On Minimax Robust Estimation Problem for Stochastic Sequences with Harmonizable Symmetric α-Stable Increments

Abstract

We consider the problem of the optimal linear estimation of the functional $A_N\xi ={\sum }_{k=0}^{N}a\left(k\right)\xi \left(k\right)$ which depends on the unknown values $\xi \left(k\right)$, $k=0,1,\dots ,N$, of a stochastic sequence with harmonizable symmetric $\alpha$-stable nth increments, $1<\alpha \le 2$. The derived estimates are based on observations at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$. Cases of observations without noise, with harmonizable symmetric $\alpha$-stable noise and with noise with harmonizable symmetric $\alpha$-stable increments are studied. Classical solutions as well as minimax robust ones are obtained.

1. Introduction

In this paper, we present the results of an investigation into the problem of estimating the missed observations of stochastic sequences with harmonizable symmetric $\alpha$-stable nth increments.

The classical methods of deriving estimates of the unobserved values of a stochastic process rely fundamentally on the existence of finite first and second moments of the underlying stochastic process. When these moments do not exist, as is the case for Cauchy-, Pareto-, and Lévy-stable processes with stability index $0<\alpha <2$, the moment-based inequalities on which the classical methods depend are formally inapplicable, and no rigorous estimate of the unobserved values can be derived within the classical moment-based framework. This limitation persists in the mathematical foundations of the classical framework and has been documented repeatedly in the authoritative literature of the field. See the canonical monograph on stable processes by Samorodnitsky and Taqqu [1], which characterizes stable processes as ‘always with infinite variance’; the most widely cited applied monograph on heavy-tailed processes by Embrechts, Klüppelberg, and Mikosch [2] that documents the infinite variance property and the breakdown of classical moment-based tools for stable distributions; and more recent publications by Nolan [3,4]. For more details and references, we recommend the bibliography on stable distributions by Nolan [5].

Symmetric $\alpha$-stable random variables are widely used in signal processing, finance and economics, since they are suitable for modeling heavy-tailed time series. For example, we refer to studies by Bidarkota et al. [6], Borak et al. [7], Molina-Munozet et al. [8], Padilla et al. [9], and Reuss et al. [10] which have been published in the past few decades.

The problem of estimating the unknown values of harmonizable random sequences and processes was investigated in papers by Cambanis [11], Cambanis and Soltani [12], Cambanis and Miamee [13], and Hosoya [14]. The interpolation problem for harmonizable symmetric $\alpha$-stable random sequences was investigated in papers by Weron [15] and Pourahmadi [16]. Some results in this field were obtained by Moklyachuk and Ostapenko [17] and Masyutka and Moklyachuk [18], who proposed using the minimax approach for the estimation of such processes. The minimax methods used for time series estimation in the case of spectral uncertainty were analyzed earlier by Franke [19] and Vastola and Poor [20]; see also the survey papers by Kassam and Poor [21] and Moklyachuk [22].

Another widely studied type of time series consists of stochastic sequences with stationary increments or integrated sequences [23]. The extrapolation, interpolation and filtering problems for stochastic sequences and random processes with nth stationary increments have been investigated by Luz and Moklyachuk [24,25,26].

In this article, we investigate the problem of the optimal estimation of the linear functional

$$A_N\xi ={\displaystyle \sum _{k=0}^N}a\left(k\right)\xi \left(k\right)$$

which depends on the unknown values $\xi \left(k\right)$, $k=0,1,\dots ,N$, of a stochastic sequence with harmonizable symmetric $\alpha$-stable nth increments. The solution to this problem relies on covariance, which is defined for $1<\alpha \le 2$. Therefore, this study is carried out only for this range of the parameter $\alpha$. Depending on the available observations, we may consider three types of problems: (i) estimation without noise, where we observe the sequence $\xi \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$; (ii) estimation with noise, where we observe the sequence $\xi \left(m\right)+\eta \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$; and (iii) estimation with semi-noise, when the sequence $\xi \left(m\right)$ is observed at points $m\le -1$ and the sequence $\xi \left(m\right)+\theta \left(m\right)$ is observed at points $m\ge N+1$. Here, $\eta \left(m\right)$ is a harmonizable symmetric $\alpha$-stable sequence independent of $\xi \left(m\right)$, and $\theta \left(m\right)$ is a sequence with harmonizable symmetric $\alpha$-stable nth increments independent of $\xi \left(m\right)$. The novelty of the proposed results lies in extending previous studies on the estimation of stationary increment sequences [24,25] and harmonizable symmetric $\alpha$-stable sequences [17] by combining these two types of non-stationarity.

The article is organized as follows. In Section 2, we describe a symmetric $\alpha$-stable random variable and a stochastic sequence, give a definition of a harmonizable symmetric $\alpha$-stable increment sequence, and provide some properties of such sequences. In Section 3, we describe a classical estimation problem as well as a minimax approach. Some other auxiliary results and definitions are provided. In Section 4, we present the solutions obtained for the stated estimation problems. In Section 5, we give some examples of the minimax estimation for particular classes of admissible spectral densities. Section 6 is dedicated to a discussion of the results.

2. Preliminaries and Problem Statement

2.1. Symmetric $\alpha$-Stable Random Sequence

In this subsection, we give a brief overview of the properties of symmetric $\alpha$-stable random sequences [11,12,17].

Definition 1

(symmetric $\alpha$-stable random variable). A real random variable ξ is said to be symmetric α-stable, $S\alpha S$, if its characteristic function has the form $E\exp \left(it\xi \right)=\exp (-c{\left|t\right|}^{\alpha })$ for some $c\ge 0$ and $0<\alpha \le 2.$ The real random variables ${\xi }_1,{\xi }_2,\dots ,{\xi }_n$ are jointly $S\alpha S$ if all linear combinations ${\sum }_{k=1}^n{a}_k{\xi }_k$ are $S\alpha S$, or, equivalently, if the characteristic function of the random vector $\overrightarrow{\xi }=({\xi }_1,\dots ,{\xi }_n)$ is of the form

$${\varphi }_{\overrightarrow{\xi }}\left(\overrightarrow{t}\right)=E\exp \left(i{\displaystyle \sum _{k=1}^n}{t}_k{\xi }_k\right)=\exp \left\{-{\int }_{S_n}{\left|{\displaystyle \sum _{k=1}^n}{t}_k{x}_k\right|}^{\alpha }d{\mathsf{\Gamma }}_{\overrightarrow{\xi }}\left(\overrightarrow{x}\right)\right\},$$

where $\overrightarrow{t}=(t_1,\dots ,t_n)$ is a real vector, and ${\mathsf{\Gamma }}_{\overrightarrow{\xi }}\left(\overrightarrow{x}\right)$ is a symmetric measure defined on the unit sphere $S_n\in R^n$, which is referred to as the spectral measure of the random vector $\overrightarrow{\xi }=({\xi }_1,\dots ,{\xi }_n)$. There is a one-to-one correspondence between the distribution of $\overrightarrow{\xi }$ and its spectral measure ${\mathsf{\Gamma }}_{\overrightarrow{\xi }}\left(\overrightarrow{x}\right)$. (Cambanis [11]).

For real jointly $S\alpha S$ random variables $\xi ,\eta$ with $1<\alpha \le 2$, the covariation of $\xi$ and $\eta$ is defined by

$${[\xi ,\eta ]}_{\alpha }={\int }_{S_2}\left(x\right){\left(y\right)}^{<\alpha -1>}d{\mathsf{\Gamma }}_{\xi ,\eta }(x,y),$$

where ${\left(y\right)}^{<\beta >}={\left|y\right|}^{\beta -1}y$.

For jointly $S\alpha S$ random variables $\xi ={\xi }_1+i{\xi }_2$ and $\eta ={\eta }_1+i{\eta }_2$ the covariation of $\xi$ with $\eta$ is defined as follows (Cambanis [11])

$${[\xi ,\eta ]}_{\alpha }={\int }_{S_4}(x_1+i{x}_2){(y_1+i{y}_2)}^{<\alpha -1>}d{\mathsf{\Gamma }}_{{\xi }_1,{\xi }_2,{\eta }_1,{\eta }_2}(x_1,x_2,y_1,y_2),$$

where $z^{<\beta >}={\left|z\right|}^{\beta -1}\overline{z}$ for a complex number z and $\beta >0.$

Note that the covariation in general is not symmetric and is not linear in the second argument. For jointly $S\alpha S$ $\xi ,{\xi }_1,{\xi }_2,\eta$ the following properties are satisfied (Cambanis [11], Weron [15]):

• linearity in the first argument ${[{\xi }_1+{\xi }_2,\eta ]}_{\alpha }={[{\xi }_1,\eta ]}_{\alpha }+{[{\xi }_2,\eta ]}_{\alpha }$,
• linearity in the second argument ${[\xi ,{\eta }_1+{\eta }_2]}_{\alpha }={[\xi ,{\eta }_1]}_{\alpha }+{[\xi ,{\eta }_2]}_{\alpha }$ for independent ${\eta }_1$ and ${\eta }_2$,
• zero covariation ${[\xi ,\eta ]}_{\alpha }=0$ for independent $\xi$ and $\eta$,
• ${[a\xi ,b\eta ]}_{\alpha }=a{\left(b\right)}^{<\alpha -1>}{[\xi ,\eta ]}_{\alpha }$,
• $\left|{[\xi ,\eta ]}_{\alpha }\right|\le {\parallel \xi \parallel }_{\alpha }{\parallel \eta \parallel }_{\alpha }^{\alpha -1}$.

The functional

$${\parallel \xi \parallel }_{\alpha }={[\xi ,\xi ]}_{\alpha }^{1/\alpha }$$

(1)

is a norm on the linear space of $S\alpha S$ random variables which is equivalent to convergence in probability. The equality

$$\parallel {\displaystyle \sum _{k=1}^n}{\xi }_k{\parallel }_{\alpha }^{\alpha }={\displaystyle \sum _{k=1}^n}{\parallel {\xi }_k\parallel }_{\alpha }^{\alpha }$$

(2)

holds for independent random variables ${\xi }_1,\dots ,{\xi }_n$.

The mapping

$$\xi \to {[\xi ,\eta ]}_{\alpha }$$

(3)

is a bounded linear functional with the norm ${\parallel \eta \parallel }_{\alpha }^{\alpha -1}$ on the linear space of $S\alpha S$ random variables, and every bounded linear functional on such a space is of this form for some $\eta$.

It should be noted that ${\parallel ·\parallel }_{\alpha }$ is not necessarily the usual $L_{\alpha }$ norm.

The following lemma provides the properties of $z^{<\beta >}.$

Lemma 1.

