# Minimax interpolation of 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 deal with the problem of optimal estimation of the linear functionals
constructed from the missed values of a continuous time stochastic process
$\xi(t)$ with periodically stationary increments at points $t\in[0;(N+1)T]$
based on observations of this process with periodically stationary noise. To
solve the problem, a sequence of stochastic functions
$\\{\xi^{(d)}_{j}(u)=\xi^{(d)}_{j}(u+jT,\tau),\,\,u\in[0,T),j\in\mathbb{Z}\\}.$
is constructed. It forms a $L_{2}([0,T);H)$-valued stationary increment
sequence $\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ or corresponding to it an
(infinite dimensional) vector stationary increment sequence
$\\{\vec{\xi}^{(d)}_{j}=(\xi^{(d)}_{kj},k=1,2,\dots)^{\top},j\in\mathbb{Z}\\}$.
In the case of a known spectral density, 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 investigate the interpolation problem for the stochastic
processes $\xi(t)$, $t\in\mathbb{R}$, with a periodically correlated
increments $\xi^{(d)}(t,\tau T)=\Delta_{T\tau}^{d}\xi(t)$ of order $d$ and
period $T$, where $\Delta_{s}\xi(t)=\xi(t)-\xi(t-s)$. The resent studies, for
example Basawa et al. [1], Dudek et al. [3], Reisen et al. [28], show a
constant interest to the non-stationary models and robust methods of
estimation.

Kolmogorov [12], Wiener [32] and Yaglom [34] developed the methods of solution
of interpolation, extrapolation and filtering problems for stationary
stochastic sequences and processes. For these problems, they considered the
estimates $\widetilde{x}(t)$ constructed as a linear combination of the
available observations. As an optimal linear estimate, they chose the one
which minimizes the mean square error
$\Delta(\widetilde{x}(t),f)=\mathsf{E}|x(t)-\widetilde{x}(t)|^{2}$ for the
fixed spectral density $f(\lambda)$ of the stationary process or sequence
$x(t)$. The problems were also studied in the presence of the noise sequence
or process.

The developed classical estimation methods are not directly applicable in
practice. Usually, the exact spectral structure of the processes isn’t
available. The estimated spectral densities can be considered instead.
However, Vastola and Poor [31] showed the concrete examples, where such
substitution can result in a significant increase of the estimate error.
Therefore, it is reasonable to consider the estimates, 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 estimation
method was proposed by Grenander [7] who considered an extrapolation of the
functional $Ax=\int_{0}^{1}a(t)x(t)dt$ as a game between two players, one of
which minimizes $\Delta(\widetilde{A}x,f)$ by $\widetilde{A}\zeta$ and another
one maximizes it by $f$. The game has a saddle point under proper conditions:

$\max_{\begin{subarray}{c}f\end{subarray}}\min_{\begin{subarray}{c}\widetilde{A}x\end{subarray}}\mathsf{\Delta}(\widetilde{A}x,f)=\min_{\begin{subarray}{c}\widetilde{A}x\end{subarray}}\max_{\begin{subarray}{c}f\end{subarray}}\Delta(\widetilde{A}x,f)=\nu.$

For more details, we refer to the further study by Franke and Poor [4] and the
survey paper by Kassam and Poor [10]. A wide range of results has been
obtained by Moklyachuk [18, 20, 22]. These results have been extended on the
vector-valued stationary processes and sequences by Moklyachuk and Masyutka
[24].

The concept of stationarity admits some generalizations, two of which are
stationary $d$th increments and periodical correlation. A combination of them
is in scope of the article. Random processes with stationary $d$th increments
$x(t)$ were introduced by Yaglom and Pinsker [27]. The increment process
$x^{(d)}(t,\tau):=\Delta_{\tau}^{d}x(t)$ generated by $x(t)$ is stationary by
the variable $t$, which means that the mathematical expectations
$\mathsf{E}\xi^{(n)}(t,\tau)$ and
$\mathsf{E}\xi^{(n)}(t+s,\tau_{1})\xi^{(n)}(t,\tau_{2})$ do not depend on $t$.
Yaglom and Pinsker described the spectral representation of such process and
the spectral density canonical factorization. They also introduced and solved
the extrapolation problem for them. The minimax-robust extrapolation,
interpolation and filtering problems for stochastic processes with stationary
increments were investigated by Luz and Moklyachuk [14].

Dubovetska and Moklyachuk [2] derived the classical and minimax-robust
estimates for another generalization of stationary processes – periodically
correlated (cyclostationary) processes, introduced by Gladyshev [6]. 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 [26] for a review of
recent works on cyclostationarity and its applications).

