be as in (5.33) with $h(\lambda)=t_{1}(\lambda)=1$, and
$t_{2}(\lambda)=\sin^{2l}(\lambda-\lambda_{0})$, where $l\in\mathbb{N}$ and
$\lambda_{0}$ is an arbitrary point from $[-\pi,\pi]$, that is,
$g(\lambda)=\sin^{-2l}(\lambda-\lambda_{0})$. Then, in view of the third
equality in (7.14) and (5.45) for the geometric mean $G(g)$ we have

$G(g)=G(\sin^{-2l}(\lambda-\lambda_{0}))=G^{-1}(\sin^{2l}(\lambda-\lambda_{0}))={4^{l}},$
(5.46)

and in view of (5.30), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)={4^{l}}.$

Thus, dividing the spectral density $f$ by the non-negative trigonometric
polynomial $\sin^{2l}(\lambda-\lambda_{0})$ of degree $2l$ ($l\in\mathbb{N}$),
yields a $4^{l}$-fold asymptotic increase of the prediction error.

Notice that the values of the geometric mean $G(g)$ obtained in (5.45) and
(5.46) do not depend on the choice of the point $\lambda_{0}\in[-\pi,\pi]$.

Putting together Examples 5.5 \- 5.9 and using Proposition 7.8(a) we have the
following summary example.

######  Example 5.10.

Let $\\{X(t),\,t\in\mathbb{Z}\\}$ be a stationary deterministic process with a
spectral density $f$ satisfying the conditions of Theorem 5.12. Let
$h(\lambda)=ce^{\varphi(\lambda)}(\lambda^{2}+1)$,
$t_{1}(\lambda)=\sin^{2k}(\lambda-\lambda_{1})$ and
$t_{2}(\lambda)=\sin^{2l}(\lambda-\lambda_{2})$, where $c$ is an arbitrary
positive constant, $\varphi(\lambda)$ is an arbitrary odd function and
$\lambda_{1},\lambda_{2}$ are arbitrary points from $[-\pi,\pi]$. Let the
function $g$ be defined as in (5.33), that is,

$g(\lambda)=h(\lambda)\cdot\frac{t_{1}(\lambda)}{t_{2}(\lambda)}=ce^{\varphi(\lambda)}(\lambda^{2}+1)\frac{\sin^{2k}(\lambda-\lambda_{1})}{\sin^{2l}(\lambda-\lambda_{2})}.$
(5.47)

Then, in view of Proposition 7.8(a) and relations (5.40)–(5.42) and
(5.45)–(5.47), we have

$\displaystyle G(g)$ $\displaystyle=$ $\displaystyle
G(h)\frac{G(t_{1})}{G(t_{2})}=G(c)G(e^{\varphi})G(\lambda^{2}+1)G(\sin^{2k}(\lambda-\lambda_{1}))G(\sin^{-2l}(\lambda-\lambda_{2}))$
(5.48) $\displaystyle=$
$\displaystyle(c)(1)\exp\\{\ln(1+\pi^{2})-2+\frac{2}{\pi}\arctan\pi\\}(4^{-k})(4^{l})\approx
3.3c4^{l-k},$

and in view of (5.30) and (5.48), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)\approx
3.3c4^{l-k}.$

######  Example 5.11.

Let the function $g(\lambda)$ ($\lambda\in\Lambda$) be as in (5.34) with
$h(\lambda)=1$ and $t(\lambda)=\sin(\lambda-\lambda_{0})$, where $\lambda_{0}$
is an arbitrary point from $[-\pi,\pi]$, that is,
$g(\lambda)=|\sin(\lambda-\lambda_{0})|^{\alpha}$, $\alpha>0$. Then, according
to Example 5.8, for the geometric mean of $\sin^{2}(\lambda-\lambda_{0})$ we
have

$G(\sin^{2}(\lambda-\lambda_{0}))=\frac{1}{4}.$ (5.49)

According to Proposition 7.8(a) and (5.49), for the geometric mean of
$g(\lambda)$, we obtain