Let $x,y$ be complex numbers, $\beta >0$. Then,

• ${\left|x\right|}^{<\beta >}=x·x^{<\beta -1>},$
• $\left|{\left|x\right|}^{<\beta >}\right|={\left|x\right|}^{<\beta >},$
• if $x^{<\beta >}=v$, then $x=v^{<1/\beta >}={\left|v\right|}^{(1-\beta )/\beta }\overline{v},$
• $x^{<1>}=\overline{x},$
• $x^{<\alpha >}{x}^{<\beta >}={\displaystyle \frac{\overline{x}}{\left|x\right|}}{x}^{<\alpha +\beta >}$, $x\ne 0$,
• ${\displaystyle \frac{x^{<\alpha >}}{x^{<\beta >}}}={\displaystyle \frac{x}{\left|x\right|}}{x}^{<\alpha -\beta >}$, $x\ne 0$,
• ${\left(cx\right)}^{<\alpha >}={\left|c\right|}^{\alpha -1}c{x}^{<\alpha >},c\in \mathbb{R},$
• ${\left(x^{<\alpha >}\right)}^{<\beta >}={\overline{x}}^{<\alpha \beta >},$
• ${\left(xy\right)}^{<\alpha >}=x^{<\alpha >}{y}^{<\alpha >},$
• ${\left(x^{\alpha }\right)}^{<\beta >}={\left(x^{<\beta >}\right)}^{\alpha },$
• ${\left(x^{<\alpha >}\right)}^{\beta }={\left(x^{\beta }\right)}^{<\alpha >},$
• ${\left|x^{<\alpha >}\right|}^{\beta }={\left|x\right|}^{\alpha \beta },$
• ${(x+y)}^{<\alpha >}=\overline{x}{|x+y|}^{\alpha -1}+\overline{y}{|x+y|}^{\alpha -1}.$

Definition 2

(symmetric $\alpha$-stable stochastic sequence). A stochastic sequence $\left\{\xi \right(n),n\in \mathbb{Z}\}$ is called symmetric α-stable, $S\alpha S$, stochastic sequence, if all linear combinations ${\sum }_{m=1}^l{a}_m\xi \left(n_m\right)$ are $S\alpha S$ random variables.

Let $Z=\left\{Z\right(t):-\infty <t<\infty \}$ be a complex $S\alpha S$ process with independent increments. The spectral measure of the process Z is defined as $\nu \left\{(s,t]\right\}={\parallel Z\left(t\right)-Z\left(s\right)\parallel }_{\alpha }^{\alpha }.$

The integrals $\int a\left(t\right)dZ\left(t\right)$ can be defined for all $a\left(t\right)\in L_{\alpha }\left(\nu \right)$ with the following properties for all $a\in L_{\alpha }\left(\nu \right),b\in L_{\alpha }\left(\nu \right)$ (see Cambanis [11], Cambanis and Soltani [12], and Hosoya [14]):

$${∥\int a(t)dZ(t)∥}_{\alpha }^{\alpha }=\int {\left|a(t)\right|}^{\alpha }d\nu (t),$$

$${\left[\int a(t)dZ(t),\int b(t)dZ(t)\right]}_{\alpha }=\int a\left(t\right){\left(b\left(t\right)\right)}^{<\alpha -1>}d\nu \left(t\right).$$

Definition 3

(Harmonizable symmetric $\alpha$-stable sequence). A symmetric α-stable, $S\alpha S$, stochastic sequence ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$ is said to be harmonizable, $HS\alpha S$, if there exists an $S\alpha S$ process $Z=\left\{Z\right(\lambda ):\lambda \in [-\pi ,\pi \left]\right\}$ with independent increments and finite spectral measure ν such that sequence $\xi \left(m\right)$ has the spectral representation

$$\xi \left(m\right)={\int }_{-\pi }^{\pi }{e}^{im\lambda }dZ\left(\lambda \right),m\in \mathbb{Z},$$

and the covariation has the representation

$${[\xi \left(k\right),\xi \left(m\right)]}_{\alpha }={\int }_{-\pi }^{\pi }{e}^{i(k-m)\lambda }d\nu \left(\lambda \right),k,m\in \mathbb{Z}.$$

2.2. Harmonizable Symmetric $\alpha$-Stable Increment Sequence

Definition 4

(stochastic nth increment sequence). For a given stochastic sequence ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$, the sequence

$${\xi }^{\left(n\right)}(m,\mu )={(1-B_{\mu })}^n\xi \left(m\right)={\displaystyle \sum _{l=0}^n}{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)\xi (m-l\mu ),$$

(4)

where $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)={\displaystyle \frac{n!}{l!(n-l)!}}$, $B_{\mu }$ denotes a backward shift operator with step $\mu \in \mathbb{Z}$, such that $B_{\mu }\xi \left(m\right)=\xi (m-\mu )$, is called a stochastic nth increment sequence with step $\mu \in \mathbb{Z}$.

Throughout the paper, we consider $\mu >0$ and $1<\alpha \le 2$.

Let us give a definition of a harmonizable symmetric $\alpha$-stable increment sequence which is of interest in this paper.

Definition 5

(harmonizable symmetric $\alpha$-stable increment sequence). The stochastic nth increment sequence ${\xi }^{\left(n\right)}(m,\mu )$ generated by a symmetric α-stable, $S\alpha S$, stochastic sequence ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$ is said to be (strongly) harmonizable symmetric α-stable, $HS\alpha S$, if there exists an $S\alpha S$ stochastic process $Z_{{\xi }^{\left(n\right)}}\left(\lambda \right)$ with independent increments on $[-\pi ,\pi )$ and a left-continuous nondecreasing bounded (spectral) function $F\left(\lambda \right)$, $F(-\pi )=0$, such that the sequence ${\xi }^{\left(n\right)}(m,\mu )$ has the spectral representation

$${\xi }^{\left(n\right)}(m,\mu )={\int }_{-\pi }^{\pi }{e}^{im\lambda }{(1-e^{-i\mu \lambda })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right),$$

(5)

where the process $Z_{{\xi }^{\left(n\right)}}\left(\lambda \right)$ is associated with the spectral function $F\left(\lambda \right)$ by the relation

$$\parallel Z_{{\xi }^{\left(n\right)}}\left(t_2\right)-Z_{{\xi }^{\left(n\right)}}\left(t_1\right){\parallel }_{\alpha }^{\alpha }=F\left(t_2\right)-F\left(t_1\right)<\infty -\pi \le t_1<t_2<\pi .$$

The stochastic sequence ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$ which determines the $HS\alpha S$ nth increment sequence ${\xi }^{\left(n\right)}(m,\mu )$ by Formula (4) is called a stochastic sequence with harmonizable symmetric α-stable nth increments.

Using equality (5) and Lemma 1, we obtain a representation of the covariation of the $HS\alpha S$ increment sequence ${\xi }^{\left(n\right)}(m,\mu )$:

$$\begin{array}{c}{[{\xi }^{\left(n\right)}(m_1,{\mu }_1),{\xi }^{\left(n\right)}(m_2,{\mu }_2)]}_{\alpha }\hfill \\ \hfill ={\int }_{-\pi }^{\pi }{e}^{i(m_1-m_2)\lambda }{(1-e^{-i{\mu }_1\lambda })}^n{\left({(1-e^{-i{\mu }_2\lambda })}^n\right)}^{<\alpha -1>}{\displaystyle \frac{1}{{\left|\lambda \right|}^{\alpha n}}}dF\left(\lambda \right),m_1,m_2\in \mathbb{Z}.\end{array}$$

(6)

Consider a sequence $\zeta \left(m\right)=\xi \left(m\right)+\eta \left(m\right)$ with the $HS\alpha S$ increments, where ${\left\{\eta \left(m\right)\right\}}_{m\in \mathbb{Z}}$ is a $HS\alpha S$ stochastic sequence, independent of $\xi \left(m\right)$, admitting a spectral representation

$$\eta \left(m\right)={\int }_{-\pi }^{\pi }{e}^{i\lambda m}d{Z}_{\eta }\left(\lambda \right),$$

$Z_{\eta }\left(\lambda \right)$, $\lambda \in [-\pi ,\pi )$, is a complex $S\alpha S$ process with independent increments, that corresponds to the spectral function $G\left(\lambda \right)$. The $HS\alpha S$ increment sequence ${\zeta }^{\left(n\right)}(m,\mu )$ admits the spectral representation

$${\zeta }^{\left(n\right)}(m,\mu )={\int }_{-\pi }^{\pi }{e}^{i\lambda m}{(1-e^{-i\mu \lambda })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)+{\int }_{-\pi }^{\pi }{e}^{i\lambda m}{(1-e^{-i\mu \lambda })}^{n}d{Z}_{\eta }\left(\lambda \right),$$

where $d{Z}_{\eta }\left(\lambda \right)={\left(i\lambda \right)}^{-n}d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right)$, $\lambda \in [-\pi ,\pi )$. If the spectral functions $F\left(\lambda \right)$ and $G\left(\lambda \right)$ have the spectral densities $f\left(\lambda \right)$ and $g\left(\lambda \right)$, then the spectral density $p\left(\lambda \right)$ of the sequence $\zeta \left(m\right)$ is given by

$$p\left(\lambda \right)=f\left(\lambda \right)+{\left|\lambda \right|}^{\alpha n}g\left(\lambda \right).$$

Let ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$ and ${\left\{\theta \left(m\right)\right\}}_{m\in \mathbb{Z}}$ be two independent sequences with $HS\alpha S$ nth increments that have spectral densities $f\left(\lambda \right)$ and $q\left(\lambda \right)$, respectively. Then, the sequence ${\{\xi \left(m\right)+\theta \left(m\right)\}}_{m\in \mathbb{Z}}$ has the spectral density $f\left(\lambda \right)+q\left(\lambda \right)$.

2.3. Linear Functionals from Sequences with Harmonizable Symmetric $\alpha$-Stable Increments

Consider the functional

$$A_N\xi ={\displaystyle \sum _{k=0}^N}a\left(k\right)\xi \left(k\right).$$

The following lemma describes its representation in terms of the functionals from the increments of the sequences $\xi \left(m\right)$ or $\zeta \left(m\right)=\xi \left(m\right)+\eta \left(m\right)$.

