# Filtering problem for sequences with periodically stationary multiseasonal
increments with spectral densities allowing canonical factorizations

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 consider a stochastic sequence $\xi(m)$ with periodically stationary
generalized multiple increments of fractional order which combines
cyclostationary, multi-seasonal, integrated and fractionally integrated
patterns. The filtering problem is solved for this type of sequences based on
observations with a periodically stationary noise. When spectral densities are
known and allow the canonical factorizations, we derive the mean square error
and the spectral characteristics of the optimal estimate of the functional
$A{\xi}=\sum_{k=0}^{\infty}{a}(k){\xi}(-k)$. Formulas that determine the least
favourable spectral densities and the minimax (robust) spectral
characteristics of the optimal linear estimate of the functional are proposed
in the case where the spectral densities are not known, but some sets of
admissible spectral densities are given.

Keywords:Periodically Stationary Sequence, SARFIMA, Fractional Integration,
Filtering, Optimal Linear Estimate, Mean Square Error, Least Favourable
Spectral Density Matrix, Minimax Spectral Characteristics

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

## Introduction

Non-stationary time series models have found wide-ranging applications in
economics, finance, climatology, air pollution, signal processing. A
fundamental example is the general multiplicative model, known as
$SARIMA(p,d,q)\times(P,D,Q)_{s}$, which was introduced in the book by Box and
Jenkins et al. [4]. This model incorporates both integrated and seasonal
factors, and can be represented as follows:

$\Psi(B^{s})\psi(B)(1-B)^{d}(1-B^{s})^{D}x_{t}=\Theta(B^{s})\theta(B)\varepsilon_{t},$
(1)

where ${\varepsilon_{t}}$ is a sequence of independent and identically
distributed (i.i.d.) random variables, and where $\Psi(z)$ and $\Theta(z)$ are
two polynomials of degrees $P$ and $Q$, respectively, with roots outside the
unit circle. The parameters $d$ and $D$ can take fractional values. The
process in equation (1) is stationary and invertible when $|d+D|<1/2$ and
$|D|<1/2$. One application of seasonal ARFIMA models to the analysis of
monetary aggregates used by the U.S. Federal Reserve is demonstrated in the
work of Porter-Hudak [32].

In the field of statistical inference for seasonal long-memory sequences,
recent research has yielded several notable results. One such contribution is
the work by Tsai, Rachinger, and Lin [36], who developed methods for
estimating model parameters when measurement errors are present. In another
study, Baillie, Kongcharoen, and Kapetanios [2] compared two commonly used
estimation procedures for prediction problem based on ARFIMA models.
Specifically, they compared the performance of maximum likelihood estimation
(MLE) to a two-step local Whittle estimator. Through a simulation study, they
found that the MLE estimator outperformed the two-step local Whittle
estimator. In addition, Hassler and Pohle [13] evaluated the predictive
performance of various forecasting methods for inflation and return volatility
time series. Their analysis provided compelling evidence in support of models
with a fractional integration component.

Another class of non-stationary processes is the periodically correlated or
cyclostationary processes, introduced by Gladyshev [9]. These processes belong
to the class of time-dependent spectrum processes and are widely used in
signal processing and communications. For recent works on cyclostationarity
and its applications, see the review by Napolitano [30]. Periodic time series
can be viewed as an extension of seasonal models [1, 3, 20, 31].

The methods used for parameter estimation and filtering of time series data
often fail to account for real-world challenges such as outliers, measurement
errors, incomplete information about spectral structure. As a result, there is
a growing interest in robust estimation methods that can effectively handle
such issues. For example, Reisen et al. [33] and Solci et al. [35] have
proposed robust estimates for SARIMA and PAR models. Other researchers,
including Grenander [11], Hosoya [14], Franke [7], Vastola and Poor [37],
Moklyachuk [26, 27], and Luz and Moklyachuk [21], Liu et al. [19], have also
investigated various aspects of minimax extrapolation, interpolation, and
filtering problems for stationary sequences and processes.

In this article, we extend our investigation of robust filtering for
stochastic sequences with periodically stationary long memory multiple
seasonal increments (or sequences with periodically stationary general
multiplicative (GM) increments) by focusing on spectral densities that allow
canonical factorizations, whereas in [21], the results were obtained using
Fourier transformations of the spectral densities.

The mentioned sequences were introduced by Luz and Moklyachuk in our earlier
work [23], motivated by an increasing interest in models with multiple
seasonal and periodic patterns (see the works of Dudek [6], Gould et al. [10],
and Hurd and Piparas [15]). This research continues previous works on minimax
filtering of stationary vector-valued processes, periodically correlated
processes, and processes with stationary increments. Specifically, Moklyachuk
and Masyutka [28], Moklyachuk and Golichenko (Dubovetska) [5], and Luz and
Moklyachuk [22] have performed research in these areas. Additionally, we
mention the works by Moklyachuk, Masyutka, and Sidei [29], which derive
minimax estimates of stationary processes from observations with missing
values.

The article is structured as follows. In Section 1, we provide a brief review
of the GM increment sequence
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$ and the stochastic
sequence $\xi(m)$ with periodically stationary (periodically correlated,
cyclostationary) GM increments, as well as the spectral theory of vector-
valued GM increment sequences. In Section 2, we address the classical
filtering problem for linear functionals $A\xi$ and $A_{N}\xi$ that are
constructed from unobserved values of the sequence $\xi(m)$. We assume that
the spectral densities of the sequence $\xi(m)$ and a noise sequence $\eta(m)$
are known and allow canonical factorization. The estimates are derived in
terms of coefficients of canonical factorizations of the spectral densities,
making use of results obtained in [24] by using the Fourier transformations of
the spectral densities. Section 3 focuses on the minimax (robust) estimation
for cases where the spectral densities of sequences are not precisely known,
but some sets of admissible spectral densities are specified. For
illustration, We propose particular types of admissible spectral density sets,
which are generalizations of the sets described in a survey article by Kassam
and Poor [17] for stationary stochastic processes.

##  1 Stochastic sequences with periodically stationary generalized multiple
increments

###  1.1 Preliminary notations and definitions

Consider a stochastic sequence $\xi(m)$, $m\in\mathbb{Z}$, and a backward
shift operator $B_{\mu}$ with the step $\mu\in\mathbb{Z}$, such that
$B_{\mu}\xi(m)=\xi(m-\mu)$; $B:=B_{1}$. Then
$B_{\mu}^{s}=B_{\mu}B_{\mu}\cdot\ldots\cdot B_{\mu}$. Define a _multiplicative
incremental operator_

$\chi_{\overline{\mu},\overline{s}}^{(d)}(B)=\prod_{i=1}^{r}(1-B_{\mu_{i}}^{s_{i}})^{d_{i}}=\sum_{k=0}^{n(\gamma)}e_{\gamma}(k)B^{k},$

where $d:=d_{1}+d_{2}+\ldots+d_{r}$,
$\overline{d}=(d_{1},d_{2},\ldots,d_{r})\in(\mathbb{N}^{*})^{r}$,
$\overline{s}=(s_{1},s_{2},\ldots,s_{r})\in(\mathbb{N}^{*})^{r}$ and
$\overline{\mu}=(\mu_{1},\mu_{2},\ldots,\mu_{r})\in(\mathbb{N}^{*})^{r}$ or
$\in(\mathbb{Z}\setminus\mathbb{N})^{r}$;
$n(\gamma):=\sum_{i=1}^{r}\mu_{i}s_{i}d_{i}$. Here
$\mathbb{N}^{*}=\mathbb{N}\setminus\\{0\\}$. The explicit representation of
the coefficients $e_{\gamma}(k)$ is given in [23]. Within the article,
$\delta_{lp}$ denotes Kronecker symbols, ${n\choose l}=\frac{n!}{l!(n-l)!}$.

######  Definition 1 ([23]).

For a stochastic sequence $\xi(m)$, $m\in\mathbb{Z}$, the sequence

$\displaystyle\chi_{\overline{\mu},\overline{s}}^{(d)}(\xi(m))$
$\displaystyle:=$
$\displaystyle\chi_{\overline{\mu},\overline{s}}^{(d)}(B)\xi(m)=(1-B_{\mu_{1}}^{s_{1}})^{d_{1}}(1-B_{\mu_{2}}^{s_{2}})^{d_{2}}\cdot\ldots\cdot(1-B_{\mu_{r}}^{s_{r}})^{d_{r}}\xi(m)$
(2) $\displaystyle=$
$\displaystyle\sum_{l_{1}=0}^{d_{1}}\ldots\sum_{l_{r}=0}^{d_{r}}(-1)^{l_{1}+\ldots+l_{r}}{d_{1}\choose
l_{1}}\cdot\ldots\cdot{d_{r}\choose
l_{r}}\xi(m-\mu_{1}s_{1}l_{1}-\cdots-\mu_{r}s_{r}l_{r})$

is called _a stochastic generalized multiple (GM) increment sequence_ of
differentiation order $d$ with a fixed seasonal vector
$\overline{s}\in(\mathbb{N}^{*})^{r}$ and a varying step
$\overline{\mu}\in(\mathbb{N}^{*})^{r}$ or
$\in(\mathbb{Z}\setminus\mathbb{N})^{r}$.

######  Definition 2 ([23]).

A stochastic GM increment sequence
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\xi(m))$ is called a wide sense
stationary if the mathematical expectations

$\displaystyle\mathsf{E}\chi_{\overline{\mu},\overline{s}}^{(d)}(\xi(m_{0}))$
$\displaystyle=$ $\displaystyle c^{(d)}_{\overline{s}}(\overline{\mu}),$
$\displaystyle\mathsf{E}\chi_{\overline{\mu}_{1},\overline{s}}^{(d)}(\xi(m_{0}+m))\chi_{\overline{\mu}_{2},\overline{s}}^{(d)}(\xi(m_{0}))$
$\displaystyle=$ $\displaystyle
D^{(d)}_{\overline{s}}(m;\overline{\mu}_{1},\overline{\mu}_{2})$

exist for all $m_{0},m,\overline{\mu},\overline{\mu}_{1},\overline{\mu}_{2}$
and do not depend on $m_{0}$. The function
$c^{(d)}_{\overline{s}}(\overline{\mu})$ is called a mean value and the
function $D^{(d)}_{\overline{s}}(m;\overline{\mu}_{1},\overline{\mu}_{2})$ is
called a structural function of the stationary GM increment sequence (of a
stochastic sequence with stationary GM increments).
The stochastic sequence $\xi(m)$, $m\in\mathbb{Z}$ determining the stationary
GM increment sequence $\chi_{\overline{\mu},\overline{s}}^{(d)}(\xi(m))$ by
(2) is called a stochastic sequence with stationary GM increments (or GM
increment sequence of order $d$).