In this article, we deal with a problem of the mean-square optimal estimation
of the linear functional $A_{NT}{\xi}=\int_{0}^{(N+1)T}a(t)\xi(t)dt$ which
depends on the unknown values of a random process $\xi(t)$ with periodically
stationary $d$th increments from observations $\xi(t)+\eta(t)$ at points
$\mathbb{R}\setminus[0;(N+1)T]$. The similar problems for discrete time
processes have been studied by Kozak and Moklyachuk [13], Luz and Moklyachuk
[15, 16]. The extrapolation problem without noise for continues time processes
with periodically stationary increments has been studied by Luz and Moklyachuk
[17]. In sections 1, we describe the periodically stationary increment process
as a stationary $H$-valued increment sequence. In section 2, the classical
interpolation problem is introduced and solved. Particularly, formulas for
calculating the mean-square error and the spectral characteristic of the
optimal linear estimates of the functional $A_{NT}{\xi}$ are derived under the
condition of spectral certainty. The results on minimax-robust interpolation
for the studied processes are presented in section 3, where the relations that
determine the least favourable spectral densities and the minimax spectral
characteristics are derived for some classes of spectral densities.

##  1 Preliminary results

###  1.1 Periodically correlated processes and generated vector stationary
sequences

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

######  Definition 1.1 (Gladyshev [6]).

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 [DubovetskaMoklyachuk2013],
[23]

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

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

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

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)

Then the stationary sequence $\\{\zeta_{j},j\in\mathbb{Z}\\}$ 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 $\\{\zeta_{j},j\in\mathbb{Z}\\},$ or a 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 [9], [18]

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

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 limitations:

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

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 functions 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

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

###  1.2 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 1.2 (Luz and Moklyachuk [17]).

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
Definition 1.2 are called the _mean value_ and the _structural function_ of
the stochastic process $\xi(t)$ with periodically stationary (periodically
correlated) increments.

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 Section 1.1 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, [9], [18]

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

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

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

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 1.1, 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 [14]. 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

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

##  2 Hilbert space projection method of interpolation

Consider an infinite dimension vector stationary increment sequence
$\\{\vec{\xi}^{(d)}_{j},j\in\mathbb{Z}\\}$ (7) generated by the increment
process $\xi^{(d)}(t,\tau T)$, $t\in\mathbb{R}$, which has the spectral
density matrix $f(\lambda)=\\{f_{ij}(\lambda)\\}_{i,j=1}^{\infty}$.

Let $\eta(t)$, $t\in\mathbb{R}$, be an uncorrelated with the process $\xi(t)$
periodically stationary stochastic process, which generates an infinite
dimension vector stationary sequence $\\{\vec{\eta}_{j},j\in\mathbb{Z}\\}$ (3)
with the spectral density matrix
$g(\lambda)=\\{g_{ij}(\lambda)\\}_{i,j=1}^{\infty}$.

We will assume that the mean values of the increment sequence
$\vec{\xi}^{(d)}_{j}$ and stationary sequence $\vec{\eta}_{j}$ equal to 0. We
will also consider the increment step $\tau>0$.

By the classical interpolation problem we understand the problem of the mean
square optimal linear estimation of the functional

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

which depends on the unknown values of the stochastic process $\xi(t)$ with
periodically correlated $d$th increments. Estimates are based on observations
of the process $\zeta(t)=\xi(t)+\eta(t)$ at points
$t\in\mathbb{R}\setminus[0;(N+1)T]$, where $\eta(t)$.

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

$\int_{-\pi}^{\pi}\text{Tr}\left[\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}(f(\lambda)+\lambda^{2d}g(\lambda))^{-1}\right]d\lambda<\infty.$
(11)

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

The functional $A_{NT}{\xi}$ does not belong to the Hilbert space
$H=L_{2}(\Omega,\mathcal{F},\mathsf{P})$ of random variables with a zero mean
value and a finite variance. To apply a Hilbert pace projection method of
estimation, represent the functional $A_{NT}{\xi}$ as a sum of a functional
with finite second moment belonging to $H$ and a functional depended on the
observed values of the process $\zeta(t)$ (“initial values”). This
representation is described by the following two lammas.

######  Lemma 2.1 ([17]).

The linear functional

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

allows the representation

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

where

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

and

$\displaystyle v^{\tau,N}(t)$ $\displaystyle=$
$\displaystyle\sum_{l=\left\lceil-\frac{t}{\tau
T}\right\rceil}^{\min\left\\{\left[\frac{{(N+1)T}-t}{\tau
T}\right],d\right\\}}(-1)^{l}{d\choose l}b^{\tau,N}(t+l\tau T),\quad
t\in[-\tau Td;0),$ (12) $\displaystyle b^{\tau,N}(t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\left[\frac{{(N+1)T}-t}{\tau T}\right]}a(t+\tau
Tk)d(k)=D^{\tau T,N}\mathbf{a}(t),\,t\in[0;{(N+1)T}].$ (13)

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

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

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

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

From Lemma 2.1, we obtain the following representation of the functional
$A_{NT}\xi$:

$A_{NT}\xi=A_{NT}\zeta-A_{NT}\eta=B_{NT}\zeta-A_{NT}\eta-
V_{NT}\zeta=H_{NT}\xi-V_{NT}\zeta,$ $H_{NT}\xi:=B_{NT}\zeta-A_{NT}\eta,$

where

$A_{NT}\zeta=\int_{0}^{(N+1)T}a(t)\zeta(t)dt,\quad
A_{NT}\eta=\int_{0}^{(N+1)T}a(t)\eta(t)dt,$
$B_{NT}\zeta=\int_{0}^{(N+1)T}b_{\tau,N}(t)\zeta^{(d)}(t,\tau T)dt,\quad
V_{NT}\zeta=\int_{-\tau Td}^{0}v_{\tau,N}(t)\zeta(t)dt,$

the functions $b_{\tau,N}(t)$, $t\in[0;(N+1)T]$, and $v_{\tau,N}(t)$,
$t\in[-\tau Td;0)$, are calculated by formulas (13) and (12) respectively.