Lemma 2

([25]). The functional $A_N\xi$ admits the representations

$$A_N\xi =B_N^{\mu }\xi -V_N^{\mu }\xi$$

(7)

and

$$A_N\xi =A_N\zeta -A_N\eta =H_N^{\mu }\xi -V_N^{\mu }\zeta ,$$

(8)

where

$$B_N^{\mu }\xi ={\displaystyle \sum _{k=0}^N}{b}_{\mu ,N}\left(k\right){\xi }^{\left(n\right)}(k,\mu )={\displaystyle \sum _{k=0}^N}{\left(D_N^{\mu }{\mathbf{a}}_N\right)}_k{\xi }^{\left(n\right)}(k,\mu ),V_N^{\mu }\xi ={\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)\xi \left(k\right)$$

and

$$H_N^{\mu }\xi :=B_N^{\mu }\zeta -A_N\eta ,$$

$$A_N\zeta ={\displaystyle \sum _{k=0}^N}a\left(k\right)\zeta \left(k\right),A_N\eta ={\displaystyle \sum _{k=0}^N}a\left(k\right)\eta \left(k\right),$$

$$B_N^{\mu }\zeta ={\displaystyle \sum _{k=0}^N}{b}_{\mu ,N}\left(k\right){\zeta }^{\left(n\right)}(k,\mu ),V_N^{\mu }\zeta ={\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)\zeta \left(k\right).$$

Coefficients $b_{\mu ,N}\left(k\right)$, $k=0,1,2,\dots ,N$, and $v_{\mu ,N}\left(k\right)$, $k=-\mu n,-\mu n+1,\dots ,-1$, are given by

$$\begin{array}{cc}\hfill v_{\mu ,N}\left(k\right)& ={\displaystyle \sum _{l=⌈-k/\mu ⌉}^{\min \left\{\left[(N-k)/\mu \right],n\right\}}}{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)b_{\mu ,N}(l\mu +k),k=-\mu n,-\mu n+1,\dots ,-1,\hfill \\ \hfill b_{\mu ,N}\left(k\right)& ={\displaystyle \sum _{m=k}^N}a\left(m\right)d_{\mu }(m-k)={\left(D_N^{\mu }{\mathbf{a}}_N\right)}_k,k=0,1,\dots ,N,\hfill \end{array}$$

where $\left[x\right]$ is the integer part of x, $\lceil x\rceil$ is the least integer greater than or equal to x, and the coefficients $\{d_{\mu }\left(k\right):k\ge 0\}$ are defined by the relation

$${\displaystyle \sum _{k=0}^{\infty }}{d}_{\mu }\left(k\right)x^k={\left({\displaystyle \sum _{j=0}^{\infty }}{x}^{\mu j}\right)}^n,$$

$D_N^{\mu }$ is an $(N+1)\times (N+1)$ matrix with the entries ${\left(D_N^{\mu }\right)}_{k,j}=d_{\mu }(j-k)$ if $0\le k\le j\le N$, and ${\left(D_N^{\mu }\right)}_{k,j}=0$ if $0\le j<k\le N$; ${\mathbf{a}}_N={(a\left(0\right),a\left(1\right),a\left(2\right),\dots ,a\left(N\right))}^{\top }$ is a vector of dimension $N+1$.

3. Estimation Problem for Stochastic Sequences with Harmonizable Stable nth Increments

Linear Estimation Problem

Consider the problem of optimal linear estimation of the functional $A_N\xi$ which depends on the unknown values $\xi \left(k\right)$, $k=0,1,\dots ,N$, of a stochastic sequence with harmonizable symmetric $\alpha$-stable nth increments. The linear estimates ${\tilde{A}}_N\xi$ are based on observations $\xi \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$ (estimation without noise), $\xi \left(m\right)+\eta \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$ (estimation with noise), or $\xi \left(m\right)$ at points $m\le -1$ and $\xi \left(m\right)+\theta \left(m\right)$ at points $m\ge N+1$ (estimation with semi-noise), where the sequences $\eta \left(m\right)$ and $\theta \left(m\right)$ are defined in Section 2.2. To derive the estimates, we consider the $\alpha$-norm errors, or variances,

$$\begin{array}{cc}\hfill {\sigma }_{\alpha }({\tilde{A}}_N\xi ;f)& =\parallel A_N\xi -{\tilde{A}}_N{\xi \parallel }_{\alpha }^{\alpha },\hfill \\ \hfill {\sigma }_{\alpha }({\tilde{A}}_N\xi ;f,g)& =\parallel A_N\xi -{\tilde{A}}_N{\xi \parallel }_{\alpha }^{\alpha },\hfill \\ \hfill {\sigma }_{\alpha }({\tilde{A}}_N\xi ;f,q)& =\parallel A_N\xi -{\tilde{A}}_N{\xi \parallel }_{\alpha }^{\alpha }\hfill \end{array}$$

for the estimation without noise, with noise or with semi-noise, respectively.

As a classical solution to the estimation problem we consider the optimal linear estimates (OLEs) ${\widehat{A}}_N\xi$ that minimize the values ${\sigma }_{\alpha }({\tilde{A}}_N\xi ;f)$, ${\sigma }_{\alpha }({\tilde{A}}_N\xi ;f,g)$ or ${\sigma }_{\alpha }({\tilde{A}}_N\xi ;f,q)$ for the given spectral densities f, g and q of the sequences $\xi \left(m\right)$, $\eta \left(m\right)$ and $\theta \left(m\right)$.

The functionals $V_N^{\mu }\xi$ and $V_N\zeta$ from representations (7) and (8) depend on the known observations at points $k\in \{-\mu n,-\mu n+1,\dots ,-1\}$. Thus, the classical solution to the estimation problem can be given as

$${\widehat{A}}_N\xi ={\widehat{B}}_N^{\mu }\xi -V_N^{\mu }\xi$$

(9)

or

$${\widehat{A}}_N\xi ={\widehat{H}}_N^{\mu }\xi -V_N^{\mu }\xi ,$$

(10)

where ${\widehat{B}}_N^{\mu }\xi$ is the OLE of the functional $B_N^{\mu }\xi$ based on either observations ${\xi }^{\left(n\right)}(m,\mu )$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$ (estimation without noise), or ${\xi }^{\left(n\right)}(m,\mu )$ at points $m\le -1$ and ${\xi }^{\left(n\right)}(m,\mu )+{\theta }^{\left(n\right)}(m,\mu )$ at points $m\ge N+\mu n+1$ (estimation with semi-noise); ${\widehat{H}}_N^{\mu }\xi$ is the optimal linear estimate of the functional $H_N^{\mu }\xi$ based on the observations ${\xi }^{\left(n\right)}(m,\mu )+{\eta }^{\left(n\right)}(m,\mu )$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$ (estimation with noise). Thus, the following relations hold:

$$\begin{array}{cc}\hfill {\sigma }_{\alpha }({\widehat{A}}_N\xi ;f)& ={\sigma }_{\alpha }({\widehat{B}}_N^{\mu }\xi ;f)={\parallel B_N^{\mu }\xi -{\widehat{B}}_N^{\mu }\xi \parallel }_{\alpha }^{\alpha },\hfill \\ \hfill {\sigma }_{\alpha }({\widehat{A}}_N\xi ;f,g)& ={\sigma }_{\alpha }({\widehat{H}}_N^{\mu }\xi ;f,g)={\parallel H_N^{\mu }\xi -{\widehat{H}}_N^{\mu }\xi \parallel }_{\alpha }^{\alpha },\hfill \\ \hfill {\sigma }_{\alpha }({\widehat{A}}_N\xi ;f,q)& ={\sigma }_{\alpha }({\widehat{B}}_N^{\mu }\xi ;f,q)={\parallel B_N^{\mu }\xi -{\widehat{B}}_N^{\mu }\xi \parallel }_{\alpha }^{\alpha }.\hfill \end{array}$$

Denote by $H^{0-}\left({\xi }_{\mu }^{\left(n\right)}\right)$ the closed linear manifold in the ${\parallel ·\parallel }_{\alpha }$ norm generated by the values $\{{\xi }^{\left(n\right)}(k,\mu ):k\le -1\}$ of the $HS\alpha S$ sequence in the space H generated by the set $\{{\xi }^{\left(n\right)}(k,\mu ):k\in \mathbb{Z}\}$, and denote by $H^{N+}\left({\xi }_{-\mu }^{\left(n\right)}\right)$ the closed linear manifold generated by the set $\{{\xi }^{\left(n\right)}(k,-\mu ):k\ge N+1\}$. Note that

$$H^{N+}\left({\xi }_{-\mu }^{\left(n\right)}\right)=H^{(N+\mu n)+}\left({\xi }_{\mu }^{\left(n\right)}\right).$$

In the same way as above, define the closed linear manifolds $H^{0-}\left({\zeta }_{\mu }^{\left(n\right)}\right)$ and $H^{(N+\mu n)+}\left({\zeta }_{\mu }^{\left(n\right)}\right)$ in the ${\parallel ·\parallel }_{\alpha }$ norm.

Define the closed linear subspace $L_{\alpha }^{0-}\left(s\right)\oplus L_{\alpha }^{(N+\mu n)+}\left(s\right)$ in the space $L_{\alpha }\left(s\right)$ generated by the functions

$$\left\{e^{i\lambda k}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}:k\le -1\right\}\bigcup \left\{e^{i\lambda k}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}:k\ge N+\mu n+1\right\},$$

where the functions $f\left(\lambda \right)$, $p\left(\lambda \right)=f\left(\lambda \right)+{\left|\lambda \right|}^{\alpha n}g\left(\lambda \right)$, and $\left(f\right(\lambda ),f(\lambda )+q(\lambda \left)\right)$ can be chosen as $s\left(\lambda \right)$ in the cases of estimation without noise, with noise and with semi-noise, respectively.

There is a map between the functions $e^{i\lambda k}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}$ of the space $L_{\alpha }\left(f\right)$ and the random variables ${\xi }^{\left(n\right)}(k,\mu )$ of the space H, and there is a map between the functions $e^{i\lambda k}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}$ of the space $L_{\alpha }{(f+|\lambda |}^{\alpha n}g)$ and the random variables ${\xi }^{\left(n\right)}(k,\mu )+{\eta }^{\left(n\right)}(k,\mu )$ of the space H.

In contrast to the case $\alpha =2$, the optimal linear estimate ${\widehat{B}}_N^{\mu }\xi$ does not need to coincide with the regression estimate $\mathsf{E}\left(B_N^{\mu }\xi \right|{\xi }^{\left(n\right)}(k,\mu ),k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\})$ since the latter does not generally belong to $L_{\alpha }^{0-}\left(s\right)\oplus L_{\alpha }^{(N+\mu n)+}\left(s\right)$. Even if the regression estimate is linear, it still does not need to coincide with the OLE ${\widehat{B}}_N^{\mu }\xi$; see the work of Cambanis and Miller [27], who discussed these two types of estimates.

Pourahmadi [16] showed that, for the interpolation problem for an $HS\alpha S$ sequence X and a closed linear observation subspace N of H, there is a unique element ${\widehat{X}}_n$ in N which minimizes the distance to $X_n$ and is uniquely determined by

$${[Y,X_n-{\widehat{X}}_n]}_{\alpha }=0,Y\in N.$$

Since the increment sequence ${\xi }^{\left(n\right)}(m,\mu )$ is a $HS\alpha S$ sequence, we obtain that the estimate ${\widehat{B}}_N^{\mu }\xi$ is uniquely determined by

$${[{\xi }^{\left(n\right)}(k,\mu ),B_N^{\mu }\xi -{\widehat{B}}_N^{\mu }\xi ]}_{\alpha }=0,k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}.$$

This conclusion follows from the map between the spaces H and $L_{\alpha }\left(s\right)$, as well as from the condition

$$\int v{(x-P_{M}x)}^{<\alpha -1>}dF=0,v\in M,$$

which uniquely determines a projection $P_{M}x={\widehat{B}}_N^{\mu }\xi$ of $x=B_N^{\mu }\xi$ onto $M=L_{\alpha }^{0-}\left(s\right)\oplus L_{\alpha }^{(N+\mu n)+}\left(s\right)$. This projection $P_{M}x$ is the unique element of M which minimizes the distance from $x\in L_{\alpha }\left(s\right)$ to the closed linear subspace $M=L_{\alpha }^{0-}\left(s\right)\oplus L_{\alpha }^{(N+\mu n)+}\left(s\right)$ [28]. The above reasoning is also valid for the cases of interpolation with noise. Thus, we may obtain the following conditions characterizing the OLEs.