######  Remark 1.

Spectral properties of one-pattern increment sequence
$\chi_{\mu,1}^{(n)}(\xi(m)):=\xi^{(n)}(m,\mu)=(1-B_{\overline{\mu}})^{n}\xi(m)$
and the continuous time increment process
$\xi^{(n)}(t,\tau)=(1-B_{\tau})^{n}\xi(t)$ are described in [38], [39].

###  1.2 Definition and spectral representation of stochastic sequences with
periodically stationary GM increment

In this subsection, we present definition, justification and a brief review of
the spectral theory of stochastic sequences with periodically stationary
multiple seasonal increments, introduced in [23].

######  Definition 3.

A stochastic sequence $\xi(m)$, $m\in\mathbb{Z}$ is called a _stochastic
sequence with periodically stationary (periodically correlated) GM increments_
with period $T$ if the mathematical expectations

$\displaystyle\mathsf{E}\chi_{\overline{\mu},T\overline{s}}^{(d)}(\xi(m+T))$
$\displaystyle=$
$\displaystyle\mathsf{E}\chi_{\overline{\mu},T\overline{s}}^{(d)}(\xi(m))=c^{(d)}_{T\overline{s}}(m,\overline{\mu}),$
$\displaystyle\mathsf{E}\chi_{\overline{\mu}_{1},T\overline{s}}^{(d)}(\xi(m+T))\chi_{\overline{\mu}_{2},T\overline{s}}^{(d)}(\xi(k+T))$
$\displaystyle=$ $\displaystyle
D^{(d)}_{T\overline{s}}(m+T,k+T;\overline{\mu}_{1},\overline{\mu}_{2})=D^{(d)}_{T\overline{s}}(m,k;\overline{\mu}_{1},\overline{\mu}_{2})$

exist for every $m,k,\overline{\mu}_{1},\overline{\mu}_{2}$ and $T>0$ is the
least integer for which these equalities hold.

It follows from Definition 3 that the sequence

$\xi_{p}(m)=\xi(mT+p-1),\quad p=1,2,\dots,T;\quad m\in\mathbb{Z}$ (3)

forms a vector-valued sequence
$\vec{\xi}(m)=\left\\{\xi_{p}(m)\right\\}_{p=1,2,\dots,T},m\in\mathbb{Z}$ with
stationary GM increments as follows:

$\chi_{\overline{\mu},\overline{s}}^{(d)}(\xi_{p}(m))=\chi_{\overline{\mu},T\overline{s}}^{(d)}(\xi(mT+p-1)),\quad
p=1,2,\dots,T,$

where $\chi_{\overline{\mu},\overline{s}}^{(d)}(\xi_{p}(m))$ is the GM
increment of the $p$-th component of the vector-valued sequence
$\vec{\xi}(m)$.

The following theorem describes the spectral structure of the vector-valued GM
increment [16], [23].

######  Theorem 1.

1\. The mean value and the structural function of the vector-valued stochastic
stationary GM increment sequence
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$ can be represented in
the form

$\displaystyle c^{(d)}_{\overline{s}}(\overline{\mu})$ $\displaystyle=$
$\displaystyle c\prod_{i=1}^{r}\mu_{i}^{d_{i}},$ (4) $\displaystyle
D^{(d)}_{\overline{s}}(m;\overline{\mu}_{1},\overline{\mu}_{2})$
$\displaystyle=$ $\displaystyle\int_{-\pi}^{\pi}e^{i\lambda
m}\chi_{\overline{\mu}_{1}}^{(d)}(e^{-i\lambda})\chi_{\overline{\mu}_{2}}^{(d)}(e^{i\lambda})\frac{1}{|\beta^{(d)}(i\lambda)|^{2}}dF(\lambda),$
(5)

where

$\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})=\prod_{j=1}^{r}(1-e^{-i\lambda\mu_{j}s_{j}})^{d_{j}},\quad\beta^{(d)}(i\lambda)=\prod_{j=1}^{r}\prod_{k_{j}=-[s_{j}/2]}^{[s_{j}/2]}(i\lambda-2\pi
ik_{j}/s_{j})^{d_{j}},$

$c$ is a vector, $F(\lambda)$ is the matrix-valued spectral function of the
stationary stochastic sequence
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$. The vector $c$ and
the matrix-valued function $F(\lambda)$ are determined uniquely by the GM
increment sequence $\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$.

2\. The stationary vector-valued GM increment sequence
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$ admits the spectral
representation

$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))=\int_{-\pi}^{\pi}e^{im\lambda}\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})\frac{1}{\beta^{(d)}(i\lambda)}d\vec{Z}_{\xi^{(d)}}(\lambda),$
(6)

where $\vec{Z}_{\xi^{(d)}}(\lambda)=\\{Z_{p}(\lambda)\\}_{p=1}^{T}$ is a
(vector-valued) stochastic process with uncorrelated increments on
$[-\pi,\pi)$ connected with the spectral function $F(\lambda)$ by the relation

$\mathsf{E}(Z_{p}(\lambda_{2})-Z_{p}(\lambda_{1}))(\overline{Z_{q}(\lambda_{2})-Z_{q}(\lambda_{1})})=F_{pq}(\lambda_{2})-F_{pq}(\lambda_{1}),$
$-\pi\leq\lambda_{1}<\lambda_{2}<\pi,\quad p,q=1,2,\dots,T.$

Consider another vector-valued stochastic sequence with the stationary GM
increments $\vec{\zeta}(m)=\vec{\xi}(m)+\vec{\eta}(m)$, where $\vec{\eta}(m)$
is a vector-valued stationary stochastic sequence, uncorrelated with
$\vec{\xi}(m)$, with the spectral representation

$\vec{\eta}(m)=\int_{-\pi}^{\pi}e^{i\lambda m}dZ_{\eta}(\lambda),$

where $Z_{\eta}(\lambda)=\\{Z_{\eta,p}(\lambda)\\}_{p=1}^{T}$,
$\lambda\in[-\pi,\pi)$, is a stochastic process with uncorrelated increments,
that corresponds to the spectral function $G(\lambda)$ [12]. The stochastic
stationary GM increment
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\zeta}(m))$ allows the spectral
representation

$\displaystyle\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\zeta}(m))$
$\displaystyle=$ $\displaystyle\int_{-\pi}^{\pi}e^{i\lambda
m}\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}dZ_{\xi^{(n)}}(\lambda)+\int_{-\pi}^{\pi}e^{i\lambda
m}\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})dZ_{\eta}(\lambda),$

while
$dZ_{\eta}(\lambda)=(\beta^{(d)}(i\lambda))^{-1}dZ_{\eta^{(n)}}(\lambda)$,
$\lambda\in[-\pi,\pi)$. Therefore, in the case where the spectral functions
$F(\lambda)$ and $G(\lambda)$ have the spectral densities $f(\lambda)$ and
$g(\lambda)$, the spectral density
$p(\lambda)=\\{p_{ij}(\lambda)\\}_{i,j=1}^{T}$ of the stochastic sequence
$\vec{\zeta}(m)$ is determined by the formula

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

For a regular stationary GM increment sequence
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$ [24], there exists an
innovation sequence
${\vec{\varepsilon}(u)=\\{\varepsilon_{k}(u)\\}_{k=1}^{q},u\in\mathbb{Z}}$ and
a sequence of matrix-valued functions
$\varphi^{(d)}(k,\overline{\mu})=\\{\varphi^{(d)}_{ij}(k,\overline{\mu})\\}_{i=\overline{1,T}}^{j=\overline{1,q}}$,
$k\geq 0$, such that

$\sum_{k=0}^{\infty}\sum_{i=1}^{T}\sum_{j=1}^{q}|\varphi^{(d)}_{ij}(k,\overline{\mu})|^{2}<\infty,\quad\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))=\sum_{k=0}^{\infty}\varphi^{(d)}(k,\overline{\mu})\vec{\varepsilon}(m-k).$
(7)

Representation (7) is called _a canonical moving average representation_ of
the stochastic stationary GM increment sequence
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$. Its spectral
function $F(\lambda)$ has the spectral density
$f(\lambda)=\\{f_{ij}(\lambda)\\}_{i,j=1}^{T}$ admitting the canonical
factorization

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

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

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

Then the following relation holds true:

$\varphi_{\overline{\mu}}(e^{-i\lambda})\varphi^{*}_{\overline{\mu}}(e^{-i\lambda})=\frac{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}{|\beta^{(d)}(i\lambda)|^{2}}f(\lambda)=\prod_{j=1}^{r}\frac{\left|1-e^{-i\lambda\mu_{j}s_{j}}\right|^{2d_{j}}}{\prod_{k_{j}=-[s_{j}/2]}^{[s_{j}/2]}|\lambda-2\pi
k_{j}/s_{j}|^{2d_{j}}}f(\lambda).$ (8)

The one-sided moving average representation (7) and relation (8) are used for
finding the mean square optimal estimates of unobserved values of vector-
valued sequences with stationary GM increments.

##  2 Hilbert space projection method of filtering

###  2.1 Filtering of vector-valued stochastic sequence with stationary GM
increments

Consider a vector-valued stochastic sequence $\vec{\xi}(m)$ with stationary GM
increments constructed from transformation (3) and a vector-valued stationary
stochastic sequence $\vec{\eta}(m)$ uncorrelated with the sequence
$\vec{\xi}(m)$. Let the stationary GM increment sequence
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))=\\{\chi_{\overline{\mu},\overline{s}}^{(d)}(\xi_{p}(m))\\}_{p=1}^{T}$
and the stationary sequence $\vec{\eta}(m)$ have absolutely continuous
spectral functions $F(\lambda)$ and $G(\lambda)$ with the spectral densities
$f(\lambda)=\\{f_{ij}(\lambda)\\}_{i,j=1}^{T}$ and
$g(\lambda)=\\{g_{ij}(\lambda)\\}_{i,j=1}^{T}$ respectively. Without loss of
generality assume that
$\mathsf{E}\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))=0$,
$\mathsf{E}\vec{\eta}(m)=0$ and $\overline{\mu}>\overline{0}$.