$G(g)=G(|\sin(\lambda-\lambda_{0})|^{\alpha})=G\left(\left(\sin^{2}(\lambda-\lambda_{0})\right)^{\alpha/2}\right)=G^{\alpha/2}(\sin^{2}(\lambda-\lambda_{0}))=\frac{1}{2^{\alpha}},$
(5.50)

and in view of (5.30), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)=\frac{1}{2^{\alpha}}.$

Thus, multiplying the spectral density $f(\lambda)$ by the function
$g(\lambda)=|\sin(\lambda-\lambda_{0})|^{\alpha}$ yields a $2^{\alpha}$-fold
asymptotic reduction of the prediction error.

######  Example 5.12.

Let the function $g(\lambda)$ be as in (5.39) with $h(\lambda)=1$ and
$q(\lambda)=\lambda$, that is, $g(\lambda)=|\lambda|^{\alpha}$,
$\alpha\in\mathbb{R}$. By direct calculation we obtain

$\displaystyle\ln
G(g)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\ln|\lambda|^{\alpha}\,d\lambda=\frac{\alpha}{\pi}\int_{0}^{\pi}\ln\lambda\,d\lambda=\alpha\ln(\pi/e).$

Therefore

$\displaystyle G(g)=\left(\pi/e\right)^{\alpha}\approx(1.156)^{\alpha},$

and in view of (5.30), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)=\left(\frac{\pi}{e}\right)^{\alpha}\approx(1.156)^{\alpha}.$

Thus, multiplying the spectral density $f(\lambda)$ by the function
$g(\lambda)=|\lambda|^{\alpha}$ multiplies the prediction error asymptotically
by $(\pi/e)^{\alpha}\approx(1.156)^{\alpha}$.

It follows from Proposition 3.1(d) that the same asymptotic is true for the
prediction error with spectral density
$\bar{g}(\lambda)=|\lambda-\lambda_{0}|^{\alpha}$, $\lambda_{0}\in[-\pi,\pi]$.

####  5.5.4 Rosenblatt’s second theorem revisited

We first analyze the Pollaczek-Szegő function $f_{a}(\lambda)$ given by (5.3)
(cf. Pollaczek [52] and Szegő [70]). We have

$f_{a}(\lambda)=\frac{2e^{2\lambda\varphi(\lambda)}e^{-\pi\varphi(\lambda)}}{e^{\pi\varphi(\lambda)}+e^{-\pi\varphi(\lambda)}}=\frac{2e^{2\lambda\varphi(\lambda)}}{e^{2\pi\varphi(\lambda)}+1},\quad
0\leq\lambda\leq\pi,\quad\varphi(\lambda):=\varphi_{a}(\lambda)=(a/2)\cot\lambda.$
(5.51)

Observe that $\varphi(\lambda)\to+\infty$ as $\lambda\to 0^{+}$, and we have

$\varphi(\lambda)\sim a/(2\lambda),\quad e^{2\lambda\varphi(\lambda)}\sim
e^{a},\quad e^{2\pi\varphi(\lambda)}+1\sim e^{a\pi/\lambda}\quad{\rm
as}\quad\lambda\to 0^{+}.$ (5.52)

Taking into account that $f_{a}(\lambda)$ is an even function, from (5.51) and
(5.52) we obtain the following asymptotic relation for $f_{a}(\lambda)$ in a
vicinity of the point $\lambda=0$.

$f_{a}(\lambda)\sim 2e^{a}\exp\left\\{-{a\pi}/{|\lambda|}\right\\}\quad{\rm
as}\quad\lambda\to 0.$ (5.53)

Next, observe that $\varphi(\lambda)\to-\infty$ as $\lambda\to\pi$, and we
have

$\varphi(\lambda)=-\varphi(\pi-\lambda)\sim(-a/2)(\pi-\lambda),\quad
2\lambda\varphi(\lambda)\sim-a\pi/(\pi-\lambda),\quad{\rm
as}\quad\lambda\to\pi.$ (5.54)

In view of (5.51) and (5.54) we obtain the following asymptotic of the
function $f_{a}(\lambda)$ in a vicinity of the point $\lambda=\pi$.

