# Prediction problem for continuous time stochastic processes with
periodically correlated increments observed with noise

Maksym Luz, Mikhail Moklyachuk BNP Paribas Cardif in Ukraine, Kyiv, Ukraine,
<EMAIL_ADDRESS>of Probability Theory, Statistics and Actuarial
Mathematics, Taras Shevchenko National University of Kyiv, Kyiv 01601,
Ukraine<EMAIL_ADDRESS>
(28th August 2024)

###### Abstract

We propose solution of the problem of the mean square optimal estimation of
linear functionals which depend on the unobserved values of a continuous time
stochastic process with periodically correlated increments based on
observations of this process with periodically stationary noise. To solve the
problem, we transform the processes to the sequences of stochastic functions
which form an infinite dimensional vector stationary sequences. In the case of
known spectral densities of these sequences, we obtain formulas for
calculating values of the mean square errors and the spectral characteristics
of the optimal estimates of the functionals. Formulas determining the least
favorable spectral densities and the minimax (robust) spectral characteristics
of the optimal linear estimates of functionals are derived in the case where
the sets of admissible spectral densities are given.

Keywords: Periodically Correlated Increments, Minimax-Robust Estimate, Mean
Square Error

AMS 2010 subject classifications. Primary: 60G10, 60G25, 60G35, Secondary:
62M20, 62P20, 93E10, 93E11

## Introduction

In this article, we study the prediction problem for a continuous time
stochastic process $\xi(t)$, $t\in\mathbb{R}$, with periodically correlated
increments $\xi^{(d)}(t,\tau T)=\Delta_{T\tau}^{d}\xi(t)$ of order $d$ and
period $T$, where $\Delta_{s}\xi(t)=\xi(t)-\xi(t-s)$, based on observations of
the process $\xi(t)$ with a periodically correlated noise stochastic process
$\eta(t)$, $t\in\mathbb{R}$. The resent studies, for example, by Basawa et al.
[1], Dudek et al. [4], Reisen et al. [30], Solci et al. [32], show a constant
interest to the non-stationary models and robust methods of estimation.

Kolmogorov [13], Wiener [34] and Yaglom [36] developed effective methods of
solution of interpolation, extrapolation (prediction) and filtering problems
for stationary stochastic sequences and processes. For a particular problem,
they developed methods of finding an estimate $\widetilde{x}(t)$ constructed
from available observations that minimizes the mean square error
$\Delta(\widetilde{x}(t),f)=\mathsf{E}|x(t)-\widetilde{x}(t)|^{2}$ in the case
where the spectral density $f(\lambda)$ of the stationary process or sequence
$x(t)$ is exactly known and fixed. Such estimates are called optimal linear
estimates within this article.

The developed classical estimation methods are not directly applicable in
practice since the exact spectral structure of the processes is not usually
available. In this case the estimated spectral densities can be considered as
the true ones. However, Vastola and Poor [33] showed with the help of the
concrete examples, that such substitution can result in a significant increase
of the estimate error. Therefore it is reasonable to consider the estimates,
called minimax-robust, which minimize the maximum of the mean-square errors
for all spectral densities from a given set of admissible spectral densities
simultaneously. The minimax-robust method of extrapolation was proposed by
Grenander [8] who considered the estimation of the functional
$Ax=\int_{0}^{1}a(t)x(t)dt$ as a game between two players, one of which
minimizes the mean square error $\Delta(f,\widetilde{A}x)$ by
$\widetilde{A}\zeta$ and another one maximizes the error by $f$. He showed
that the game has a saddle point solution under proper conditions. For more
details see the further study by Franke and Poor [5], Hosoya [9] and the
survey paper by Kassam and Poor [11]. A wide range of results has been
obtained by Moklyachuk [20, 21, 22, 23, 24]. These results have been extended
on the vector-valued stationary processes and sequences by Moklyachuk, Sidei
and Masyutka [26, 27].

The concept of stationarity admits some generalizations, a combination of two
of which – stationary $d$th increments and periodical correlation – is in
scope of this article. Random processes with stationary $d$th increments
$x(t)$ were introduced by Yaglom and Pinsker [29]. The increment sequence
$x^{(d)}(t,\tau)=\Delta_{\tau}^{d}x(t)$ generated by such process is
stationary by the variable $t$, namely, the mathematical expectations
$\mathsf{E}\xi^{(d)}(t,\tau)$ and
$\mathsf{E}\xi^{(d)}(t+s,\tau_{1})\xi^{(d)}(t,\tau_{2})$ do not depend on $t$.
Yaglom and Pinsker [29] described the spectral representation of such process
and the spectral density canonical factorization, and they also solved the
extrapolation problem for these processes. The minimax-robust extrapolation,
interpolation and filtering problems for stochastic processes with stationary
increments were investigated by Luz and Moklyachuk [16].

Dubovetska and Moklyachuk [3] derived the classical and minimax-robust
estimates for another generalization of stationary processes – periodically
correlated (cyclostationary) processes, introduced by Gladyshev [7]. The
correlation function $K(t,s)={\textsf{E}}{x(t)\overline{x(s)}}$ of such
processes is a $T$-periodic function: $K(t,s)=K(t+T,s+T)$, which implies a
time-dependent spectrum. Periodically correlated processes are widely used in
signal processing and communications (see Napolitano [28] for a review of
recent works on cyclostationarity and its applications).

In this article, we deal with the problem of the mean-square optimal
estimation of the linear functionals $A\xi=\int_{0}^{\infty}a(t)\xi(t)dt$ and
$A_{NT}{\xi}=\int_{0}^{(N+1)T}a(t)\xi(t)dt$ which depend on the unobserved
values of a continuous time stochastic process $\xi(t)$ with periodically
stationary $d$-th increments based on observations of this process with
periodically stationary noise at points $t<0$.

Similar problems for discrete time processes have been studied by Kozak, Luz
and Moklyachuk [14], Luz and Moklyachuk [15, 17, 18]. The problem of
estimation of continuous time stochastic process $\xi(t)$ with periodically
stationary $d$-th increments based on observations of the process without
noise at points $t<0$ was studied by Luz and Moklyachuk [19].

The article is organized as follows. In section 1, we describe a presentation
of a continuous time periodically stationary process as a stationary H-valued
sequence. This approach is extended on the periodically stationary increments
in the section 2. The traditional Hilbert space projection method of
prediction is developed in section 3. Particularly, formulas for calculating
the mean-square errors and the spectral characteristics of the optimal linear
estimates of the functionals $A\xi$ and $A_{NT}{\xi}$ are derived under some
conditions on spectral densities. An approach to solution of the prediction
problem which is based on factorizations of spectral densities is developed in
section 4. In section 5 we present our results on minimax-robust prediction
for the studied processes: relations that determine the least favourable
spectral densities and the minimax spectral characteristics are derived for
some classes of spectral densities.

##  1 Continuous time periodically correlated processes and generated vector
stationary sequences

In this section, we present a brief review of properties of periodically
correlated processes and describe an approach to presenting it as stationary
$H$-valued sequences. In the next section, this approach is applied to develop
the spectral theory for periodically correlated increment processes.

######  Definition 1.1 (Gladyshev [7]).

A mean-square continuous stochastic process $\eta:\mathbb{R}\to
H=L_{2}(\Omega,\mathcal{F},\mathbb{P})$, with $\textsf{E}\eta(t)=0,$ is called
periodically correlated (PC) with period $T$, if its correlation function
$K(t,s)={\textsf{E}}{\eta(t)\overline{\eta(s)}}$ for all $t,s\in\mathbb{R}$
and some fixed $T>0$ is such that

$K(t,s)={\textsf{E}}{\eta(t)\overline{\eta(s)}}={\textsf{E}}{\eta(t+T)\overline{\eta(s+T)}}=K(t+T,s+T).$

For a periodically correlated stochastic process $\eta(t)$, one can construct
the following sequence of stochastic functions [2], [25]

$\\{\eta_{j}(u)=\eta(u+jT),u\in[0,T),j\in\mathbb{Z}\\}.$ (1)

Sequence (1) forms a $L_{2}([0,T);H)$-valued stationary sequence
$\\{\eta_{j},j\in\mathbb{Z}\\}$ with the correlation function

$B_{\eta}(l,j)=\langle\eta_{l},\eta_{j}\rangle_{H}=\int_{0}^{T}\textsf{E}[\eta(u+lT)\overline{\eta(u+jT)}]du\\\
=\int_{0}^{T}K_{\eta}(u+(l-j)T,u)du=B_{\eta}(l-j),$

where $K_{\eta}(t,s)=\textsf{E}{\eta}(t)\overline{{\eta}(s)}$ is the
correlation function of the PC process $\eta(t)$. Chose the following
orthonormal basis in the space $L_{2}([0,T);\mathbb{R})$

$\\{\widetilde{e}_{k}=\frac{1}{\sqrt{T}}e^{2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}u/T},k=1,2,3,\dots\\},\quad\langle\widetilde{e}_{j},\widetilde{e}_{k}\rangle=\delta_{kj}.$
(2)

Making use of this basis the stationary sequence
$\\{\eta_{j},j\in\mathbb{Z}\\}$ can be represented in the form

$\eta_{j}=\sum_{k=1}^{\infty}\eta_{kj}\widetilde{e}_{k},$ (3)

where

$\eta_{kj}=\langle\eta_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}\eta_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv.$

The sequence $\\{\eta_{j},j\in\mathbb{Z}\\},$ or the corresponding to it
vector sequence

$\\{\vec{\eta}_{j}=(\eta_{kj},k=1,2,\dots)^{\top},j\in\mathbb{Z}\\},$

is called a generated by the process $\\{\eta(t),t\in\mathbb{R}\\}$ vector
stationary sequence. The components
$\\{\eta_{kj}\\}:k=1,2,\dots;j\in\mathbb{Z}$ of the generated stationary
sequence $\\{\eta_{j},j\in\mathbb{Z}\\}$ satisfy the relations [10], [20]

$\textsf{E}{\eta_{kj}}=0,\quad\|{\eta}_{j}\|^{2}_{H}=\sum_{k=1}^{\infty}\textsf{E}|\eta_{kj}|^{2}\leq
P_{\eta}=B_{\eta}(0),\quad\textsf{E}\eta_{kl}\overline{\eta}_{nj}=\langle
R_{\eta}(l-j)\widetilde{e}_{k},\widetilde{e}_{n}\rangle.$

The correlation function $R_{\eta}(j)$ of the generated stationary sequence
$\\{\eta_{j},j\in\mathbb{Z}\\}$ is a correlation operator function. The
correlation operator $R_{\eta}(0)=R_{\eta}$ is a kernel operator and its
kernel norm satisfies the following properties:

$\|{\eta}_{j}\|^{2}_{H}=\sum_{k=1}^{\infty}\langle
R_{\eta}\widetilde{e}_{k},\widetilde{e}_{k}\rangle\leq P_{\eta}.$

The generated stationary sequence $\\{\eta_{j},j\in\mathbb{Z}\\}$ has the
spectral density function $g(\lambda)=\\{g_{kn}(\lambda)\\}_{k,n=1}^{\infty}$,
that is positive valued operator function of variable $\lambda\in[-\pi,\pi)$,
if its correlation function $R_{\eta}(j)$ can be represented in the form

$\langle
R_{\eta}(j)\widetilde{e}_{k},\widetilde{e}_{n}\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ij\lambda}\langle
g(\lambda)\widetilde{e}_{k},\widetilde{e}_{n}\rangle d\lambda.$

We finish our review by the statement, that for almost all
$\lambda\in[-\pi,\pi)$ the spectral density $f(\lambda)$ is a kernel operator
with an integrable kernel norm

$\sum_{k=1}^{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}\langle
g(\lambda)\widetilde{e}_{k},\widetilde{e}_{k}\rangle
d\lambda=\sum_{k=1}^{\infty}\langle
R_{\eta}\widetilde{e}_{k},\widetilde{e}_{k}\rangle=\|{\eta}_{j}\|^{2}_{H}\leq
P_{\eta}.$

##  2 Stochastic processes with periodically correlated $d$th increments

For a given stochastic process $\\{\xi(t),t\in\mathbb{R}\\}$, consider the
stochastic $d$th increment process

$\xi^{(d)}(t,\tau)=(1-B_{\tau})^{d}\xi(t)=\sum_{l=0}^{d}(-1)^{l}{d\choose
l}\xi(t-l\tau),$ (4)

with the step $\tau\in\mathbb{R}$, generated by the stochastic process
$\xi(t)$. Here $B_{\tau}$ is the backward shift operator:
$B_{\tau}\xi(t)=\xi(t-\tau)$, $\tau\in\mathbb{R}$.

We prefer to use the notation $\xi^{(d)}(t,\tau)$ instead of widely used
$\Delta_{\tau}^{d}\xi(t)$ to avoid a duplicate with the mean square error
notation.

######  Definition 2.1.

A stochastic process $\\{\xi(t),t\in\mathbb{R}\\}$ is called a stochastic
process with periodically stationary (periodically correlated) increments with
the step $\tau\in\mathbb{Z}$ and the period $T>0$ if the mathematical
expectations

$\displaystyle\mathsf{E}\xi^{(d)}(t+T,\tau T)$ $\displaystyle=$
$\displaystyle\mathsf{E}\xi^{(d)}(t,\tau T)=c^{(d)}(t,\tau T),$
$\displaystyle\mathsf{E}\xi^{(d)}(t+T,\tau_{1}T)\xi^{(d)}(s+T,\tau_{2}T)$
$\displaystyle=$ $\displaystyle D^{(d)}(t+T,s+T;\tau_{1}T,v_{2}T)$
$\displaystyle=$ $\displaystyle D^{(d)}(t,s;\tau_{1}T,\tau_{2}T)$

exist for every $t,s\in\mathbb{R}$, $\tau_{1},\tau_{2}\in\mathbb{Z}$ and for
some fixed $T>0$.

The functions $c^{(d)}(t,\tau T)$ and $D^{(d)}(t,s;\tau_{1}T,\tau_{2}T)$ from
the Definition 2.1 are called the _mean value_ and the _structural function_
of the stochastic process $\xi(t)$ with periodically stationary (periodically
correlated) increments.

######  Remark 2.1.

For spectral properties of one-pattern increment sequence
$\chi_{\mu,1}^{(n)}(\xi(m)):=\xi^{(n)}(m,\mu)=(1-B_{\mu})^{n}\xi(m)$ see,
e.g., [16], pp. 1-8; [6], pp. 48–60, 261–268; [36], pp. 390–430. The
corresponding results for continuous time increment process
$\xi^{(n)}(t,\tau)=(1-B_{\tau})^{n}\xi(t)$ are described in [35], [36].

For a review of the properties of periodically correlated processes, we refer
to [2], [25]. Here we present a generalization of these properties on the
processes with periodically correlated increments from Definition 2.1.

For the stochastic process $\\{\xi(t),t\in\mathbb{R}\\}$ with periodically
correlated increments $\xi^{(d)}(t,\tau T)$ and the integer step $\tau$, we
follow the procedure described in the Section 1 and construct a sequence of
stochastic functions

$\\{\xi^{(d)}_{j}(u):=\xi^{(d)}_{j,\tau}(u)=\xi^{(d)}_{j}(u+jT,\tau
T),\,\,u\in[0,T),j\in\mathbb{Z}\\}.$ (5)

Sequence (5) forms a $L_{2}([0,T);H)$-valued stationary increment sequence
$\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ with the structural function

$\displaystyle B_{\xi^{(d)}}(l,j)$ $\displaystyle=$
$\displaystyle\langle\xi^{(d)}_{l},\xi^{(d)}_{j}\rangle_{H}=\int_{0}^{T}\textsf{E}[\xi^{(d)}_{j}(u+lT,\tau_{1}T)\overline{\xi^{(d)}_{j}(u+jT,\tau_{2}T)}]du$
$\displaystyle=$
$\displaystyle\int_{0}^{T}D^{(d)}(u+(l-j)T,u;\tau_{1}T,\tau_{2}T)du=B_{\xi^{(d)}}(l-j).$

Making use of the orthonormal basis (2) the stationary increment sequence
$\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ can be represented in the form

$\xi^{(d)}_{j}=\sum_{k=1}^{\infty}\xi^{(d)}_{kj}\widetilde{e}_{k},$ (6)

where

$\xi^{(d)}_{kj}=\langle\xi^{(d)}_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}\xi^{(d)}_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv.$

We call this sequence $\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\},$ or the
corresponding to it vector sequence

$\\{\vec{\xi}^{(d)}(j,\tau)=\vec{\xi}^{(d)}_{j}=(\xi^{(d)}_{kj},k=1,2,\dots)^{\top}=(\xi^{(d)}_{k}(j,\tau),k=1,2,\dots)^{\top},j\in\mathbb{Z}\\},$
(7)

_an infinite dimension vector stationary increment sequence_ generated by the
increment process $\\{\xi^{(d)}(t,\tau T),t\in\mathbb{R}\\}$. Further, we will
omit the word vector in the notion generated vector stationary increment
sequence.