The OLE ${\widehat{B}}_N^{\mu }\xi$ is characterised by the conditions

A1:

${[{\xi }^{\left(n\right)}(k,\mu ),B_N^{\mu }\xi -{\widehat{B}}_N^{\mu }\xi ]}_{\alpha }=0$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$;

A2:

${\widehat{B}}_N^{\mu }\xi \in H^{0-}\left({\xi }_{\mu }^{\left(n\right)}\right)\oplus H^{(N+\mu n)+}\left({\xi }_{\mu }^{\left(n\right)}\right)$.

The OLE ${\widehat{H}}_N^{\mu }\xi$ is characterised by the conditions

B1:

${[{\xi }^{\left(n\right)}(k,\mu )+{\eta }^{\left(n\right)}(k,\mu ),H_N^{\mu }\xi -{\widehat{H}}_N^{\mu }\xi ]}_{\alpha }=0$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$;

B2:

${\widehat{H}}_N^{\mu }\xi \in H^{0-}({\xi }_{\mu }^{\left(n\right)}+{\eta }_{\mu }^{\left(n\right)})\oplus H^{(N+\mu n)+}({\xi }_{\mu }^{\left(n\right)}+{\eta }_{\mu }^{\left(n\right)})$.

The OLE ${\widehat{B}}_N^{\mu }\xi$ is characterised by the conditions

C1.1:

${[{\xi }^{\left(n\right)}(k,\mu ),B_N^{\mu }\xi -{\widehat{B}}_N^{\mu }\xi ]}_{\alpha }=0$, $k\le -1$;

C1.2:

${[{\xi }^{\left(n\right)}(k,\mu )+{\theta }^{\left(n\right)}(k,\mu ),B_N^{\mu }\xi -{\widehat{B}}_N^{\mu }\xi ]}_{\alpha }=0$, $k\ge N+\mu n+1$;

C2:

${\widehat{B}}_N^{\mu }\xi \in H^{0-}\left({\xi }_{\mu }^{\left(n\right)}\right)\oplus H^{(N+\mu n)+}({\xi }_{\mu }^{\left(n\right)}+{\theta }_{\mu }^{\left(n\right)})$.

For further analysis, we consider the following minimality conditions for the functions $f\left(\lambda \right)>0$, $q\left(\lambda \right)>0$ and $g\left(\lambda \right)>0$ [15,16]

$${\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}f\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}d\lambda <\infty ,{\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}q\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}d\lambda <\infty ,$$

(11)

$${\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}{(f\left(\lambda \right)+|\lambda |}^{\alpha n}g\left(\lambda \right))}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}d\lambda <\infty .$$

(12)

These conditions are necessary and sufficient for $\alpha$-norm errors ${\sigma }_{\alpha }({\widehat{A}}_N\xi ;f)$, ${\sigma }_{\alpha }({\widehat{A}}_N\xi ;f,q)$, and ${\sigma }_{\alpha }({\widehat{A}}_N\xi ;f,g)$ to be nonzero when $A_N\xi \not\equiv 0$.

Remark 1.

Minimality conditions (11) and (12) contain the factor $|1-e^{-i\lambda \mu }|$ which has zeros at $\lambda =2\pi k/\mu \in (-\pi ;\pi )$. Let us show that these conditions are relevant for all $\mu >0$. Since the increment sequence ${\xi }^{\left(n\right)}(m,\mu )$ is harmonizable, it admits a representation

$${\xi }^{\left(n\right)}(m,\mu )={\int }_{-\pi }^{\pi }{e}^{im\lambda }d{Z}_{\mu }\left(\lambda \right),m\in \mathbb{Z},\mu \in \mathbb{N},$$

where, for each μ, $Z_{\mu }=\{Z_{\mu }\left(\lambda \right):\lambda \in [-\pi ,\pi ]\}$ is an $S\alpha S$ process with independent increments and finite spectral measure $F_{\mu }$ such that

$${[{\xi }^{\left(n\right)}(m_0+m,\mu ),{\xi }^{\left(n\right)}(m_0,\mu )]}_{\alpha }={\int }_{-\pi }^{\pi }{e}^{im\lambda }d{F}_{\mu }\left(\lambda \right),m_0,m\in \mathbb{Z}.$$

Taking into account (6), obtain the following relations for the increment sequences ${\xi }^{\left(n\right)}(m,\mu )$ and ${\xi }^{\left(n\right)}(m,1)$:

$${\int }_{-\pi }^{\pi }{e}^{im\lambda }d{F}_{\mu }\left(\lambda \right)={\int }_{-\pi }^{\pi }{e}^{im\lambda }{|1-e^{-i\lambda \mu }|}^{\alpha n}{\displaystyle \frac{1}{{\left|\lambda \right|}^{\alpha n}}}dF\left(\lambda \right),m\in \mathbb{Z},$$

$${\int }_{-\pi }^{\pi }{e}^{im\lambda }d{F}_1\left(\lambda \right)={\int }_{-\pi }^{\pi }{e}^{im\lambda }{|1-e^{-i\lambda }|}^{\alpha n}{\displaystyle \frac{1}{{\left|\lambda \right|}^{\alpha n}}}dF\left(\lambda \right),m\in \mathbb{Z}.$$

Thus,

$$F_{\mu }\left(\lambda \right)={\int }_{-\pi }^{\lambda }{|1-e^{-i\lambda \mu }|}^{\alpha n}{\displaystyle \frac{1}{{\left|\lambda \right|}^{\alpha n}}}dF\left(\lambda \right),$$

$$F_1\left(\lambda \right)={\int }_{-\pi }^{\lambda }{|1-e^{-i\lambda }|}^{\alpha n}{\displaystyle \frac{1}{{\left|\lambda \right|}^{\alpha n}}}dF\left(\lambda \right),$$

which implies the equality

$${\int }_{-\pi }^{\lambda }{\displaystyle \frac{{|\lambda |}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}}}d{F}_{\mu }\left(\lambda \right)={\int }_{-\pi }^{\lambda }{\displaystyle \frac{{|\lambda |}^{\alpha n}}{{|1-e^{-i\lambda }|}^{\alpha n}}}d{F}_1\left(\lambda \right).$$

(13)

The right side of equality (13) does not depend on μ, thus the function

$$F\left(\lambda \right)={\int }_{-\pi }^{\lambda }{\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}}}d{F}_{\mu }\left(\lambda \right)$$

is well defined once the right side of equality (13) is well defined. In the right side of equality (13), the factor ${|1-e^{-i\lambda }|}^{\alpha n}$ has zero on $(-\pi ;\pi )$ at $\lambda =0$, which is vanished by the factor ${\left|\lambda \right|}^{\alpha n}$. The sufficiency of the factor ${\lambda }^2$ for random processes in the case $\alpha =2$ was established by Yaglom [29].

As a minimax solution to the linear estimation problem, we consider the estimates ${\widehat{A}}_N\xi$ which minimize the maximum values of $\alpha$-norm errors ${\sigma }_{\alpha }({\tilde{A}}_N\xi ;f)$, ${\sigma }_{\alpha }({\tilde{A}}_N\xi ;f,g)$, or ${\sigma }_{\alpha }({\tilde{A}}_N\xi ;f,q)$ for all spectral densities f, g, and q from the given classes $\mathcal{D}={\mathcal{D}}_f$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_g$, and $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_q$ of admissible spectral densities simultaneously. Taking into account the maps between the subspaces of the spaces H and $L_{\alpha }\left(s\right)$ stated above, this approach is formalized by the following definitions.

Definition 6.

For the given classes of spectral densities $\mathcal{D}={\mathcal{D}}_f$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_g$, and $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_q$, the spectral densities $f^0\left(\lambda \right)\in {\mathcal{D}}_f$, $g^0\left(\lambda \right)\in {\mathcal{D}}_g$, and $q^0\left(\lambda \right)\in {\mathcal{D}}_q$ are called least favorable spectral densities, LFSDs, in the classes $\mathcal{D}$ for the optimal linear estimation of $A_N\xi$ if the following relations hold:

$$\begin{array}{cc}\hfill {\sigma }_{\alpha }\left(f^0\right)& ={\sigma }_{\alpha }(h\left(f^0\right);f^0)=\underset{f\in {\mathcal{D}}_f}{\max }{\sigma }_{\alpha }(h\left(f\right);f),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\sigma }_{\alpha }(f^0,g^0)& ={\sigma }_{\alpha }(h(f^0,g^0);f^0,g^0)=\underset{(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g}{\max }{\sigma }_{\alpha }(h(f,g);f,g),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\sigma }_{\alpha }(f^0,q^0)& ={\sigma }_{\alpha }(h(f^0,q^0);f^0,q^0)=\underset{(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q}{\max }{\sigma }_{\alpha }(h(f,q);f,q),\hfill \end{array}$$

(16)

where $h\left(f^0\right)$, $h(f^0,g^0)$, and $h(f^0,q^0)$ are the spectral characteristics of the OLEs ${\widehat{A}}_N\xi$ for the estimation without noise, estimation with noise, and estimation with semi-noise problems, respectively.

Definition 7.

For the given classes of spectral densities $\mathcal{D}={\mathcal{D}}_f$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_g$, and $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_q$, the spectral characteristics $h^0\left(\lambda \right)$ of the OLEs of the functional $A_N\xi$ are called minimax spectral characteristics, MSCs, if the following conditions are satisfied:

• for the estimation without noise:

  $$h^0\left(\lambda \right)\in H_{\mathcal{D}}=\bigcap _{f\in {\mathcal{D}}_f}{L}_{\alpha }^{0-}\left(f\right)\oplus L_{\alpha }^{(N+\mu n)+}\left(f\right),$$

  $$\underset{h\in H_{\mathcal{D}}}{\min }\underset{f\in {\mathcal{D}}_f}{\max }{\sigma }_{\alpha }(h;f)=\underset{f\in {\mathcal{D}}_f}{\max }{\sigma }_{\alpha }(h^0;f);$$
• for the estimation with noise:

  $$h^0\left(\lambda \right)\in H_{\mathcal{D}}=\bigcap _{(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g}{L}_{\alpha }^{0-}{(f+|\lambda |}^{\alpha n}g)\oplus L_{\alpha }^{(N+\mu n)+}{(f+|\lambda |}^{\alpha n}g),$$

  $$\underset{h\in H_{\mathcal{D}}}{\min }\underset{(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g}{\max }{\sigma }_{\alpha }(h;f,g)=\underset{(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g}{\max }{\sigma }_{\alpha }(h^0;f,g);$$
• for the estimation with semi-noise:

  $$h^0\left(\lambda \right)\in H_{\mathcal{D}}=\bigcap _{(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q}{L}_{\alpha }^{0-}\left(f\right)\oplus L_{\alpha }^{(N+\mu n)+}(f+q),$$

  $$\underset{h\in H_{\mathcal{D}}}{\min }\underset{(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q}{\max }{\sigma }_{\alpha }(h;f,q)=\underset{(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q}{\max }{\sigma }_{\alpha }(h^0;f,q).$$

In the next section, we provide the solutions to the classical and minimax estimation problems.

4. Main Results

4.1. Solution to Classical Estimation Problem Without Noise: Projection Method of Estimation

Theorem 1.