$f_{a}(\lambda)\sim 2e^{2\lambda\varphi(\lambda)}\sim
2\exp\left\\{-{a\pi}/{(\pi-\lambda)}\right\\}\quad{\rm as}\quad\lambda\to\pi.$
(5.55)

Putting together (5.53) and (5.55), and taking into account evenness of
$f_{a}(\lambda)$, we conclude that

$f_{a}(\lambda)\sim\left\\{\begin{array}[]{ll}2e^{a}\exp\left\\{-{a\pi}/{|\lambda|}\right\\}&\mbox{as
$\lambda\to 0$},\\\ 2\exp\left\\{-{a\pi}/{(\pi-|\lambda|)}\right\\}&\mbox{as
$\lambda\to\pm\pi$},\end{array}\right.$ (5.56)

Thus, the function $f_{a}(\lambda)$ is positive everywhere except for points
$\lambda=0,\pm\pi,$ and has a very high order of contact with zero at these
points, so that Szegő’s condition (2.23) is violated implying that
$G(f_{a})=0$. Also, observe that $f_{a}(\lambda)$ is infinitely differentiable
at all points of the segment $[-\pi,\pi]$ including the points
$\lambda=0,\pm\pi,$ and attains it maximum value of 1 at the points
$\pm\pi/2$. For some specific values of the parameter $a$ the graph of the
function $f_{a}(\lambda)$ is represented in Figure 3a).

Figure 3: a) Graph of the function $f_{a}(\lambda)$. b) Graph of the function
$\hat{f}_{a}(\lambda)$.

For $a>0$ and $\lambda\in[-\pi,\pi]$, consider the pair of functions
$\hat{f}_{1}(\lambda)$ and $\hat{f}_{2}(\lambda)$ defined by formulas:

$\hat{f}_{1}(\lambda):=\exp\left\\{-{a\pi}/{|\lambda|}\right\\},\quad\hat{f}_{2}(\lambda):=\exp\left\\{-{a\pi}/{(\pi-|\lambda|)}\right\\}.$
(5.57)

Observe that the function $\hat{f}_{1}(\lambda)$ is positive everywhere except
for point $\lambda=0$ at which it has the same order of contact with zero as
$f_{a}(\lambda)$, and hence $G(\hat{f}_{1})=0$. Also, $\hat{f}_{1}(\lambda)$
is infinitely differentiable at all points of the segment $[-\pi,\pi]$ except
for the points $\lambda=\pm\pi,$ where it attains its maximum value equal to
$e^{-a}$. As for the function $\hat{f}_{2}(\lambda)$, it is positive
everywhere except for points $\lambda=\pm\pi,$ at which it has the same order
of contact with zero as $f_{a}(\lambda)$, and hence $G(\hat{f}_{2})=0$. Also,
$\hat{f}_{2}(\lambda)$ is infinitely differentiable at all points of the
segment $[-\pi,\pi]$ except for the point $\lambda=0$, where it attains its
maximum value equal to $e^{-a}$. For some specific values of the parameter $a$
the graphs of functions $\hat{f}_{1}(\lambda)$ and $\hat{f}_{2}(\lambda)$ are
represented in Figure 4.

Figure 4: a) Graph of the function $\hat{f}_{1}(\lambda)$. b) Graph of the
function $\hat{f}_{2}(\lambda)$.