Components $\\{\xi^{(d)}_{kj}\\}:k=1,2,\dots;j\in\mathbb{Z}$ of the generated
stationary increment sequence $\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ are such
that, [10], [20]

$\textsf{E}{\xi^{(d)}_{kj}}=0,\quad\|\xi^{(d)}_{j}\|^{2}_{H}=\sum_{k=1}^{\infty}\textsf{E}|\xi^{(d)}_{kj}|^{2}\leq
P_{\xi^{(d)}}=B_{\xi^{(d)}}(0),$

and

$\textsf{E}\xi^{(d)}_{kl}\overline{\xi^{(d)}_{nj}}=\langle
R_{\xi^{(d)}}(l-j;\tau_{1},\tau_{2})\widetilde{e}_{k},\widetilde{e}_{n}\rangle.$

The _structural function_
$R_{\xi^{(d)}}(j):=R_{\xi^{(d)}}(j;\tau_{1},\tau_{2})$ of the generated
stationary increment sequence $\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ is a
correlation operator function. The correlation operator
$R_{\xi^{(d)}}(0)=R_{\xi^{(d)}}$ is a kernel operator and its kernel norm
satisfies the following limitations:

$\|\xi^{(d)}_{j}\|^{2}_{H}=\sum_{k=1}^{\infty}\langle
R_{\xi^{(d)}}\widetilde{e}_{k},\widetilde{e}_{k}\rangle\leq P_{\xi^{(d)}}.$

Suppose that the structural function $R_{\xi^{(d)}}(j)$ admits a
representation

$\langle
R_{\xi^{(d)}}(j;\tau_{1},\tau_{2})\widetilde{e}_{k},\widetilde{e}_{n}\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ij\lambda}(1-e^{-i\tau_{1}\lambda})^{d}(1-e^{i\tau_{2}\lambda})^{d}\frac{1}{\lambda^{2d}}\langle
f(\lambda)\widetilde{e}_{k},\widetilde{e}_{n}\rangle d\lambda.$

Then $f(\lambda)=\\{f_{kn}(\lambda)\\}_{k,n=1}^{\infty}$ is a _spectral
density function_ of the generated stationary increment sequence
$\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$. It is a positive valued operator
functions of variable $\lambda\in[-\pi,\pi)$, and for almost all
$\lambda\in[-\pi,\pi)$ it is a kernel operator with an integrable kernel norm

$\sum_{k=1}^{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}(1-e^{-i\tau_{1}\lambda})^{d}(1-e^{i\tau_{2}\lambda})^{d}\frac{1}{\lambda^{2d}}\langle
f(\lambda)\widetilde{e}_{k},\widetilde{e}_{k}\rangle d\lambda\\\
=\sum_{k=1}^{\infty}\langle
R_{\xi^{(d)}}\widetilde{e}_{k},\widetilde{e}_{k}\rangle=\|{\zeta}_{j}\|^{2}_{H}\leq
P_{\xi^{(d)}}.$ (8)

The stationary $d$th increment sequence $\vec{\xi}^{(d)}_{j}$ admits the
spectral representation [12]

$\vec{\xi}^{(d)}_{j}=\int_{-\pi}^{\pi}e^{i\lambda
j}(1-e^{-i\tau\lambda})^{d}\frac{1}{(i\lambda)^{d}}d\vec{Z}_{\xi^{(d)}}(\lambda),$

where $\vec{Z}_{\xi^{(d)}}(\lambda)=\\{Z_{k}(\lambda)\\}_{k=1}^{\infty}$ is a
vector-valued random process with uncorrelated increments on $[-\pi,\pi)$.

Consider the generated stationary stochastic sequence $\vec{\eta}_{j}$ defined
in Section LABEL:section_PC_process_i_n_c, which is uncorrelated with the
increment sequence $\vec{\xi}^{(d)}_{j}$. It admits the spectral
representation

$\vec{\eta}_{j}=\int_{-\pi}^{\pi}e^{i\lambda j}d\vec{Z}_{\eta}(\lambda),$ (9)

where $\vec{Z}_{\eta}(\lambda)$ is a vector-valued random process with
uncorrelated increments on $[-\pi,\pi)$. The spectral representation of the
sequence $\vec{\zeta}^{(d)}_{j}$, generated by the process
$\zeta(t)=\xi(t)+\eta(t)$, is determined by the spectral densities
$f(\lambda)$ and $g(\lambda)$ by the relation

$\vec{\zeta}^{(d)}_{j}=\int_{-\pi}^{\pi}e^{i\lambda
j}(1-e^{-i\tau\lambda})^{d}\frac{1}{(i\lambda)^{d}}d\vec{Z}_{\xi^{(d)}+\eta^{(d)}}(\lambda).$
(10)

The random processes $\vec{Z}_{\eta}(\lambda)$ and
$\vec{Z}_{\eta^{(d)}}(\lambda)$ are connected by the relation
$d\vec{Z}_{\eta^{(d)}}(\lambda)=(i\lambda)^{d}d\vec{Z}_{\eta}(\lambda)$,
$\lambda\in[-\pi,\pi)$, see [16]. The spectral density
$p(\lambda)=\\{p_{kn}(\lambda)\\}_{k,n=1}^{\infty}$ of the sequence
$\vec{\zeta}^{(d)}_{j}$ is determined by the spectral densities $f(\lambda)$
and $g(\lambda)$ by the relation

$p(\lambda)=f(\lambda)+\lambda^{2d}g(\lambda).$

In the space $H=L_{2}(\Omega,\cal F,\mathsf{P})$, consider a closed linear
subspace

$H(\vec{\xi}^{\,(d)})=\overline{span}\\{\xi^{(d)}_{kj}:k=1,2,\dots;\,j\in\mathbb{Z}\\}$

generated by the components of the generated stationary increment sequence
$\vec{\xi}^{\,(d)}=\\{\xi^{(d)}_{kj}=\xi^{(d)}_{k}(j,\tau),\,\tau>0\\}$. For
$q\in\mathbb{Z}$, consider also a closed linear subspace

$H^{q}(\vec{\xi}^{\,(d)})=\overline{span}\\{\xi^{(d)}_{kj}:k=1,2,\dots;\,j\leq
q\\}.$

Define a subspace

$S(\vec{\xi}^{\,(d)})=\bigcap_{q\in\mathbb{Z}}H^{q}(\vec{\xi}^{\,(d)})$

of the Hilbert space $H(\vec{\xi}^{\,(d)})$. The space $H(\vec{\xi}^{\,(d)})$
admits a decomposition $H(\vec{\xi}^{\,(d)})=S(\vec{\xi}^{\,(d)})\oplus
R(\vec{\xi}^{\,(d)})$ where $R(\vec{\xi}^{\,(d)})$ is the orthogonal
complement of the subspace $S(\vec{\xi}^{\,(d)})$ in the space
$H(\vec{\xi}^{\,(d)})$.

######  Definition 2.2.

A stationary (wide sense) increment sequence
$\vec{\xi}^{(d)}_{j}=\\{\xi^{(d)}_{kj}\\}_{k=1}^{\infty}$ is called regular if
$H(\vec{\xi}^{\,(d)})=R(\vec{\xi}^{\,(d)})$, and it is called singular if
$H(\vec{\xi}^{\,(d)})=S(\vec{\xi}^{\,(d)})$.

######  Theorem 2.1.

A stationary increment sequence $\xi^{(d)}_{j}$ is uniquely represented in the
form

$\xi^{(d)}_{kj}=\xi^{(d)}_{S,kj}+\xi^{(d)}_{R,kj}$ (11)

where $\xi^{(d)}_{R,kj},k=1,\ldots,\infty$, is a regular stationary increment
sequence and $\xi^{(d)}_{S,kj},k=1,\dots,\infty$, is a singular stationary
increment sequence. The increment sequences $\xi^{(d)}_{R,kj}$ and
$\xi^{(d)}_{S,kj}$ are orthogonal for all $j\in\mathbb{Z}$. They are defined
by the formulas

$\displaystyle\xi^{(d)}_{S,kj}$
$\displaystyle=\mathsf{E}[\xi^{(d)}_{kj}|S(\vec{\xi}^{\,(d)})],$
$\displaystyle\xi^{(d)}_{R,kj}$
$\displaystyle=\xi^{(d)}_{kj}-\xi^{(d)}_{S,kj}.$

Consider an innovation sequence
${\vec{\varepsilon}(u)=\\{\varepsilon_{m}(u)\\}_{m=1}^{M},u\in\mathbb{Z}}$ for
a regular stationary increment, namely, the sequence of uncorrelated random
variables such that
$\mathsf{E}\varepsilon_{m}(u)\overline{\varepsilon}_{j}(v)=\delta_{mj}\delta_{uv}$,
$\mathsf{E}|\varepsilon_{m}(u)|^{2}=1,m,j=1,\dots,M;u\in\mathbb{Z}$, and
$H^{r}(\vec{\xi}^{(d)})=H^{r}(\vec{\varepsilon})$ holds true for all
$r\in\mathbb{Z}$, where $H^{r}(\vec{\varepsilon})$ is the Hilbert space
generated by elements $\\{\varepsilon_{m}(u):m=1,\dots,M;u\leq r\\}$,
$\delta_{mj}$ and $\delta_{uv}$ are Kronecker symbols.

######  Theorem 2.2.

A stationary increment sequence $\vec{\xi}^{(d)}_{j}$ is regular if and only
if there exists an innovation sequence
${\vec{\varepsilon}(u)=\\{\varepsilon_{m}(u)\\}_{m=1}^{M},u\in\mathbb{Z}}$ and
a sequence of matrix-valued functions
$\varphi^{(d)}(l,\tau)=\\{\varphi^{(d)}_{km}(l,\tau)\\}_{k=\overline{1,\infty}}^{m=\overline{1,M}}$,
$l\geq 0$, such that

$\sum_{l=0}^{\infty}\sum_{k=1}^{\infty}\sum_{m=1}^{M}|\varphi^{(d)}_{km}(l,\tau)|^{2}<\infty,\quad\xi^{(d)}_{kj}=\sum_{l=0}^{\infty}\sum_{m=0}^{M}\varphi^{(d)}_{km}(l,\tau)\vec{\varepsilon}_{m}(j-l).$
(12)

Representation (12) is called the canonical moving average representation of
the generated stationary increment sequence $\vec{\xi}^{\,(d)}_{j}$.

The spectral function $F(\lambda)$ of a stationary increment sequence
$\vec{\xi}^{\,(d)}_{j}$ which admits canonical representation
$(\ref{odnostRuhSer_cont})$ has the spectral density
$f(\lambda)=\\{f_{ij}(\lambda)\\}_{i,j=1}^{\infty}$ admitting the canonical
factorization

$f(\lambda)=\varphi(e^{-i\lambda})\varphi^{*}(e^{-i\lambda}),$ (13)

where the function $\varphi(z)=\sum_{k=0}^{\infty}\varphi(k)z^{k}$ has
analytic in the unit circle $\\{z:|z|\leq 1\\}$ components
$\varphi_{ij}(z)=\sum_{k=0}^{\infty}\varphi_{ij}(k)z^{k};i=1,\dots,\infty;j=1,\dots,M$.
Based on moving average representation $(\ref{odnostRuhSer_cont})$ define

$\varphi_{\tau}(z)=\sum_{k=0}^{\infty}\varphi^{(d)}(k,\tau)z^{k}=\sum_{k=0}^{\infty}\varphi_{\tau}(k)z^{k}.$

Then the following factorization holds true:

$\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f(\lambda)=\varphi_{\tau}(e^{-i\lambda})\varphi^{*}_{\tau}(e^{-i\lambda}),\quad\varphi_{\tau}(e^{-i\lambda})=\sum_{k=0}^{\infty}\varphi_{\tau}(k)e^{-i\lambda
k}.$ (14)

##  3 Hilbert space projection method of prediction

Let a periodically correlated increment process $\xi^{(d)}(t,\tau T)$,
$t\in\mathbb{R}$, generates by formula (6) an infinite dimension vector
stationary increment sequence $\\{\vec{\xi}^{(d)}_{j},j\in\mathbb{Z}\\}$ which
has the spectral density matrix
$f(\lambda)=\\{f_{ij}(\lambda)\\}_{i,j=1}^{\infty}$. As a noise process,
consider a periodically stationary stochastic process $\eta(t)$,
$t\in\mathbb{R}$, uncorrelated with the process $\xi(t)$. Let the process
$\eta(t)$ generates by formula (3) an infinite dimension vector stationary
sequence $\\{\vec{\eta}_{j},j\in\mathbb{Z}\\}$ with the spectral density
matrix $g(\lambda)=\\{g_{ij}(\lambda)\\}_{i,j=1}^{\infty}$.

By the classical prediction problem we understand the problem of the mean
square optimal linear estimation of the functionals

$A{\xi}=\int_{0}^{\infty}a(t)\xi(t)dt,\quad
A_{NT}{\xi}=\int_{0}^{(N+1)T}a(t)\xi(t)dt$

which depend on the unknown values of the stochastic process $\xi(t)$.
Estimates are based on observations of the process $\zeta(t)=\xi(t)+\eta(t)$
at points $t<0$.

Assumptions:

  * •

the mean values of the increment sequence $\vec{\xi}^{(d)}_{j}$ and stationary
sequence $\vec{\eta}_{j}$ equal to 0; the increment step $\tau>0$;

  * •

the spectral densities $f(\lambda)$ and $g(\lambda)$ satisfy the _minimality
condition_

$\int_{-\pi}^{\pi}\text{Tr}\left[\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}(f(\lambda)+\lambda^{2d}g(\lambda))^{-1}\right]d\lambda<\infty.$
(15)

The latter assumption is the necessary and sufficient condition under which
the mean square errors of the optimal estimates of the functional $A\vec{\xi}$
to be defined below is not equal to $0$.

The Hilbert space projection method of estimation may be applied under the
condition that the element, which we want to estimate, belongs to the Hilbert
space $H=L_{2}(\Omega,\mathcal{F},\mathsf{P})$ of random variables with a zero
mean value and a finite variance. That is not the case for the functional
$A{\xi}$. To overcome this difficulty we find a representation of the
functional $A{\xi}$ as a sum of a functional with finite second moment from
the space $H$ and a functional depended on the observed values of the process
$\zeta(t)=\xi(t)+\eta(t)$. This representation is described by the following
two lammas.

######  Lemma 3.1 ([19]).

The linear functional

$A\zeta=\int_{0}^{\infty}a(t)\zeta(t)dt$

allows the representation

$A\zeta=B\zeta-V\zeta,$

where

$B\zeta=\int_{0}^{\infty}b^{\tau}(t)\zeta^{(d)}(t,\tau T)dt,\quad
V\zeta=\int_{-\tau Td}^{0}v^{\tau}(t)\zeta(t)dt,$

and

$\displaystyle v^{\tau}(t)$ $\displaystyle=$
$\displaystyle\sum_{l=\left\lceil-\frac{t}{\tau
T}\right\rceil}^{d}(-1)^{l}{d\choose l}b^{\tau}(t+l\tau T),\quad t\in[-\tau
Td;0),$ (16) $\displaystyle b^{\tau}(t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}a(t+\tau Tk)d(k)=D^{\tau
T}\mathbf{a}(t),\,\,t\geq 0,$ (17)

Here $\lceil x\rceil$ denotes the least integer greater than or equal to $x$,
$[x]$ denotes the integer part of $x$, coefficients $\\{d(k):k\geq 0\\}$ are
determined by the relation

$\sum_{k=0}^{\infty}d(k)x^{k}=\left(\sum_{j=0}^{\infty}x^{j}\right)^{d},$

$D^{\tau T}$ is the linear transformation acting on an arbitrary function
$x(t)$, $t\geq 0$, as follows:

$D^{\tau T}\mathbf{x}(t)=\sum_{k=0}^{\infty}x(t+\tau Tk)d(k).$

From Lemma 3.1, we obtain the following representation of the functional
$A\xi$:

$A\xi=A\zeta-A\eta=B\zeta-A\eta-V\zeta=H\xi-V\zeta,$

where

$H\xi:=B\zeta-A\eta,$

and

$A\zeta=\int_{0}^{\infty}a(t)\zeta(t)dt,\quad
A\eta=\int_{0}^{\infty}a(t)\eta(t)dt,$
$B\zeta=\int_{0}^{\infty}b^{\tau}(t)\zeta^{(d)}(t,\tau T)dt,\quad
V\zeta=\int_{-\tau Td}^{0}v^{\tau}(t)\zeta(t)dt,$

the functions $b_{\tau}(t)$, $t\in[0;\infty)$, and $v_{\tau}(t)$, $t\in[-\tau
Td;0)$, are calculated by formulas (17) and (16) respectively.