Let a stochastic sequence ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$ generate an $HS\alpha S$ nth increment sequence ${\left\{{\xi }^{\left(n\right)}(m,\mu )\right\}}_{m\in \mathbb{Z}}$, and let the spectral function $F\left(\lambda \right)$ of ${\xi }^{\left(n\right)}(m,\mu )$ be absolutely continuous with spectral density $f\left(\lambda \right)$ satisfying minimality condition (11). The OLE ${\widehat{A}}_N\xi$ of the unknown value of the functional $A_N\xi$ based on observations $\left\{\xi \right(m):m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\left\}\right\}$ is given by

$${\widehat{A}}_N\xi =-{\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)\xi \left(k\right)+{\int }_{-\pi }^{\pi }{h}_{\mu }\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right).$$

(17)

The spectral characteristic $h_{\mu }\left(\lambda \right)$ of the OLE ${\widehat{A}}_N\xi$ is given by

$$h_{\mu }\left(\lambda \right)=B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-{\left({\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>},$$

(18)

where

$$B_N^{\mu }\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^N}{\left(D_N^{\mu }{\mathbf{a}}_N\right)}_k{e}^{i\lambda k},C_{\mu ,N}\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^{N+\mu n}}{c}_{\mu }\left(k\right)e^{-i\lambda k},$$

and the coefficients $c_{\mu }\left(k\right)$, $k=0,1,2,\dots ,N+\mu n$ are determined by the system of equations

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right)-{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}f\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{C}_{\mu ,N}{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right)e^{-i\lambda l}d\lambda =0,\hfill \\ \hfill l=0,1,\dots ,N+\mu n.\end{array}$$

(19)

The α-norm error of the OLE ${\widehat{A}}_N\xi$ is given by

$${\sigma }_{\alpha }({\widehat{A}}_N\xi ;f)={\int }_{-\pi }^{\pi }{\left|{\left({\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }f\left(\lambda \right)d\lambda .$$

(20)

Proof. 

Equality (9) implies representation (17) of the OLE ${\widehat{A}}_N\xi$, where the $h_{\mu }\left(\lambda \right)$ is a spectral characteristic of the estimate ${\widehat{B}}_N^{\mu }\xi$.

Using Condition A1 and the spectral representation

$$B_N^{\mu }\xi ={\int }_{-\pi }^{\pi }{B}_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right),B_N^{\mu }\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^N}{\left(D_N^{\mu }{\mathbf{a}}_N\right)}_k{e}^{i\lambda k},$$

(21)

we obtain the following equations for all $l\le -1$ and $l\ge N+\mu n+1$:

$${\int }_{-\pi }^{\pi }{e}^{i\lambda l}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}f\left(\lambda \right){\left(B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-h_{\mu }\left(\lambda \right)\right)}^{<\alpha -1>}d\lambda =0,$$

from which one can derive representation (18) of $h_{\mu }\left(\lambda \right)$ with the unknown coefficients $c_{\mu }\left(k\right)$, $k=0,1,2,\dots ,N+\mu n$.

Using Condition A2, we conclude that $h_{\mu }\left(\lambda \right)$ is given by

$$h_{\mu }\left(\lambda \right)=H_{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}},$$

where

$$H_{\mu }\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{-1}}{h}_{\mu }\left(k\right)e^{i\lambda k}+{\displaystyle \sum _{k=N+\mu n+1}^{\infty }}{h}_{\mu }\left(k\right)e^{i\lambda k},$$

which yields the following system of equations determining the coefficients $c_{\mu }\left(k\right)$, $k=0,1,2,\dots ,N+\mu n$:

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right)-{\displaystyle \frac{{\left(i\lambda \right)}^n}{{(1-e^{-i\lambda \mu })}^n}}{\left({\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right)e^{-i\lambda l}d\lambda =0,\hfill \\ \hfill l=0,1,\dots ,N+\mu n.\end{array}$$

Thus, using Lemma 1, we obtain the set (19) from the statement of the theorem.

Spectral representation (21) of the functional $B_N^{\mu }\xi$ and the derived expressions for the OLE ${\widehat{A}}_N\xi$ yield Formula (20) for the $\alpha$-norm error ${\sigma }_{\alpha }({\widehat{A}}_N\xi ;f)$. □

4.2. Solution to Classical Estimation Problem with Noise: Projection Method of Estimation

Theorem 2.

Let a stochastic sequence ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$ generate an $HS\alpha S$ nth increment sequence ${\left\{{\xi }^{\left(n\right)}(m,\mu )\right\}}_{m\in \mathbb{Z}}$, and let the spectral function $F\left(\lambda \right)$ of ${\xi }^{\left(n\right)}(m,\mu )$ be absolutely continuous with spectral density $f\left(\lambda \right)$. Let ${\left\{\eta \left(m\right)\right\}}_{m\in \mathbb{Z}}$ be an $HS\alpha S$ stochastic sequence independent from ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$, and let the spectral function $G\left(\lambda \right)$ of $\eta \left(m\right)$ be absolutely continuous with spectral density $g\left(\lambda \right)$. Assume that $f\left(\lambda \right)$ and $g\left(\lambda \right)$ satisfy minimality condition (12). The OLE ${\widehat{A}}_N\xi$ of the unknown value of the functional $A_N\xi$ based on observations $\left\{\xi \right(m)+\eta (m):m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\left\}\right\}$ is given by

$${\widehat{A}}_N\xi =-{\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)(\xi \left(k\right)+\eta \left(k\right))+{\int }_{-\pi }^{\pi }{h}_{\mu }\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right).$$

(22)

The spectral characteristic $h_{\mu }\left(\lambda \right)$ of the OLE is given by

$$h_{\mu }\left(\lambda \right)=H_{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}},$$

(23)

where

$$H_{\mu }\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{-1}}{h}_{\mu }\left(k\right)e^{i\lambda k}+{\displaystyle \sum _{k=N+\mu n+1}^{\infty }}{h}_{\mu }\left(k\right)e^{i\lambda k}.$$

The coefficients $h_{\mu }\left(k\right)$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$ are determined by the system of equations

$$\begin{array}{c}{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\left(B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }\left(e^{i\lambda }\right)\right)}^{<\alpha -1>}{\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda \hfill \\ +{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\left(B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }\left(e^{i\lambda }\right)\right)}^{<\alpha -1>}{|1-e^{-i\lambda \mu }|}^{\alpha n}g\left(\lambda \right)d\lambda =0,\\ \hfill l\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\},\end{array}$$

(24)

where $B_N^{\mu }\left(e^{i\lambda }\right)$ is defined in Theorem 1,

$$A_N\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^N}a\left(k\right)e^{i\lambda k}.$$

The α-norm error of the OLE ${\widehat{A}}_N\xi$ is given by

$$\begin{array}{cc}\hfill {\sigma }_{\alpha }({\widehat{A}}_N\xi ;f,g)& ={\int }_{-\pi }^{\pi }{\left|B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }\left(e^{i\lambda }\right)\right|}^{\alpha }{\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda \hfill \\ & +{\int }_{-\pi }^{\pi }{\left|B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }\left(e^{i\lambda }\right)\right|}^{\alpha }{|1-e^{-i\lambda \mu }|}^{\alpha n}g\left(\lambda \right)d\lambda .\hfill \end{array}$$

(25)

Proof. 

Condition B2 implies representation (23) of the spectral characteristic $h_{\mu }\left(\lambda \right)$ with unknown coefficients $h_{\mu }\left(k\right)$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$, to be found.

Equality (10) implies representation (22) of the OLE ${\widehat{A}}_N\xi$, where $h_{\mu }\left(\lambda \right)$ is a spectral characteristic of the OLE ${\widehat{H}}_N^{\mu }\xi$. The functional $H_N^{\mu }\xi =B_N^{\mu }\zeta -A_N\eta$ admits the spectral representation

$$H_N^{\mu }\xi ={\int }_{-\pi }^{\pi }{B}_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}+{\eta }^{\left(n\right)}}\left(\lambda \right)-{\int }_{-\pi }^{\pi }{A}_N\left(e^{i\lambda }\right)d{Z}_{\eta }\left(\lambda \right).$$

(26)

Thus,

$$\begin{array}{cc}\hfill H_N^{\mu }\xi -{\widehat{H}}_N^{\mu }\xi & ={\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-h_{\mu }\left(\lambda \right)\right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)\hfill \\ & +{\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-h_{\mu }\left(\lambda \right)\right)d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right)\hfill \\ & =:{\int }_{-\pi }^{\pi }{U}_{\xi }^{\mu }\left(e^{i\lambda }\right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)+{\int }_{-\pi }^{\pi }{U}_{\eta }^{\mu }\left(e^{i\lambda }\right)d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right).\hfill \end{array}$$

Recall some covariation properties from Lemma 1, namely linearity in the first argument, linearity in the second argument for independent random variables, and zero covariation value for the independent random variables. Then, using the independence of the sequences $\xi \left(m\right)$ and $\eta \left(m\right)$, as well as Condition B1, we obtain the following equations for all $l\le -1$ and $l\ge N+\mu n+1$:

$$\begin{array}{cc}\hfill & {\left[{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right);{\int }_{-\pi }^{\pi }{U}_{\xi }^{\mu }\left(e^{i\lambda }\right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)\right]}_{\alpha }\hfill \\ & +{\left[{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right);{\int }_{-\pi }^{\pi }{U}_{\eta }^{\mu }\left(e^{i\lambda }\right)d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right)\right]}_{\alpha }\hfill \\ & ={\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}{\left(U_{\xi }^{\mu }\left(e^{i\lambda }\right)\right)}^{<\alpha -1>}f\left(\lambda \right)d\lambda \hfill \\ & +{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}{\left(U_{\eta }^{\mu }\left(e^{i\lambda }\right)\right)}^{<\alpha -1>}{\left|\lambda \right|}^{\alpha n}g\left(\lambda \right)d\lambda \hfill \\ & =0.\hfill \end{array}$$

These equations for the unknown coefficients $h_{\mu }\left(k\right)$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$, of the spectral characteristic $h_{\mu }\left(\lambda \right)$ are rewritten in the form of system (24) from the theorem statement.

Spectral representation (26) of the functional $H_N^{\mu }\xi$ and the derived expressions for the OLE ${\widehat{A}}_N\xi$ yield Formula (25) for the $\alpha$-norm error ${\sigma }_{\alpha }({\widehat{A}}_N\xi ;f,g)$. □

4.3. Solution to Classical Estimation Problem with Semi-Noise: Projection Method of Estimation

Theorem 3.