Denote by $\hat{f}_{a}(\lambda)$ the product of functions
$\hat{f}_{1}(\lambda)$ and $\hat{f}_{2}(\lambda)$ defined in (5.57) and
normalized by the factor $e^{4a}$:

$\hat{f}_{a}(\lambda):=e^{4a}\hat{f}_{1}(\lambda)\hat{f}_{2}(\lambda)=e^{4a}\exp\left\\{-{a\pi^{2}}/{(|\lambda|(\pi-|\lambda|))}\right\\},$
(5.58)

and observe that $\hat{f}_{a}(\lambda)$ behaves similar to $f_{a}(\lambda)$.
Indeed, the function $\hat{f}_{a}(\lambda)$ also is positive everywhere except
for points $\lambda=0,\pm\pi,$ it is infinitely differentiable at all points
of the segment $[-\pi,\pi]$ including the points $\lambda=0,\pm\pi,$ and
attains it maximum value of 1 at the points $\pm\pi/2$. Also, in view of
(5.56) and (5.58), at points $\lambda=0,\pm\pi$ the function
$\hat{f}_{a}(\lambda)$ has the same order of zeros as $f_{a}(\lambda)$, and
hence $G(\hat{f}_{a})=0$. Thus, the process $X(t)$ with spectral density
$\hat{f}_{a}(\lambda)$ is deterministic. For some specific values of the
parameter $a$ the graph of the function $\hat{f}_{a}(\lambda)$ is represented
in Figure 3b).

The functions $f_{a}(\lambda)$ and $\hat{f}_{a}(\lambda)$ defined by (5.3) and
(5.58), respectively, satisfy the conditions of Theorem 5.15. Therefore, we
have (see (5.38))

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f}_{a})}{\sigma_{n}^{2}(f_{a})}=G(\hat{f}_{a}/f_{a}):=\hat{C}(a)>0.$
(5.59)

In view of (5.4) and (5.59) we have

$\sigma^{2}_{n}(\hat{f}_{a})\sim C(a)\cdot n^{-a}\quad{\rm as}\quad
n\to\infty.$ (5.60)

where

$C(a):=\frac{\Gamma^{2}\left(({a+1)}/2\right)\hat{C}(a)}{\pi 2^{2-a}}.$ (5.61)

The values of the constants $\hat{C}(a)$ and $C(a)$ for some specific values
of the parameter $a$ are given in Table 1.

Table 1: The values of constants $\hat{C}(a)$ and $C(a)$ $a$ | $\frac{\Gamma^{2}\left((a+1)/2\right)}{\pi 2^{2-a}}$ | $\hat{C}(a)$ | $C(a)$
---|---|---|---
0.1 | 0.223 | 0.797 | 0.178
0.5 | 0.169 | 1.113 | 0.188
1.0 | 0.159 | 2.545 | 0.406
1.5 | 0.185 | 6.446 | 1.193
2.0 | 0.250 | 16.830 | 4.214
3.0 | 0.637 | 119.220 | 76.379
3.3 | 0.902 | 215.715 | 194.656
3.4 | 1.020 | 263.173 | 268.375
5.0 | 10.186 | 6128.990 | 62429.000
10.0 | 223256 | 1.104 $\cdot 10^{8}$ | 2.428 $\cdot 10^{13}$

Now we compare the prediction errors $\sigma^{2}_{n}(\hat{f}_{1})$ and
$\sigma^{2}_{n}(\hat{f}_{2})$ with $\sigma^{2}_{n}(f_{a})$. To this end,
observe first that the function
$g_{1}(\lambda):=f_{a}(\lambda)/\hat{f}_{1}(\lambda)$ has a very high order of
contact with zero at points $\lambda=\pm\pi,$ so that Szegő’s condition (2.23)
is violated implying that $G(g_{1})=0$. Besides, the function $g_{1}(\lambda)$
is continuous on $[-\pi,\pi]$, and hence $g_{1}\in B^{-}$. Therefore,
according to Proposition 5.4 b), we have

$\sigma^{2}_{n}(f_{a})=o\left(\sigma^{2}_{n}(\hat{f}_{1})\right)\quad{\rm
as}\quad n\to\infty.$ (5.62)

Similar arguments applied to the function $f_{2}(\lambda)$ yield

$\sigma^{2}_{n}(f_{a})=o\left(\sigma^{2}_{n}(\hat{f}_{2})\right)\quad{\rm
as}\quad n\to\infty.$ (5.63)

The relations (5.62) and (5.63) show that the rate of convergence to zero of
the prediction errors $\sigma^{2}_{n}(\hat{f}_{1})$ and
$\sigma^{2}_{n}(\hat{f}_{2})$ is less than the one for
$\sigma^{2}_{n}(f_{a})$, that is, the power rate of convergence $n^{-a}$ (see
(5.60)). Thus, the rate of convergence $n^{-a}$ is due to the joint
contribution of all zeros $\lambda=0,\pm\pi$ of the function $f_{a}(\lambda)$,
whereas each of these zeros separately does not guarantee the rate of
convergence $n^{-a}$.