The following lemma represent the functional $H_{NT}\xi$ in terms of the
sequences $\vec{\eta}_{j}=(\eta_{kj},k=1,2,\dots)^{\top}$ and
$\vec{\zeta}^{(d)}_{j}=\vec{\xi}^{(d)}_{j}+\vec{\eta}^{(d)}_{j}=(\zeta^{(d)}_{kj},k=1,2,\dots)^{\top}$,
$j\in\mathbb{Z}$.

######  Lemma 2.2.

The functional $H_{NT}\xi$ 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

$\vec{b}^{\tau,N}_{j}=(b^{\tau,N}_{kj},k=1,2,\dots)^{\top}=(b^{\tau,N}_{1j},b^{\tau,N}_{3j},b^{\tau,N}_{2j},\dots,b^{\tau,N}_{2k+1,j},b^{\tau,N}_{2k,j},\dots)^{\top},$
$b^{\tau,N}_{kj}=\langle
b^{\tau,N}_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}b^{\tau,N}_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv,$

and

$\vec{a}_{j}=(a_{kj},k=1,2,\dots)^{\top}=(a_{1j},a_{3j},a_{2j},\dots,a_{2k+1,j},a_{2k,j},\dots)^{\top},$
$a_{kj}=\langle
a_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}a_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv,\quad
k=1,2,\dots,\,j=0,1,\ldots,N.$

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

$\vec{b}_{j}^{\tau,N}=\sum_{m=j}^{N}\mathsf{diag}_{\infty}(d_{\tau}(m-j))\vec{a}_{m}=(D^{\tau}_{N}{\mathbf{a}_{N}})_{j},\quad
j=0,1,\dots,N.$ (14)

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

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

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

###### Proof.

Define

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

and

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

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

$(-1)^{k}\left[\frac{k}{2}\right]+(-1)^{m}\left[\frac{m}{2}\right]=0$ (15)

of two variables $(k,m)$, which is given by pairs $(1,1)$, $(2l+1,2l)$ and
$(2l,2l+1)$ for $l=2,3,\dots$, the functional $B_{NT}\zeta$ can be rewritten
in the form [23]

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

In the same way we obtain the representation of the the functional
$V_{NT}\eta$

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

Then from Lemma 2.1 obtain

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

and

$b^{\tau,N}_{kj}=\sum_{l=0}^{\left[\frac{N-j}{\tau}\right]}a_{kj+\tau
l}d(l),\,j=0,1,\ldots,N,$

which finalizes the proof. ∎

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

$\|\vec{a}_{j}\|<\infty,\quad\|\vec{a}_{j}\|^{2}=\sum_{k=1}^{\infty}|{a}_{kj}|^{2},\quad
j=0,1,\dots,N.$ (16)

and

$\|\vec{b}^{\tau,N}_{j}\|<\infty,\quad\|\vec{b}^{\tau,N}_{j}\|^{2}=\sum_{k=1}^{\infty}|{b}^{\tau,N}_{kj}|^{2},\quad
j=0,1,\dots,N.$ (17)

It follows from conditions (16) - (17) that the functional
$H_{NT}\xi=H_{N}\vec{\xi}$ has a finite second moment and the functional
$V_{NT}\zeta$ depends on the observations of the process $\xi(t)+\eta(t)$ at
points $t\in[\tau Td;0)$. Therefore, the estimates $\widehat{A}_{N}\vec{\xi}$
and $\widehat{H}_{N}\vec{\xi}$ of the functionals $A_{N}\vec{\xi}$ and
$H_{N}\vec{\xi}$, as well as the mean-square errors
$\Delta(f,g;\widehat{A}_{NT}\xi)=\Delta(f,g;\widehat{A}_{N}\vec{\xi})=\mathsf{E}|A_{N}\vec{\xi}-\widehat{A}_{N}\vec{\xi}|^{2}$
and
$\Delta(f,g;\widehat{H}_{NT}\xi)=\Delta(f,g;\widehat{H}_{N}\vec{\xi})=\mathsf{E}|H_{N}\vec{\xi}-\widehat{H}_{N}\vec{\xi}|^{2}$
of the estimates $\widehat{A}_{N}\vec{\xi}$ and $\widehat{H}_{N}\vec{\xi}$
satisfy the following relations

$\widehat{A}_{N}\vec{\xi}=\widehat{H}_{N}\vec{\xi}-V_{NT}\zeta,$ (18)
$\Delta(f,g;\widehat{A}_{N}\vec{\xi})=\mathsf{E}|A_{N}\vec{\xi}-\widehat{A}_{N}\vec{\xi}|^{2}=\mathsf{E}|H_{N}\vec{\xi}-V_{NT}\zeta-\widehat{H}_{N}\vec{\xi}+V_{NT}\zeta|^{2}$
$=\mathsf{E}|H_{N}\vec{\xi}-\widehat{H}_{N}\vec{\xi}|^{2}=\Delta(f,g;\widehat{H}_{N}\vec{\xi}).$

Thus, it is enough to find the estimate of the functional $H_{N}\vec{\xi}$.