Let stochastic sequences ${\left\{\xi \left(m\right)\right\}}_{m\in \mathbb{Z}}$ and ${\left\{\theta \left(m\right)\right\}}_{m\in \mathbb{Z}}$ generate $HS\alpha S$ nth increment sequences ${\left\{{\xi }^{\left(n\right)}(m,\mu )\right\}}_{m\in \mathbb{Z}}$ and ${\left\{{\theta }^{\left(n\right)}(m,\mu )\right\}}_{m\in \mathbb{Z}}$, and let the spectral functions $F\left(\lambda \right)$ and $Q\left(\lambda \right)$ of ${\xi }^{\left(n\right)}(m,\mu )$ and ${\theta }^{\left(n\right)}(m,\mu )$ be absolutely continuous with spectral densities $f\left(\lambda \right)$ and $q\left(\lambda \right)$ satisfying minimality condition (11). The OLE ${\widehat{A}}_N\xi$ of the unknown value of the functional $A_N\xi$ based on observations $\xi \left(m\right)$ at points $m\le -1$ and $\xi \left(m\right)+\theta \left(m\right)$ at points $m\ge N+1$ is given by

$${\widehat{A}}_N\xi =-{\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)\xi \left(k\right)+{\int }_{-\pi }^{\pi }{h}_{\mu }^1\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)+{\int }_{-\pi }^{\pi }{h}_{\mu }^2\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}+{\theta }^{\left(n\right)}}\left(\lambda \right).$$

(27)

The spectral characteristic $h_{\mu }\left(\lambda \right)=(h_{\mu }^1\left(\lambda \right),h_{\mu }^2\left(\lambda \right))$ of the estimate ${\widehat{A}}_N\xi$ is given by

$$\begin{array}{cc}\hfill h_{\mu }^1\left(\lambda \right)& =B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-U_{\mu ,N}^f{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}-U_{\mu ,N}^q{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill h_{\mu }^2\left(\lambda \right)& =U_{\mu ,N}^q{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>},\hfill \end{array}$$

(29)

where $B_N^{\mu }\left(e^{i\lambda }\right)$ is defined in Theorem 1,

$$U_{\mu ,N}^f\left(e^{i\lambda }\right)={\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}^1\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}},U_{\mu ,N}^q\left(e^{i\lambda }\right)={\displaystyle \frac{{\left(i\lambda \right)}^n(C_{\mu ,N}^1\left(e^{i\lambda }\right)-C_{\mu ,N}^2\left(e^{i\lambda }\right))}{{(1-e^{-i\lambda \mu })}^{n}q\left(\lambda \right)}},$$

and

$$C_{\mu ,N}^1\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^{\infty }}{c}_{\mu }^1\left(k\right)e^{-i\lambda k},C_{\mu ,N}^2\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{N+\mu n}}{c}_{\mu }^2\left(k\right)e^{-i\lambda k}.$$

The coefficients $c_{\mu }^1\left(k\right)$, $k\ge 0$, and $c_{\mu }^2\left(k\right)$, $k\le N+\mu n$, are determined from by the system of equations

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right)-{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}f\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{C}_{\mu ,N}^1{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right.\hfill \\ \left.-{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}q\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{(C_{\mu ,N}^1\left(e^{i\lambda }\right)-C_{\mu ,N}^2\left(e^{i\lambda }\right))}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right)e^{-i\lambda l}d\lambda =0,\\ \hfill l\ge 0,\end{array}$$

(30)

and

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left({\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}q\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{(C_{\mu ,N}^1\left(e^{i\lambda }\right)-C_{\mu ,N}^2\left(e^{i\lambda }\right))}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right)e^{-i\lambda l}d\lambda =0,\hfill \\ \hfill l\le N+\mu n.\end{array}$$

(31)

The α-norm error of the OLE ${\widehat{A}}_N\xi$ is given by

$${\sigma }_{\alpha }({\widehat{A}}_N\xi ;f,q)={\int }_{-\pi }^{\pi }{\left|U_{\mu ,N}^f{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }f\left(\lambda \right)d\lambda +{\int }_{-\pi }^{\pi }{\left|U_{\mu ,N}^q{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }q\left(\lambda \right)d\lambda .$$

(32)

Proof. 

Let us consider the spectral representation of ${\widehat{B}}_N^{\mu }\xi$ in the form

$${\widehat{B}}_N^{\mu }\xi ={\int }_{-\pi }^{\pi }{h}_{\mu }^1\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)+{\int }_{-\pi }^{\pi }{h}_{\mu }^2\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}+{\theta }^{\left(n\right)}}\left(\lambda \right).$$

Then equality (9) implies representation (27) of the optimal estimate ${\widehat{A}}_N\xi$.

Using spectral representation (21) of the functional $B_N^{\mu }\xi$, we obtain

$$\begin{array}{cc}\hfill B_N^{\mu }\xi -{\widehat{B}}_N^{\mu }\xi & ={\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-h_{\mu }^1\left(\lambda \right)-h_{\mu }^2\left(\lambda \right)\right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)\hfill \\ & -{\int }_{-\pi }^{\pi }{h}_{\mu }^2\left(\lambda \right)d{Z}_{{\theta }^{\left(n\right)}}\left(\lambda \right).\hfill \end{array}$$

By Lemma 1, the covariation is linear in the second argument for independent random variables, and the covariation of independent random variables is zero. Thus, using the independence of the sequences $\xi \left(m\right)$ and $\theta \left(m\right)$, as well as Condition C1.1, we obtain the equations for all $l\le -1$

$${\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}f\left(\lambda \right){\left(B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-h_{\mu }^1\left(\lambda \right)-h_{\mu }^2\left(\lambda \right)\right)}^{<\alpha -1>}d\lambda =0,$$

from which we derive the representation

$$h_{\mu }^1\left(\lambda \right)+h_{\mu }^2\left(\lambda \right)=B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-{\left({\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}^1\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}$$

(33)

with the unknown coefficients $c_{\mu }^1\left(k\right)$, $k\ge 0$, defining $C_{\mu ,N}^1\left(e^{i\lambda }\right)$ from the theorem statement.

Using the covariation linearity in the first argument, covariation properties stated above, and Condition C1.2, we obtain the equations for all $l\ge N+\mu n+1$

$$\begin{array}{c}{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}f\left(\lambda \right){\left(B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-h_{\mu }^1\left(\lambda \right)-h_{\mu }^2\left(\lambda \right)\right)}^{<\alpha -1>}\hfill \\ \hfill -{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}q\left(\lambda \right){\left(h_{\mu }^2\left(\lambda \right)\right)}^{<\alpha -1>}d\lambda =0,\end{array}$$

from which we derive the representation

$$C_{\mu ,N}^1\left(e^{i\lambda }\right)-{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}q\left(\lambda \right){\left(h_{\mu }^2\left(\lambda \right)\right)}^{<\alpha -1>}=C_{\mu ,N}^2\left(e^{i\lambda }\right)$$

with the unknown coefficients $c_{\mu }^2\left(k\right)$, $k\le N+\mu n$, defining $C_{\mu ,N}^2\left(e^{i\lambda }\right)$ from the theorem statement. Thus,

$$h_{\mu }^2\left(\lambda \right)={\left({\displaystyle \frac{{\left(i\lambda \right)}^n(C_{\mu ,N}^1\left(e^{i\lambda }\right)-C_{\mu ,N}^2\left(e^{i\lambda }\right))}{{(1-e^{-i\lambda \mu })}^{n}q\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}.$$

(34)

Representations (33) and (34) imply Formulas (28) and (29) for the spectral characteristic $h_{\mu }\left(\lambda \right)=(h_{\mu }^1\left(\lambda \right),h_{\mu }^2\left(\lambda \right))$ of the OLE ${\widehat{A}}_N\xi$.

Using Condition C2, we conclude that $h_{\mu }\left(\lambda \right)=(h_{\mu }^1\left(\lambda \right),h_{\mu }^2\left(\lambda \right))$ is given by

$$\begin{array}{cc}\hfill h_{\mu }^1\left(\lambda \right)& =H_{\mu }^1\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}},H_{\mu }^1\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{-1}}{h}_{\mu }^1\left(k\right)e^{i\lambda k},\hfill \\ \hfill h_{\mu }^2\left(\lambda \right)& =H_{\mu }^2\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}},H_{\mu }^2\left(e^{i\lambda }\right)={\displaystyle \sum _{k=N+\mu n+1}^{\infty }}{h}_{\mu }^2\left(k\right)e^{i\lambda k},\hfill \end{array}$$

which allows us to derive the following set of equations determining the coefficients $c_{\mu }^1\left(k\right)$, $k\ge 0$, and $c_{\mu }^2\left(k\right)$, $k\le N+\mu n$, respectively:

$$\begin{array}{cc}\hfill {\int }_{-\pi }^{\pi }{h}_{\mu }^1\left(\lambda \right){\displaystyle \frac{{\left(i\lambda \right)}^n}{{(1-e^{-i\lambda \mu })}^n}}{e}^{-i\lambda l}d\lambda & =0,l\ge 0,\hfill \\ \hfill {\int }_{-\pi }^{\pi }{h}_{\mu }^2\left(\lambda \right){\displaystyle \frac{{\left(i\lambda \right)}^n}{{(1-e^{-i\lambda \mu })}^n}}{e}^{-i\lambda l}d\lambda & =0,l\le N+\mu n.\hfill \end{array}$$

Thus, using Lemma 1 we obtain the set of Equations (30) and (31) from the theorem statement.

The spectral representation (21) of the functional $B_N^{\mu }\xi$, the derived expressions for the OLE ${\widehat{A}}_N\xi$, as well as covariation properties stated above, yield Formula (32) for the $\alpha$-norm error ${\sigma }_{\alpha }({\widehat{A}}_N\xi ;f,q)$. □

4.4. Examples of Classical Solutions to the Interpolation Problem

In contrast to the case $\alpha =2$, the derived equations lead to systems of nonlinear equations that are difficult to solve for $1<\alpha <2$. Thus, the methods developed for the stationary increment sequences cannot be directly extended to the $HS\alpha S$ increment sequences for $1<\alpha <2$. We illustrate this point with the examples presented in this subsection.

In the examples presented in this subsection, we consider two independent sequences $\xi \left(m\right)$ and $\theta \left(m\right)$ with $HS\alpha S$ increments of order $n=1$ and $\alpha =4/3$. Suppose that the increment sequences ${\xi }^{\left(1\right)}(m,1)$ and ${\theta }^{\left(1\right)}(m,1)$ with a step $\mu =1$ are described by AR(1) models with parameters $-1<\varphi <1$, $-1<\psi <1$, respectively, and have spectral densities

$$f_1\left(\lambda \right)={\displaystyle \frac{1}{{|1+\varphi e^{-i\lambda }|}^{4/3}}},q_1\left(\lambda \right)={\displaystyle \frac{1}{{|1+\psi e^{-i\lambda }|}^{4/3}}}.$$

Then, the spectral densities of the increment sequences ${\xi }^{\left(1\right)}(m,\mu )$ and ${\theta }^{\left(1\right)}(m,\mu )$ are

$$f\left(\lambda \right)={\displaystyle \frac{{|\lambda |}^{4/3}}{{|1+e^{-i\lambda }|}^{4/3}}}{\displaystyle \frac{1}{{|1+\varphi e^{-i\lambda }|}^{4/3}}},q\left(\lambda \right)={\displaystyle \frac{{|\lambda |}^{4/3}}{{|1+e^{-i\lambda }|}^{4/3}}}{\displaystyle \frac{1}{{|1+\psi e^{-i\lambda }|}^{4/3}}}.$$

Example 1.

Consider a semi-noise interpolation problem for the functional $A_1\xi =\xi \left(0\right)+\xi \left(1\right)$ based on observations $\left\{\xi \right(m):m\le -1\}\cup \left\{\xi \right(m)+\theta (m):m\ge 2\}$.