Filtering problem. Consider the problem of mean square optimal linear
estimation of the functional

$A\vec{\xi}=\sum_{k=0}^{\infty}(\vec{a}(k))^{\top}\vec{\xi}(-k),$ (9)

which depends on unobserved values of a stochastic sequence
$\vec{\xi}(k)=\\{\xi_{p}(k)\\}_{p=1}^{T}$ with stationary GM increments.
Estimates are based on observations of the sequence
$\vec{\zeta}(k)=\vec{\xi}(k)+\vec{\eta}(k)$ at points $k=0,-1,-2,\ldots$.

We suppose that the conditions on coefficients
$\vec{a}(k)=\\{a_{p}(k)\\}_{p=1}^{T}$, $k\geq 0$

$\sum_{k=0}^{\infty}\|\vec{a}(k)\|<\infty,\quad\sum_{k=0}^{\infty}(k+1)\|\vec{a}(k)\|^{2}<\infty,$
(10)

and _the minimality condition_ on the spectral densities $f(\lambda)$ and
$g(\lambda)$

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

are satisfied. The second condition (11) is the necessary and sufficient one
under which the mean square error of the optimal estimate of functional
$A\vec{\xi}$ is not equal to $0$.

Any linear estimate $\widehat{A}\vec{\xi}$ of the functional $A\vec{\xi}$
allows the representation

$\widehat{A}\vec{\xi}=\sum_{k=0}^{\infty}(\vec{a}(k))^{\top}(\vec{\xi}(-k)+\vec{\eta}(-k))-\int_{-\pi}^{\pi}(\vec{h}_{\overline{\mu}}(\lambda))^{\top}dZ_{\xi^{(n)}+\eta^{(n)}}(\lambda),$
(12)

where $\vec{h}_{\overline{\mu}}(\lambda)=\\{h_{p}(\lambda)\\}_{p=1}^{T}$ is
the spectral characteristic of the estimate $\widehat{A}\vec{\eta}$.

In the Hilbert space $L_{2}(p)$, define

$L_{2}^{0}(p)=\overline{span}\\{e^{i\lambda
k}\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})(\beta^{(d)}(i\lambda))^{-1}\vec{\delta}_{l}:\,\vec{\delta}_{l}=\\{\delta_{lp}\\}_{p=1}^{T},\,l=1,\ldots,T,\,k\leq
0\\}.$

Define the following matrix-valued Fourier coefficients:

$\displaystyle S_{\overline{\mu}}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-i\lambda
k}\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\left[g(\lambda)(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))^{-1}\right]^{\top}d\lambda,\quad
k\in\mathbb{Z},$ $\displaystyle P_{\overline{\mu}}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-i\lambda
k}\dfrac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\left[(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))^{-1}\right]^{\top}d\lambda,\quad
k\in\mathbb{Z},$ $\displaystyle Q(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-i\lambda
k}\left[f(\lambda)(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))^{-1}g(\lambda)\right]^{\top}d\lambda,\quad
k\in\mathbb{Z}.$

Define the vectors
$\mathbf{a}=((\vec{a}(0))^{\top},(\vec{a}(1))^{\top},(\vec{a}(2))^{\top},\ldots)^{\top}$
and
$\mathbf{a}_{\overline{\mu}}=((\vec{a}_{\overline{\mu}}(0))^{\top},(\vec{a}_{\overline{\mu}}(1))^{\top},(\vec{a}_{\overline{\mu}}(2))^{\top},\ldots)^{\top}$,
where the coefficients
$\vec{a}_{\overline{\mu}}(k)=\vec{a}_{-{\overline{\mu}}}(k-n(\gamma))$, $k\geq
0$,

$\vec{a}_{-\overline{\mu}}(m)=\sum_{l=\max\left\\{m,0\right\\}}^{m+n(\gamma)}e_{\gamma}(l-m)\vec{a}(l),\quad
m\geq-n(\gamma).$ (13)

Define the matrices $\mathbf{S}_{\overline{\mu}}$,
$\mathbf{P}_{\overline{\mu}}$ and $\mathbf{Q}$ by the matrix-valued entries
$(\mathbf{S}_{\overline{\mu}})_{l,k}=S_{\overline{\mu}}(l+1+k-n(\gamma))$,
$(\mathbf{P}_{\overline{\mu}})_{l,k}=P_{\overline{\mu}}(l-k)$ and
$(\mathbf{Q})_{l,k}=Q(l-k)$, $l,k\geq 0$.

The solution to the filtering problem is described by the following theorem in
terms of Fourier coefficients
$\\{S_{\overline{\mu}}(k),P_{\overline{\mu}}(k),Q(k):k\in\mathbb{Z}\\}$ .

######  Theorem 2 ([24]).

A solution $\widehat{A}\vec{\xi}$ to the filtering problem for the linear
functional $A\vec{\xi}$ of the values of a vector-valued stochastic sequence
$\vec{\xi}(m)$ with stationary GM increments under conditions (10) and (11) is
calculated by formula (12). The spectral characteristic
$\vec{h}_{\overline{\mu}}(\lambda)$ and the value of the mean square error
$\Delta(f,g;\widehat{A}\vec{\xi})$ are calculated by the formulas

$(\vec{h}_{\overline{\mu}}(\lambda))^{\top}=\left[\chi_{\overline{\mu}}^{(d)}(e^{i\lambda})(A(e^{-i\lambda}))^{\top}g(\lambda)-(C_{\overline{\mu}}(e^{i\lambda}))^{\top}\right]p^{-1}(\lambda)\frac{\overline{\beta^{(d)}(i\lambda)}}{\chi_{\overline{\mu}}^{(d)}(e^{i\lambda})},$
(14)

where

$A(e^{-i\lambda})=\sum_{k=0}^{\infty}\vec{a}(k)e^{-i\lambda k},\quad
C_{\overline{\mu}}(e^{i\lambda})=\sum_{k=0}^{\infty}\left(\mathbf{P}_{\overline{\mu}}^{-1}\mathbf{S}_{\overline{\mu}}\mathbf{a}_{\overline{\mu}}\right)_{k}e^{i\lambda(k+1)},$

and

$\displaystyle\Delta\left(f,g;\widehat{A}\vec{\xi}\right)=\mathsf{E}\left|A\vec{\xi}-\widehat{A}\vec{\xi}\right|^{2}=\left\langle\mathbf{S}_{\overline{\mu}}\mathbf{a}_{\overline{\mu}},\mathbf{P}_{\overline{\mu}}^{-1}\mathbf{S}_{\overline{\mu}}\mathbf{a}_{\overline{\mu}}\right\rangle+\left\langle\mathbf{Q}\mathbf{a},\mathbf{a}\right\rangle.$
(15)

######  Remark 2.

The filtering problem in the presence of fractional integration is considered
in [24].

###  2.2 Filtering based on factorizations of the spectral densities

The main goal of the article is to derive the classical and minimax estimates
of the functional $A\vec{\xi}$ in terms of the coefficients of the canonical
factorizations of the spectral densities $f(\lambda)$, $g(\lambda)$ and
$f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda)$.

Let the following canonical factorizations take place

$\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))=\Theta_{\overline{\mu}}(e^{-i\lambda})\Theta_{\overline{\mu}}^{*}(e^{-i\lambda}),\quad\Theta_{\overline{\mu}}(e^{-i\lambda})=\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k},$ (16) $g(\lambda)=\sum_{k=-\infty}^{\infty}g(k)e^{i\lambda
k}=\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda}),\quad\Phi(e^{-i\lambda})=\sum_{k=0}^{\infty}\phi(k)e^{-i\lambda
k}.$ (17)

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

$\Psi_{\overline{\mu}}(e^{-i\lambda})\Theta_{\overline{\mu}}(e^{-i\lambda})=E_{q},$

where $E_{q}$ is an identity $q\times q$ matrix. One can check that the
following factorization takes place

$\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))^{-1}=\Psi_{\overline{\mu}}^{*}(e^{-i\lambda})\Psi_{\overline{\mu}}(e^{-i\lambda}),\quad\Psi_{\overline{\mu}}(e^{-i\lambda})=\sum_{k=0}^{\infty}\psi_{\overline{\mu}}(k)e^{-i\lambda
k},$ (18)

######  Remark 3.

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 representations of $\mathbf{P}_{\overline{\mu}}$
and
$\mathbf{P}_{\overline{\mu}}^{-1}\mathbf{S}_{\overline{\mu}}\mathbf{a}_{\overline{\mu}}$,
which contain coefficients of factorizations (16) – (18).

######  Lemma 1.

Let factorization (16) takes place and let $q\times T$ matrix function
$\Psi_{\overline{\mu}}(e^{-i\lambda})$ satisfy an equation
$\Psi_{\overline{\mu}}(e^{-i\lambda})\Phi_{\overline{\mu}}(e^{-i\lambda})=E_{q}$.
Define the linear operators $\Psi_{\overline{\mu}}$ and
$\Theta_{\overline{\mu}}$ in the space $\ell_{2}$ by the matrices with the
matrix entries $(\Psi_{\overline{\mu}})_{k,j}=\psi_{\overline{\mu}}(k-j)$,
$(\Theta_{\overline{\mu}})_{k,j}=\theta_{\overline{\mu}}(k-j)$ for $0\leq
j\leq k$, $(\Psi_{\overline{\mu}})_{k,j}=0$,
$(\Theta_{\overline{\mu}})_{k,j}=0$ for $0\leq k<j$. Then:
a) the linear operator $\mathbf{P}_{\overline{\mu}}$ admits the factorization

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

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

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

###### Proof.

See [25]. ∎

######  Lemma 2.

Let factorizations (16) and (17) take place. Define by
$\widetilde{e}_{\overline{\mu}}(m)=\left(\Theta^{\top}_{\overline{\mu}}\mathbf{S}_{\overline{\mu}}\widetilde{\mathbf{a}}_{\overline{\mu}}\right)_{m}$,
$m\geq 0$, the $m$th element of the vector
$\widetilde{\mathbf{e}}_{\overline{\mu}}=\Theta^{\top}_{\overline{\mu}}\mathbf{S}_{\overline{\mu}}\widetilde{\mathbf{a}}_{\overline{\mu}}$.
Then

$\widetilde{e}_{\overline{\mu}}(m)=\sum_{j=-n(\gamma)}^{\infty}Z_{\overline{\mu}}(m+j+1)a_{-\mu}(j),$

where $Z_{\overline{\mu}}(j)$, $j\in\mathbb{Z}$, are defined as

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

###### Proof.

See Appendix. ∎

Define the linear operators $\mathbf{G}^{-}$,
$\mathbf{G}^{+}$,$\widetilde{\Phi}^{+}$ in the space $\ell_{2}$ by matrices
with the matrix entries $(\mathbf{G}^{-})_{l,k}=\overline{g}(l-k)$,
$(\mathbf{G}^{+})_{l,k}=\overline{g}(l+k)$, $\phi^{\top}(k+j)$
$l,k\in\mathbb{Z}$. And the linear operator $\widetilde{\Phi}$ in the space
$\ell_{2}$ determined by a matrix with the matrix entries
$(\widetilde{\Phi})_{k,j}=\phi^{\top}(k-j)$ for $0\leq j\leq k$,
$(\widetilde{\Phi})_{k,j}=0$ for $0\leq k<j$.