Making use of spectral representations (10) and (9), onr can obtain the
following spectral representation of the functional $H_{N}\vec{\xi}$

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

where

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

The next step is to define the subspace of
$H=L_{2}(\Omega,\mathcal{F},\mathsf{P})$ generated by the observations.
Consider the closed linear subspaces

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

and

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

The equality $\xi^{(d)}_{k}(j,-\tau)=(-1)^{d}\xi^{(d)}_{k}(j+\tau d,\tau)$
implies

$H^{N+}(\xi_{-\tau}^{(d)}+\eta_{-\tau}^{(d)})=H^{(N+\tau
d)+}(\xi_{\tau}^{(d)}+\eta_{\tau}^{(d)}).$

Define the following closed linear subspaces of the Hilbert space
$L_{2}(f(\lambda)+\lambda^{2d}g(\lambda))$ of vector-valued functions endowed
by the inner product $\langle
g_{1};g_{2}\rangle=\int_{-\pi}^{\pi}(g_{1}(\lambda))^{\top}(f(\lambda)+\lambda^{2d}g(\lambda))\overline{g_{2}(\lambda)}d\lambda$
which is generated by the functions

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

and

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

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

Representation (10) yields a map between the elements $e^{i\lambda
j}(1-e^{-i\lambda\tau})^{d}(i\lambda)^{-d}$ of the space

$L_{2}^{0-}(f(\lambda)+\lambda^{2d}g(\lambda))\oplus
L_{2}^{N+}(f(\lambda)+\lambda^{2d}g(\lambda))$

and the elements $\vec{\xi}^{(d)}_{j}+\vec{\eta}^{(d)}_{j}$ of the space

$H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})\oplus
H^{N+}(\xi^{(d)}_{-\tau}+\eta^{(d)}_{-\tau})=H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})\oplus
H^{(N+\tau d)+}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau}).$

Relation (18) implies that a linear estimate $\widehat{A}_{N}\vec{\xi}$ of
$A_{N}\vec{\xi}$ can be represented as

$\widehat{A}_{N}\vec{\xi}=\int_{-\pi}^{\pi}(\vec{h}_{\tau,N}(\lambda))^{\top}d\vec{Z}_{\xi^{(d)}+\eta^{(d)}}(\lambda)-\int_{-\tau
Td}^{0}v_{\tau,N}(t)\zeta(t)dt,$ (19)

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

Now we can apply the Hilbert space projection method proposed by Kolmogorov
[12]. The mean square optimal estimate $\widehat{H}_{N}\vec{\xi}$ is found as
a projection of the element $H_{N}\vec{\xi}$ on the subspace
$H^{0-}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})\oplus H^{(N+\tau
d)+}(\xi^{(d)}_{\tau}+\eta^{(d)}_{\tau})$. This projection is characterized by
two conditions:

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

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

The following relation follows from condition 2) and holds true for all
$j\leq-1$ and $j\geq N+\tau d+1$

$\int_{-\pi}^{\pi}\left[\left(\vec{B}_{\tau,N}(e^{i\lambda})\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}-\vec{h}_{\tau,N}(\lambda)\right)^{\top}p(\lambda)-(\vec{A}_{N}(e^{i\lambda}))^{\top}g(\lambda)(-i\lambda)^{d}\right]\times\\\
\times\frac{(1-e^{i\lambda\tau})^{d}}{(-i\lambda)^{d}}e^{-i\lambda
j}d\lambda=0.$ (20)

Thus, the spectral characteristic $\vec{h}_{\tau,N}(\lambda)$ of the estimate
$\widehat{H}_{N}\vec{\xi}$ can be represented as

$(\vec{h}_{\tau,N}(\lambda))^{\top}=(\vec{B}_{\tau,N}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}\\\
-\left((\vec{A}_{\tau,N}(e^{i\lambda}))^{\top}g(\lambda)+(\vec{C}_{\tau,N}(e^{i\lambda}))^{\top}\right)\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}(f(\lambda)+\lambda^{2d}g(\lambda))^{-1},$
(21)

where

$\vec{A}_{\tau,N}(e^{i\lambda})=(1-e^{i\lambda\tau})^{d}\vec{A}_{N}(e^{i\lambda})=\sum_{j=0}^{N+\tau
d}\vec{a}_{j}^{\tau,N}e^{i\lambda
j},\quad\vec{C}_{\tau,N}(e^{i\lambda})=\sum_{j=0}^{N+\tau
d}\vec{c}_{j}^{\tau,N}e^{i\lambda j},$

the coefficients $\\{\vec{a}_{j}^{\tau,N}:0\leq j\leq N+\tau d\\}$ are
calculated by the formula

$\vec{a}_{j}^{\tau,N}=\sum_{l=\max\left\\{\left\lceil\frac{j-N}{\tau}\right\rceil,0\right\\}}^{\min\left\\{\left[\frac{j}{\tau}\right],d\right\\}}(-1)^{l}{d\choose
l}\vec{a}(j-\tau l),\quad 0\leq j\leq N+\tau d,$ (22)