The functional $H\xi$ allows a representation in terms of the sequences
$\vec{\eta}_{j}=(\eta_{kj},k=1,2,\dots)^{\top}$ and
$\vec{\zeta}^{(d)}_{j}=\vec{\xi}^{(d)}_{j}+\vec{\eta}^{(d)}_{j}=(\zeta^{(d)}_{kj},k=1,2,\dots)^{\top}$,
$j\in\mathbb{Z}$, which is described in the following lemma.

######  Lemma 3.2.

The functional $H\xi=B\zeta-A\eta$ can be represented in the form

$H\xi=\sum_{j=0}^{\infty}{(\vec{b}^{\tau}_{j})}^{\top}\vec{\zeta}^{(d)}_{j}-\sum_{j=0}^{\infty}{(\vec{a}_{j})}^{\top}\vec{\eta}_{j}=B\vec{\zeta}-V\vec{\eta}=:H\vec{\xi},$

where the vector

$\vec{b}^{\tau}_{j}=(b^{\tau}_{kj},k=1,2,\dots)^{\top}=(b^{\tau}_{1j},b^{\tau}_{3j},b^{\tau}_{2j},\dots,b^{\tau}_{2k+1,j},b^{\tau}_{2k,j},\dots)^{\top},$

with the entries

$b^{\tau}_{kj}=\langle
b^{\tau}_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}b^{\tau}_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv,$

and the vector

$\vec{a}_{j}=(a_{kj},k=1,2,\dots)^{\top}=(a_{1j},a_{3j},a_{2j},\dots,a_{2k+1,j},a_{2k,j},\dots)^{\top},$

with the entries

$a_{kj}=\langle
a_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}a_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv,\\\
k=1,2,\dots,\,j=0,1,\ldots,\infty.$

The coefficients $\\{\vec{a}_{j},j=0,1,\dots,\infty\\}$ and
$\\{\vec{b}^{\tau}_{j},j=0,1,\dots,\infty\\}$ are related as

$\vec{b}_{j}^{\tau}=\sum_{m=j}^{\infty}\mathsf{diag}_{\infty}(d_{\tau}(m-j))\vec{a}_{m}=(D^{\tau}{\mathbf{a}})_{j},\quad
j=0,1,\dots,\infty.$ (18)

where $\mathbf{a}=((\vec{a}_{0})^{\top},(\vec{a}_{1})^{\top},\ldots)^{\top}$,
the coefficients $\\{d_{\tau}(k):k\geq 0\\}$ are determined by the
relationship

$\sum_{k=0}^{\infty}d_{\tau}(k)x^{k}=\left(\sum_{j=0}^{\infty}x^{\tau
j}\right)^{d},$

$D^{\tau}$ is a linear transformation determined by a matrix with the infinite
dimension matrix entries $D^{\tau}(k,j),k,j=0,1,\dots$ such that
$D^{\tau}(k,j)=\mathsf{diag}_{\infty}(d_{\tau}(j-k))$ if $0\leq k\leq
j\leq\infty$ and $D^{\tau}(k,j)=\mathsf{diag}_{\infty}(0)$ for $0\leq
j<k\leq\infty$; $\mathsf{diag}_{\infty}(x)$ denotes an infinite dimensional
diagonal matrix with the entry $x$ on its diagonal.

###### Proof.

See Appendix. ∎

Assume, that coefficients $\\{\vec{a}_{j},j=0,1,\dots\\}$ and
$\\{\vec{b}^{\tau}_{j},j=0,1,\dots\\}$, that determine the functional
$H\vec{\xi}$, satisfy the conditions

$\sum_{j=0}^{\infty}\|\vec{a}_{j}\|<\infty,\quad\sum_{j=0}^{\infty}(j+1)\|\vec{a}_{j}\|<\infty,\quad\|\vec{a}_{j}\|^{2}=\sum_{k=1}^{\infty}|{a}_{kj}|^{2},$
(19)
$\sum_{j=0}^{\infty}\|\vec{b}^{\tau}_{j}\|<\infty,\quad\sum_{j=0}^{\infty}(j+1)\|\vec{b}^{\tau}_{j}\|<\infty,\quad\|\vec{b}^{\tau}_{j}\|^{2}=\sum_{k=1}^{\infty}|{b}^{\tau}_{kj}|^{2}.$
(20)

Under conditions (19) - (20) the functional $H\vec{\xi}$ has finite second
moment. Since the functional $V\zeta$ depends on the observations
$\\{\xi(t)+\eta(t):t\in[\tau Td;0)\\}$, the estimates $\widehat{A}\xi$ and
$\widehat{H}\vec{\xi}$ of the functionals $A\xi$ and $H\vec{\xi}$, as well as
the mean-square errors
$\Delta(f,g;\widehat{A}\xi)=\mathsf{E}|A\xi-\widehat{A}\xi|^{2}$ and
$\Delta(f,g;\widehat{H}\vec{\xi})=\Delta(f,g;\widehat{H}\vec{\xi})=\mathsf{E}|H\vec{\xi}-\widehat{H}\vec{\xi}|^{2}$
of the estimates $\widehat{A}\xi$ and $\widehat{H}\vec{\xi}$ satisfy the
relations

$\widehat{A}\xi=\widehat{H}\vec{\xi}-V\zeta,$ (21)
$\displaystyle\Delta(f,g;\widehat{A}\xi)$
$\displaystyle=\mathsf{E}|A\xi-\widehat{A}\xi|^{2}$
$\displaystyle=\mathsf{E}|H\vec{\xi}-V\zeta-\widehat{H}\vec{\xi}+V\zeta|^{2}$
$\displaystyle=\mathsf{E}|H\vec{\xi}-\widehat{H}\vec{\xi}|^{2}$
$\displaystyle=\Delta(f,g;\widehat{H}\vec{\xi}).$

Thus, the functional $H\vec{\xi}$ is a target element to be estimated. Let us
describe its spectral representation. Making use of representations (10) and
(9), we obtain

$H\vec{\xi}=\int_{-\pi}^{\pi}(\vec{B}_{\tau}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}d\vec{Z}_{\xi^{(d)}+\eta^{(d)}}(\lambda)-\int_{-\pi}^{\pi}(\vec{A}(e^{i\lambda}))^{\top}d\vec{Z}_{\eta}(\lambda),$

where

$\vec{B}_{\tau}(e^{i\lambda})=\sum_{j=0}^{\infty}\vec{b}_{j}^{\tau}e^{i\lambda
j}=\sum_{j=0}^{\infty}(D^{\tau}\mathbf{a})_{j}e^{i\lambda
j},\quad\vec{A}(e^{i\lambda})=\sum_{j=0}^{\infty}\vec{a}_{j}e^{i\lambda j}.$

The classical approach of estimation consists in finding a projection of the
element $H\vec{\xi}$ on the closed linear subspace of
$H=L_{2}(\Omega,\mathcal{F},\mathsf{P})$ generated by the observations. Let us
define this subspace as

$H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})=\overline{span}\\{\vec{\xi}^{(d)}_{kj}+\vec{\eta}^{(d)}_{kj}:k=1,\dots,\infty;j=-1,-2,-3,\dots\\}.$

Define also the closed linear subspaces of the Hilbert space
$L_{2}(f(\lambda)+\lambda^{2d}g(\lambda))$ of vector-valued functions endowed
by the inner product

$\langle
g_{1};g_{2}\rangle=\int_{-\pi}^{\pi}(g_{1}(\lambda))^{\top}(f(\lambda)+\lambda^{2d}g(\lambda))\overline{g_{2}(\lambda)}d\lambda$

as

$L_{2}^{0-}(f(\lambda)+\lambda^{2d}g(\lambda))=\overline{span}\\{e^{i\lambda
j}(1-e^{-i\lambda\tau})^{d}\frac{1}{(i\lambda)^{d}}\vec{\delta}_{k},\\\
k=1,2,3,\dots;\,\,j=-1,-2,-3,\dots\\},$

where $\vec{\delta}_{k}=\\{\delta_{kl}\\}_{l=1}^{\infty}$, $\delta_{kl}$ are
Kronecker symbols.

######  Remark 3.1.

Representation (10) yields a map between the elements $e^{i\lambda
j}(1-e^{-i\lambda\tau})^{d}(i\lambda)^{-d}\vec{\delta}_{k}$ of the space
$L_{2}^{0-}(f(\lambda)+\lambda^{2d}g(\lambda))$ and the elements
$\vec{\xi}^{(d)}_{kj}+\vec{\eta}^{(d)}_{kj}$ of the space
$H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})$.

The mean square optimal estimate $\widehat{H}\vec{\xi}$ is found as a
projection of the element $H\vec{\xi}$ on the subspace
$H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})$:

$\widehat{H}\vec{\xi}=\mathsf{Proj}_{H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})}H\vec{\xi}.$

Relation (21) let us write the optimal estimate $\widehat{A}\xi$ in the form

$\widehat{A}\xi=\mathsf{Proj}_{H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})}H\vec{\xi}-V\zeta$

or in the form

$\widehat{A}\xi=\int_{-\pi}^{\pi}(\vec{h}_{\tau}(\lambda))^{\top}d\vec{Z}_{\xi^{(d)}+\eta^{(d)}}(\lambda)-\int_{-\tau
Td}^{0}v^{\tau}(t)\zeta(t)dt,$ (22)

where $\vec{h}_{\tau}(\lambda)=\\{h_{k}^{\tau}(\lambda)\\}_{k=1}^{\infty}$ is
the spectral characteristic of the estimate $\widehat{H}\vec{\xi}$.

Denote

$\vec{A}_{\tau}(e^{i\lambda})=(1-e^{i\lambda\tau})^{d}\vec{A}(e^{i\lambda})=\sum_{j=0}^{\infty}\vec{a}_{j}^{\tau}e^{i\lambda
j},$

where

$\vec{a}_{j}^{\tau}=\sum_{l=0}^{\min\left\\{\left[j/\tau\right],d\right\\}}(-1)^{l}{d\choose
l}\vec{a}(j-\tau l),\quad j\geq 0.$ (23)

Define the vector

$\mathbf{a}^{\tau}=((\vec{a}^{\tau}_{0})^{\top},(\vec{a}^{\tau}_{1})^{\top},(\vec{a}^{\tau}_{2})^{\top},\ldots)^{\top}.$

With the help of the Fourier coefficients

$\displaystyle T^{\tau}_{l,j}$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i\lambda(j-l)}\dfrac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\left[g(\lambda)(f(\lambda)+\lambda^{2d}g(\lambda))^{-1}\right]^{\top}d\lambda,\quad
l,j\geq 0,$ $\displaystyle P_{l,j}^{\tau}$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i\lambda(j-l)}\dfrac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\left[(f(\lambda)+\lambda^{2d}g(\lambda))^{-1}\right]^{\top}d\lambda,\quad
l,j\geq 0,$ $\displaystyle Q_{l,j}$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i\lambda(j-l)}\left[f(\lambda)(f(\lambda)+\lambda^{2d}g(\lambda))^{-1}g(\lambda)\right]^{\top}d\lambda,\quad
l,j\geq 0,$

of the corresponding matrix functions, define the linear operators
$\mathbf{P}_{\tau}$, $\mathbf{T}_{\tau}$ and $\mathbf{Q}$ in the space
$\ell_{2}$ by matrices with the infinite dimensional matrix entries
$(\mathbf{P}_{\tau})_{l,j}=P_{l,j}^{\tau}$,
$(\mathbf{T}_{\tau})_{l,j}=T^{\tau}_{l,j}$ and $(\mathbf{Q})_{l,j}=Q_{l,j}$,
$l,j\geq 0$.

Notation:
$\langle\vec{x},\vec{y}\rangle=\sum_{j=0}^{\infty}(\vec{x}_{j})^{\top}\overline{\vec{y}}_{j}$
for vectors
$\vec{x}=((\vec{x}_{0})^{\top},(\vec{x}_{1})^{\top},(\vec{x}_{2})^{\top},\ldots)^{\top}$,
$\vec{y}=((\vec{y}_{0})^{\top},(\vec{y}_{1})^{\top},(\vec{y}_{2})^{\top},\ldots)^{\top}$.

######  Theorem 3.1.

Consider two uncorrelated processes: a stochastic process $\xi(t)$,
$t\in\mathbb{R}$ with a periodically stationary increments, which determines a
generated stationary $d$th increment sequence $\vec{\xi}^{(d)}_{j}$ with the
spectral density matrix $f(\lambda)=\\{f_{kn}(\lambda)\\}_{k,n=1}^{\infty}$,
and a periodically stationary stochastic process $\eta(t)$, $t\in\mathbb{R}$,
which determines a generated stationary sequence $\vec{\eta}_{j}$ with the
spectral density matrix $g(\lambda)=\\{g_{kn}(\lambda)\\}_{k,n=1}^{\infty}$.
Let the coefficients $\vec{a}_{j}$, $\vec{b}^{\tau}_{j}$, $j=0,1,\ldots,$
generated by the function $a(t)$, $t\geq 0$, satisfy conditions (19) – (20).
Let minimality condition (15) be satisfied. The optimal linear estimate
$\widehat{A}\xi$ of the functional $A\xi$ based on observations of the process
$\xi(t)+\eta(t)$ at points $t<0$ is calculated by formula (22). The spectral
characteristic $\vec{h}_{\tau}(\lambda)$ is calculated by formula

$\displaystyle(\vec{h}_{\tau}(\lambda))^{\top}$
$\displaystyle=(\vec{B}_{\tau}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}$
$\displaystyle\quad-\left((\vec{A}_{\tau}(e^{i\lambda}))^{\top}g(\lambda)+(\vec{C}_{\tau}(e^{i\lambda}))^{\top}\right)\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}(f(\lambda)+\lambda^{2d}g(\lambda))^{-1},$
(24)

where

$\vec{C}_{\tau}(e^{i\lambda})=\sum_{j=0}^{\infty}(\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau})_{j}e^{i\lambda
j}.$

The value of the mean-square error is calculated by the formula

$\displaystyle\Delta(f,g;\widehat{A}\vec{\xi})$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}(\mathbf{C}^{f}_{\tau}(e^{i\lambda}))^{\top}(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}f(\lambda)$
$\displaystyle\quad\quad\times(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}\overline{\mathbf{C}^{f}_{\tau}(e^{i\lambda})}d\lambda$
$\displaystyle\quad+\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\lambda^{4n}}{|1-e^{i\lambda\tau}|^{4n}}(\mathbf{C}^{g}_{\tau}(e^{i\lambda}))^{\top}(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}g(\lambda)$
$\displaystyle\quad\quad\times(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}\overline{\mathbf{C}^{g}_{\tau}(e^{i\lambda})}d\lambda$
$\displaystyle=\langle
D^{\tau}\mathbf{a}-\mathbf{T}_{\tau}\mathbf{a}^{\tau},\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau}\rangle+\langle\mathbf{Q}\mathbf{a},\mathbf{a}\rangle.$
(25)

where

$\displaystyle{\mathbf{C}^{f}_{\tau}(e^{i\lambda})}$
$\displaystyle:=\overline{g(\lambda)}\vec{A}_{\tau}(e^{i\lambda})+\vec{C}_{\tau}(e^{i\lambda}),$
$\displaystyle{\mathbf{C}}^{g}_{\tau}(e^{i\lambda})$
$\displaystyle:={|1-e^{i\lambda\tau}|^{2d}}\lambda^{-2d}\overline{f^{0}(\lambda)}\vec{A}(e^{i\lambda})-{(1-e^{-i\lambda\tau})^{d}}\vec{C}_{\tau}(e^{i\lambda}).$

###### Proof.

See Appendix. ∎

######  Corollary 3.1.

The spectral characteristics $\vec{h}_{\tau}^{1}(\lambda)$ and
$\vec{h}_{\tau}^{2}(\lambda)$ of the optimal estimates
$\widehat{B}\vec{\zeta}$ and $\widehat{A}\vec{\eta}$ of the functionals
$B\vec{\zeta}$ and $A\vec{\eta}$ based on observations $\xi(t)+\eta(t)$ at
points $t<0$ are calculated by the formulas

$\displaystyle(\vec{h}_{\tau}^{1}(\lambda))^{\top}$
$\displaystyle=(\vec{B}_{\tau}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}$
$\displaystyle\quad-\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}\left(\sum_{j=0}^{\infty}(\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a})_{j}e^{i\lambda
j}\right)^{\top}(f(\lambda)+{\lambda}^{2d}g(\lambda)^{-1},$
$\displaystyle(\vec{h}_{\tau}^{2}(\lambda))^{\top}$
$\displaystyle=(\vec{A}_{\tau}(e^{i\lambda}))^{\top}{g(\lambda)}\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}$
$\displaystyle\quad-\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}\left(\sum_{j=0}^{\infty}(\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau})_{j}e^{i\lambda
j}\right)^{\top}(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1},$

respectively.

The optimal estimate $\widehat{A}_{NT}\xi$ of the functional $A_{NT}\xi$ which
depend on the unknown values of the process $\xi(t)$ at points
$t\in[0,T(N+1)]$, based on observations of the process $\xi(t)+\eta(t)$ at
points $t<0$ can be obtained by using Theorem 3.1.