Define also the coefficients $\\{\vec{b}_{-\mu}(k):k\geq 0\\}$ as follows:
$\vec{b}_{-\mu}(0)=0$, $\vec{b}_{-\mu}(k)=\vec{a}_{-\mu}(-k)$ for $1\leq k\leq
n(\gamma)$, $\vec{b}_{-\mu}(k)=0$ for $k>n(\gamma)$, where coefficients
$\vec{a}_{-\mu}(k)$ are calculated by formula (13), and the vectors

$\mathbf{a}_{-\mu}=((\vec{a}_{-\mu}(0))^{\top},(\vec{a}_{-\mu}(1))^{\top},(\vec{a}_{-\mu}(2))^{\top},\ldots)^{\top},\quad\mathbf{b}_{-\mu}=((\vec{b}_{-\mu}(0))^{\top},(\vec{b}_{-\mu}(1))^{\top},(\vec{b}_{-\mu}(2))^{\top},\ldots)^{\top}.$

The following theorem describes a solution to the filtering problem in the
case when the spectral densities $f(\lambda)$ and $g(\lambda)$ admit canonical
factorizations (16) – (18) .

######  Theorem 3.

A solution $\widehat{A}\vec{\xi}$ to the filtering problem for the linear
functional $A\vec{\xi}$ of the values of a vector-valued stochastic sequence
$\vec{\xi}(m)$ with stationary GM increments under condition (10) and provided
that the spectral densities $f(\lambda)$ and $g(\lambda)$ of the stochastic
vector sequences $\vec{\xi}(m)$ and $\vec{\eta}(m)$ admit canonical
factorizations (16) – (18) is calculated by formula (12). The spectral
characteristic $\vec{h}_{\overline{\mu}}(\lambda)$ is calculated by the
formulas

$\displaystyle\vec{h}_{\overline{\mu}}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}((\widetilde{\Psi}_{\overline{\mu}})^{*}\mathbf{G}^{-}\mathbf{a}_{-\mu}+(\widetilde{\Psi}_{\overline{\mu}})^{*}\mathbf{G}^{+}\mathbf{b}_{-\mu})_{m}e^{-i\lambda
m}$ (19) $\displaystyle=$
$\displaystyle\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}(\overline{\psi}_{\overline{\mu}}\mathbf{C}^{-}_{\overline{\mu},g}+\overline{\psi}_{\overline{\mu}}\mathbf{C}^{+}_{\overline{\mu},g})_{m}e^{-i\lambda
m},$

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

$(\overline{\psi}_{\overline{\mu}}\mathbf{C}^{\pm}_{\overline{\mu},g})_{m}=\sum_{k=0}^{\infty}\overline{\psi}_{\overline{\mu}}(k)\mathbf{c}^{\pm}_{\overline{\mu},g}(k+m),$
$\mathbf{c}^{-}_{\overline{\mu},g}(m)=\sum_{k=0}^{\infty}\overline{g}(m-k)a_{-\overline{\mu}}(k)=\sum_{l=0}^{\infty}\overline{\phi}(l)\sum_{k=0}^{l+m}\phi^{\top}(l+m-k)a_{-\overline{\mu}}(k)=\sum_{l=0}^{\infty}\overline{\phi}(l)(\widetilde{\Phi}\mathbf{a}_{-\overline{\mu}})_{l+m}=(\widetilde{\Phi}^{*}\widetilde{\Phi}\mathbf{a}_{-\overline{\mu}})_{m},$
$\mathbf{c}^{+}_{\overline{\mu},g}(m)=\sum_{k=0}^{\infty}\overline{g}(m+k)b_{-\overline{\mu}}(k)=\sum_{l=0}^{\infty}\overline{\phi}(l)\sum_{k=0}^{\infty}\phi^{\top}(l+m+k)b_{-\overline{\mu}}(k)=\sum_{l=0}^{\infty}\overline{\phi}(l)(\widetilde{\Phi}^{+}\mathbf{b}_{-\overline{\mu}})_{l+m}=(\widetilde{\Phi}^{*}\widetilde{\Phi}^{+}\mathbf{b}_{-\overline{\mu}})_{m}.$

The the value of the mean square error $\Delta(f,g;\widehat{A}\vec{\xi})$ is
calculated by the formulas

$\displaystyle\Delta\left(f,g;\widehat{A}\xi\right)=\|\widetilde{\Phi}\mathbf{a}\|^{2}-\|\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\|^{2}.$
(20)

###### Proof.

See Appendix. ∎

######  Remark 4.

The following factorizations hold true:

$\mathbf{G}^{-}=\widetilde{\Phi}^{*}\widetilde{\Phi},\quad\mathbf{G}^{+}=\widetilde{\Phi}^{*}\widetilde{\Phi}^{+}.$

The filtering problem for the functional $A_{N}\vec{\xi}$ is solved directly
by Theorem 3 by putting $\vec{a}(k)=\vec{0}$ for $k>N$. To solve the filtering
problem for the $p$th coordinate of the single vector $\vec{\xi}(-N)$, we put
$\vec{a}(N)=\vec{\delta}_{p}$, $\vec{a}(k)=\vec{0}$ for $k\neq N$.

The following corollaries take place.

######  Corollary 1.

A solution $\widehat{A}_{N}\vec{\xi}$ to the filtering problem for the linear
functional $A_{N}\vec{\xi}$ of the values of a vector-valued stochastic
sequence $\vec{\xi}(m)$ with stationary GM increments under condition (10) is
calculated by the formula

$\widehat{A}_{N}\vec{\xi}=\sum_{k=0}^{N}(\vec{a}(k))^{\top}(\vec{\xi}(-k)+\vec{\eta}(-k))-\int_{-\pi}^{\pi}(\vec{h}_{\overline{\mu},N}(\lambda))^{\top}dZ_{\xi^{(n)}+\eta^{(n)}}(\lambda).$
(21)

The spectral characteristic $\vec{h}_{\overline{\mu},N}(\lambda)$ and the
value of the mean square error $\Delta(f,g;\widehat{A}_{N}\vec{\xi})$ of the
optimal estimate $\widehat{A}_{N}\vec{\xi}$ are calculated by formulas (19)
and (20) for the vectors $\mathbf{a}$, $\mathbf{a}_{-\mu}$,
$\mathbf{b}_{-\mu}$ calculated as

$\mathbf{a}_{N}=((\vec{a}(0))^{\top},(\vec{a}(1))^{\top},(\vec{a}(2))^{\top},\ldots,(\vec{a}(N))^{\top},0,\ldots)^{\top},$
$\mathbf{a}_{-\mu,N}=((\vec{a}_{-\mu,N}(0))^{\top},(\vec{a}_{-\mu,N}(1))^{\top},...,(\vec{a}_{-\mu,N}(N))^{\top},0,\ldots)^{\top},$
$\mathbf{b}_{-\mu,N}=(0,(\vec{a}_{-\mu,N}(-1))^{\top},(\vec{a}_{-\mu,N}(-2))^{\top},...,(\vec{a}_{-\mu,N}(-n(\gamma)))^{\top},0\ldots)^{\top}$

where

$\displaystyle\vec{a}_{-\overline{\mu},N}(m)=\sum_{l=\max\left\\{m,0\right\\}}^{\min\\{m+n(\gamma),N\\}}e_{\gamma}(l-m)\vec{a}(l),\quad-n(\gamma)\leq
m\leq N.$ (22)

######  Corollary 2.

The optimal linear estimate $\widehat{\xi}_{p}(-N)$ of an unobserved value
$\xi_{p}(-N)$, $N\geq 0$, of the stochastic vector sequence $\vec{\xi}(m)$
with GM stationary increments based on observations of the sequence
$\vec{\xi}(m)+\vec{\eta}(m)$ at points $m=0,-1,-2,\ldots$, where the noise
sequence $\vec{\eta}(m)$ is uncorrelated with $\vec{\xi}(m)$, is calculated by
the formula

$\widehat{\xi}_{p}(-N)=(\xi_{p}(-N)+\eta_{p}(-N))-\int_{-\pi}^{\pi}(\vec{h}_{\overline{\mu},N,p}(\lambda))^{\top}dZ_{\xi^{(n)}+\eta^{(n)}}(\lambda).$
(23)

Put

$\displaystyle\vec{a}_{-\overline{\mu},N,p}(m)=e_{\gamma}(N-m)\vec{\delta}_{p},\quad
N-n(\gamma)\leq m\leq N.$

If $N<n(\gamma)$, the spectral characteristic
$\vec{h}_{\overline{\mu},N,p}(\lambda)$ and the value of the mean square error
$\Delta(f,g;\widehat{\xi}_{p}(-N))$ of the optimal estimate
$\widehat{\xi}_{p}(-N)$ are calculated by formulas (19) and (20) for the
vectors $\mathbf{a}$, $\mathbf{a}_{-\mu}$, $\mathbf{b}_{-\mu}$ calculated as