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

Condition 1) implies that the following equations should be satisfied

$\int_{-\pi}^{\pi}\biggl{[}(\vec{B}_{\tau,N}(e^{i\lambda}))^{\top}-(\vec{A}_{\tau,N}(e^{i\lambda}))^{\top}g(\lambda)\frac{\lambda^{2d}}{(1-e^{-i\lambda\tau})^{d}(1-e^{i\lambda\tau})^{d}}(f(\lambda)+\lambda^{2d}g(\lambda))^{-1}-$
$-\frac{\lambda^{2d}(\vec{C}_{\tau,N}(e^{i\lambda}))^{\top}}{(1-e^{-i\lambda\tau})^{d}(1-e^{i\lambda\tau})^{d}}(f(\lambda)+\lambda^{2d}g(\lambda))^{-1}\biggr{]}e^{-i\lambda
j}d\lambda=0,\,0\leq j\leq N+\tau d.$ (23)

Define for $0\leq l,j\leq N+\tau d$ the Fourier coefficients of the
corresponding functions

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

Making use of the defined Fourier coefficients, relation (23) can be presented
as a system of $N+\tau d+1$ linear equations determining the unknown
coefficients $\vec{c}_{j}^{\tau,N}$, $0\leq j\leq N+\tau d$.

$\displaystyle\vec{b}_{l}^{\tau,N}-\sum_{j=0}^{N+\tau
d}T^{\tau}_{l,j}\vec{a}_{j}^{\tau,N}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N+\tau d}P_{l,j}^{\tau}\vec{c}_{j}^{\tau,N},\quad
0\leq l\leq N,$ $\displaystyle-\sum_{j=0}^{N+\tau
d}T^{\tau}_{l,j}\vec{a}_{j}^{\tau,N}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N+\tau d}P_{l,j}^{\tau}\vec{c}_{j}^{\tau,N},\quad
N+1\leq l\leq N+\tau d.$

Denote by $[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau d}$ a vector of dimension
$N+\tau d+1$ with infinite dimensional entries, which is constructed by adding
$\tau d$ infinite dimensional zero vectors to the vector
$D_{N}^{\tau}\mathbf{a}_{N}$ of dimension $N+1$. Then the system (2) – (2) can
be represented in the matrix form

$[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-\mathbf{T}^{\tau}_{N}\mathbf{a}^{\tau}_{N}=\mathbf{P}^{\tau}_{N}\mathbf{c}^{\tau}_{N},$

where

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

are vectors of dimension $N+\tau d+1$ with the infinite dimensional entries;
$\mathbf{P}^{\tau}_{N}$ and $\mathbf{T}^{\tau}_{N}$ are matrices of dimension
$(N+\tau d+1)\times(N+\tau d+1)$ with the infinite dimensional matrix entries
$(\mathbf{P}^{\tau}_{N})_{l,j}=P_{j,l}^{\tau}$ and
$(\mathbf{T}^{\tau}_{N})_{l,j}=T^{\tau}_{l,j}$, $0\leq l,j\leq N+\tau d$.

Thus, the coefficients $\vec{c}_{j}^{\tau,N}$, $0\leq j\leq N+\tau d$, are
determined by the formula

$\vec{c}_{j}^{\tau,N}=\left((\mathbf{P}^{\tau}_{N})^{-1}[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-(\mathbf{P}^{\tau}_{N})^{-1}\mathbf{T}^{\tau}_{N}\mathbf{a}^{\tau}_{N}\right)_{j},\quad
0\leq j\leq N+\tau d,$

where $\left((\mathbf{P}^{\tau}_{N})^{-1}[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-(\mathbf{P}^{\tau}_{N})^{-1}\mathbf{T}^{\tau}_{N}\mathbf{a}^{\tau}_{N}\right)_{j}$,
$0\leq j\leq N+\tau d$, is the $j$th element of the vector
$(\mathbf{P}^{\tau}_{N})^{-1}[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-(\mathbf{P}^{\tau}_{N})^{-1}\mathbf{T}^{\tau}_{N}\mathbf{a}^{\tau}_{N}$.

The existence of the inverse matrix $(\mathbf{P}^{\tau}_{N})^{-1}$ was
justified in [14] under condition (11).

The spectral characteristic $\vec{h}_{\tau,N}(\lambda)$ of the estimate
$\widehat{H}_{N}\xi$ of the functional $H_{N}\xi$ is calculated by formula
(21), where

$\vec{C}_{\tau,N}(e^{i\lambda})=\sum_{j=0}^{N+\tau
d}\left((\mathbf{P}^{\tau}_{N})^{-1}[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-(\mathbf{P}^{\tau}_{N})^{-1}\mathbf{T}^{\tau}_{N}\mathbf{a}^{\tau}_{N}\right)_{j}e^{i\lambda
j}.$ (24)

Denote

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

The value of the mean-square errors of the estimates $\widehat{A}_{NT}\xi$ and
$\widehat{H}_{N}\vec{\xi}$ can be calculated by the formula

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

where $\mathbf{Q}_{N}$ is a matrix of the dimension $(N+1)\times(N+1)$ with
the infinite dimensional matrix entries $(\mathbf{Q}_{N})_{j,l}=Q_{j,l}$,
$0\leq j,l\leq N$. Thus, the following theorem holds true.