We first formulate the following corollaries from Lemma 3.1 and Lemma 3.2.

######  Corollary 3.2.

The linear functional

$A_{NT}\xi=\int_{0}^{(N+1)T}a(t)\xi(t)dt$

allows the representation

$A_{NT}\xi=B_{NT}\xi-V_{NT}\xi,$

where

$B_{NT}\xi=\int_{0}^{(N+1)T}b^{\tau,N}(t)\xi^{(d)}(t,\tau T)dt,\quad
V_{NT}\xi=\int_{-\tau Td}^{0}v^{\tau,N}(t)\xi(t)dt,$

and

$\displaystyle v^{\tau,N}(t)$ $\displaystyle=\sum_{l=\left\lceil-\frac{t}{\tau
T}\right\rceil}^{\min\left\\{\left[\frac{{(N+1)T}-t}{\tau
T}\right],d\right\\}}(-1)^{l}{d\choose l}b^{\tau,N}(t+l\tau T),\quad
t\in[-\tau Td;0),$ (26) $\displaystyle b^{\tau,N}(t)$
$\displaystyle=\sum_{k=0}^{\left[\frac{{(N+1)T}-t}{\tau T}\right]}a(t+\tau
Tk)d(k)=D^{\tau T,N}\mathbf{a}(t).t\in[0;{(N+1)T}],$ (27)

The linear transformation $D^{\tau T,N}$ acts on an arbitrary function $x(t)$,
$t\in[0;{(N+1)T}]$, as follows

$D^{\tau T,N}\mathbf{x}(t)=\sum_{k=0}^{\left[\frac{{(N+1)T}-t}{\tau
T}\right]}x(t+\tau Tk)d(k).$

######  Corollary 3.3.

The functional

$H_{NT}\xi=B_{NT}\zeta-A_{NT}\eta$
$B_{NT}\zeta=\int_{0}^{T(N+1)}b^{\tau,N}(t)\zeta^{(d)}(t,\tau T)dt,\quad
A_{NT}\eta=\int_{0}^{T(N+1)}a(t)\eta(t)dt,$

can be represented in the form

$H_{NT}\xi=\sum_{j=0}^{N}{(\vec{b}^{\tau,N}_{j})}^{\top}\vec{\zeta}^{(d)}_{j}-\sum_{j=0}^{N}{(\vec{a}_{j})}^{\top}\vec{\eta}_{j}=B_{N}\vec{\zeta}-V_{N}\vec{\eta}=:H_{N}\vec{\xi},$

where the vector

$\vec{b}^{\tau,N}_{j}=(b^{\tau,N}_{kj},k=1,2,\dots)^{\top}=(b^{\tau,N}_{1j},b^{\tau,N}_{3j},b^{\tau,N}_{2j},\dots,b^{\tau,N}_{2k+1,j},b^{\tau,N}_{2k,j},\dots)^{\top},$

with the entries

$b^{\tau,N}_{kj}=\langle
b^{\tau,N}_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}b^{\tau,N}_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv.$

The coefficients $\\{\vec{a}_{j},j=0,1,\dots,N\\}$ and
$\\{\vec{b}^{\tau,N}_{j},j=0,1,\dots,N\\}$ are related as

$\vec{b}_{j}^{\tau,N}=\sum_{m=j}^{N}\mathsf{diag}_{\infty}(d_{\tau}(m-j))\vec{a}_{m}=(D^{\tau}_{N}{\mathbf{a}_{N}})_{j},\quad
j=0,1,\dots,N.$ (28)

where
$\mathbf{a}_{N}=((\vec{a}_{0})^{\top},(\vec{a}_{1})^{\top},\ldots,(\vec{a}_{N})^{\top},0,\ldots)^{\top}$,
$D^{\tau}_{N}$ is a linear transformation determined by a matrix with the
infinite dimension matrix entries $D^{\tau}_{N}(k,j),k,j\geq 0$ such that
$D^{\tau}_{N}(k,j)=\mathsf{diag}_{\infty}(d_{\tau}(j-k))$ if $0\leq k\leq
j\leq N$ and $D^{\tau}_{N}(k,j)=\mathsf{diag}_{\infty}(0)$ for $0\leq j<k$ or
$j,k>N$.

Put $a(t)=0$, $t>T(N+1)$. Define vector coefficients
$\\{\vec{a}_{j}^{\tau,N}:0\leq j\leq N+\tau d\\}$ by the formula

$\vec{a}_{j}^{\tau,N}=\sum_{l=\max\left\\{\left\lceil\frac{j-N}{\tau}\right\rceil,0\right\\}}^{\min\left\\{\left[\frac{j}{\tau}\right],d\right\\}}(-1)^{l}{d\choose
l}\vec{a}(j-\tau l),\quad 0\leq j\leq N+\tau d,$

and a vector

$\mathbf{a}^{\tau}_{N}=((\vec{a}_{0}^{\tau,N})^{\top},(\vec{a}_{1}^{\tau,N})^{\top},(\vec{a}_{2}^{\tau,N})^{\top},\ldots,(\vec{a}_{N+\tau
d}^{\tau,N})^{\top},0\ldots)^{\top}.$

The following theorem holds true.

######  Theorem 3.2.

Consider two uncorrelated processes: a stochastic process $\xi(t)$,
$t\in\mathbb{R}$ with periodically stationary increments, which determines a
generated stationary $d$th increment sequence $\vec{\xi}^{(d)}_{j}$ with the
spectral density matrix $f(\lambda)=\\{f_{kn}(\lambda)\\}_{k,n=1}^{\infty}$,
and a periodically stationary stochastic process $\eta(t)$, $t\in\mathbb{R}$,
which determines a generated stationary sequence $\vec{\eta}_{j}$ with the
spectral density matrix $g(\lambda)=\\{g_{kn}(\lambda)\\}_{k,n=1}^{\infty}$.
Let the coefficients $\vec{a}_{j}$, $\vec{b}^{\tau}_{j}$, $j=0,1,\ldots,N$
generated by the function $a(t)$, $0\leq t\leq T(N+1)$, satisfy conditions

$\|\vec{a}_{j}\|<\infty,\quad\|\vec{a}_{j}\|^{2}=\sum_{k=1}^{\infty}|{a}_{kj}|^{2},\quad
j=0,1,\dots,N,$

and

$\|\vec{b}^{\tau,N}_{j}\|<\infty,\quad\|\vec{b}^{\tau,N}_{j}\|^{2}=\sum_{k=1}^{\infty}|{b}^{\tau,N}_{kj}|^{2},\quad
j=0,1,\dots,N.$

Let minimality condition (15) be satisfied. The optimal linear estimate
$\widehat{A}_{NT}\xi$ of the functional $A_{NT}\xi$ based on observations of
the process $\xi(t)+\eta(t)$ at points $t<0$ is calculated by formula

$\widehat{A}_{NT}\xi=\int_{-\pi}^{\pi}(\vec{h}_{\tau,N}(\lambda))^{\top}d\vec{Z}_{\xi^{(d)}+\eta^{(d)}}(\lambda)-\int_{-\tau
Td}^{0}v_{\tau,N}(t)\zeta(t)dt,$

where the spectral characteristic $\vec{h}_{\tau,N}(\lambda)$ of the optimal
estimate $\widehat{A}_{NT}\xi$ is calculated by formula

$\displaystyle(\vec{h}_{\tau,N}(\lambda))^{\top}$
$\displaystyle=\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\left(\sum_{j=0}^{N}(D^{\tau}_{N}\mathbf{a}_{N})_{j}e^{i\lambda
j}\right)^{\top}$
$\displaystyle\quad-\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}\left(\sum_{j=0}^{N+\tau
d}\vec{a}^{\tau,N}_{j}e^{i\lambda
j}\right)^{\top}g(\lambda)(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}$
$\displaystyle\quad-\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}\left(\sum_{j=0}^{\infty}(\mathbf{P}_{\tau}^{-1}D_{N}^{\tau}\mathbf{a}_{N}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}_{\tau,N})_{j}e^{i\lambda
j}\right)^{\top}$
$\displaystyle\quad\quad\times(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1},$

The value of the mean-square error is calculated by formula

$\displaystyle\Delta(f,g;\widehat{A}_{NT}\xi)$
$\displaystyle=\Delta(f,g;\widehat{H}_{N}\vec{\xi})=\mathsf{E}|H_{N}\vec{\xi}-\widehat{H}_{N}\vec{\xi}|^{2}=$
$\displaystyle=\langle
D_{N}^{\tau}\mathbf{a}_{N}-\mathbf{T}_{\tau}\mathbf{a}_{\tau,N},\mathbf{P}_{\tau}^{-1}D_{N}^{\tau}\mathbf{a}_{N}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}_{\tau,N}\rangle$
$\displaystyle\quad+\langle\mathbf{Q}_{N}\mathbf{a}_{N},\mathbf{a}_{N}\rangle,$

where $\mathbf{Q}_{N}$ is a linear operator in the space $\ell_{2}$ defined by
the matrix with the infinite dimensional matrix entries
$(\mathbf{Q}_{N})_{l,j}=Q_{l,j}$, $0\leq l,j\leq N$, and
$(\mathbf{Q}_{N})_{l,j}=0$ otherwise.

##  4 Prediction based on factorizations of spectral densities

Assume that the following canonical factorizations take place

$\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f(\lambda)+\lambda^{2d}g(\lambda))=\Theta_{\tau}(e^{-i\lambda})\Theta_{\tau}^{*}(e^{-i\lambda}),\\\
\Theta_{\tau}(e^{-i\lambda})=\sum_{k=0}^{\infty}\Theta_{\tau}(k)e^{-i\lambda
k},$ (29)

$g(\lambda)=\sum_{k=-\infty}^{\infty}g(k)e^{i\lambda
k}=\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda}),\quad\Phi(e^{-i\lambda})=\sum_{k=0}^{\infty}\phi(k)e^{-i\lambda
k}.$ (30)

Define the matrix-valued function
$\Psi_{\tau}(e^{-i\lambda})=\\{\Psi_{ij}(e^{-i\lambda})\\}_{i=\overline{1,M}}^{j=\overline{1,\infty}}$
by the equation

$\Psi_{\tau}(e^{-i\lambda})\Theta_{\tau}(e^{-i\lambda})=E_{M},$

where $E_{M}$ is an identity $M\times M$ matrix. Then the following
factorization takes place

$\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}(f(\lambda)+\lambda^{2d}g(\lambda))^{-1}=\Psi_{\tau}^{*}(e^{-i\lambda})\Psi_{\tau}(e^{-i\lambda}),\,\Psi_{\tau}(e^{-i\lambda})=\sum_{k=0}^{\infty}\psi_{\tau}(k)e^{-i\lambda
k},$ (31)

######  Remark 4.1.

Any spectral density matrix $f(\lambda)$ is self-adjoint:
$f(\lambda)=f^{*}(\lambda)$. Thus,
$(f(\lambda))^{\top}=\overline{f(\lambda)}$. One can check that an inverse
spectral density $f^{-1}(\lambda)$ is also self-adjoint
$f^{-1}(\lambda)=(f^{-1}(\lambda))^{*}$ and
$(f^{-1}(\lambda))^{\top}=\overline{f^{-1}(\lambda)}$.

The following Lemmas provide factorizations of the operators
$\mathbf{P}_{\tau}$ and $\mathbf{T}_{\tau}$, which contain coefficients of
factorizations (29) – (31).

######  Lemma 4.1.

Let factorization (29) takes place and let $M\times\infty$ matrix function
$\Psi_{\tau}(e^{-i\lambda})$ satisfy equation
$\Psi_{\tau}(e^{-i\lambda})\Theta_{\tau}(e^{-i\lambda})=E_{M}$. Define the
linear operators $\Psi_{\tau}$ and $\Theta_{\tau}$ in the space $\ell_{2}$ by
the matrices with the matrix entries $(\Psi_{\tau})_{k,j}=\psi_{\tau}(k-j)$,
$(\Theta_{\tau})_{k,j}=\Theta_{\tau}(k-j)$ for $0\leq j\leq k$,
$(\Psi_{\tau})_{k,j}=0$, $(\Theta_{\tau})_{k,j}=0$ for $0\leq k<j$. Then:
a) the linear operator $\mathbf{P}_{\tau}$ admits the factorization

$\mathbf{P}_{\tau}=(\Psi_{\tau})^{\top}\overline{\Psi}_{\tau};$

b) the inverse operator $(\mathbf{P}_{\tau})^{-1}$ admits the factorization

$(\mathbf{P}_{\tau})^{-1}=\overline{\Theta}_{\tau}(\Theta_{\tau})^{\top}.$

######  Lemma 4.2.

Let factorizations (29) and (30) take place. Then the operator
$\mathbf{T}_{\tau}$ admits the representation

$\mathbf{T}_{\tau}=(\Psi_{\tau})^{\top}\mathbf{Z}_{\tau},$

where $\mathbf{Z}_{\tau}$ is a linear operator in the space $\ell_{2}$ defined
by a matrix with the entries

$\displaystyle(\mathbf{Z}_{\tau})_{k,j}$ $\displaystyle=$
$\displaystyle\sum_{l=j}^{\infty}\overline{\psi}_{\tau}(l-j)\overline{g}(l-k),\quad
k,j\geq 0$ $\displaystyle g(k)$ $\displaystyle=$
$\displaystyle\sum_{m=\max\\{0,-k\\}}^{\infty}\phi(m)\phi^{*}(k+m),\quad
k\in\mathbb{Z}.$

######  Remark 4.2.

Lemma 4.1 and Lemma 4.2 imply the factorization

$(\mathbf{P}_{\tau})^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau}=\overline{\Theta}_{\tau}(\Theta_{\tau})^{\top}(\Psi_{\tau})^{\top}\mathbf{Z}_{\tau}\mathbf{a}^{\tau}=\overline{\Theta}_{\tau}\mathbf{Z}_{\tau}\mathbf{a}^{\tau}=\overline{\Theta}_{\tau}\mathbf{e}_{\tau},$

where $\mathbf{e}_{\tau}:=\mathbf{Z}_{\tau}\mathbf{a}^{\tau}$.

The proofs of Lemma 4.1 and Lemma 4.2, as well as the justification of the
following representations of the spectral characteristics
$\vec{h}_{\tau}^{1}(\lambda)$ and $\vec{h}_{\tau}^{2}(\lambda)$, correspond to
the ones in [18] for the finite-dimensional vector stationary increment
sequences.

The spectral characteristic $\vec{h}_{\tau}^{2}(\lambda)$ of the optimal
estimate $\widehat{{A}}\vec{\eta}$ from Corollary 3.1 can be presented as

$\vec{h}_{\tau}^{2}(\lambda)=\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\Psi^{\top}_{\tau}(e^{-i\lambda})\vec{C}_{\tau,g}(e^{-i\lambda}),$

where
$\overline{\psi}_{\tau}=(\overline{\psi}_{\tau}(0),\overline{\psi}_{\tau}(1),\overline{\psi}_{\tau}(2),\ldots)$,

$\vec{C}_{\tau,g}(e^{-i\lambda})=\sum_{m=1}^{\infty}(\overline{\psi}_{\tau}\mathbf{C}_{\tau,g})_{m}e^{-i\lambda
m},$ $\displaystyle(\overline{\psi}_{\tau}\mathbf{C}_{\tau,g})_{m}$
$\displaystyle=\sum_{k=0}^{\infty}\overline{\psi}_{\tau}(k)\mathbf{c}_{\tau,g}(k+m),$
$\displaystyle\mathbf{c}_{\tau,g}(m)$
$\displaystyle=\sum_{k=0}^{\infty}\overline{g}(m+k)\vec{a}^{\tau}_{k}=\sum_{l=0}^{\infty}\overline{\phi}(l)\sum_{k=0}^{\infty}\phi^{\top}(l+m+k)\vec{a}^{\tau}_{k}$
$\displaystyle=\sum_{l=0}^{\infty}\overline{\phi}(l)(\widetilde{\Phi}\mathbf{a}^{\tau})_{l+m},$
$\displaystyle(\widetilde{\Phi}\mathbf{a}^{\tau})_{m}$
$\displaystyle=\sum_{k=0}^{\infty}\phi^{\top}(m+k)\vec{a}^{\tau}_{k}.$

The spectral characteristic $\vec{h}_{\tau}^{2}(\lambda)$ of the optimal
estimate $\widehat{B}\vec{\xi}$ from Corollary 3.1 can be presented as

$\displaystyle\vec{h}_{\tau}^{1}(\lambda)$
$\displaystyle=\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\left(\vec{B}_{\tau}(e^{i\lambda})-\Psi^{\top}_{\tau}(e^{-i\lambda})\vec{r}_{\tau}(e^{i\lambda})\right)$
$\displaystyle=\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\Psi^{\top}_{\tau}(e^{-i\lambda})\vec{C}_{\tau,1}(e^{-i\lambda}),$

where

$\displaystyle\vec{r}_{\tau}(e^{i\lambda})$
$\displaystyle=\sum_{m=0}^{\infty}(\theta^{\top}_{\tau}D^{\tau}\mathbf{A})_{m}e^{i\lambda
m}=\sum_{m=0}^{\infty}\sum_{p=0}^{\infty}\theta^{\top}_{\tau}(p)\vec{b}^{\tau}_{p+m}e^{i\lambda
m},$ $\displaystyle\vec{C}_{\tau,1}(e^{-i\lambda})$
$\displaystyle=\sum_{m=1}^{\infty}(\theta^{\top}_{\tau}\widetilde{\mathbf{B}}_{\tau})_{m}e^{-i\lambda
m}=\sum_{m=1}^{\infty}\sum_{p=m}^{\infty}\theta^{\top}_{\tau}(p)\vec{b}^{\tau}_{p-m}e^{-i\lambda
m}$
$\displaystyle=\sum_{m=1}^{\infty}\sum_{p=0}^{\infty}\theta^{\top}_{\tau}(m+p)\vec{b}^{\tau}_{p}e^{-i\lambda
m},$

a vector
$\theta^{\top}_{\tau}=((\Theta_{\tau}(0))^{\top},(\Theta_{\tau}(1))^{\top},(\Theta_{\tau}(2))^{\top},\ldots)$;
$\mathbf{A}$ is a linear symmetric operator determined by the matrix with the
vector entries $(\mathbf{A})_{k,j}=\vec{a}_{k+j}$, $k,j\geq 0$;
$\widetilde{\mathbf{B}}_{\tau}$ is a linear operator, which is determined by a
matrix with the vector entries
$(\widetilde{\mathbf{B}}_{\tau})_{k,j}=\vec{b}^{\tau}_{k-j}$ for $0\leq j\leq
k$, $(\widetilde{\mathbf{B}}_{\tau})_{k,j}=0$ for $0\leq k<j$.