##  6 An Application. Asymptotic behavior of the extreme eigenvalues of
truncated Toeplitz matrices

In this section we analyze the relationship between the rate of convergence to
zero of the prediction error $\sigma_{n}^{2}(f)$ and the minimal eigenvalue of
a truncated Toeplitz matrix generated by the spectral density $f$, by showing
how it is possible to obtain information in both directions.

The problem of asymptotic behavior of the extreme eigenvalues of truncated
(finite sections) Toeplitz matrices goes back to the classical works by Kac,
Murdoch and Szegő [43], Parter [50], Widom [75], and Chan [14], where the
problem was studied for truncated Toeplitz matrices generated by continuous
and continuously differentiable functions (symbols). Since then the problem
for various classes of symbols was studied by many authors. For instance,
Pourahmadi [53], Serra [63, 64], and Babayan and Ginovyan [7] considered the
problem in the case where the symbol of Toeplitz matrix is not (necessarily)
continuous nor differentiable (see also Böttcher and Grudsky [12]). In this
section, we review and summarize some known results from the above cited
references and state some new results.

###  6.1 Extreme eigenvalues of truncated Toeplitz matrices

Let $f(\lambda)$ be a real-valued Lebesgue integrable function defined on
$\Lambda:=[-\pi,\pi]$, $T_{n}(f):=||r_{k-j}||,$ ${j,k=0,1,...,n}$, be the
truncated Toeplitz matrix generated by the Fourier coefficients of $f$, and
let $\lambda_{1,n}(f)\leq\lambda_{2,n}(f)\leq\cdots\lambda_{n+1,n}(f)$ be the
eigenvalues of $T_{n}(f)$. We denote by $m_{f}:={\rm ess\ inf}f$ and
$M_{f}:={\rm ess\ sup}f$ the essential minimum and the essential maximum of
$f$, respectively. We will refer to $f(\lambda)$ as a symbol for the Toeplitz
matrix $T_{n}(f)$.

We first recall Szegő’s distribution theorem (see, e.g., Grenander and Szegő
[32], p. 64-65).

######  Theorem 6.1.

For every continuous function $F$ defined in $[m_{f},M_{f}]$ the following
asymptotic relation holds:

$\lim_{n\to\infty}\frac{1}{n+1}\sum_{k=1}^{n+1}F(\lambda_{k,n}(f))=\frac{1}{2\pi}\int_{-\pi}^{\pi}F(f(u))\,du.$
(6.1)

Moreover, the spectrum of $T_{n}(f)$ is contained in $(m_{f},M_{f})$, and

$\lim_{n\to\infty}\lambda_{1,n}(f)=m_{f}\quad{\rm
and}\quad\lim_{n\to\infty}\lambda_{n+1,n}(f)=M_{f}.$ (6.2)

The problem of interest is to describe the rate of convergence in (6.2),
depending on the properties of the symbol $f$. In the following, without loos
of generality, we assume that $m_{f}:={\rm ess\ inf}f=0$. Also, we study the
asymptotic behavior of the minimum eigenvalue of $T_{n}(f)$, for the maximum
eigenvalue it is sufficient to consider the minimum eigenvalue of the matrix
$T_{n}(-f)$.

The rate of convergence of extreme eigenvalues has been studied by Kac,
Murdoch and Szegő [43], Parter [50], Widom [75] and Chan [14], under the
following regularity condition on $f$ (see, e.g., Grenander and Szegő [32],
Section 5.4(a), p. 72).