######  Theorem 2.1.

Consider uncorrelated a stochastic process $\xi(t)$, $t\in\mathbb{R}$ with a
periodically stationary increments and a periodically stationary stochastic
process $\eta(t)$, $t\in\mathbb{R}$, which determine 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 generated
stationary sequence $\vec{\eta}_{j}$ with the spectral density matrix
$g(\lambda)=\\{g_{kn}(\lambda)\\}_{k,n=1}^{\infty}$ respectively. Let the
coefficients $\vec{a}_{j}$, $\vec{b}^{\tau,N}_{j}$, $j=0,1,\ldots,N$ generated
by the function $a(t)$, $t\geq 0$, satisfy conditions (16) - (17). Let
minimality condition (11) 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 $t\in\mathbb{R}\setminus[0;(N+1)T]$ is
calculated by formula (19). The spectral characteristic
$\vec{h}_{\tau,N}(\lambda)$ of the optimal estimate $\widehat{A}_{NT}\xi$ and
the mean-square error $\Delta(f,g;\widehat{A}_{NT}\xi)$ are calculated by
formulas (21) – (24) and (25) respetively.

######  Corollary 2.1.

The spectral characteristic $\vec{h}_{\tau,N}(\lambda)$ (21) admits the
representation
$\vec{h}_{\tau,N}(\lambda)=\vec{h}_{\tau,N}^{1}(\lambda)-\vec{h}_{\tau,N}^{2}(\lambda)$,
where

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

Here $\vec{h}_{\tau,N}^{1}(\lambda)$ and $\vec{h}_{\tau,N}^{2}(\lambda)$ are
spectral characteristics of the optimal estimates $\widehat{B}_{N}\vec{\zeta}$
and $\widehat{A}_{N}\vec{\eta}$ of the functionals $B_{N}\vec{\zeta}$ and
$A_{N}\vec{\eta}$ respectively based on observations $\xi(t)+\eta(t)$ at
points of the set $t\in\mathbb{R}\setminus[0;(N+1)T]$.

##  3 Minimax-robust method of estimation

The values of the mean square errors and the spectral characteristics of the
optimal estimate of the functional ${A}_{NT}\xi$ depending on the unobserved
values of a stochastic process ${\xi}(t)$ which determine a generated
stationary stochastic $d$th increment sequence $\vec{\xi}^{(d)}_{j}$ with the
spectral density matrix $f(\lambda)$ based on observations of the process
$\xi(t)+\eta(t)$ at points $\mathbb{R}\setminus[0;(N+1)T]$ can be calculated
by formulas (21), (24), (25) respectively, under the condition that spectral
densities $f(\lambda)$ and $g(\lambda)$ of stochastic sequences
$\vec{\xi}_{j}$ and $\vec{\eta}_{j}$ are exactly known.

In practical cases, however, spectral densities usually are not exactly known.
If in such cases a set $\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ of
admissible spectral densities is defined, the minimax-robust approach to
estimation of linear functionals depending on unobserved values of stochastic
sequences with stationary increments may be applied. This method consists in
finding an estimate that minimizes the maximal values of the mean square
errors for all spectral densities from a given class
$\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ of admissible spectral
densities simultaneously.

To formalize this approach we present the following definitions.

######  Definition 3.1.

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
estimation of the functional $A_{N}\vec{\xi}$ 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 3.2.

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
$A_{N}\vec{\xi}$ 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 lemma holds true.

######  Lemma 3.1.

The spectral densities $f^{0}\in\mathcal{D}_{f}$, $g^{0}\in\mathcal{D}_{g}$
which satisfy minimality condition (11) are least favorable in the class
$\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ for the optimal linear
estimation of the functional $A_{N}\vec{\xi}$ based on observations of the
process $\xi(t)+\eta(t)$ at points $t\in\mathbb{R}\setminus[0;(N+1)T]$ if the
matrices $(\mathbf{P}^{\tau}_{N})^{0}$, $(\mathbf{T}^{\tau}_{N})^{0}$,
$(\mathbf{Q}_{N})^{0}$ whose entries are defined by the Fourier coefficients
of the functions

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

determine a solution of the constraint optimisation problem

$\max_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}\left(\left\langle[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-\mathbf{T}^{\tau}_{N}\mathbf{a}_{\tau},(\mathbf{P}^{\tau}_{N})^{-1}[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-(\mathbf{P}^{\tau}_{N})^{-1}\mathbf{T}^{\tau}_{N}\mathbf{a}^{\tau}_{N}\right\rangle\right.\\\
+\left.\langle\mathbf{Q}_{N}\mathbf{a}_{N},\mathbf{a}_{N}\rangle\right)=$
$=\left\langle[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-(\mathbf{T}^{\tau}_{N})^{0}\mathbf{a}^{\tau}_{N},((\mathbf{P}^{\tau}_{N})^{0})^{-1}[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-((\mathbf{P}^{\tau}_{N})^{0})^{-1}(\mathbf{T}^{\tau}_{N})^{0}\mathbf{a}^{\tau}_{N}\right\rangle\\\
+\langle\mathbf{Q}^{0}_{N}\mathbf{a}_{N},\mathbf{a}_{N}\rangle.$ (26)

The minimax spectral characteristic $h^{0}=h_{\tau,N}(f^{0},g^{0})$ is
calculated by formula (21) if $h_{\tau,N}(f^{0},g^{0})\in H_{\mathcal{D}}$.