$\mathbf{a}_{N,p}=(0,0,\ldots,(\mathbf{\delta}_{p})^{\top},0,\ldots)^{\top},$
$\mathbf{a}_{-\mu,N,p}=((\vec{a}_{-\mu,N,p}(0))^{\top},(\vec{a}_{-\mu,N,p}(1))^{\top},...,(\vec{a}_{-\mu,N,p}(N))^{\top},0,\ldots)^{\top},$
$\mathbf{b}_{-\mu,N,p}=(0,(\vec{a}_{-\mu,N,p}(-1))^{\top},(\vec{a}_{-\mu,N,p}(-2))^{\top},...,(\vec{a}_{-\mu,N,p}(N-n(\gamma)))^{\top},0\ldots)^{\top}.$

If $N\geq n(\gamma)$, the spectral characteristic
$\vec{h}_{\overline{\mu},N,p}(\lambda)$ and the value of the mean square error
$\Delta(f,g;\widehat{\xi}_{p}(-N))$ of the optimal estimate
$\widehat{\xi}_{p}(-N)$ are calculated by formulas

$\displaystyle\vec{h}_{\overline{\mu},N,p}(\lambda)=\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}((\widetilde{\Psi}_{\overline{\mu}})^{*}\mathbf{G}^{-}\mathbf{a}_{-\mu,N,p})_{m}e^{-i\lambda
m}$ (24)

and

$\displaystyle\Delta\left(f,g;\widehat{\xi}_{p}(-N)\right)=\left\langle\overline{g}(0)\vec{\delta}_{p},\vec{\delta}_{p}\right\rangle-\|(\widetilde{\Psi}_{\overline{\mu}})^{*}\mathbf{G}^{-}\mathbf{a}_{-\mu,N,p}\|^{2},$

where

$\mathbf{a}_{-\mu,N,p}=(0,\ldots,0,(\vec{a}_{-\mu,N,p}(N-n(\gamma)))^{\top},...,(\vec{a}_{-\mu,N,p}(N))^{\top},0,\ldots)^{\top}.$

###  2.3 Filtering of stochastic sequences with periodically stationary GM
increment

Consider the filtering problem for the functionals

$A{\xi}=\sum_{k=0}^{\infty}{a}^{(\xi)}(k)\xi(-k),\quad
A_{M}{\xi}=\sum_{k=0}^{N}{a}^{(\xi)}(k)\xi(-k)$ (25)

which depend on unobserved values of a stochastic sequence $\xi(m)$ with
periodically stationary GM increments. Estimates are based on observations of
the sequence $\xi(m)+\eta(m)$ at points $m=0,-1,-2,\ldots$, where the
periodically stationary noise sequence $\eta(m)$ is uncorrelated with
$\xi(m)$.

The functional $A{\xi}$ can be represented in the form

$\displaystyle A{\xi}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}{a}^{(\xi)}(k)\xi(-k)=\sum_{m=0}^{\infty}\sum_{p=1}^{T}{a}^{(\xi)}(mT+p-1)\xi(-mT-p+1)$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\sum_{p=1}^{T}a_{p}(m)\xi_{p}(-m)=\sum_{m=0}^{\infty}(\vec{a}(m))^{\top}\vec{\xi}(-m)=A\vec{\xi},$

where

$\vec{\xi}(m)=({\xi}_{1}(m),{\xi}_{2}(m),\dots,{\xi}_{T}(m))^{\top},\,{\xi}_{p}(m)=\xi(mT+p-1);\,p=1,2,\dots,T;$
(26)

and

$\vec{a}(m)=({a}_{1}(m),{a}_{2}(m),\dots,{a}_{T}(m))^{\top},\,{a}_{p}(m)=a^{(\xi)}(mT+p-1);\,p=1,2,\dots,T.$
(27)

In the same way, the functional $A{\eta}$ is represented as

$A{\eta}=\sum_{k=0}^{\infty}{a}^{(\xi)}(k)\eta(-k)=\sum_{m=0}^{\infty}(\vec{a}(m))^{\top}\vec{\eta}(-m)=A\vec{\eta},$

where

$\vec{\eta}(m)=({\eta}_{1}(m),{\eta}_{2}(m),\dots,{\eta}_{T}(m))^{\top},\,{\eta}_{p}(m)=\eta(mT+p-1);\,p=1,2,\dots,T.$
(28)

Making use of the introduced notations and statements of Theorem 2 we can
claim that the following theorem holds true.

######  Theorem 4.

Let a stochastic sequence ${\xi}(m)$ with periodically stationary GM
increments and a stochastic periodically stationary sequence ${\eta}(m)$
generate by formulas (26) and (28) vector-valued stochastic sequences
$\vec{\xi}(m)$ and $\vec{\eta}(m)$ with the spectral densities matrices
$f(\lambda)=\\{f_{ij}(\lambda)\\}_{i,j=1}^{T}$ and
$g(\lambda)=\\{g_{ij}(\lambda)\\}_{i,j=1}^{T}$ admitting canonical
factorizations (16) – (18). A solution $\widehat{A}\xi$ to the filtering
problem for the functional $A\xi=A\vec{\xi}$ under conditions (10) is
calculated by formula (12) for the coefficients $\vec{a}(m)$, $m\geq 0$,
defined in (27). The spectral characteristic
$\vec{h}_{\overline{\mu}}(\lambda)=\\{h_{p}(\lambda)\\}_{p=1}^{T}$ and the
value of the mean square error $\Delta(f;\widehat{A}\xi)$ of the estimate
$\widehat{A}\xi$ are calculated by formulas (19) and (20) respectively.

The functional $A_{M}{\xi}$ can be represented in the form

$\displaystyle A_{M}{\xi}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{M}{a}^{(\xi)}(k)\zeta(-k)=\sum_{m=0}^{N}\sum_{p=1}^{T}{a}^{(\xi)}(mT+p-1)\xi(-mT-p+1)$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{N}\sum_{p=1}^{T}a_{p}(m)\xi_{p}(-m)=\sum_{m=0}^{N}(\vec{a}(m))^{\top}\vec{\xi}(-m)=A_{N}\vec{\xi},$

where $N=[\frac{M}{T}]$, the sequence $\vec{\xi}(m)$ is determined by formula
(26),

$\displaystyle(\vec{a}(m))^{\top}$ $\displaystyle=$
$\displaystyle({a}_{1}(m),{a}_{2}(m),\dots,{a}_{T}(m))^{\top},$
$\displaystyle{a}_{p}(m)$ $\displaystyle=$ $\displaystyle
a^{\zeta}(mT+p-1);\,0\leq m\leq N;1\leq p\leq T;\,mT+p-1\leq M;$
$\displaystyle{a}_{p}(N)$ $\displaystyle=$ $\displaystyle 0;\quad M+1\leq
NT+p-1\leq(N+1)T-1;1\leq p\leq T.$ (29)

An estimate of a single unobserved value $\xi(-M)$, $M\geq 0$ of a stochastic
sequence ${\xi}(m)$ with periodically stationary GM increments is obtained by
making use of the notations
$\xi(-M)=\xi_{p}(-N)=(\vec{\delta}_{p})^{\top}\vec{\xi}(N)$,
$N=[\frac{M}{T}]$, $p=M+1-NT$. We can conclude that the following corollaries
hold true.

######  Corollary 3.

Let a stochastic sequence ${\xi}(m)$ with periodically stationary GM
increments and a stochastic periodically stationary sequence ${\eta}(m)$
generate by formulas (26) and (28) vector-valued stochastic sequences
$\vec{\xi}(m)$ and $\vec{\eta}(m)$. A solution $\widehat{A}_{M}\xi$ to the
filtering problem for the functional $A_{M}\xi=A_{N}\vec{\xi}$ under condition
(11) is calculated by formula (21) for the coefficients $\vec{a}(m)$, $0\leq
m\leq N$, defined in (29). The spectral characteristic and the value of the
mean square error of the estimate $\widehat{A}_{M}\xi$ are calculated by
formulas (19) and (20) respectively.

######  Corollary 4.

Let a stochastic sequence ${\xi}(m)$ with periodically stationary GM
increments and a stochastic periodically stationary sequence ${\eta}(m)$
generate by formulas (26) and (28) vector-valued stochastic sequences
$\vec{\xi}(m)$ and $\vec{\eta}(m)$. A solution $\widehat{\xi}(-M)$ to the
filtering problem for an unobserved value
$\xi(-M)=\xi_{p}(-N)=(\vec{\delta}_{p})^{\top}\vec{\xi}(-N)$,
$N=[\frac{M}{T}]$, $p=M+1-NT$, under condition (11) is calculated by formula
(23). The spectral characteristic and the value of the mean square error of
the estimate $\widehat{\xi}(-M)$ are calculated by formulas (19) and (20) or
(24) and (2) respectively.

##  3 Minimax (robust) method of filtering

Solutions of the problem of estimating the functionals ${A}\vec{\xi}$ and
${A}_{N}\vec{\xi}$ constructed from unobserved values of the stochastic
sequence $\vec{\xi}(m)$ with stationary GM increments
$\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$ having the spectral
density matrix $f(\lambda)$ based on its observations with stationary noise
$\vec{\xi}(m)+\vec{\eta}(m)$ at points $m=0,-1,-2,\dots$ are proposed in
Theorem 3 and Corollary 1 in the case where the spectral density matrices
$f(\lambda)$ and $g(\lambda)$ of the target sequence and the noise are exactly
known.

In this section, we study the case where the complete information about the
spectral density matrices is not available, while some sets of admissible
spectral densities $\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ is
known. The minimax approach of estimation of the functionals from unobserved
values of stochastic sequences is considered, which consists in finding an
estimate that minimizes the maximal values of the mean square errors for all
spectral densities from a class $\mathcal{D}$ simultaneously. This method will
be applied for the concrete classes of spectral densities.

The proceed with the stated problem, we recall the following definitions [26].

######  Definition 4.

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 the least favourable densities in the class $\mathcal{D}$ for optimal
linear filtering 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.

For a given class of spectral densities
$\mathcal{D}=\mathcal{D}_{f}\times\mathcal{D}_{g}$ the spectral characteristic
$\vec{h}^{0}(\lambda)$ of the optimal estimate of the functional $A\xi$ is
called minimax (robust) if the following relations hold true

$\displaystyle\vec{h}^{0}(\lambda)\in H_{\mathcal{D}}$ $\displaystyle=$
$\displaystyle\bigcap_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}L_{2}^{0}(p),$
$\displaystyle\min_{\vec{h}\in
H_{\mathcal{D}}}\max_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}\Delta(\vec{h};f,g)$
$\displaystyle=$
$\displaystyle\max_{(f,g)\in\mathcal{D}_{f}\times\mathcal{D}_{g}}\Delta(\vec{h}^{0};f,g).$

Taking into account the introduced definitions and the relations derived in
the previous sections we can verify that the following lemma holds true.