Condition A. Let $f(\lambda)$ be real, continuous and periodic with period
$2\pi$. Let $\min f(\lambda)=f(\lambda_{0})=0$ and let $\lambda=\lambda_{0}$
be the only value of $\lambda$ (mod $2\pi)$ for which this minimum is
attained. Moreover, let $f(\lambda)$ have continuous derivatives of order $2k$
$(k\in\mathbb{N})$ in some neighborhood of $\lambda=\lambda_{0}$ with
$f^{(2k)}(\lambda_{0})\neq 0$.

######  Theorem 6.2 (Kac, Murdoch and Szegő [43]).

Under Condition A the following asymptotic relation holds:

$\lambda_{1,n}(f)\simeq n^{-2k}\quad{\rm as}\quad n\to\infty.$ (6.3)

Observe that Condition A is too restrictive which may be hard to verify or may
even be unsatisfied in some areas of application such as prediction theory of
stationary processes and signal processing, where $f$ is viewed as a spectral
density of a stationary process. Another limitation is the possibility that
$f$ has more than one zero.

By using new linear algebra tools, Serra [63] has extended Theorem 6.2, by
proving that the rate of convergence of $\lambda_{1,n}(f)$ depends only on the
order of the zero of $f$, but not (necessarily) on the smoothness of $f$ as it
is required in Theorem 6.2. Moreover, in Serra [65] Theorem 6.2 was further
extended to the case of a function $f\in L^{1}[-\pi,\pi]$ having several
global minima by showing that the maximal order of the zeros of function $f$
is the only parameter which characterizes the rate of convergence of
$\lambda_{1,n}(f)$. In particular, in Serra [63] was proved the following
result.

######  Proposition 6.1 (Serra [63]).

Let $f$ be a nonnegative integrable function on $[-\pi,\pi]$. If $f$ has a
zero of order $2k$ at a point $\lambda=\lambda_{0}$, that is,
$f(\lambda)\asymp(\lambda-\lambda_{0})^{2k}$, then $\lambda_{1,n}(f)\asymp
n^{-2k}$.

###  6.2 The relationship between the prediction error and the minimal
eigenvalue.

Let $X(t),$ $t=0,\pm 1,\ldots,$ be a stationary sequence possessing a spectral
density function $f(\lambda),$ $\lambda\in[-\pi,\pi],$ and let
$\sigma_{n}^{2}(f)$ be the prediction error in predicting $X(0)$ by the past
of $X(t)$ of length $n$ (see formula (3.1)).

The next proposition provides a relationship between the minimal eigenvalue
$\lambda_{1,n}(f)$ of a truncated Toeplitz matrix $T_{n}(f)$ generated by
spectral density $f$ and the prediction error $\sigma_{n}^{2}(f)$ (see
Pourahmadi [53] and Serra [63]).

######  Proposition 6.2.

Let $f$, $\lambda_{1,n}(f)$ and $\sigma_{n}^{2}(f)$ be as above. Then for any
$n\in\mathbb{N}$ the following inequalities hold:

$\lambda_{1,n}(f)\leq\sigma_{n}^{2}(f)\leq
M_{f}\frac{\lambda_{1,n}(f)}{\lambda_{1,n-1}(f)}.$ (6.4)

The first inequality in (6.4) was proved in Pourahmadi [53], while the proof
of the second inequality in (6.4) can be found in Serra [63].

Recall that for a stationary process $X(t)$ with spectral density $f(\lambda)$
by $E_{f}$ we denote the spectrum of $X(t)$, that is,
$E_{f}:=\\{\lambda:\,f(\lambda)>0\\}$ (see (5.6)). Thus, the closure
$\overline{E}_{f}$ of $E_{f}$ is the support of the spectral density $f$.
Also, by $\tau^{*}(E_{f})$ we denote the outer transfinite diameter of the set
$E_{f}$ (defined by (7.8)).

The following theorem is an immediate consequence of Theorem 5.8(b) and
Proposition 6.2 (cf. Pourahmadi [53].)