Then the spectral characteristic $\vec{h}_{\tau}(\lambda)$ of the estimate
$\widehat{A}\xi$ can be calculated by the formula

$\displaystyle\vec{h}_{\tau}(\lambda)$
$\displaystyle=\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\tau}(k)e^{-i\lambda
k}\right)\sum_{m=1}^{\infty}\left(\theta^{\top}_{\tau}\widetilde{\mathbf{B}}_{\tau}-\overline{\psi}_{\tau}\mathbf{C}_{\tau,g}\right)_{m}e^{-i\lambda
m}$
$\displaystyle=\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\Psi^{\top}_{\tau}(e^{-i\lambda})\left(\vec{C}_{\tau,1}(e^{-i\lambda})-\vec{C}_{\tau,g}(e^{-i\lambda})\right)$
$\displaystyle=\vec{B}_{\tau}(e^{i\lambda})\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}-\widetilde{h}_{\tau}(\lambda),$
(32)

where

$\displaystyle\widetilde{\vec{h}}_{\tau}(\lambda)$
$\displaystyle=\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\tau}(k)e^{-i\lambda
k}\right)$
$\displaystyle\quad\times\left(\sum_{m=0}^{\infty}(\theta^{\top}_{\tau}D^{\tau}\mathbf{A})_{m}e^{i\lambda
m}+\sum_{m=1}^{\infty}(\overline{\psi}_{\tau}\mathbf{C}_{\tau,g})_{m}e^{-i\lambda
m}\right)$
$\displaystyle=\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\Psi^{\top}_{\tau}(e^{-i\lambda})\left(\vec{r}_{\tau}(e^{i\lambda})+\vec{C}_{\tau,g}(e^{-i\lambda})\right),$

The value of the mean square error of the estimate $\widehat{A}\xi$ is
calculated by the formula

$\displaystyle\Delta(f,g;\widehat{A}\xi)$
$\displaystyle=\Delta(f,g;\widehat{H}\vec{\xi})=\mathsf{E}|H\vec{\xi}-\widehat{H}\vec{\xi}|^{2}$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}(\vec{A}(e^{i\lambda}))^{\top}g(\lambda)\overline{\vec{A}(e^{i\lambda})}d\lambda$
$\displaystyle\quad+\frac{1}{2\pi}\int_{-\pi}^{\pi}(\widetilde{\vec{h}}_{\tau}(e^{i\lambda}))^{\top}(f(\lambda)+\lambda^{2d}g(\lambda))\overline{\widetilde{\vec{h}}_{\tau}(e^{i\lambda})}d\lambda$
$\displaystyle\quad-\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(i\lambda)^{d}}{(1-e^{-i\lambda\tau})^{d}}(\widetilde{h}_{\tau}(e^{i\lambda}))^{\top}g(\lambda)\overline{\vec{A}^{\tau}(e^{i\lambda})}d\lambda$
$\displaystyle\quad-\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}(\vec{A}^{\tau}(e^{i\lambda}))^{\top}g(\lambda)\overline{\widetilde{\vec{h}}_{\tau}(e^{i\lambda})}d\lambda$
$\displaystyle=\|\Phi^{\top}\mathbf{a}^{\tau}\|^{2}+\|\widetilde{\Phi}\mathbf{a}^{\tau}\|_{1}^{2}+\left\langle\theta^{\top}_{\tau}D^{\tau}\mathbf{A}-\overline{\psi}_{\tau}\mathbf{C}_{\tau,g},\theta^{\top}_{\tau}D^{\tau}\mathbf{A}\right\rangle$
$\displaystyle\quad-\left\langle\theta^{\top}_{\tau}D^{\tau}\mathbf{A},\mathbf{Z}_{\tau}\mathbf{a}^{\tau}\right\rangle-\left\langle\mathbf{Z}_{\tau}\mathbf{a}^{\tau},\overline{\psi}_{\tau}\mathbf{C}_{\tau,g}\right\rangle_{1},$
(33)

where $\|\vec{x}\|_{1}^{2}=\langle\vec{x},\vec{x}\rangle_{1}$,
$\langle\vec{x},\vec{y}\rangle_{1}=\sum_{j=1}^{\infty}(\vec{x}_{j})^{\top}\overline{\vec{y}}_{j}$
for the vectors

$\vec{x}=((\vec{x}_{0})^{\top},(\vec{x}_{1})^{\top},(\vec{x}_{2})^{\top},\ldots)^{\top},\quad\vec{y}=((\vec{y}_{0})^{\top},(\vec{y}_{1})^{\top},(\vec{y}_{2})^{\top},\ldots)^{\top}.$

The obtained results are summarized in the form of the following theorem.

######  Theorem 4.1.

Let the conditions of Theorem 3.1 be fulfilled and the spectral densities
$f(\lambda)$ and $g(\lambda)$ of the stochastic processes $\xi(t)$ and
$\eta(t)$ admit canonical factorizations (29) – (31). Then the spectral
characteristic $\vec{h}_{\tau}(\lambda)$ and the value of the mean square
error $\Delta(f,g;\widehat{A}\xi)$ of the optimal estimate $\widehat{A}\xi$ of
the functional $A\xi$ based on observations of the processes $\xi(t)+\eta(t)$
at points $t<0$ can be calculated by formulas (32) and (33) respectively.

##  5 Minimax (robust) method of prediction

Consider the estimation problem for the functional ${A}\xi$ based on the
observations $\xi(t)+\eta(t)$ at points $t<0$ when the spectral densities of
sequences are not exactly known while a set
$\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ of admissible spectral
densities is defined. The minimax (robust) approach of estimation is applied.
It is formalized by the following two definitions.

######  Definition 5.1.

For a given class of spectral densities
$\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ the spectral densities
$f^{0}(\lambda)\in\mathcal{D}_{f}$, $g^{0}(\lambda)\in\mathcal{D}_{g}$ are
called least favorable in the class $\mathcal{D}$ for the optimal linear
prediction of the functional $A\xi$ if the following relation holds true:

$\Delta(f^{0},g^{0})=\Delta(h(f^{0},g^{0});f^{0},g^{0})=\max_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}\Delta(h(f,g);f,g).$

######  Definition 5.2.

For a given class of spectral densities
$\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ the spectral characteristic
$h^{0}(\lambda)$ of the optimal linear estimate of the functional $A\xi$ is
called minimax-robust if there are satisfied the conditions

$h^{0}(\lambda)\in
H_{\mathcal{D}}=\bigcap_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}L_{2}^{0-}(f(\lambda)+\lambda^{2d}g(\lambda)),$

and

$\min_{h\in
H_{\mathcal{D}}}\max_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}\Delta(h;f,g)=\max_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}\Delta(h^{0};f,g).$

Taking into account the introduced definitions and the derived relations we
can verify that the following lemmas hold true.

######  Lemma 5.1.

Spectral densities $f^{0}\in\mathcal{D}_{f}$, $g^{0}\in\mathcal{D}_{g}$ which
satisfy condition (15) are least favorable in the class
$\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ for the optimal linear
prediction of the functional $A\vec{\xi}$ if operators
$\mathbf{P}_{\tau}^{0}$, $\mathbf{T}_{\tau}^{0}$, $\mathbf{Q}^{0}$ defined by
the Fourier coefficients of the functions

$\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\left[g^{0}(\lambda)(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))^{-1}\right]^{\top},$
$\dfrac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\left[(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))^{-1}\right]^{\top},$
$\left[f^{0}(\lambda)(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))^{-1}g^{0}(\lambda)\right]^{\top}$

determine a solution of the constrained optimization problem

$\max_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}(\langle
D^{\tau}\mathbf{a}-\mathbf{T}_{\tau}\mathbf{a}^{\tau},\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau}\rangle+\langle\mathbf{Q}\mathbf{a},\mathbf{a}\rangle)\\\
=\langle
D^{\tau}\mathbf{a}-\mathbf{T}^{0}_{\tau}\mathbf{a}^{\tau},(\mathbf{P}^{0}_{\tau})^{-1}D^{\tau}\mathbf{a}-(\mathbf{P}^{0}_{\tau})^{-1}\mathbf{T}^{0}_{\tau}\mathbf{a}^{\tau}\rangle+\langle\mathbf{Q}^{0}\mathbf{a},\mathbf{a}\rangle.$
(34)

The minimax spectral characteristic $h^{0}=h_{\tau}(f^{0},g^{0})$ is
calculated by formula (24) if $h_{\tau}(f^{0},g^{0})\in H_{\mathcal{D}}$.

######  Lemma 5.2.

The spectral densities $f^{0}\in\mathcal{D}_{f}$, $g^{0}\in\mathcal{D}_{g}$
which admit canonical factorizations (14), (29) and (30) are least favourable
densities in the class $\mathcal{D}$ for the optimal linear prediction of the
functional $A\xi$ based on observations of the process $\xi(t)+\eta(t)$ at
points $t<0$ if the matrix coefficients of canonical factorizations (29) and
(30) determine a solution to the constrained optimization problem

$\displaystyle\|\Phi^{\top}\mathbf{a}^{\tau}\|^{2}$
$\displaystyle+\|\widetilde{\Phi}\mathbf{a}^{\tau}\|_{1}^{2}+\left\langle\theta^{\top}_{\tau}D^{\tau}\mathbf{A}-\overline{\psi}_{\tau}\mathbf{C}_{\tau,g},\theta^{\top}_{\tau}D^{\tau}\mathbf{A}\right\rangle$
$\displaystyle-\left\langle\theta^{\top}_{\tau}D^{\tau}\mathbf{A},\mathbf{Z}_{\tau}\mathbf{a}^{\tau}\right\rangle-\left\langle\mathbf{Z}_{\tau}\mathbf{a}^{\tau},\overline{\psi}_{\tau}\mathbf{C}_{\tau,g}\right\rangle_{1}\rightarrow\sup,$
(35)

for

$\displaystyle f(\lambda)$
$\displaystyle=\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\Theta_{\tau}(e^{-i\lambda})\Theta_{\tau}^{*}(e^{-i\lambda})-\lambda^{2d}\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda})\in\mathcal{D}_{f},$
$\displaystyle g(\lambda)$
$\displaystyle=\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda})\in\mathcal{D}_{g}.$

The minimax spectral characteristic $\vec{h}^{0}=\vec{h}_{\tau}(f^{0},g^{0})$
is calculated by formula (32) if $\vec{h}_{\tau}(f^{0},g^{0})\in
H_{\mathcal{D}}$.

######  Lemma 5.3.

The spectral density $g^{0}\in\mathcal{D}_{g}$ which admits canonical
factorizations (29), (30) with the known spectral density $f(\lambda)$ is the
least favourable in the class $\mathcal{D}_{g}$ for the optimal linear
prediction of the functional $A\xi$ based on observations of the process
$\xi(t)+\eta(t)$ at points $t<0$ if the matrix coefficients of the canonical
factorizations

$\displaystyle f(\lambda)+\lambda^{2d}g^{0}(\lambda)$
$\displaystyle=\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\left(\sum_{k=0}^{\infty}\theta^{0}_{\tau}(k)e^{-i\lambda
k}\right)\left(\sum_{k=0}^{\infty}\theta^{0}_{\tau}(k)e^{-i\lambda
k}\right)^{*},$ (36) $\displaystyle g^{0}(\lambda)$
$\displaystyle=\left(\sum_{k=0}^{\infty}\phi^{0}(k)e^{-i\lambda
k}\right)\left(\sum_{k=0}^{\infty}\phi^{0}(k)e^{-i\lambda k}\right)^{*}$ (37)

and the equation
$\Psi^{0}_{\tau}(e^{-i\lambda})\Theta^{0}_{\tau}(e^{-i\lambda})=E_{M}$
determine a solution to the constrained optimization problem

$\displaystyle\|\Phi^{\top}\mathbf{a}^{\tau}\|^{2}$
$\displaystyle+\|\widetilde{\Phi}\mathbf{a}^{\tau}\|_{1}^{2}+\left\langle\theta^{\top}_{\tau}D^{\tau}\mathbf{A}-\overline{\psi}_{\tau}\mathbf{C}_{\tau,g},\theta^{\top}_{\tau}D^{\tau}\mathbf{A}\right\rangle$
$\displaystyle-\left\langle\theta^{\top}_{\tau}D^{\tau}\mathbf{A},\mathbf{Z}_{\tau}\mathbf{a}^{\tau}\right\rangle-\left\langle\mathbf{Z}_{\tau}\mathbf{a}^{\tau},\overline{\psi}_{\tau}\mathbf{C}_{\tau,g}\right\rangle_{1}\rightarrow\sup,$
(38)

for

$g(\lambda)=\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda})\in\mathcal{D}_{g}.$

The minimax spectral characteristic $\vec{h}^{0}=\vec{h}_{\tau}(f,g^{0})$ is
calculated by formula (32) if $\vec{h}_{\tau}(f,g^{0})\in H_{\mathcal{D}}$.

######  Lemma 5.4.

The spectral density $f^{0}\in\mathcal{D}_{f}$ which admits canonical
factorizations (14), (29) with the known spectral density $g(\lambda)$ is the
least favourable spectral density in the class $\mathcal{D}_{f}$ for the
optimal linear prediction of the functional $A\xi$ based on observations of
the process $\xi(t)+\eta(t)$ at points $t<0$ if matrix coefficients of the
canonical factorization

$f^{0}(\lambda)+\lambda^{2d}g(\lambda)=\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\left(\sum_{k=0}^{\infty}\theta^{0}_{\tau}(k)e^{-i\lambda
k}\right)\left(\sum_{k=0}^{\infty}\theta^{0}_{\tau}(k)e^{-i\lambda
k}\right)^{*},$ (39)

and the equation
$\Psi^{0}_{\tau}(e^{-i\lambda})\Theta^{0}_{\tau}(e^{-i\lambda})=E_{M}$
determine a solution to the constrained optimization problem

$\displaystyle\left\langle\theta^{\top}_{\tau}D^{\tau}\mathbf{A}-\overline{\psi}_{\tau}\mathbf{C}_{\tau,g},\theta^{\top}_{\tau}D^{\tau}\mathbf{A}\right\rangle-\left\langle\theta^{\top}_{\tau}D^{\tau}\mathbf{A},\mathbf{Z}_{\tau}\mathbf{a}^{\tau}\right\rangle-\left\langle\mathbf{Z}_{\tau}\mathbf{a}^{\tau},\overline{\psi}_{\tau}\mathbf{C}_{\tau,g}\right\rangle_{1}\rightarrow\sup,$
(40)

for

$\displaystyle f(\lambda)$
$\displaystyle=\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\Theta_{\tau}(e^{-i\lambda})\Theta_{\tau}^{*}(e^{-i\lambda})-\lambda^{2d}\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda})\in\mathcal{D}_{f}$

for the fixed matrix coefficients $\\{\phi(k):k\geq 0\\}$. The minimax
spectral characteristic $\vec{h}^{0}=\vec{h}_{\tau}(f^{0},g)$ is calculated by
formula (32) if $\vec{h}_{\tau}(f^{0},g)\in H_{\mathcal{D}}$.