######  Lemma 3.

The spectral densities $f^{0}\in\mathcal{D}_{f}$, $g^{0}\in\mathcal{D}_{g}$
which admit canonical factorizations (8), (16) and (17) are least favourable
densities in the class $\mathcal{D}$ for the optimal linear filtering of the
functional $A\vec{\xi}$ based on observations of the sequence
$\vec{\xi}(m)+\vec{\eta}(m)$ at points $m=0,-1,-2,\ldots$ if the matrix
coefficients of canonical factorizations (16) and (17) determine a solution to
the constrained optimization problem

$\displaystyle\|\widetilde{\Phi}\mathbf{a}\|^{2}-\|\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\|^{2}\rightarrow\sup,$
(30) $\displaystyle f(\lambda)$ $\displaystyle=$
$\displaystyle\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\Theta_{\overline{\mu}}(e^{-i\lambda})\Theta_{\overline{\mu}}^{*}(e^{-i\lambda})-|\beta^{(d)}(i\lambda)|^{2}\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda})\in\mathcal{D}_{f},$
$\displaystyle g(\lambda)$ $\displaystyle=$
$\displaystyle\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda})\in\mathcal{D}_{g}.$

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

######  Lemma 4.

The spectral density $g^{0}\in\mathcal{D}_{g}$ which admits canonical
factorizations (16), (17) with the known spectral density $f(\lambda)$ is the
least favourable in the class $\mathcal{D}_{g}$ for the optimal linear
filtering of the functional $A\xi$ based on observations of the sequence
$\vec{\xi}(m)+\vec{\eta}(m)$ at points $m=0,-1,-2,\ldots$ if the matrix
coefficients of the canonical factorizations

$\displaystyle f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g^{0}(\lambda)$
$\displaystyle=$
$\displaystyle\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\left(\sum_{k=0}^{\infty}\theta^{0}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\left(\sum_{k=0}^{\infty}\theta^{0}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)^{*},$ $\displaystyle g^{0}(\lambda)$ $\displaystyle=$
$\displaystyle\left(\sum_{k=0}^{\infty}\phi^{0}(k)e^{-i\lambda
k}\right)\left(\sum_{k=0}^{\infty}\phi^{0}(k)e^{-i\lambda k}\right)^{*}$

are determined by the equation
$\Psi^{0}_{\overline{\mu}}(e^{-i\lambda})\Theta^{0}_{\overline{\mu}}(e^{-i\lambda})=E_{q}$
and a solution $\\{\psi^{0}_{\overline{\mu}}(k),\phi^{0}(k):k\geq 0\\}$ to the
constrained optimization problem

$\displaystyle\|\widetilde{\Phi}\mathbf{a}\|^{2}-\|\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\|^{2}\rightarrow\sup,$
(31)
$g(\lambda)=\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda})\in\mathcal{D}_{g}.$

The minimax spectral characteristic
$\vec{h}^{0}=\vec{h}_{\overline{\mu}}(f,g^{0})$ is calculated by formula (19)
if $\vec{h}_{\overline{\mu}}(f,g^{0})\in H_{\mathcal{D}}$.

######  Lemma 5.

The spectral density $f^{0}\in\mathcal{D}_{f}$ which admits canonical
factorizations (8), (16) with the known spectral density $g(\lambda)$ is the
least favourable spectral density in the class $\mathcal{D}_{f}$ for the
optimal linear filtering of the functional $A\vec{\xi}$ based on observations
of the sequence $\vec{\xi}(m)+\vec{\eta}(m)$ at points $m=0,-1,-2,\ldots$ if
matrix coefficients of the canonical factorization

$\displaystyle
f^{0}(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda)=\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\left(\sum_{k=0}^{\infty}\theta^{0}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\left(\sum_{k=0}^{\infty}\theta^{0}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)^{*}$

are determined by the equation
$\Psi^{0}_{\overline{\mu}}(e^{-i\lambda})\Theta^{0}_{\overline{\mu}}(e^{-i\lambda})=E_{q}$
and a solution $\\{\psi^{0}_{\overline{\mu}}(k):k\geq 0\\}$ to the constrained
optimization problem

$\displaystyle\|\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\|^{2}\rightarrow\inf,$
(32) $\displaystyle f(\lambda)$ $\displaystyle=$
$\displaystyle\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\Theta_{\overline{\mu}}(e^{-i\lambda})\Theta_{\overline{\mu}}^{*}(e^{-i\lambda})-|\beta^{(d)}(i\lambda)|^{2}\Phi(e^{-i\lambda})\Phi^{*}(e^{-i\lambda})\in\mathcal{D}_{f}$

for the fixed matrix coefficients $\\{\phi(k):k\geq 0\\}$. The minimax
spectral characteristic $\vec{h}^{0}=\vec{h}_{\overline{\mu}}(f^{0},g)$ is
calculated by formula (19) if $\vec{h}_{\overline{\mu}}(f^{0},g)\in
H_{\mathcal{D}}$.

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

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

hold true if $h^{0}=h_{\overline{\mu}}(f^{0},g^{0})$,
$h_{\overline{\mu}}(f^{0},g^{0})\in H_{\mathcal{D}}$ and $(f^{0},g^{0})$ is a
solution of the constrained optimization problem

$\widetilde{\Delta}(f,g)=-\Delta(h_{\overline{\mu}}(f^{0},g^{0});f,g)\to\inf,\quad(f,g)\in\mathcal{D},$
(33)

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

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

where

$\displaystyle\mathbf{h}^{0}_{\overline{\mu},f}(e^{-i\lambda})$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}(\overline{\psi}_{\overline{\mu}}\mathbf{C}^{-}_{\overline{\mu},g}+\overline{\psi}_{\overline{\mu}}\mathbf{C}^{+}_{\overline{\mu},g})_{m}e^{-i\lambda
m},$ $\displaystyle\mathbf{h}^{0}_{\overline{\mu},g}(e^{-i\lambda})$
$\displaystyle=$
$\displaystyle\left(\sum_{k=0}^{\infty}\theta^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\left(\sum_{k=0}^{\infty}a(k)e^{-i\lambda
k}\right)-\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})\left(\sum_{m=0}^{\infty}(\overline{\psi}_{\overline{\mu}}\mathbf{C}^{-}_{\overline{\mu},g}+\overline{\psi}_{\overline{\mu}}\mathbf{C}^{+}_{\overline{\mu},g})_{m}e^{-i\lambda
m}\right).$

The constrained optimization problem (33) is equivalent to the unconstrained
optimization problem

$\Delta_{\mathcal{D}}(f,g)=\widetilde{\Delta}(f,g)+\delta(f,g|\mathcal{D})\to\inf,$
(34)

where $\delta(f,g|\mathcal{D})$ is the indicator function of the set
$\mathcal{D}$, namely $\delta(f,g|\mathcal{D})=0$ if $(f,g)\in\mathcal{D}$ and
$\delta(f|\mathcal{D})=+\infty$ if $(f,g)\notin\mathcal{D}$. The condition
$0\in\partial\Delta_{\mathcal{D}}(f^{0},g^{0})$ characterizes a solution
$(f^{0},g^{0})$ of the stated unconstrained optimization problem. This
condition is the necessary and sufficient condition that the point
$(f^{0},g^{0})$ belongs to the set of minimums of the convex functional
$\Delta_{\mathcal{D}}(f,g)$ [27, 34]. Thus, it allows us to find the
equalities for the least favourable spectral densities in some special classes
of spectral densities $\mathcal{D}$.

The form of the functional $\Delta(h_{\overline{\mu}}(f^{0},g^{0});f,g)$ is
suitable for application of the Lagrange method of indefinite multipliers to
the constrained optimization problem (33). Thus, the complexity of the problem
is reduced to finding the subdifferential of the indicator function of the set
of admissible spectral densities. We illustrate the solving of the problem
(34) for concrete sets admissible spectral densities in the following
subsections. A semi-uncertain filtering problem, when the spectral density
$f(\lambda)$ is known and the spectral density $g(\lambda)$ belongs to in
class $\mathcal{D}_{g}$, is considered as well.

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

Consider the minimax filtering problem for the functional $A\vec{\xi}$ for
sets of admissible spectral densities $\mathcal{D}_{0}^{k}$, $k=1,2,3,4$ of
the sequence with GM increments $\vec{\xi}(m)$

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

where $p,p_{k},k=\overline{1,T}$ are given numbers, $P,B_{1}$ are given
positive-definite Hermitian matrices, and sets of admissible spectral
densities $\mathcal{D}_{V}^{U}$, $k=1,2,3,4$ for the stationary noise sequence
$\vec{\eta}(m)$

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

where $g_{1}(\lambda)=\\{g_{ij}^{1}(\lambda)\\}_{i,j=1}^{T}$ is a fixed
spectral density, $B_{2}$ is a given positive-definite Hermitian matrix,
$\delta,\delta_{k},k=\overline{1,T}$, $\delta_{i}^{j},i,j=\overline{1,T}$, are
given numbers.