Lemma 2 implies $A_1\xi =B_1^1\xi -V_1^1\xi$, $B_1^1\xi =b_{1,1}\left(0\right){\xi }^{\left(1\right)}(0,1)+b_{1,1}\left(1\right){\xi }^{\left(1\right)}(1,1)$, $b_{1,1}\left(0\right)=2$, $b_{1,1}\left(1\right)=1$, and $V_1^1\xi =v_{1,1}(-1)\xi (-1)$, $v_{1,1}(-1)=-2$. Put $b_{1,1}\left(2\right):=0$.

Equations (30) and (31), which determine

$$C_{1,1}^1\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^{\infty }}{c}_1^1\left(k\right)e^{-i\lambda k},C_{1,1}^2\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^2}{c}_1^2\left(k\right)e^{-i\lambda k}$$

in Theorem 3 are of the form

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left(2+e^{i\lambda }-{|1+\varphi e^{-i\lambda }|}^4{C}_{1,1}^1{\left(e^{i\lambda }\right)}^{<3>}\right.\hfill \\ \hfill \left.{-|1+\psi e^{-i\lambda }|}^4{(C_{1,1}^1\left(e^{i\lambda }\right)-C_{1,1}^2\left(e^{i\lambda }\right))}^{<3>}\right)e^{-i\lambda l}d\lambda =0,l\ge 0,\end{array}$$

(35)

and

$${\int }_{-\pi }^{\pi }\left({|1+\psi e^{-i\lambda }|}^4{(C_{1,1}^1\left(e^{i\lambda }\right)-C_{1,1}^2\left(e^{i\lambda }\right))}^{<3>}\right)e^{-i\lambda l}d\lambda =0,l\le 2.$$

(36)

Denote $C_{1,1}^{12}\left(e^{i\lambda }\right)=C_{1,1}^1\left(e^{i\lambda }\right)-C_{1,1}^2\left(e^{i\lambda }\right)$. Then,

$$C_{1,1}^{12}\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{\infty }}{c}_1^{12}\left(k\right)e^{-i\lambda k},$$

where $c_1^{12}\left(k\right)=c_1^1\left(k\right)$, $k\ge 3$, $c_1^{12}\left(k\right)=c_1^1\left(k\right)-c_1^2\left(k\right)$, $k=0,1,2$, and $c_1^{12}\left(k\right)=-c_1^2\left(k\right)$, $\le -1$. Note that $C_{1,1}^1{\left(e^{i\lambda }\right)}^{<3>}=C_{1,1}^1\left(e^{i\lambda }\right){\overline{C_{1,1}^1\left(e^{i\lambda }\right)}}^2$, $C_{1,1}^{12}{\left(e^{i\lambda }\right)}^{<3>}=C_{1,1}^{12}\left(e^{i\lambda }\right){\overline{C_{1,1}^{12}\left(e^{i\lambda }\right)}}^2$.

Equations (35) are equivalent to the following system of nonlinear equations:

$$\begin{array}{cc}\hfill b_{1,1}\left(l\right)& =\sum _{\begin{array}{c}m=-2,\dots ,2,k_i\ge 0,\\ m-k_1+k_2+k_3=l\end{array}}f\left(m\right)c_1^1\left(k_1\right)\overline{c_1^1}\left(k_2\right)\overline{c_1^1}\left(k_3\right),l=0,1,2,\hfill \\ \hfill 0& =\sum _{\begin{array}{c}m=-2,\dots ,2,k_i\ge 0,\\ m-k_1+k_2+k_3=l\end{array}}f\left(m\right)c_1^1\left(k_1\right)\overline{c_1^1}\left(k_2\right)\overline{c_1^1}\left(k_3\right)\hfill \\ \hfill & +\sum _{\begin{array}{c}m=-2,\dots ,2,k_i\in \mathbb{Z},\\ m+k_1+k_2+k_3=l\end{array}}q\left(m\right)c_1^{12}\left(k_1\right)\overline{c_1^{12}}(-k_2)\overline{c_1^{12}}(-k_3),l\ge 3,\hfill \end{array}$$

and Equations (36) are equivalent to the following system of nonlinear equations:

$$0=\sum _{\begin{array}{c}m=-2,\dots ,2,k_i\in \mathbb{Z},\\ m+k_1+k_2+k_3=l\end{array}}q\left(m\right)c_1^{12}\left(k_1\right)\overline{c_1^{12}}(-k_2)\overline{c_1^{12}}(-k_3),l\le 2,$$

where $f\left(m\right)$, $q\left(m\right)$, $m=-2,\dots ,2$, are determined by

$$\begin{array}{cc}\hfill {|1+\varphi e^{-i\lambda }|}^4& =f(-2)e^{-2i\lambda }+f(-1)e^{-i\lambda }+f\left(0\right)+f\left(1\right)e^{i\lambda }+f\left(2\right)e^{2i\lambda },\hfill \\ \hfill {|1+\psi e^{-i\lambda }|}^4& =q(-2)e^{-2i\lambda }+q(-1)e^{-i\lambda }+q\left(0\right)+q\left(1\right)e^{i\lambda }+q\left(2\right)e^{2i\lambda }.\hfill \end{array}$$

The derived systems of nonlinear equations determine the coefficients $c_1^1\left(k\right)$, $k\ge 0$, and $c_1^2\left(k\right)$, $k\le 2$, which define the spectral characteristic (28) and (29) of the OLE ${\widehat{A}}_1\xi$.

The interpolation problem becomes simpler in the absence of the noise sequence $\theta \left(m\right)$, as demonstrated in the following examples.

Example 2.

Consider an interpolation problem without noise for the functional $A_1\xi =\xi \left(0\right)+\xi \left(1\right)$ based on observations $\left\{\xi \right(m):m\in \mathbb{Z}\setminus \{0,1\left\}\right\}$. Using the results of the previous example, in the absence of noise, we have $C_{1,1}^1\left(e^{i\lambda }\right)=C_{1,1}^2\left(e^{i\lambda }\right)$. Thus, omitting the indices for simplicity,

$$C_{1,1}\left(e^{i\lambda }\right)=C_{1,1}^1\left(e^{i\lambda }\right)=c\left(0\right)+c\left(1\right)e^{-i\lambda }+c\left(2\right)e^{-2i\lambda },C_{1,1}{\left(e^{i\lambda }\right)}^{<3>}={\displaystyle \sum _{k=-2}^4}\omega \left(k\right)e^{i\lambda k},$$

where

$$\begin{array}{cc}\hfill \omega (-2)& =c\left(2\right)\overline{c}{\left(0\right)}^2,\hfill \\ \hfill \omega (-1)& =c\left(1\right)\overline{c}{\left(0\right)}^2+2c\left(2\right)\overline{c}\left(0\right)\overline{c}\left(1\right),\hfill \\ \hfill \omega \left(0\right)& =c\left(0\right)\overline{c}{\left(0\right)}^2+2c\left(1\right)\overline{c}\left(0\right)\overline{c}\left(1\right)+c\left(2\right)(2\overline{c}\left(0\right)\overline{c}\left(2\right)+\overline{c}{\left(1\right)}^2),\hfill \\ \hfill \omega \left(1\right)& =2c\left(0\right)\overline{c}\left(0\right)\overline{c}\left(1\right)+2c\left(1\right)(\overline{c}\left(0\right)\overline{c}\left(2\right)+\overline{c}{\left(1\right)}^2)+2c\left(2\right)\overline{c}\left(1\right)\overline{c}\left(2\right),\hfill \\ \hfill \omega \left(2\right)& =2c\left(0\right)(\overline{c}\left(0\right)\overline{c}\left(2\right)+\overline{c}{\left(1\right)}^2)+2c\left(1\right)\overline{c}\left(1\right)\overline{c}\left(2\right)+c\left(2\right)\overline{c}{\left(2\right)}^2,\hfill \\ \hfill \omega \left(3\right)& =2c\left(0\right)\overline{c}\left(1\right)\overline{c}\left(2\right)+c\left(1\right)\overline{c}{\left(2\right)}^2,\hfill \\ \hfill \omega \left(4\right)& =c\left(0\right)\overline{c}{\left(2\right)}^2.\hfill \end{array}$$

Recall that $b_{1,1}\left(0\right)=2$, $b_{1,1}\left(1\right)=1$, $b_{1,1}\left(2\right):=0$. Then the coefficients $c\left(0\right)$, $c\left(1\right)$, $c\left(2\right)$ are obtained as a solution of the following system of equations:

$$\begin{array}{cc}\hfill 2& =f(-2)\omega \left(2\right)+f(-1)\omega \left(1\right)+f\left(0\right)\omega \left(0\right)+f\left(1\right)\omega (-1)+f\left(2\right)\omega (-2),\hfill \\ \hfill 1& =f(-2)\omega \left(3\right)+f(-1)\omega \left(2\right)+f\left(0\right)\omega \left(1\right)+f\left(1\right)\omega \left(0\right)+f\left(2\right)\omega (-1),\hfill \\ \hfill 0& =f(-2)\omega \left(4\right)+f(-1)\omega \left(3\right)+f\left(0\right)\omega \left(2\right)+f\left(1\right)\omega \left(1\right)+f\left(2\right)\omega \left(0\right),\hfill \end{array}$$

where $f(-2)={\varphi }^2$, $f(-1)=2\varphi (1+{\varphi }^2)$, $f\left(0\right)=1+4{\varphi }^2+{\varphi }^4$, $f\left(1\right)=2\varphi (1+{\varphi }^2)$, $f\left(2\right)={\varphi }^2$. The OLE ${\widehat{A}}_1\xi$ is defined by the formula

$${\widehat{A}}_1\xi =2\xi (-1)+{\int }_{-\pi }^{\pi }{\tilde{h}}_1\left(\lambda \right){\displaystyle \frac{(1-e^{-i\lambda })}{\left(i\lambda \right)}}d{Z}_{{\xi }^{\left(1\right)}}\left(\lambda \right),$$

where

$${\tilde{h}}_1\left(\lambda \right)=2+e^{i\lambda }-{|1+\varphi e^{-i\lambda }|}^4{(c\left(0\right)+c\left(1\right)e^{-i\lambda }+c\left(2\right)e^{-2i\lambda })}^{<3>}.$$

The α-norm error of the OLE ${\widehat{A}}_1\xi$ is given by

$${\sigma }_{4/3}({\widehat{A}}_1\xi ;f)={\int }_{-\pi }^{\pi }{|1+\varphi e^{-i\lambda }|}^4{|c\left(0\right)+c\left(1\right)e^{-i\lambda }+c\left(2\right)e^{-2i\lambda }|}^{4}d\lambda .$$

Example 3.