For more detailed analysis of properties of the least favorable spectral
densities and minimax-robust spectral characteristics we observe that the
minimax spectral characteristic $h^{0}$ and the least favourable spectral
densities $(f^{0},g^{0})$ form a saddle point of the function $\Delta(h;f,g)$
on the set $H_{\mathcal{D}}\times\mathcal{D}$.

The saddle point inequalities

$\Delta(h;f^{0},g^{0})\geq\Delta(h^{0};f^{0},g^{0})\geq\Delta(h^{0};f,g)\quad\forall
f\in\mathcal{D}_{f},\forall g\in\mathcal{D}_{g},\forall h\in H_{\mathcal{D}}$

hold true if $h^{0}=\vec{h}_{\tau}(f^{0},g^{0})$ and
$\vec{h}_{\tau}(f^{0},g^{0})\in H_{\mathcal{D}}$, where $(f^{0},g^{0})$ is a
solution of the constrained optimisation problem

$\widetilde{\Delta}(f,g)=-\Delta(\vec{h}_{\tau}(f^{0},g^{0});f,g)\to\inf,\quad(f,g)\in\mathcal{D},$
(41)

where the functional $\Delta(\vec{h}_{\tau}(f^{0},g^{0});f,g)$ is calculated
by the formula

$\displaystyle\Delta(\vec{h}_{\tau}(f^{0},g^{0});f,g)=$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}(\mathbf{C}^{f0}_{\tau}(e^{i\lambda}))^{\top}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))^{-1}f(\lambda)$
$\displaystyle\quad\quad\times(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))^{-1}\overline{\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}d\lambda$
$\displaystyle\quad+\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\lambda^{4n}}{|1-e^{i\lambda\tau}|^{4n}}(\mathbf{C}^{g0}_{\tau}(e^{i\lambda}))^{\top}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))^{-1}g(\lambda)$
$\displaystyle\quad\quad\times(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))^{-1}\overline{\mathbf{C}^{g0}_{\tau}(e^{i\lambda})}d\lambda,$

where

$\displaystyle{\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}$
$\displaystyle:=\overline{g^{0}(\lambda)}\vec{A}_{\tau}(e^{i\lambda})+\sum_{j=0}^{\infty}((\mathbf{P}^{0}_{\tau})^{-1}D^{\tau}\mathbf{a}-(\mathbf{P}^{0}_{\tau})^{-1}\mathbf{T}^{0}_{\tau}\mathbf{a}^{\tau})_{j}e^{i\lambda
j},$ $\displaystyle{\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})$
$\displaystyle:={|1-e^{i\lambda\tau}|^{2d}}\lambda^{-2d}\overline{f^{0}(\lambda)}\vec{A}(e^{i\lambda})$
$\displaystyle\quad-{(1-e^{-i\lambda\tau})^{d}}\sum_{j=0}^{\infty}((\mathbf{P}^{0}_{\tau})^{-1}D^{\tau}\mathbf{a}-(\mathbf{P}^{0}_{\tau})^{-1}\mathbf{T}^{0}_{\tau}\mathbf{a}^{\tau})_{j}e^{i\lambda
j}.$

or it is calculated by the formula

$\displaystyle\Delta\left(\vec{h}_{\tau}(f^{0},g^{0});f,g\right)=$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(\mathbf{r}^{0}_{\tau,f}(e^{-i\lambda}))^{\top}\Psi^{0}_{\tau}(e^{-i\lambda})f(\lambda)(\Psi^{0}_{\tau}(e^{-i\lambda}))^{*}\overline{\mathbf{r}^{0}_{\tau,f}(e^{-i\lambda})}d\lambda$
$\displaystyle\quad+\frac{1}{2\pi}\int_{-\pi}^{\pi}(\mathbf{r}^{0}_{\tau,g}(e^{-i\lambda}))^{\top}\Psi^{0}_{\tau}(e^{-i\lambda})g(\lambda)(\Psi^{0}_{\tau}(e^{-i\lambda}))^{*}\overline{\mathbf{r}^{0}_{\tau,g}(e^{-i\lambda})}d\lambda,$

where

$\displaystyle\mathbf{r}^{0}_{\tau,f}(e^{-i\lambda})$
$\displaystyle=\sum_{m=0}^{\infty}((\theta^{0}_{\tau})^{\top}D^{\tau}\mathbf{A})_{m}e^{i\lambda
m}+\sum_{m=1}^{\infty}(\overline{\psi}^{0}_{\tau}\mathbf{C}^{0}_{\tau,g})_{m}e^{-i\lambda
m},$ $\displaystyle\mathbf{r}^{0}_{\tau,g}(e^{-i\lambda})$
$\displaystyle=(1-e^{-i\lambda\tau})^{d}\left(\sum_{m=0}^{\infty}((\theta^{0}_{\tau})^{\top}D^{\tau}\mathbf{A})_{m}e^{i\lambda
m}+\sum_{m=1}^{\infty}(\overline{\psi}_{\tau}^{0}\mathbf{C}^{0}_{\tau,g})_{m}e^{-i\lambda
m}\right)$
$\displaystyle\quad-(\Theta^{0}_{\tau}(e^{-i\lambda}))^{\top}A(e^{i\lambda}).$

The constrained optimization problem (41) is equivalent to the unconstrained
optimisation problem

$\Delta_{\mathcal{D}}(f,g)=\widetilde{\Delta}(f,g)+\delta(f,g|\mathcal{D}_{f}\times\mathcal{D}_{g})\to\inf,$
(42)

where $\delta(f,g|\mathcal{D}_{f}\times\mathcal{D}_{g})$ is the indicator
function of the set $\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$.
Solution $(f^{0},g^{0})$ of this unconstrained optimization problem is
characterized by the condition
$0\in\partial\Delta_{\mathcal{D}}(f^{0},g^{0})$, where
$\partial\Delta_{\mathcal{D}}(f^{0},g^{0})$ is the subdifferential of the
functional $\Delta_{\mathcal{D}}(f,g)$ at point
$(f^{0},g^{0})\in\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$, that is
the set of all continuous linear functionals $\Lambda$ on $L_{1}\times L_{1}$
which satisfy the inequality
$\Delta_{\mathcal{D}}(f,g)-\Delta_{\mathcal{D}}(f^{0},g^{0})\geq\Lambda((f,g)-(f^{0},g^{0})),(f,g)\in\mathcal{D}$
(see [24, 31] for more details). This condition makes it possible to find the
least favourable spectral densities in some special classes of spectral
densities $\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$.

The form of the functionals $\Delta(\vec{h}_{\tau}(f^{0},g^{0});f,g)$,
$\Delta(\vec{h}_{\tau}(f^{0},p^{0});f,p)$ is convenient for application the
Lagrange method of indefinite multipliers for finding solution to the problem
(42). Making use of the method of Lagrange multipliers and the form of
subdifferentials of the indicator functions
$\delta(f,g|\mathcal{D}_{f}\times\mathcal{D}_{g})$,
$\delta(f,p|\mathcal{D}_{f}\times\mathcal{D}_{p})$ of the sets
$\mathcal{D}_{f}\times\mathcal{D}_{g}$, $\mathcal{D}_{f}\times\mathcal{D}_{p}$
of spectral densities, we describe relations that determine least favourable
spectral densities in some special classes of spectral densities (see [16, 24]
for additional details).

###  5.1 Least favorable spectral densities in classes
$\mathcal{D}_{0}\times\mathcal{D}_{1\delta}$

Consider the prediction problem for the functional $A{\xi}$ which depends on
unobserved values of a process $\xi(t)$ with stationary increments based on
observations of the process $\xi(t)+\eta(t)$ at points $t<0$ under the
condition that the sets of admissible spectral densities
$\mathcal{D}_{0}^{k},{D_{1\delta}^{k}},k=1,2,3,4$ are defined as follows:

$\displaystyle\mathcal{D}_{0}^{1}$
$\displaystyle=\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f(\lambda)d\lambda=P\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{0}^{2}$
$\displaystyle=\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}{\rm{Tr}}\,[f(\lambda)]d\lambda=p\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{0}^{3}$
$\displaystyle=\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{kk}(\lambda)d\lambda=p_{k},k=\overline{1,\infty}\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{0}^{4}$
$\displaystyle=\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}\left\langle
B_{1},f(\lambda)\right\rangle d\lambda=p\right.\bigg{\\}},$

and

$\displaystyle\mathcal{D}_{1\delta}^{1}$
$\displaystyle=\left\\{g(\lambda)\biggl{|}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|g_{ij}(\lambda)-g_{ij}^{1}(\lambda)\right|d\lambda\leq\delta_{i}^{j},i,j=\overline{1,\infty}\right\\}.$
$\displaystyle\mathcal{D}_{1\delta}^{2}$
$\displaystyle=\left\\{g(\lambda)\biggl{|}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|{\rm{Tr}}(g(\lambda)-g_{1}(\lambda))\right|d\lambda\leq\delta\right\\};$
$\displaystyle\mathcal{D}_{1\delta}^{3}$
$\displaystyle=\left\\{g(\lambda)\biggl{|}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|g_{kk}(\lambda)-g_{kk}^{1}(\lambda)\right|d\lambda\leq\delta_{k},k=\overline{1,\infty}\right\\};$
$\displaystyle\mathcal{D}_{1\delta}^{4}$
$\displaystyle=\left\\{g(\lambda)\biggl{|}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\left\langle
B_{2},g(\lambda)-g_{1}(\lambda)\right\rangle\right|d\lambda\leq\delta\right\\};$

Here $g_{1}(\lambda)=\\{g_{ij}^{1}(\lambda)\\}_{i,j=1}^{\infty}$ is a fixed
spectral density, $p,p_{k},k=\overline{1,\infty}$, are given numbers, $P$,
$B_{2}$ are a given positive-definite Hermitian matrices,
$\delta,\delta_{k},k=\overline{1,\infty}$,
$\delta_{i}^{j},i,j=\overline{1,\infty}$, are given numbers.

From the condition $0\in\partial\Delta_{\mathcal{D}}(f^{0},g^{0})$ we find the
following equations which determine the least favourable spectral densities
for these given sets of admissible spectral densities.

For the first set of admissible spectral densities
$\mathcal{D}_{0}^{1}\times\mathcal{D}_{1\delta}^{1}$, we have equations

$\left({\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}\right)\left({\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}\right)^{*}=\\\
=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)\vec{\alpha}_{f}\cdot\vec{\alpha}_{f}^{*}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(43)
$\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)\left\\{\beta_{ij}\gamma_{ij}(\lambda)\right\\}_{i,j=1}^{\infty}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(44)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|g^{0}_{ij}(\lambda)-g_{ij}^{1}(\lambda)\right|d\lambda=\delta_{i}^{j},$
(45)

where $\vec{\alpha}_{f}$, $\beta_{ij}$ are Lagrange multipliers, functions
$\left|\gamma_{ij}(\lambda)\right|\leq 1$ and

$\gamma_{ij}(\lambda)=\frac{g_{ij}^{0}(\lambda)-g_{ij}^{1}(\lambda)}{\left|g_{ij}^{0}(\lambda)-g_{ij}^{1}(\lambda)\right|}:\;g_{ij}^{0}(\lambda)-g_{ij}^{1}(\lambda)\neq
0,\;i,j=\overline{1,\infty}.$

For the second set of admissible spectral densities
$\mathcal{D}_{0}^{2}\times\mathcal{D}_{1\delta}^{2}$, we have equations

$\left({\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}\right)\left({\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}\right)^{*}=\alpha_{f}^{2}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)^{2},$
(46)
$\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)^{*}=\beta^{2}\gamma_{2}(\lambda)\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)^{2},$
(47)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|{\mathrm{Tr}}\,(g_{0}(\lambda)-g_{1}(\lambda))\right|d\lambda=\delta,$
(48)

where $\alpha_{f}^{2}$, $\beta^{2}$ are Lagrange multipliers, the function
$\left|\gamma_{2}(\lambda)\right|\leq 1$ and

$\gamma_{2}(\lambda)={\mathrm{sign}}\;({\mathrm{Tr}}\,(g_{0}(\lambda)-g_{1}(\lambda))):\;{\mathrm{Tr}}\,(g_{0}(\lambda)-g_{1}(\lambda))\neq
0.$

For the third set of admissible spectral densities
$\mathcal{D}_{0}^{3}\times\mathcal{D}_{1\delta}^{3}$, we have equations

$\left({\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}\right)\left({\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}\right)^{*}=\\\
=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)\left\\{\alpha_{fk}^{2}\delta_{kl}\right\\}_{k,l=1}^{\infty}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(49)
$\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)\left\\{\beta_{k}^{2}\gamma^{2}_{k}(\lambda)\delta_{kl}\right\\}_{k,l=1}^{\infty}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(50)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|g^{0}_{kk}(\lambda)-g_{kk}^{1}(\lambda)\right|d\lambda=\delta_{k},$
(51)

where $\alpha_{fk}^{2}$, $\beta_{k}^{2}$ are Lagrange multipliers,
$\delta_{kl}$ are Kronecker symbols, functions
$\left|\gamma^{2}_{k}(\lambda)\right|\leq 1$ and

$\gamma_{k}^{2}(\lambda)={\mathrm{sign}}\;(g_{kk}^{0}(\lambda)-g_{kk}^{1}(\lambda)):\;g_{kk}^{0}(\lambda)-g_{kk}^{1}(\lambda)\neq
0,\;k=\overline{1,\infty}.$

For the fourth set of admissible spectral densities
$\mathcal{D}_{0}^{4}\times\mathcal{D}_{1\delta}^{4}$, we have equations

$\left({\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}\right)\left({\mathbf{C}^{f0}_{\tau}(e^{i\lambda})}\right)^{*}=\\\
=\alpha_{f}^{2}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)B_{1}^{\top}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(52)
$\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=\beta^{2}\gamma_{2}^{\prime}(\lambda)\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)B_{2}^{\top}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(53) $\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\left\langle
B_{2},g^{0}(\lambda)-g_{1}(\lambda)\right\rangle\right|d\lambda=\delta,$ (54)

where $\alpha_{f}^{2}$, $\beta^{2}$ are Lagrange multipliers, function
$\left|\gamma_{2}^{\prime}(\lambda)\right|\leq 1$ and

$\gamma_{2}^{\prime}(\lambda)={\mathrm{sign}}\;\left\langle
B_{2},g^{0}(\lambda)-g_{1}(\lambda)\right\rangle:\;\left\langle
B_{2},g^{0}(\lambda)-g_{1}(\lambda)\right\rangle\neq 0.$

The derived results are summarized in the following theorem.

######  Theorem 5.1.

The least favorable spectral densities $f^{0}(\lambda),$ $g^{0}(\lambda),$ in
the classes $\mathcal{D}_{0}^{k}\times\mathcal{D}_{1\delta}^{k},k=1,2,3,4$ for
the optimal linear predictionof the functional $A{\xi}$ from observations of
the process ${\xi}(t)+{\eta}(t)$ at points $t<0$ are determined by equations
(43)–(45), (46)–(48), (49)–(51), (52)–(54), respectively, the minimality
condition (15), the constrained optimization problem (34) and restrictions on
densities from the corresponding classes
$\mathcal{D}_{0}^{k},\mathcal{D}_{1\delta}^{k},k=1,2,3,4$. The minimax-robust
spectral characteristic of the optimal estimate of the functional $A{\xi}$ is
determined by formula (24).

In the case where the spectral densities $f(\lambda)$ and $g(\lambda)$ admit
canonical factorizations (29) and (30), the equation for the least favourable
spectral densities are described below.