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 $h^{0}=h_{\tau,N}(f^{0},g^{0})$ and $h_{\tau,N}(f^{0},g^{0})\in
H_{\mathcal{D}}$, where $(f^{0},g^{0})$ is a solution of the constraint
optimization problem

$\widetilde{\Delta}(f,g)=-\Delta(h_{\tau,N}(f^{0},g^{0});f,g)\to\inf,\quad(f,g)\in\mathcal{D},$
(27)

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

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

where

$\displaystyle\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})$ $\displaystyle=$
$\displaystyle\overline{g^{0}(\lambda)}\vec{A}_{\tau,N}(e^{i\lambda})+\vec{C}_{\tau,N}^{0}(e^{i\lambda}),$
$\displaystyle{\mathbf{C}}^{g0}_{\tau,N}(e^{i\lambda})$ $\displaystyle=$
$\displaystyle{|1-e^{i\lambda\tau}|^{2n}}\lambda^{-2n}\overline{f^{0}(\lambda)}\vec{A}_{N}(e^{i\lambda})-{(1-e^{-i\lambda\tau})^{n}}\vec{C}_{\tau,N}^{0}(e^{i\lambda}).$

and

$\vec{C}^{0}_{\tau,N}(e^{i\lambda})=\sum_{j=0}^{N+\tau
d}(((\mathbf{P}^{\tau}_{N})^{0})^{-1}[D_{N}^{\tau}\mathbf{a}_{N}]_{+\tau
d}-((\mathbf{P}^{\tau}_{N})^{0})^{-1}(\mathbf{T}^{\tau}_{N})^{0}\mathbf{a}^{\tau}_{N})_{j}e^{i\lambda
j}.$

The constrained optimisation problem (27) is equivalent to the unconstrained
optimisation problem

$\Delta_{\mathcal{D}}(f,g)=\widetilde{\Delta}(f,g)+\delta(f,g|\mathcal{D}_{f}\times\mathcal{D}_{g})\to\inf,$
(28)

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 $(f^{0},g^{0})$ to this unconstrained optimisation 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
$(f^{0},g^{0})\in\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$. 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 functional $\Delta(h_{\tau,N}(f^{0},g^{0});f,g)$ is convenient
for application the Lagrange method of indefinite multipliers for finding
solution to the problem (28). Making use of the method of Lagrange multipliers
and the form of subdifferentials of the indicator functions
$\delta(f,g|\mathcal{D}_{f}\times\mathcal{D}_{g})$ of the set
$\mathcal{D}_{f}\times\mathcal{D}_{g}$ of spectral densities we describe
relations that determine least favourable spectral densities in some special
classes of spectral densities (see [22, 14] for additional details).

###  3.1 Least favorable spectral density in classes
$\mathcal{D}_{0}\times\mathcal{\mathcal{missing}}D_{V}^{U}$

Consider the problem of optimal linear estimation of the functional
$A_{NT}{\xi}$ which depends on unobserved values of the process $\xi(t)$ with
periodically stationary increments based on observations of the process
$\xi(t)+\eta(t)$ at points $\mathbb{R}\setminus[0;(N+1)T]$ under the condition
that the sets of admissible spectral densities
$\mathcal{D}_{f0}^{k},\mathcal{D}_{Vg}^{Uk},k=1,2,3,4$ are defined as follows:

$\displaystyle\mathcal{D}_{f0}^{1}$ $\displaystyle=$
$\displaystyle\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f(\lambda)d\lambda=P\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{f0}^{2}$ $\displaystyle=$
$\displaystyle\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}{\rm{Tr}}\,[f(\lambda)]d\lambda=p\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{f0}^{3}$ $\displaystyle=$
$\displaystyle\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{kk}(\lambda)d\lambda=p_{k},k=\overline{1,\infty}\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{f0}^{4}$ $\displaystyle=$
$\displaystyle\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}\left\langle
B_{1},f(\lambda)\right\rangle d\lambda=p\right.\bigg{\\}},$

and

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

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

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
$\mathcal{D}_{f0}^{1}\times\mathcal{D}_{Vg}^{U1}$ we have equations

$\left(\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})\right)\left(\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})\right)^{*}=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+\lambda^{2d}g^{0}(\lambda))\right)\\\
\times\vec{\alpha}_{f}\cdot\vec{\alpha}_{f}^{*}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+\lambda^{2d}g^{0}(\lambda))\right),$
(29)
$\left(\mathbf{C}^{g0}_{\tau,N}(e^{i\lambda})\right)\left(\mathbf{C}^{g0}_{\tau,N}(e^{i\lambda})\right)^{*}=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2n}g^{0}(\lambda))\right)\\\
\times(\vec{\beta}\cdot\vec{\beta}^{*}+\Gamma_{1}(\lambda)+\Gamma_{2}(\lambda))\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2n}g^{0}(\lambda))\right),$
(30)

where $\vec{\alpha}_{f}$ and $\vec{\beta}$ are vectors of Lagrange
multipliers, 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
$\mathcal{D}_{f0}^{2}\times\mathcal{D}_{Vg}^{U2}$ we have equation