######  Theorem 6.3.

Let $f$, $\lambda_{1,n}(f)$ and $\tau^{*}(E_{f})$ be as above. Then the
following inequality holds:

$\displaystyle\limsup_{n\to\infty}\sqrt[2n]{\lambda_{1,n}(f)}\leq\tau^{*}(E_{f}).$
(6.5)

Thus, in order that the minimal eigenvalue $\lambda_{1,n}(f)$ should decrease
to zero at least exponentially as $n\to\infty$, it is sufficient that the
outer transfinite diameter of the spectrum of the process $X(t)$ be less than
1. As such the continuity and differentiability of spectral density $f$ are
not required for the exponential rate of convergence of the minimal eigenvalue
$\lambda_{1,n}(f)$ to zero.

Now we proceed to discuss two specific models of deterministic processes, for
which can be obtained more information on the rate of convergence of the
minimal eigenvalue $\lambda_{1,n}(f)$ to zero as $n\to\infty$ from that of
prediction error $\sigma_{n}^{2}(f)$. Notice that, for the first model, the
spectral density $f$ of the process is discontinuous and has uncountably many
zeros, while, for the second model, the function $f$ has a zero of exponential
order at points $0,\pm\pi$. Therefore, in both cases, Condition A of Kac,
Murdoch and Szegő is violated.

####  6.2.1 A model with spectral density $f$ which is discontinuous, zero on
an interval, and positive elsewhere.

Let $X(t)$ be a stationary process for which the support $\overline{E}_{f}$ of
the spectral density $f(\lambda)$ is as in Examples 5.1 \- 5.4. Then, we can
apply Theorems 5.9 and 6.3 to obtain asymptotic estimates for the the minimal
eigenvalue $\lambda_{1,n}(f)$.

In the next theorem we state the result in the cases where the support
$\overline{E}_{f}$ of $f$ is as in Examples 5.1 and 5.3, similar estimates can
be stated in the cases where $\overline{E}_{f}$ is as in Examples 5.2 and 5.4.

######  Theorem 6.4 (Babayan and Ginovyan [7]).

Let $\overline{E}_{f}$ be the support of the spectral density $f$ of a
stationary process $X(t)$. Then for the the minimal eigenvalue
$\lambda_{1,n}(f)$ of $T_{n}(f)$ the following asymptotic estimates hold.

  * (a)

If $\overline{E}_{f}=\Gamma_{2\alpha}(\theta_{0})$, where
$\Gamma_{2\alpha}(\theta_{0})$ is as in Example 5.1, then

$\limsup_{n\to\infty}\sqrt[n]{\lambda_{1,n}(f)}\leq\sin^{2}\left(\alpha/2\right).$
(6.6)

  * (b)

If $\overline{E}_{f}=\Gamma_{\alpha,\delta}(\theta_{0})$, where
$\Gamma_{\alpha,\delta}(\theta_{0})$ is as in Example 5.3, then

$\limsup_{n\to\infty}\sqrt[n]{\lambda_{1,n}(f)}\leq\sin(\alpha/2)\sin(\alpha/2+\delta).$
(6.7)

It is important to note that in Theorem 6.4 the essential infimum $m_{f}=0$ is
attained at uncountably many points, and the spectral density $f$, in general,
is not continuous on $E_{f}$ or at the endpoints of $E_{f}$. Thus, $f$ does
not satisfy Condition A and yet the rate of convergence of $\lambda_{1,n}(f)$
to zero is much faster than in (6.3).

Using Davisson’s theorem (Theorem 5.10) and its extension (Theorem 5.11) we
obtain exact upper bounds for the minimal eigenvalue $\lambda_{1,n}(f)$ rather
than the asymptotic estimates (6.6) and (6.7).

######  Theorem 6.5 (Babayan and Ginovyan [7]).

Let $\overline{E}_{f}$ be the support of the spectral density $f$ of a
stationary process $X(t)$. Then for the the minimal eigenvalue
$\lambda_{1,n}(f)$ the following inequalities hold.