For the first set of admissible spectral densities
$\mathcal{D}_{0}^{1}\times\mathcal{D}_{1\delta}^{1}$:

$\displaystyle\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\Theta_{\tau}(e^{-i\lambda}))^{\top}\vec{\alpha}_{f}\cdot\vec{\alpha}_{f}^{*}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(55)
$\displaystyle\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\Theta_{\tau}(e^{-i\lambda}))^{\top}\left\\{\beta_{ij}\gamma_{ij}(\lambda)\right\\}_{i,j=1}^{T}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(56)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|g^{0}_{ij}(\lambda)-g_{ij}^{1}(\lambda)\right|d\lambda=\delta_{i}^{j},$
(57)

where $\vec{\alpha}_{f}$, $\beta_{ij}$ are Lagrange multipliers, functions
$\left|\gamma_{ij}(\lambda)\right|\leq 1$ and

$\gamma_{ij}(\lambda)=\frac{g_{ij}^{0}(\lambda)-g_{ij}^{1}(\lambda)}{\left|g_{ij}^{0}(\lambda)-g_{ij}^{1}(\lambda)\right|}:\;g_{ij}^{0}(\lambda)-g_{ij}^{1}(\lambda)\neq
0,\;i,j=\overline{1,\infty}.$

For the second set of admissible spectral densities
$\mathcal{D}_{0}^{2}\times\mathcal{D}_{1\delta}^{2}$:

$\displaystyle\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=\alpha_{f}^{2}(\Theta_{\tau}(e^{-i\lambda}))^{\top}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(58)
$\displaystyle\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=\beta^{2}\gamma_{2}(\lambda)(\Theta_{\tau}(e^{-i\lambda}))^{\top}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(59)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|{\mathrm{Tr}}\,(g_{0}(\lambda)-g_{1}(\lambda))\right|d\lambda=\delta,$
(60)

where $\alpha_{f}^{2}$, $\beta^{2}$ are Lagrange multipliers, function
$\left|\gamma_{2}(\lambda)\right|\leq 1$ and

$\gamma_{2}(\lambda)={\mathrm{sign}}\;({\mathrm{Tr}}\,(g_{0}(\lambda)-g_{1}(\lambda))):\;{\mathrm{Tr}}\,(g_{0}(\lambda)-g_{1}(\lambda))\neq
0.$

For the third set of admissible spectral densities
$\mathcal{D}_{0}^{3}\times\mathcal{D}_{1\delta}^{3}$:

$\displaystyle\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\Theta_{\tau}(e^{-i\lambda}))^{\top}\left\\{\alpha_{fk}^{2}\delta_{kl}\right\\}_{k,l=1}^{T}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(61)
$\displaystyle\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\Theta_{\tau}(e^{-i\lambda}))^{\top}\left\\{\beta_{k}^{2}\gamma^{2}_{k}(\lambda)\delta_{kl}\right\\}_{k,l=1}^{T}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(62)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|g^{0}_{kk}(\lambda)-g_{kk}^{1}(\lambda)\right|d\lambda=\delta_{k},$
(63)

where $\alpha_{fk}^{2}$, $\beta_{k}^{2}$ are Lagrange multipliers,
$\delta_{kl}$ are Kronecker symbols, functions
$\left|\gamma^{2}_{k}(\lambda)\right|\leq 1$ and

$\gamma_{k}^{2}(\lambda)={\mathrm{sign}}\;(g_{kk}^{0}(\lambda)-g_{kk}^{1}(\lambda)):\;g_{kk}^{0}(\lambda)-g_{kk}^{1}(\lambda)\neq
0,\;k=\overline{1,\infty}.$

For the fourth set of admissible spectral densities
$\mathcal{D}_{0}^{4}\times\mathcal{D}_{1\delta}^{4}$:

$\displaystyle\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=\alpha_{f}^{2}(\Theta_{\tau}(e^{-i\lambda}))^{\top}B_{1}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(64)
$\displaystyle\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=\beta^{2}\gamma_{2}^{\prime}(\lambda)(\Theta_{\tau}(e^{-i\lambda}))^{\top}B_{2}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(65) $\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\left\langle
B_{2},g^{0}(\lambda)-g_{1}(\lambda)\right\rangle\right|d\lambda=\delta,$ (66)

where $\alpha_{f}^{2}$, $\beta^{2}$ are Lagrange multipliers, function
$\left|\gamma_{2}^{\prime}(\lambda)\right|\leq 1$ and

$\gamma_{2}^{\prime}(\lambda)={\mathrm{sign}}\;\left\langle
B_{2},g^{0}(\lambda)-g_{1}(\lambda)\right\rangle:\;\left\langle
B_{2},g^{0}(\lambda)-g_{1}(\lambda)\right\rangle\neq 0.$

The following theorems hold true.

######  Theorem 5.2.

The least favorable spectral densities $f^{0}(\lambda)$, $g^{0}(\lambda)$ in
the classes $\mathcal{D}_{0}^{k}\times\mathcal{D}_{1\delta}^{k}$, $k=1,2,3,4$
for the optimal linear prediction of the functional $A{\xi}$ from observations
of the process $\xi(t)+\eta(t)$ at points $t<0$ are determined by canonical
factorizations (29) and (30), equations (55)–(57), (58)–(60), (61)–(63),
(64)–(66), respectively, constrained optimization problem (35) and
restrictions on densities from the corresponding classes
$\mathcal{D}_{0}^{k},\mathcal{D}_{1\delta}^{k},k=1,2,3,4$. The minimax-robust
spectral characteristic of the optimal estimate of the functional $A\vec{\xi}$
is determined by formula (32).

######  Theorem 5.3.

In the case where the spectral density $g(\lambda)$ is known, the least
favorable spectral density $f^{0}(\lambda)$ in the classes
$\mathcal{D}_{0}^{k}$, $k=1,2,3,4$ for the optimal linear predictionof the
functional $A{\xi}$ from observations of the process ${\xi}(t)+{\eta}(t)$ at
points $t<0$ is determined by canonical factorizations (30) and (29),
equations (55), (58), (61), (64), respectively, constrained optimization
problem (40) and restrictions on density from the corresponding classes
$\mathcal{D}_{0}^{k}$, $k=1,2,3,4$. The minimax-robust spectral characteristic
of the optimal estimate of the functional $A{\xi}$ is determined by formula
(32).

###  5.2 Least favorable spectral densities in classes
$\mathcal{D}_{\varepsilon}\times\mathcal{D}_{V}^{U}$

Consider the prediction problem for the functional $A{\xi}$ depending on
unobserved values of the process $\vec{\xi}(m)$ with stationary increments
based on observations of the process $\xi(t)+\eta(t)$ at points $t<0$ under
the condition that the sets of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{k},\mathcal{D}_{V}^{Uk},k=1,2,3,4$ are defined as
follows:

$\mathcal{D}_{\varepsilon}^{1}=\bigg{\\{}f(\lambda)\bigg{|}f(\lambda)=(1-\varepsilon)f_{1}(\lambda)+\varepsilon
W(\lambda),\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f(\lambda)d\lambda=P\bigg{\\}},$
$\mathcal{D}_{\varepsilon}^{2}=\bigg{\\{}f(\lambda)\bigg{|}{\mathrm{Tr}}\,[f(\lambda)]=(1-\varepsilon){\mathrm{Tr}}\,[f_{1}(\lambda)]+\varepsilon{\mathrm{Tr}}\,[W(\lambda)],\\\
\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}{\mathrm{Tr}}\,[f(\lambda)]d\lambda=p\bigg{\\}};$
$\mathcal{D}_{\varepsilon}^{3}=\bigg{\\{}f(\lambda)\bigg{|}f_{kk}(\lambda)=(1-\varepsilon)f_{kk}^{1}(\lambda)+\varepsilon
w_{kk}(\lambda),\\\
\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{kk}(\lambda)d\lambda=p_{k},k=\overline{1,\infty}\bigg{\\}};$
$\mathcal{D}_{\varepsilon}^{4}=\bigg{\\{}f(\lambda)\bigg{|}\left\langle
B_{1},f(\lambda)\right\rangle=(1-\varepsilon)\left\langle
B_{1},f_{1}(\lambda)\right\rangle+\varepsilon\left\langle
B_{1},W(\lambda)\right\rangle,\\\
\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}\left\langle
B_{1},f(\lambda)\right\rangle d\lambda=p\bigg{\\}};$

and

$\displaystyle{\mathcal{D}_{V}^{U1}}$
$\displaystyle=\bigg{\\{}g(\lambda)\bigg{|}V(\lambda)\leq g(\lambda)\leq
U(\lambda),\frac{1}{2\pi}\int_{-\pi}^{\pi}g(\lambda)d\lambda=Q\bigg{\\}},$
$\displaystyle{\mathcal{D}_{V}^{U2}}$
$\displaystyle=\bigg{\\{}g(\lambda)\bigg{|}{\mathrm{Tr}}\,[V(\lambda)]\leq{\mathrm{Tr}}\,[g(\lambda)]\leq{\mathrm{Tr}}\,[U(\lambda)],\frac{1}{2\pi}\int_{-\pi}^{\pi}{\mathrm{Tr}}\,[g(\lambda)]d\lambda=q\bigg{\\}},$
$\displaystyle{\mathcal{D}_{V}^{U3}}$
$\displaystyle=\bigg{\\{}g(\lambda)\bigg{|}v_{kk}(\lambda)\leq
g_{kk}(\lambda)\leq
u_{kk}(\lambda),\frac{1}{2\pi}\int_{-\pi}^{\pi}g_{kk}(\lambda)d\lambda=q_{k},k=\overline{1,\infty}\bigg{\\}},$
$\displaystyle{\mathcal{D}_{V}^{U4}}$
$\displaystyle=\bigg{\\{}g(\lambda)\bigg{|}\left\langle
B_{2},V(\lambda)\right\rangle\leq\left\langle
B_{2},g(\lambda)\right\rangle\leq\left\langle
B_{2},U(\lambda)\right\rangle,\frac{1}{2\pi}\int_{-\pi}^{\pi}\left\langle
B_{2},g(\lambda)\right\rangle d\lambda=q\bigg{\\}}.$

Here $f_{1}(\lambda)$, $V(\lambda)$, $U(\lambda)$ are known and fixed spectral
densities, $W(\lambda)$ is an unknown spectral density,
$p,p_{k},q,q_{k},k=\overline{1,\infty}$, are given numbers, $P,B_{1},Q,B_{2}$
are given positive-definite Hermitian matrices.

The condition $0\in\partial\Delta_{\mathcal{D}}(f^{0},g^{0})$ implies the
following equations determining the least favourable spectral densities for
these given sets of admissible spectral densities.

For the first set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{1}\times\mathcal{D}_{V}^{U1}$, we have equations

$\left({\mathbf{C}}^{f0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{f0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)(\vec{\alpha}_{f}\cdot\vec{\alpha}^{*}+\Gamma(\lambda))\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(67)
$\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)(\vec{\beta}\cdot\vec{\beta}^{*}+\Gamma_{1}(\lambda)+\Gamma_{2}(\lambda))\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(68)

where $\vec{\alpha}_{f}$ and $\vec{\beta}$ are vectors of Lagrange
multipliers, function $\Gamma(\lambda)\leq 0$ and $\Gamma(\lambda)=0$ if
$f_{0}(\lambda)>(1-\varepsilon)f_{1}(\lambda)$, the matrix
$\Gamma_{1}(\lambda)\leq 0$ and $\Gamma_{1}(\lambda)=0$ if
$g_{0}(\lambda)>V(\lambda),$ the matrix $\Gamma_{2}(\lambda)\geq 0$ and
$\Gamma_{2}(\lambda)=0$ if $g_{0}(\lambda)<U(\lambda).$

For the second set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{2}\times\mathcal{D}_{V}^{U2}$, we have equations

$\left({\mathbf{C}}^{f0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{f0}_{\tau}(e^{i\lambda})\right)^{*}=(\alpha_{f}^{2}+\gamma(\lambda))\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)^{2},$
(69)
$\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=(\beta^{2}+\gamma_{1}(\lambda)+\gamma_{2}(\lambda))\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)^{2},$
(70)

where $\alpha_{f}^{2}$, $\beta^{2}$ are Lagrange multipliers, function
$\gamma(\lambda)\leq 0$ and $\gamma(\lambda)=0$ if
${\mathrm{Tr}}\,[f^{0}(\lambda)]>(1-\varepsilon){\mathrm{Tr}}\,[f_{1}(\lambda)]$,
function $\gamma_{1}(\lambda)\leq 0$ and $\gamma_{1}(\lambda)=0$ if
${\mathrm{Tr}}\,[g^{0}(\lambda)]>{\mathrm{Tr}}\,[V(\lambda)],$ the function
$\gamma_{2}(\lambda)\geq 0$ and $\gamma_{2}(\lambda)=0$ if
${\mathrm{Tr}}\,[g^{0}(\lambda)]<{\mathrm{Tr}}\,[U(\lambda)].$

For the third set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{3}\times\mathcal{D}_{V}^{U3}$, we have equation

$\left({\mathbf{C}}^{f0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{f0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)\left\\{(\alpha_{fk}^{2}+\gamma_{k}(\lambda))\delta_{kl}\right\\}_{k,l=1}^{\infty}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(71)
$\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)\left\\{(\beta_{k}^{2}+\gamma_{1k}(\lambda)+\gamma_{2k}(\lambda))\delta_{kl}\right\\}_{k,l=1}^{\infty}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(72)

where $\alpha_{fk}^{2}$, $\beta_{k}^{2}$ are Lagrange multipliers,
$\delta_{kl}$ are Kronecker symbols, functions $\gamma_{k}(\lambda)\leq 0$ and
$\gamma_{k}(\lambda)=0$ if
$f_{kk}^{0}(\lambda)>(1-\varepsilon)f_{kk}^{1}(\lambda)$, functions
$\gamma_{1k}(\lambda)\leq 0$ and $\gamma_{1k}(\lambda)=0$ if
$g_{kk}^{0}(\lambda)>v_{kk}(\lambda),$ functions $\gamma_{2k}(\lambda)\geq 0$
and $\gamma_{2k}(\lambda)=0$ if $g_{kk}^{0}(\lambda)<u_{kk}(\lambda).$

For the forth set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{4}\times\mathcal{D}_{V}^{U4}$, we have equation

$\left({\mathbf{C}}^{f0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{f0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=(\alpha_{f}^{2}+\gamma^{\prime}(\lambda))\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)B_{1}^{\top}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(73)
$\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)\left({\mathbf{C}}^{g0}_{\tau}(e^{i\lambda})\right)^{*}=\\\
=(\beta^{2}+\gamma^{\prime}_{1}(\lambda)+\gamma^{\prime}_{2}(\lambda))\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right)B_{2}^{\top}\times\\\
\times\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2d}g^{0}(\lambda))\right),$
(74)

where $\alpha_{f}^{2}$, $\beta^{2}$, are Lagrange multipliers, function
$\gamma^{\prime}(\lambda)\leq 0$ and $\gamma^{\prime}(\lambda)=0$ if $\langle
B_{1},f^{0}(\lambda)\rangle>(1-\varepsilon)\langle
B_{1},f_{1}(\lambda)\rangle$, functions $\gamma^{\prime}_{1}(\lambda)\leq 0$
and $\gamma^{\prime}_{1}(\lambda)=0$ if $\langle
B_{2},g^{0}(\lambda)\rangle>\langle B_{2},V(\lambda)\rangle,$ functions
$\gamma^{\prime}_{2}(\lambda)\geq 0$ and $\gamma^{\prime}_{2}(\lambda)=0$ if
$\langle B_{2},g^{0}(\lambda)\rangle<\langle B_{2},U(\lambda)\rangle.$

The following theorem holds true.

######  Theorem 5.4.

The least favorable spectral densities $f^{0}(\lambda)$, $g^{0}(\lambda)$ in
classes $\mathcal{D}_{\varepsilon}^{k}\times\mathcal{D}_{V}^{Uk},k=1,2,3,4$
for the optimal linear extrapolation of the functional $A{\xi}$ from
observations of the process $\xi(t)+\eta(t)$ at points $t<0$ are determined by
equations (67)–(68), (69)–(70), (71)–(72), (73)–(74), respectively, the
minimality condition (15), the constrained optimization problem (34) and
restrictions on densities from the corresponding classes
$\mathcal{D}_{\varepsilon}^{k},\mathcal{D}_{V}^{Uk},k=1,2,3,4$. The minimax-
robust spectral characteristic of the optimal estimate of the functional
$A{\xi}$ is determined by formula (24).

Let the spectral densities $f(\lambda)$ and $g(\lambda)$ admit the canonical
factorizations (29) and (30). The equations for the least favourable spectral
densities are described below.