The condition $0\in\partial\Delta_{\mathcal{D}}(f^{0},g^{0})$ implies 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
$\mathcal{D}_{f0}^{1}\times\mathcal{D}_{g1\delta}^{1}$:

$\displaystyle\left({\mathbf{h}^{0}_{\overline{\mu},f}(e^{i\lambda})}\right)\left({\mathbf{h}^{0}_{\overline{\mu},f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=$
$\displaystyle(\Theta_{\overline{\mu}}(e^{-i\lambda}))^{\top}\vec{\alpha}_{f}\cdot\vec{\alpha}_{f}^{*}\overline{\Theta_{\overline{\mu}}(e^{-i\lambda})},$
(35)
$\displaystyle\left({\mathbf{h}^{0}_{\overline{\mu},g}(e^{i\lambda})}\right)\left({\mathbf{h}^{0}_{\overline{\mu},g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=$
$\displaystyle(\Theta_{\overline{\mu}}(e^{-i\lambda}))^{\top}\left\\{\beta_{ij}\gamma_{ij}(\lambda)\right\\}_{i,j=1}^{T}\overline{\Theta_{\overline{\mu}}(e^{-i\lambda})},$
(36)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|g^{0}_{ij}(\lambda)-g_{ij}^{1}(\lambda)\right|d\lambda=\delta_{i}^{j},$
(37)

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

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

For the second set of admissible spectral densities
$\mathcal{D}_{f0}^{2}\times\mathcal{D}_{g1\delta}^{2}$ we have equations

$\displaystyle\left({\mathbf{h}^{0}_{\overline{\mu},f}(e^{i\lambda})}\right)\left({\mathbf{h}^{0}_{\overline{\mu},f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=$
$\displaystyle\alpha_{f}^{2}(\Theta_{\overline{\mu}}(e^{-i\lambda}))^{\top}\overline{\Theta_{\overline{\mu}}(e^{-i\lambda})},$
(38)
$\displaystyle\left({\mathbf{h}^{0}_{\overline{\mu},g}(e^{i\lambda})}\right)\left({\mathbf{h}^{0}_{\overline{\mu},g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=$
$\displaystyle\beta^{2}\gamma_{2}(\lambda)(\Theta_{\overline{\mu}}(e^{-i\lambda}))^{\top}\overline{\Theta_{\overline{\mu}}(e^{-i\lambda})},$
(39)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|{\mathrm{Tr}}\,(g_{0}(\lambda)-g_{1}(\lambda))\right|d\lambda=\delta,$
(40)

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
$\mathcal{D}_{f0}^{3}\times\mathcal{D}_{g1\delta}^{3}$ we have equations

$\displaystyle\left({\mathbf{h}^{0}_{\overline{\mu},f}(e^{i\lambda})}\right)\left({\mathbf{h}^{0}_{\overline{\mu},f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=$
$\displaystyle(\Theta_{\overline{\mu}}(e^{-i\lambda}))^{\top}\left\\{\alpha_{fk}^{2}\delta_{kl}\right\\}_{k,l=1}^{T}\overline{\Theta_{\overline{\mu}}(e^{-i\lambda})},$
(41)
$\displaystyle\left({\mathbf{h}^{0}_{\overline{\mu},g}(e^{i\lambda})}\right)\left({\mathbf{h}^{0}_{\overline{\mu},g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=$
$\displaystyle(\Theta_{\overline{\mu}}(e^{-i\lambda}))^{\top}\left\\{\beta_{k}^{2}\gamma^{2}_{k}(\lambda)\delta_{kl}\right\\}_{k,l=1}^{T}\overline{\Theta_{\overline{\mu}}(e^{-i\lambda})},$
(42)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|g^{0}_{kk}(\lambda)-g_{kk}^{1}(\lambda)\right|d\lambda=\delta_{k},$
(43)

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

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

For the fourth set of admissible spectral densities
$\mathcal{D}_{f0}^{4}\times\mathcal{D}_{g1\delta}^{4}$ we have equations

$\displaystyle\left({\mathbf{h}^{0}_{\overline{\mu},f}(e^{i\lambda})}\right)\left({\mathbf{h}^{0}_{\overline{\mu},f}(e^{i\lambda})}\right)^{*}$
$\displaystyle=$
$\displaystyle\alpha_{f}^{2}(\Theta_{\overline{\mu}}(e^{-i\lambda}))^{\top}B_{1}\overline{\Theta_{\overline{\mu}}(e^{-i\lambda})},$
(44)
$\displaystyle\left({\mathbf{h}^{0}_{\overline{\mu},g}(e^{i\lambda})}\right)\left({\mathbf{h}^{0}_{\overline{\mu},g}(e^{i\lambda})}\right)^{*}$
$\displaystyle=$
$\displaystyle\beta^{2}\gamma_{2}^{\prime}(\lambda)(\Theta_{\overline{\mu}}(e^{-i\lambda}))^{\top}B_{2}\overline{\Theta_{\overline{\mu}}(e^{-i\lambda})},$
(45) $\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\left\langle
B_{2},g_{0}(\lambda)-g_{1}(\lambda)\right\rangle\right|d\lambda=\delta,$ (46)

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 theorem holds true.

######  Theorem 5.

The least favorable spectral densities $f^{0}(\lambda)$, $g^{0}(\lambda)$ in
the classes $\mathcal{D}_{f0}^{k}\times\mathcal{D}_{g1\delta}^{k}$,
$k=1,2,3,4$ for the optimal linear filtering of the functional $A\vec{\xi}$
from observations of the sequence $\vec{\xi}(m)+\vec{\eta}(m)$ at points
$m=0,-1,-2,\ldots$ are determined by canonical factorizations (8), (16) and
(17), equations (35)–(37), (38)–(40), (41)–(43), (44)–(46), respectively,
constrained optimization problem (30) and restrictions on densities from the
corresponding classes
$\mathcal{D}_{f0}^{k},\mathcal{D}_{g1\delta}^{k},k=1,2,3,4$. The minimax-
robust spectral characteristic of the optimal estimate of the functional
$A\vec{\xi}$ is determined by the formula (19).

###  3.2 Semi-uncertain filtering problem in classes
$\mathcal{D}_{\varepsilon}$ of least favorable noise spectral density

Consider a semi-uncertain filtering problem for the functional $A\vec{\xi}$,
where the spectral density $g(\lambda)$ of the stationary noise sequence
$\vec{\eta}(m)$ is known and $f(\lambda)$ of the sequence with GM increments
$\vec{\xi}(m)$ the spectral density belongs to the sets of admissible spectral
densities $\mathcal{D}_{\varepsilon}^{k},k=1,2,3,4$

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

where $f_{1}(\lambda)$ is a fixed spectral density, $W(\lambda)$ is an unknown
spectral density, $p,p_{k},k=\overline{1,T}$, are given numbers, $P,B_{2}$ are
given positive-definite Hermitian matrices.

The condition $0\in\partial\Delta_{\mathcal{D}}(f^{0},g)$ implies the
equations which determine the least favourable spectral densities of the noise
sequence $\vec{\xi}(m)$. Note that the elements
$\mathbf{C}^{-}_{\overline{\mu},g}$ and $\mathbf{C}^{+}_{\overline{\mu},g}$
are known and determined by the coefficients $\\{\phi(k),k\geq 0\\}$ of the
canonical factorization of the spectral density matrix $g(\lambda)$.

For the first set of admissible spectral density
$\mathcal{D}_{\varepsilon}^{1}$ we have an equation

$\left({\sum_{m=0}^{\infty}\left(\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\right)_{m}e^{-i\lambda
m}}\right)\left({\sum_{m=0}^{\infty}\left(\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\right)_{m}e^{-i\lambda
m}}\right)^{*}\\\
=\left(\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k}\right)^{\top}(\vec{\alpha}_{f}\cdot\vec{\alpha}_{f}^{*}+\Gamma(\lambda))\left(\overline{\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k}}\right),$ (47)

where $\vec{\alpha}_{f}$ ia a vector of Lagrange multipliers, matrix
$\Gamma(\lambda)\leq 0$ and $\Gamma(\lambda)=0$ if
$f_{0}(\lambda)>(1-\varepsilon)f_{1}(\lambda)$.

For the second set of admissible spectral densities
$\mathcal{D}_{\varepsilon}^{2}$ we have an equation

$\left({\sum_{m=0}^{\infty}\left(\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\right)_{m}e^{-i\lambda
m}}\right)\left({\sum_{m=0}^{\infty}\left(\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\right)_{m}e^{-i\lambda
m}}\right)^{*}\\\
=(\alpha_{f}^{2}+\gamma(\lambda))\left(\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k}\right)^{\top}\left(\overline{\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k}}\right),$ (48)

where $\alpha_{f}^{2}$ is a Lagrange multiplier, function $\gamma(\lambda)\leq
0$ and $\gamma(\lambda)=0$ if
${\mathrm{Tr}}\,[f_{0}(\lambda)]>(1-\varepsilon){\mathrm{Tr}}\,[f_{1}(\lambda)]$.

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

$\left({\sum_{m=0}^{\infty}\left(\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\right)_{m}e^{-i\lambda
m}}\right)\left({\sum_{m=0}^{\infty}\left(\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\right)_{m}e^{-i\lambda
m}}\right)^{*}\\\
=\left(\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k}\right)^{\top}\left\\{(\alpha_{fk}^{2}+\gamma_{k}(\lambda))\delta_{kl}\right\\}_{k,l=1}^{T}\left(\overline{\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k}}\right),$ (49)

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

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

$\left({\sum_{m=0}^{\infty}\left(\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\right)_{m}e^{-i\lambda
m}}\right)\left({\sum_{m=0}^{\infty}\left(\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\right)_{m}e^{-i\lambda
m}}\right)^{*}\\\
=(\alpha_{f}^{2}+\gamma^{\prime}(\lambda))\left(\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k}\right)^{\top}B_{1}\left(\overline{\sum_{k=0}^{\infty}\theta_{\overline{\mu}}(k)e^{-i\lambda
k}}\right),$ (50)

where $\alpha_{f}^{2}$ is a Lagrange multiplier, function
$\gamma^{\prime}(\lambda)\leq 0$ and $\gamma^{\prime}(\lambda)=0$ if $\langle
B_{1},f_{0}(\lambda)\rangle>(1-\varepsilon)\langle
B_{1},f_{1}(\lambda)\rangle$.

The following theorems hold true.

######  Theorem 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 foltering of the functional $A\vec{\xi}$
from observations of the sequence $\vec{\xi}(m)+\vec{\eta}(m)$ at points
$m=0,-1,-2,\ldots$ is determined by canonical factorizations (8) and (16),
equations (47), (48), (49), (50), respectively, constrained optimization
problem (32) and restrictions on density from the corresponding classes
$\mathcal{D}_{\varepsilon}^{k}$, $k=1,2,3,4$. The minimax-robust spectral
characteristic of the optimal estimate of the functional $A\vec{\xi}$ is
determined by the formula (19).