$\left(\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})\right)\left(\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})\right)^{*}=\alpha_{f}^{2}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+\lambda^{2d}g^{0}(\lambda))\right)^{2},$
(31)
$\left(\mathbf{C}^{g0}_{\tau,N}(e^{i\lambda})\right)\left(\mathbf{C}^{g0}_{\tau,N}(e^{i\lambda})=\right)^{*}\\\
=(\beta^{2}+\gamma_{1}(\lambda)+\gamma_{2}(\lambda))\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2n}g^{0}(\lambda))\right)^{2},$
(32)

where $\alpha_{f}^{2}$, $\beta^{2}$ are Lagrange multipliers, the 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
$\mathcal{D}_{f0}^{3}\times\mathcal{D}_{Vg}^{U3}$ we have equation

$\left(\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})\right)\left(\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})\right)^{*}=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+\lambda^{2d}g^{0}(\lambda))\right)\\\
\times\left\\{\alpha_{fk}^{2}\delta_{kl}\right\\}_{k,l=1}^{\infty}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+\lambda^{2d}g^{0}(\lambda))\right),$
(33)
$\left(\mathbf{C}^{g0}_{\tau,N}(e^{i\lambda})\right)\left(\mathbf{C}^{g0}_{\tau,N}(e^{i\lambda})\right)^{*}=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2n}g^{0}(\lambda))\right)\\\
\times\left\\{(\beta_{k}^{2}+\gamma_{1k}(\lambda)+\gamma_{2k}(\lambda))\delta_{kl}\right\\}_{k,l=1}^{\infty}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2n}g^{0}(\lambda))\right),$
(34)

where $\alpha_{fk}^{2}$, $\beta_{k}^{2}$ are Lagrange multipliers,
$\delta_{kl}$ are Kronecker symbols, the functions $\gamma_{1k}(\lambda)\leq
0$ and $\gamma_{1k}(\lambda)=0$ if $g_{kk}^{0}(\lambda)>v_{kk}(\lambda),$ the
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
$\mathcal{D}_{f0}^{4}\times\mathcal{D}_{Vg}^{U4}$ we have equations

$\left(\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})\right)\left(\mathbf{C}^{f0}_{\tau,N}(e^{i\lambda})\right)^{*}=\alpha_{f}^{2}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+\lambda^{2d}g^{0}(\lambda))\right)\\\
\times
B_{1}^{\top}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+\lambda^{2d}g^{0}(\lambda))\right),$
(35)
$\left(\mathbf{C}^{g0}_{\tau,N}(e^{i\lambda})\right)\left(\mathbf{C}^{g0}_{\tau,N}(e^{i\lambda})\right)^{*}\\\
=(\beta^{2}+\gamma^{\prime}_{1}(\lambda)+\gamma^{\prime}_{2}(\lambda))\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2n}g^{0}(\lambda))\right)\\\
\times
B_{2}^{\top}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}(f^{0}(\lambda)+{\lambda}^{2n}g^{0}(\lambda))\right),$
(36)

where $\alpha_{f}^{2}$, $\beta^{2}$, are Lagrange multipliers, the 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 3.1.

Let minimality condition (11) hold true. The least favorable spectral
densities $f_{0}(\lambda)$, $g_{0}(\lambda)$ in the classes
$\mathcal{D}_{f0}^{k}\times\mathcal{D}_{Vg}^{Uk},k=1,2,3,4$ for the optimal
linear estimation of the functional $A_{N}\vec{\xi}$ from observations of the
process ${\xi}(t)+{\eta}(t)$ at points $\mathbb{R}\setminus[0;(N+1)T]$ are
determined by equations (29)–(30), (31)–(32), (33)–(34), (35)–(36),
respectively, the constrained optimization problem (26) and restrictions on
densities from the corresponding classes
$\mathcal{D}_{f0}^{k},\mathcal{D}_{Vg}^{Uk},k=1,2,3,4$. The minimax-robust
spectral characteristic of the optimal estimate of the functional
$A_{NT}{\xi}$ is determined by formula (21).

##  4 Conclusions

In this article, we present results of investigation of the stochastic
processes with periodically stationary increments. These non-stationary
stochastic processes combine periodic structure of covariation functions of
sequences as well as integrating one.

We describe methods of solution of the classical interpolation problem for a
linear functional constructed from unobserved values of a process with
periodically stationary increments. Estimates are based on observations of
this process with a periodically stationary noise process. Estimates are
obtained by representing the process under investigation as a generated
infinite dimensional vector sequence with stationary increments. The problem
is investigated in the case of spectral certainty, where spectral densities of
the generates 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 $L_{1}$ space 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
characteristic of the optimal linear estimates of the functionals are derived
for a collection of specific classes of admissible spectral densities.

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