For the first set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{1}\times\mathcal{D}_{V}^{U1}$, we have equation

$\displaystyle\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\Theta_{\tau}(e^{-i\lambda}))^{\top}(\vec{\alpha}_{f}\cdot\vec{\alpha}_{f}^{*}+\Gamma(\lambda))\overline{\Theta_{\tau}(e^{-i\lambda})},$
(75)
$\displaystyle\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\Theta_{\tau}(e^{-i\lambda}))^{\top}(\vec{\beta}\cdot\vec{\beta}^{*}+\Gamma_{1}(\lambda)+\Gamma_{2}(\lambda))\overline{\Theta_{\tau}(e^{-i\lambda})}$
(76)

where $\vec{\alpha}_{f}$ and $\vec{\beta}$ are vectors of Lagrange
multipliers, matrix $\Gamma(\lambda)\leq 0$ and $\Gamma(\lambda)=0$ if
$f^{0}(\lambda)>(1-\varepsilon)f_{1}(\lambda)$, matrix
$\Gamma_{1}(\lambda)\leq 0$ and $\Gamma_{1}(\lambda)=0$ if
$g^{0}(\lambda)>V(\lambda),$ matrix $\Gamma_{2}(\lambda)\geq 0$ and
$\Gamma_{2}(\lambda)=0$ if $g^{0}(\lambda)<U(\lambda).$

For the second set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{2}\times\mathcal{D}_{V}^{U2}$, we have equations

$\displaystyle\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\alpha_{f}^{2}+\gamma(\lambda))(\Theta_{\tau}(e^{-i\lambda}))^{\top}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(77)
$\displaystyle\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\beta^{2}+\gamma_{1}(\lambda)+\gamma_{2}(\lambda))(\Theta_{\tau}(e^{-i\lambda}))^{\top}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(78)

where $\alpha_{f}^{2}$, $\beta^{2}$ are Lagrange multipliers, function
$\gamma(\lambda)\leq 0$ and $\gamma(\lambda)=0$ if
${\mathrm{Tr}}\,[f^{0}(\lambda)]>(1-\varepsilon){\mathrm{Tr}}\,[f_{1}(\lambda)]$,
function $\gamma_{1}(\lambda)\leq 0$ and $\gamma_{1}(\lambda)=0$ if
${\mathrm{Tr}}\,[g^{0}(\lambda)]>{\mathrm{Tr}}\,[V(\lambda)],$ the function
$\gamma_{2}(\lambda)\geq 0$ and $\gamma_{2}(\lambda)=0$ if
${\mathrm{Tr}}\,[g^{0}(\lambda)]<{\mathrm{Tr}}\,[U(\lambda)].$

For the third set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{3}\times\mathcal{D}_{V}^{U3}$, we have equation

$\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)^{*}=(\Theta_{\tau}(e^{-i\lambda}))^{\top}\left\\{(\alpha_{fk}^{2}+\gamma_{k}(\lambda))\delta_{kl}\right\\}_{k,l=1}^{\infty}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(79)
$\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)^{*}=(\Theta_{\tau}(e^{-i\lambda}))^{\top}\times\\\
\times\left\\{(\beta_{k}^{2}+\gamma_{1k}(\lambda)+\gamma_{2k}(\lambda))\delta_{kl}\right\\}_{k,l=1}^{\infty}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(80)

where $\alpha_{fk}^{2}$, $\beta_{k}^{2}$ are Lagrange multipliers,
$\delta_{kl}$ are Kronecker symbols, functions $\gamma_{k}(\lambda)\leq 0$ and
$\gamma_{k}(\lambda)=0$ if
$f_{kk}^{0}(\lambda)>(1-\varepsilon)f_{kk}^{1}(\lambda)$, functions
$\gamma_{1k}(\lambda)\leq 0$ and $\gamma_{1k}(\lambda)=0$ if
$g_{kk}^{0}(\lambda)>v_{kk}(\lambda),$ functions $\gamma_{2k}(\lambda)\geq 0$
and $\gamma_{2k}(\lambda)=0$ if $g_{kk}^{0}(\lambda)<u_{kk}(\lambda).$

For the fourth set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{4}\times\mathcal{D}_{V}^{U4}$, we have equation

$\displaystyle\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\alpha_{f}^{2}+\gamma^{\prime}(\lambda))(\Theta_{\tau}(e^{-i\lambda}))^{\top}B_{1}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(81)
$\displaystyle\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)\left({\mathbf{r}^{0}_{\tau,g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=(\beta^{2}+\gamma^{\prime}_{1}(\lambda)+\gamma^{\prime}_{2}(\lambda))(\Theta_{\tau}(e^{-i\lambda}))^{\top}B_{2}\overline{\Theta_{\tau}(e^{-i\lambda})},$
(82)

where $\alpha_{f}^{2}$, $\beta^{2}$, are Lagrange multipliers, function
$\gamma^{\prime}(\lambda)\leq 0$ and $\gamma^{\prime}(\lambda)=0$ if $\langle
B_{1},f^{0}(\lambda)\rangle>(1-\varepsilon)\langle
B_{1},f_{1}(\lambda)\rangle$, functions $\gamma^{\prime}_{1}(\lambda)\leq 0$
and $\gamma^{\prime}_{1}(\lambda)=0$ if $\langle
B_{2},g^{0}(\lambda)\rangle>\langle B_{2},V(\lambda)\rangle,$ functions
$\gamma^{\prime}_{2}(\lambda)\geq 0$ and $\gamma^{\prime}_{2}(\lambda)=0$ if
$\langle B_{2},g^{0}(\lambda)\rangle<\langle B_{2},U(\lambda)\rangle.$

The following theorems hold true.

######  Theorem 5.5.

The least favorable spectral densities $f^{0}(\lambda)$, $g^{0}(\lambda)$ in
the classes
$\mathcal{D}_{\varepsilon}^{k}\times\mathcal{D}_{V}^{Uk},k=1,2,3,4$ for the
optimal linear prediction of the functional $A{\xi}$ from observations of the
process $\xi(t)+\eta(t)$ at points $t<0$ by canonical factorizations (29) and
(30), equations (75)–(76), (77)–(78), (79)–(80), (81)–(82), respectively,
constrained optimization problem (35) and restrictions on densities from the
corresponding classes
$\mathcal{D}_{\varepsilon}^{k},\mathcal{D}_{V}^{Uk},k=1,2,3,4$. The minimax-
robust spectral characteristic of the optimal estimate of the functional
$A{\xi}$ is determined by formula (32).

######  Theorem 5.6.

If the spectral density $g(\lambda)$ is known, the least favorable spectral
density $f^{0}(\lambda)$ in the classes $\mathcal{D}_{\varepsilon}^{k}$,
$k=1,2,3,4$ for the optimal linear prediction of the functional $A{\xi}$ from
observations of the process $\xi(t)+\eta(t)$ at points $t<0$ is determined by
canonical factorizations (30) and (29), equations (75), (77), (79), (81),
respectively, constrained optimization problem (40) and restrictions on
density from the corresponding classes $\mathcal{D}_{f\varepsilon}^{k}$,
$k=1,2,3,4$. The minimax-robust spectral characteristic of the optimal
estimate of the functional $A{\xi}$ is determined by formula (32).

## Conclusions

In this article, we dealt with continuous time stochastic processes with
periodically correlated $d$th increments. These stochastic processes form a
class of non-stationary stochastic processes that combine periodic structure
of covariation functions of processes as well as integrating one.

We derived solutions of the problem of estimation of the linear functionals
constructed from the unobserved values of a continuous time stochastic process
with periodically correlated $d$th increments. Estimates are based on
observations of this process with periodically stationary noise at points
$t<0$. We obtained the estimates by representing the process under
investigation as a vector-valued sequence with stationary increments. Based on
the solutions for these type of sequences, we solved the corresponding problem
for the considered class of continuous time stochastic processes. The problem
is investigated in the case of spectral certainty, where spectral densities of
sequences are exactly known. In this case we propose an approach based on the
Hilbert space projection method. We derive formulas for calculating the
spectral characteristics and the mean-square errors of the optimal estimates
of the functionals. In the case of spectral uncertainty where the spectral
densities are not exactly known while, instead, some sets of admissible
spectral densities are specified, the minimax-robust method is applied. We
propose a representation of the mean square error in the form of a linear
functional in $L_{1}$ with respect to spectral densities, which allows us to
solve the corresponding constrained optimization problem and describe the
minimax-robust estimates of the functionals. Formulas that determine the least
favorable spectral densities and minimax-robust spectral characteristics of
the optimal linear estimates of the functionals are derived for a wide list of
specific classes of admissible spectral densities.

## Appendix

_Proof of Lemma 3.2._

Define

$b^{\tau}_{j}(u)=b^{\tau}(u+jT),\quad\zeta^{(d)}_{j}(u)=\zeta^{(d)}_{j}(u+jT,\tau
T),\,\,u\in[0,T),$

and

$a_{j}(u)=a(u+jT),\quad\eta_{j}(u)=\eta_{j}(u+jT),\,\,u\in[0,T).$

Making use of decomposition (6) for the increment sequence
$\\{\zeta^{(d)}_{j},j\in\mathbb{Z}\\}$ and the solution of equation

$(-1)^{k}\left[\frac{k}{2}\right]+(-1)^{m}\left[\frac{m}{2}\right]=0$ (83)

of two variables $(k,m)$, which is given by pairs $(1,1)$, $(2l+1,2l)$ and
$(2l,2l+1)$ for $l=2,3,\dots$, rewrite the functional $B\zeta$ as [25]

$\displaystyle B\zeta$
$\displaystyle=\int_{0}^{\infty}b^{\tau}(t)\zeta^{(d)}(t,\tau
T)dt=\sum_{j=0}^{\infty}\int_{0}^{T}b^{\tau}_{j}(u)\zeta^{(d)}_{j}(u)du$
$\displaystyle=\sum_{j=0}^{\infty}\frac{1}{T}\int_{0}^{T}\left(\sum_{k=1}^{\infty}b^{\tau}_{kj}e^{2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}u/T}\right)\left(\sum_{m=1}^{\infty}\zeta^{(d)}_{mj}e^{2\pi
i\\{(-1)^{m}\left[\frac{m}{2}\right]\\}u/T}\right)du$
$\displaystyle=\sum_{j=0}^{\infty}\sum_{k=1}^{\infty}\sum_{m=1}^{\infty}b^{\tau}_{kj}\zeta^{(d)}_{mj}\frac{1}{T}\int_{0}^{T}e^{2\pi
i\left\\{(-1)^{k}\left[\frac{k}{2}\right]+(-1)^{m}\left[\frac{m}{2}\right]\right\\}u/T}du$
$\displaystyle=\sum_{j=0}^{\infty}\sum_{k=1}^{\infty}b^{\tau}_{kj}\zeta^{(d)}_{kj}=\sum_{j=0}^{\infty}{(\vec{b}^{\tau}_{j})}^{\top}\vec{\zeta}^{(d)}_{j}$
$\displaystyle=B\vec{\zeta}.$

The representation of the functional $V\eta$ is obtained in the same way:

$\displaystyle V\eta$
$\displaystyle=\int_{0}^{\infty}a(t)\eta(t)dt=\sum_{j=0}^{\infty}\int_{0}^{T}a_{j}(u)\eta_{j}(u)du$
$\displaystyle=\sum_{j=0}^{\infty}\sum_{k=1}^{\infty}a_{kj}\eta_{kj}=\sum_{j=0}^{\infty}{(\vec{a}_{j})}^{\top}\vec{\eta}_{j}$
$\displaystyle=V\vec{\eta}.$

From Lemma 3.1 we obtain

$b^{\tau}_{j}(u)=\sum_{l=0}^{\infty}a(u+jT+\tau Tl)d(l)=D^{\tau
T}a(u),\,u\in[0;T),\,j=0,1,\ldots,$

and

$b^{\tau}_{kj}=\sum_{l=0}^{\infty}a_{kj+\tau l}d(l),\,j=0,1,\ldots,$

which finalizes the proof of Lemma 3.2.

_Proof of Theorem 3.1._

A projection $\widehat{H}\vec{\xi}$ of the element $H\vec{\xi}$ on the
subspace $H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})$ is characterized by two
conditions:

1) $\widehat{H}\vec{\xi}\in H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})$;

2) $(H\vec{\xi}-\widehat{H}\vec{\xi})\perp
H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})$.

The second condition implies the following relation which holds true for all
$j\leq-1$ and for all $k\geq 1$

$\int_{-\pi}^{\pi}\bigg{(}(\vec{B}_{\tau}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}(f(\lambda)+\lambda^{2d}g(\lambda))-(\vec{h}_{\tau}(\lambda))^{\top}(f(\lambda)+\lambda^{2d}g(\lambda))-\bigg{.}\\\
\bigg{.}-(\vec{A}(e^{i\lambda}))^{\top}g(\lambda)(-i\lambda)^{d}\bigg{)}\vec{\delta}_{k}\frac{(1-e^{i\lambda\tau})^{d}}{(-i\lambda)^{d}}e^{-i\lambda
j}d\lambda=0.$

From these relations, we conclude that the spectral characteristic
$\vec{h}_{\tau}(\lambda)$ of the estimate $\widehat{H}\vec{\xi}$ allow a
representation in the form (24) where

$\vec{C}_{\tau}(e^{i\lambda})=\sum_{j=0}^{\infty}\vec{c}_{j}^{\tau}e^{i\lambda
j},$

and $\vec{c}_{j}^{\tau}=\\{c_{kj}^{\tau}\\}_{k=1}^{\infty}$, $j=0,1,2,\dots$,
are unknown coefficients to be found.

Condition 1) implies
${(i\lambda)^{d}}{(1-e^{-i\lambda\tau})^{-d}}\vec{h}_{\tau}(\lambda)\in
L_{2}^{0-},$ and thus,

$\int_{-\pi}^{\pi}\biggl{[}(\vec{B}_{\tau}(e^{i\lambda}))^{\top}-(\vec{A}_{\tau}(e^{i\lambda}))^{\top}g(\lambda)\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}\\\
-(\vec{C}_{\tau}(e^{i\lambda}))^{\top}\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}\biggr{]}e^{-i\lambda
j}d\lambda=\vec{0},\quad j\geq 0,$

which can be presented as a system of linear equations

$\vec{b}_{l}^{\tau}-\sum_{j=0}^{\infty}T^{\tau}_{l,j}\vec{a}_{j}^{\tau}=\sum_{j=0}^{\infty}P_{l,j}^{\tau}\vec{c}_{j}^{\tau},\quad
l\geq 0,$ (84)

determining the unknown coefficients ${\vec{c}}^{\tau}_{j}$, $j\geq 0$.

Rewrite the system of equations (84) in the matrix form

$D^{\tau}\mathbf{a}-\mathbf{T}_{\tau}\mathbf{a}^{\tau}=\mathbf{P}_{\tau}\mathbf{c}^{\tau},$

where

$\mathbf{c}^{\tau}=((\vec{c}^{\tau}_{0})^{\top},(\vec{c}^{\tau}_{1})^{\top},(\vec{c}^{\tau}_{2})^{\top},\ldots)^{\top}.$

Consequently, the unknown coefficients $\vec{c}^{\tau}_{j}$, $j\geq 0$,
determining the spectral characteristic $\vec{h}_{\tau}(\lambda)$ are as
follows

$\vec{c}^{\tau}_{j}=(\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau})_{j},\quad
j\geq 0,$

where
$(\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau})_{j}$,
$j\geq 0$, is the $j$th infinite dimension vector entry of the vector
$\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau}$.
The existence of the inverse matrix $(\mathbf{P}^{\tau})^{-1}$ is justified in
[16] under condition (15). Thus, the function $\vec{C}_{\tau}(e^{i\lambda})$
is calculated as

$\vec{C}_{\tau}(e^{i\lambda})=\sum_{j=0}^{\infty}(\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau})_{j}e^{i\lambda
j}$

and the spectral characteristic $\vec{h}_{\tau}(\lambda)$ is calculated by the
formula

$\displaystyle(\vec{h}_{\tau}(\lambda))^{\top}$
$\displaystyle=(\vec{B}_{\tau}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}$
$\displaystyle\quad-(\vec{A}_{\tau}(e^{i\lambda}))^{\top}g(\lambda)\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}$
$\displaystyle\quad-\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}\left(\sum_{k=0}^{\infty}(\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau})_{k}e^{i\lambda
j}\right)^{\top}$
$\displaystyle\quad\quad\times(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}.$

The value of the mean square error of the estimate $\widehat{A}\vec{\xi}$ is
calculated by the formula

$\displaystyle\Delta(f,g;\widehat{A}\vec{\xi})=\Delta(f,g;\widehat{H}\vec{\xi})=\mathsf{E}|H\vec{\xi}-\widehat{H}\vec{\xi}|^{2}=$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}\left[(\vec{A}_{\tau}(e^{i\lambda}))^{\top}g(\lambda)+(\vec{C}_{\tau}(e^{i\lambda}))^{\top}\right]$
$\displaystyle\quad\quad\times(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}\,f(\lambda)\,(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}$
$\displaystyle\quad\quad\times\left[g(\lambda)\overline{\vec{A}_{\tau}(e^{i\lambda})}+\overline{\vec{C}_{\tau}(e^{i\lambda})}\right]d\lambda$
$\displaystyle\quad+\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\lambda^{4d}}{|1-e^{i\lambda\tau}|^{4d}}\left[\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(\vec{A}(e^{i\lambda}))^{\top}f(\lambda)-{(1-e^{-i\lambda\tau})^{d}}(\vec{C}_{\tau}(e^{i\lambda}))^{\top}\right]$
$\displaystyle\quad\quad\times(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}\,g(\lambda)\,(f(\lambda)+{\lambda}^{2d}g(\lambda))^{-1}$
$\displaystyle\quad\quad\times\left[\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f(\lambda)\overline{\vec{A}(e^{i\lambda})}-{(1-e^{i\lambda\tau})^{d}}\overline{\vec{C}_{\tau}(e^{i\lambda}})\right]d\lambda$
$\displaystyle=\langle
D^{\tau}\mathbf{a}-\mathbf{T}_{\tau}\mathbf{a}^{\tau},\mathbf{P}_{\tau}^{-1}D^{\tau}\mathbf{a}-\mathbf{P}_{\tau}^{-1}\mathbf{T}_{\tau}\mathbf{a}^{\tau}\rangle+\langle\mathbf{Q}\mathbf{a},\mathbf{a}\rangle,$

which finalizes the proof of Theorem 3.1.

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