##  4 Conclusions

In this article, we present a solution of the filtering problem for stochastic
sequences with periodically stationary multiple seasonal increments, or
sequences with periodically stationary general multiplicative (GM) increments,
introduced in the article by Luz and Moklyachuk [23]. We propose a solution of
the filtering problem in the case where the spectral densities of the sequence
$\xi(m)$ and a noise sequence $\eta(m)$ are exactly known. The estimates are
derived in terms of coefficients of canonical factorizations of the spectral
densities, making use of results obtained in [24] by using the Fourier
transformations of the spectral densities. The minimax-robust approach to
filtering problem is applied in the case of spectral uncertainty where
densities of sequences are not exactly known while, instead, sets of
admissible spectral densities allowing canonical factorizations are specified.
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. Described relations determine
the least favourable spectral densities and the minimax spectral
characteristics of the optimal estimates of linear functionals for a list of
specific classes of admissible spectral densities.

## Appendix

_Proof of Lemma 2_

Factorizations (17), (18) and Remark 3 imply

$\displaystyle\sum_{k\in\mathbb{Z}}s_{\overline{\mu}}(k)e^{i\lambda k}$
$\displaystyle=$
$\displaystyle\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\left[g(\lambda)(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))^{-1}\right]^{\top}=(\Psi_{\overline{\mu}}(e^{-i\lambda}))^{\top}\overline{\Psi}_{\overline{\mu}}(e^{-i\lambda})\overline{g}(\lambda)$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\psi_{\overline{\mu}}^{\top}(l)e^{-i\lambda
l}\sum_{j\in\mathbb{Z}}Z_{\overline{\mu}}(j)e^{i\lambda
j}=\sum_{k\in\mathbb{Z}}\sum_{l=0}^{\infty}\psi_{\overline{\mu}}^{\top}(l)Z_{\overline{\mu}}(l+k)e^{i\lambda
k}.$

Then

$\displaystyle(\Theta^{\top}_{\overline{\mu}}\mathbf{S}_{\overline{\mu}}\widetilde{\mathbf{a}}_{\overline{\mu}})_{m}$
$\displaystyle=$
$\displaystyle\sum_{j=-n(\gamma)}^{\infty}\sum_{p=m}^{\infty}\theta^{\top}_{\overline{\mu}}(p-m)S_{\overline{\mu}}(p+j+1)a_{-\mu}(j)$
$\displaystyle=$
$\displaystyle\sum_{j=-n(\gamma)}^{\infty}\sum_{p=m}^{\infty}\sum_{l=0}^{\infty}\theta^{\top}_{\overline{\mu}}(p-m)\psi_{\overline{\mu}}^{\top}(l)Z_{\overline{\mu}}(l+p+j+1)a_{-\mu}(j)$
$\displaystyle=$
$\displaystyle\sum_{j=-n(\gamma)}^{\infty}\sum_{p=m}^{\infty}\sum_{k=p}^{\infty}(\psi_{\overline{\mu}}(k-p)\theta_{\overline{\mu}}(p-m))^{\top}Z_{\overline{\mu}}(k+j+1)a_{-\mu}(j)$
$\displaystyle=$
$\displaystyle\sum_{j=-n(\gamma)}^{\infty}\sum_{k=m}^{\infty}\mathsf{diag}_{q}(\delta_{k,m})Z_{\overline{\mu}}(k+j+1)a_{-\mu}(j)$
$\displaystyle=$
$\displaystyle\sum_{j=-n(\gamma)}^{\infty}Z_{\overline{\mu}}(m+j+1)a_{-\mu}(j).$

The representation for $Z_{\overline{\mu}}(j)$ follows from

$\sum_{j\in\mathbb{Z}}Z_{\overline{\mu}}(j)e^{i\lambda
j}=\overline{\Psi}_{\overline{\mu}}(e^{-i\lambda})\overline{g}(\lambda)=\sum_{k\in\mathbb{Z}}\sum_{l=0}^{\infty}\overline{\Psi}_{\overline{\mu}}(l)\overline{g}_{\overline{\mu}}(l-k)e^{i\lambda
k}.\quad\square$

_Proof of Theorem 3_

Under the conditions of Lemmas 1 and 2 on the spectral densities $f(\lambda)$
and $g(\lambda)$, formulas (14) and (15) can be rewritten as follows. Make the
following transformations:

$\displaystyle\frac{|\beta^{(d)}(i\lambda)|^{2}}{|\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\left[(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))^{-1}\right]^{\top}\left(\sum_{k=0}^{\infty}\left(\mathbf{P}_{\overline{\mu}}^{-1}\mathbf{S}_{\overline{\mu}}\mathbf{a}_{\overline{\mu}}\right)_{k}e^{i\lambda(k+1)}\right)$
$\displaystyle=$
$\displaystyle\left(\sum_{k=0}^{\infty}\psi_{\overline{\mu}}^{\top}(k)e^{-i\lambda
k}\right)\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\overline{\psi}_{\overline{\mu}}(j)(\overline{\Theta}_{\overline{\mu}}\widetilde{\mathbf{e}}_{\overline{\mu}})_{k}e^{i\lambda(k+j+1)}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=0}^{\infty}\psi_{\overline{\mu}}^{\top}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}\sum_{p=0}^{m}\sum_{k=p}^{m}\overline{\psi}_{\overline{\mu}}(m-k)\overline{\theta}_{\overline{\mu}}(k-p)\widetilde{e}_{\overline{\mu}}(p)e^{i\lambda(m+1)}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=0}^{\infty}\psi_{\overline{\overline{\mu}}}^{\top}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}\sum_{p=0}^{m}\mathsf{diag}_{q}(\delta_{m,p})\widetilde{e}_{\overline{\mu}}(m)e^{i\lambda(m+1)}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=0}^{\infty}\psi_{\overline{\overline{\mu}}}^{\top}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}\widetilde{e}_{\overline{\mu}}(m)e^{i\lambda(m+1)},$

and

$\displaystyle\frac{|\beta^{(d)}(i\lambda)|^{2}}{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})|^{2}}\left[(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))^{-1}\right]^{\top}(g(\lambda))^{\top}A(e^{-i\lambda})(1-e^{i\lambda\mu})^{n}$
$\displaystyle=$
$\displaystyle\Psi_{\overline{\mu}}^{\top}(e^{-i\lambda})\overline{\Psi_{\overline{\mu}}(e^{-i\lambda})}\overline{g(\lambda)}A(e^{-i\lambda})(1-e^{i\lambda\mu})^{n}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\sum_{m\in\mathbb{Z}}^{\infty}\sum_{j=-n(\gamma)}^{\infty}Z_{\overline{\mu}}(m+j)a_{-\mu}(j)e^{i\lambda
m}.$

Then obtain:

$\displaystyle\vec{h}_{\overline{\mu}}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}\sum_{j=-\mu
n}^{\infty}\theta^{\top}_{\overline{\mu}}(l)Z_{\overline{\mu}}(j-m)a_{-\mu}(j)e^{-i\lambda
m}$ $\displaystyle=$
$\displaystyle\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)$
$\displaystyle\times\left(\sum_{m=0}^{\infty}\sum_{j=0}^{\infty}Z_{\overline{\mu}}(j-m)a_{-\mu}(j)e^{-i\lambda
m}+\sum_{m=0}^{\infty}\sum_{j=1}^{n(\gamma)}Z_{\overline{\mu}}(-j-m)b_{-\mu}(j)e^{-i\lambda
m}\right)$ $\displaystyle=$
$\displaystyle\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)$
$\displaystyle\times\left(\sum_{m=0}^{\infty}\sum_{j=0}^{\infty}\sum_{p=m}^{\infty}\overline{\psi}_{\overline{\mu}}(p-m)\overline{g}(p-j)a_{-\mu}(j)e^{-i\lambda
m}+\sum_{m=0}^{\infty}\sum_{j=1}^{n(\gamma)}\sum_{p=m}^{\infty}\overline{\psi}_{\overline{\mu}}(p-m)\overline{g}(p+j)b_{-\mu}(j)e^{-i\lambda
m}\right)$ $\displaystyle=$
$\displaystyle\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}((\widetilde{\Psi}_{\overline{\mu}})^{*}\mathbf{G}^{-}\mathbf{a}_{-\mu}+(\widetilde{\Psi}_{\overline{\mu}})^{*}\mathbf{G}^{+}\mathbf{b}_{-\mu})_{m}e^{-i\lambda
m}$ $\displaystyle=$
$\displaystyle\frac{\chi_{\overline{\mu}}^{(d)}(e^{-i\lambda})}{\beta^{(d)}(i\lambda)}\left(\sum_{k=0}^{\infty}\psi^{\top}_{\overline{\mu}}(k)e^{-i\lambda
k}\right)\sum_{m=0}^{\infty}(\overline{\psi}_{\overline{\mu}}\mathbf{C}^{-}_{\overline{\mu},g}+\overline{\psi}_{\overline{\mu}}\mathbf{C}^{+}_{\overline{\mu},g})_{m}e^{-i\lambda
m}.$

The value of the mean square error $\Delta(f,g;\widehat{A}\xi)$ is calculated
by the formula

$\displaystyle\Delta\left(f,g;\widehat{A}\xi\right)$ $\displaystyle=$
$\displaystyle\Delta\left(f,g;\widehat{A}\eta\right)=\mathsf{E}\left|A\eta-\widehat{A}\eta\right|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}(\vec{A}(e^{i\lambda}))^{\top}g(\lambda)\overline{\vec{A}(e^{i\lambda})}d\lambda+\frac{1}{2\pi}\int_{-\pi}^{\pi}(\vec{h}_{\overline{\mu}}(\lambda))^{\top}(f(\lambda)+|\beta^{(d)}(i\lambda)|^{2}g(\lambda))\overline{\vec{h}_{\overline{\mu}}(\lambda)}d\lambda$
$\displaystyle-\frac{1}{2\pi}\int_{-\pi}^{\pi}(\vec{h}_{\overline{\mu}}(\lambda))^{\top}\beta^{(d)}(i\lambda)g(\lambda)\overline{A(e^{-i\lambda})}d\lambda-\frac{1}{2\pi}\int_{-\pi}^{\pi}(A(e^{-i\lambda}))^{\top}\overline{\beta^{(d)}(i\lambda)}g(\lambda)\overline{\vec{h}_{\overline{\mu}}(\lambda)}d\lambda$
$\displaystyle=$
$\displaystyle\|\widetilde{\Phi}\mathbf{a}\|^{2}-\|\overline{\psi}_{\overline{\mu}}(\mathbf{C}^{-}_{\overline{\mu},g}+\mathbf{C}^{+}_{\overline{\mu},g})\|^{2}.\quad\square$

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