  * (a)

If $\overline{E}_{f}=\Gamma_{2\alpha}(\theta_{0})$, where
$\Gamma_{2\alpha}(\theta_{0})$ is as in Example 5.1, then in view of (5.26)
and the first inequality in (6.4) we have

$\lambda_{1,n}(f)\leq 4c\left(\sin(\alpha/2)\right)^{2n-2},$ (6.8)

where $c=r(0)$ and $r(\cdot)$ is the covariance function of $X(t)$.

  * (b)

If $\overline{E}_{f}=\Gamma_{\alpha,\delta}(\theta_{0})$, where
$\Gamma_{\alpha,\delta}(\theta_{0})$ is as in Example 5.3, then in view of
(5.27) and the first inequality in (6.4) we have

$\lambda_{1,n}(f)\leq
4c\left(\sin(\alpha/2)\right)^{n-1}\left(\sin(\alpha/2+\delta)\right)^{n-1}.$
(6.9)

####  6.2.2 A model with spectral density $f$ possessing exponential order
zeros.

Let $X(t)$ be a stationary process with spectral density $f_{a}$ given by
formula (5.3), that is, $f_{a}$ is the Pollaczek-Szegő function. As it was
observed (see (5.5))

$f_{a}(\lambda)\sim\left\\{\begin{array}[]{ll}2e^{a}\exp\left\\{-{a\pi}/{|\lambda|}\right\\}&\mbox{as
$\lambda\to 0$},\\\ 2\exp\left\\{-{a\pi}/{(\pi-|\lambda|)}\right\\}&\mbox{as
$\lambda\to\pm\pi$}.\end{array}\right.$

Thus, the function $f_{a}$ in (6.2.2) has a zero at points $x=0,\pm\pi$ of
exponential order and is positive elsewhere ($m_{f}=0<M_{f}$). Then by Theorem
5.2 we have

$\displaystyle\delta_{n}(f_{a})=\sigma^{2}_{n}(f_{a})\sim n^{-a}\quad{\rm
as}\quad n\to\infty.$ (6.10)

Now, by using the first inequality in (6.4), from (6.10) we conclude that

$\displaystyle\lambda_{1,n}(f_{a})=O\left(n^{-a}\right)\quad{\rm as}\quad
n\to\infty.$ (6.11)

Thus, by choosing $a$ large enough one can obtain very fast rate of
convergence of $\lambda_{1,n}(f_{a})$ to zero as $n\to\infty$.

######  Remark 6.1.

The asymptotic relation (6.11) remains valid for more general models. Indeed,
let $X(t)$ be a stationary process with spectral density given by
$f(\lambda)=f_{a}(\lambda)g(\lambda)$, where $f_{a}(\lambda)$ is as in (5.3),
and $g(\lambda)$ is a function satisfying the conditions of one of Theorems
5.12, 5.13, 5.14, 5.16 or Corollary 5.4. Then for the minimal eigenvalue
$\lambda_{1,n}(f)$ of a truncated Toeplitz matrix $T_{n}(f)$ generated by
spectral density $f$, we have

$\displaystyle\lambda_{1,n}(f)=O\left(n^{-a}\right)\quad{\rm as}\quad
n\to\infty.$

More results concerning asymptotic behavior of the extreme eigenvalues of
truncated Toeplitz matrices can be found in Böttcher and Grudsky [12], and
Serra [63]–[65]).

Observe that in the above models information from the theory of stationary
processes was used to find linear-algebra results. It is of interest to study
the inverse problem.

##  7 Tools

In this section we briefly discuss the tools, used to prove the results stated
in Sections 4 and 5 (see Babayan and Ginovyan [5] – [7], and Babayan et al.
[8].

###  7.1 Some metric characteristics of bounded closed sets in the plane

We introduce here some metric characteristics of bounded closed sets in the
plane, such as, the transfinite diameter, the Chebyshev constant, and the
capacity, and discuss some properties of these characteristics. Then, we state
the theorems of Fekete and Robinson on the transfinite diameters of related
sets, as well as, an extension of Robinson’s theorem.

####  7.1.1 The transfinite diameter, the Chebyshev constant and the capacity