Consider an interpolation problem without noise for the single value $\xi \left(0\right)$ based on observations $\left\{\xi \right(m):m\in \mathbb{Z}\setminus \{0\left\}\right\}$. Let the spectral density of the increment sequence ${\xi }^{\left(1\right)}(m,\mu )$ be of the form

$$f\left(\lambda \right)={\displaystyle \frac{{\left|\lambda \right|}^{4/3}}{{|1+e^{-i\lambda }|}^{4/3}}}{\displaystyle \frac{1}{{|1+0.5e^{-i\lambda }|}^{4/3}}}.$$

Then,

$${|1+0.5e^{-i\lambda }|}^3={\displaystyle \frac{1}{4}}{e}^{-2i\lambda }+{\displaystyle \frac{5}{4}}{e}^{-i\lambda }+{\displaystyle \frac{33}{16}}+{\displaystyle \frac{5}{4}}{e}^{i\lambda }+{\displaystyle \frac{1}{4}}{e}^{2i\lambda }.$$

The estimate is given by $\widehat{\xi }\left(0\right)={\widehat{\xi }}^{\left(1\right)}(0,1)+\xi (-1)$. Using the results of the previous example, we obtain

$$C_{1,0}\left(e^{i\lambda }\right)=c\left(0\right)+c\left(1\right)e^{-i\lambda },C_{1,0}^1{\left(e^{i\lambda }\right)}^{<3>}={\displaystyle \sum _{k=-1}^2}\omega \left(k\right)e^{i\lambda k},$$

where $\omega (-1)=c\left(1\right)\overline{c}{\left(0\right)}^2$, $\omega \left(0\right)=c\left(0\right)\overline{c}{\left(0\right)}^2+2c\left(1\right)\overline{c}\left(0\right)\overline{c}\left(1\right)$, $\omega \left(1\right)=2c\left(0\right)\overline{c}\left(0\right)\overline{c}\left(1\right)+c\left(1\right)\overline{c}{\left(1\right)}^2$, $\omega \left(2\right)=c\left(0\right)\overline{c}{\left(1\right)}^2$. The coefficients $c\left(0\right)$, $c\left(1\right)$ are determined by the following two equations:

$$\begin{array}{cc}\hfill 1& ={\displaystyle \frac{1}{4}}c\left(0\right)\overline{c}{\left(1\right)}^2+{\displaystyle \frac{5}{4}}(2c\left(0\right)\overline{c}\left(0\right)\overline{c}\left(1\right)+c\left(1\right)\overline{c}{\left(1\right)}^2)\hfill \\ \hfill & +{\displaystyle \frac{33}{16}}(c\left(0\right)\overline{c}{\left(0\right)}^2+2c\left(1\right)\overline{c}\left(0\right)\overline{c}\left(1\right))+{\displaystyle \frac{5}{4}}c\left(1\right)\overline{c}{\left(0\right)}^2,\hfill \\ \hfill 0& ={\displaystyle \frac{5}{4}}c\left(0\right)\overline{c}{\left(1\right)}^2+{\displaystyle \frac{33}{16}}(2c\left(0\right)\overline{c}\left(0\right)\overline{c}\left(1\right)+c\left(1\right)\overline{c}{\left(1\right)}^2)\hfill \\ \hfill & +{\displaystyle \frac{5}{4}}(c\left(0\right)\overline{c}{\left(0\right)}^2+2c\left(1\right)\overline{c}\left(0\right)\overline{c}\left(1\right))+{\displaystyle \frac{1}{4}}c\left(1\right)\overline{c}{\left(0\right)}^2.\hfill \end{array}$$

The approximate solutions are $c\left(0\right)\approx 0.93$, $c\left(1\right)\approx -0.31$. Then the OLE $\widehat{\xi }\left(0\right)$ is given by the formula

$$\widehat{\xi }\left(0\right)=\xi (-1)+{\int }_{-\pi }^{\pi }{\tilde{h}}_1\left(\lambda \right){\displaystyle \frac{(1-e^{-i\lambda })}{\left(i\lambda \right)}}d{Z}_{{\xi }^{\left(1\right)}}\left(\lambda \right),$$

where

$$\begin{array}{cc}\hfill {\tilde{h}}_1\left(\lambda \right)& {=1-|1+0.5e^{-i\lambda }|}^4{(c\left(0\right)+c\left(1\right)e^{-i\lambda })}^{<3>}\hfill \\ & \approx 0.07e^{-3i\lambda }+0.09e^{-2i\lambda }-0.53e^{-i\lambda }+0.28e^{2i\lambda }+0.03e^{3i\lambda }-0.02e^{4i\lambda }.\hfill \end{array}$$

Finally, the OLE $\widehat{\xi }\left(0\right)$ can be written as

$$\begin{array}{cc}\hfill \widehat{\xi }\left(0\right)& \approx \xi (-1)+0.07\left(\xi \right(-3)-\xi (-4\left)\right)+0.09\left(\xi \right(-2)-\xi (-3\left)\right)-0.53\left(\xi \right(-1)-\xi (-2\left)\right)\hfill \\ \hfill & +0.28\left(\xi \right(2)-\xi (1\left)\right)+0.03\left(\xi \right(3)-\xi (2\left)\right)-0.02\left(\xi \right(4)-\xi (3\left)\right)\hfill \\ \hfill & \approx -0.07\xi (-4)-0.02\xi (-3)+0.62\xi (-2)+0.47\xi (-1)-0.28\xi \left(1\right)+0.25\xi \left(2\right)\hfill \\ \hfill & +0.05\xi \left(3\right)-0.02\xi \left(4\right).\hfill \end{array}$$

4.5. Solution to Minimax Estimation Problems

Taking into account the definitions of the minimax estimation approach and the derived solutions to the classical estimation problems we can conclude that the following lemma holds true.

Lemma 3.

The spectral densities $f^0\in {\mathcal{D}}_f$, $g^0\in {\mathcal{D}}_g$, and $q^0\in {\mathcal{D}}_q$ which satisfy minimality conditions (11) for $f^0$, (12) for $f^0$ and $g^0$, and (11) for $f^0$ and $q^0$ are LFSDs in the classes $\mathcal{D}={\mathcal{D}}_f$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_g$, and $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_q$ for the optimal linear estimation of the functional $A_N\overrightarrow{\xi }$ for the estimation without noise, the estimation with noise, the estimation with semi-noise problems, respectively, if the functions $f^0\left(\lambda \right)$, $g^0\left(\lambda \right)$, and $q^0\left(\lambda \right)$ determine the solutions to constraint optimization problems (14)–(16), where ${\sigma }_{\alpha }(h\left(f\right);f)$, ${\sigma }_{\alpha }(h(f,g);f,g)$, and ${\sigma }_{\alpha }(h(f,q);f,q)$ are defined by Formulas (20), (25), and (32), respectively. The MSCs $h^0=h_{\mu }\left(f^0\right)$, $h^0=h_{\mu }(f^0,g^0)$, and $h^0=h_{\mu }(f^0,q^0)$ are given by (18), (23), and (28)–(29) if $h_{\mu }\left(f^0\right)\in H_{{\mathcal{D}}_f}$, $h_{\mu }(f^0,g^0)\in H_{{\mathcal{D}}_f\times {\mathcal{D}}_g}$, and $h_{\mu }(f^0,q^0)\in H_{{\mathcal{D}}_f\times {\mathcal{D}}_q}$, respectively.

The MSC $h^0$ and the LFSDs $f^0$, $(f^0,g^0)$, or $(f^0,q^0)$ form the saddle points of the functions ${\sigma }_{\alpha }(h;f)$, ${\sigma }_{\alpha }(h;f,g)$, or ${\sigma }_{\alpha }(h;f,q)$ on the set $H_{\mathcal{D}}\times \mathcal{D}$.

The saddle point inequalities

$${\sigma }_{\alpha }(h;f^0)\ge {\sigma }_{\alpha }(h^0;f^0)\ge {\sigma }_{\alpha }(h^0;f)\forall f\in {\mathcal{D}}_f,\forall h\in H_{\mathcal{D}},$$

hold true if $h^0=h_{\mu }\left(f^0\right)$ and $h_{\mu }\in H_{\mathcal{D}}$, where $f^0$ is a solution to the constraint optimization problem

$$\widetilde{{\sigma }_{\alpha }}\left(f\right)=-{\sigma }_{\alpha }(h_{\mu }\left(f^0\right);f)\to \inf ,f\in {\mathcal{D}}_f,$$

(37)

where the functional ${\sigma }_{\alpha }(h_{\mu }\left(f^0\right);f)$ is given by

$${\sigma }_{\alpha }(h_{\mu }\left(f^0\right);f)={\int }_{-\pi }^{\pi }{\left|{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}{f}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{C}_{\mu ,N}^0{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }{\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda ,$$

and $C_{\mu ,N}^0\left(e^{i\lambda }\right)$ is derived from the set of Equations (19) for $f^0\left(\lambda \right)$.

The saddle point inequalities

$${\sigma }_{\alpha }(h;f^0,g^0)\ge {\sigma }_{\alpha }(h^0;f^0,g^0)\ge {\sigma }_{\alpha }(h^0;f,g)\forall f\in {\mathcal{D}}_f,\forall g\in {\mathcal{D}}_g,\forall h\in H_{\mathcal{D}}$$

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

$$\widetilde{{\sigma }_{\alpha }}(f,g)=-{\sigma }_{\alpha }(h_{\mu }(f^0,g^0);f,g)\to \inf ,(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g,$$

(38)

where the functional ${\sigma }_{\alpha }(h_{\mu }(f^0,g^0);f,g)$ is given by

$$\begin{array}{cc}\hfill {\sigma }_{\alpha }(h_{\mu }(f^0,g^0);f,g)& ={\int }_{-\pi }^{\pi }{\left|B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{\alpha }{\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda \hfill \\ & +{\int }_{-\pi }^{\pi }{\left|B_N^{\mu }\left(e^{i\lambda }\right)-{\displaystyle \frac{A_N\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{\alpha }{|1-e^{-i\lambda \mu }|}^{\alpha n}g\left(\lambda \right)d\lambda ,\hfill \end{array}$$

and $H_{\mu }^0\left(e^{i\lambda }\right)$ is derived from the set of Equations (24) for $(f^0\left(\lambda \right),g^0\left(\lambda \right))$.

The saddle point inequalities

$${\sigma }_{\alpha }(h;f^0,q^0)\ge {\sigma }_{\alpha }(h^0;f^0,q^0)\ge {\sigma }_{\alpha }(h^0;f,q)\forall f\in {\mathcal{D}}_f,\forall q\in {\mathcal{D}}_q,\forall h\in H_{\mathcal{D}}$$

hold true if $h^0=h_{\mu }(f^0,q^0)$ and $h_{\mu }\in H_{\mathcal{D}}$, where $(f^0,q^0)$ is a solution to the constraint optimization problem

$$\widetilde{{\sigma }_{\alpha }}(f,q)=-{\sigma }_{\alpha }(h_{\mu }(f^0,q^0);f,q)\to \inf ,(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q,$$

(39)

where the functional ${\sigma }_{\alpha }(h_{\mu }(f^0,q^0);f,q)$ is given by

$$\begin{array}{cc}\hfill & {\sigma }_{\alpha }(h_{\mu }(f^0,q^0);f,q)\hfill \\ & ={\int }_{-\pi }^{\pi }{\left|{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}{f}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{C}_{\mu ,N}^{0,1}{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }{\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda \hfill \\ & +{\int }_{-\pi }^{\pi }{\left|{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}{q}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{(C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)-C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right))}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }{\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}q\left(\lambda \right)d\lambda ,\hfill \end{array}$$

and $C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)$, $C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right)$ are derived from the set of Equations (30) and (31) for $(f^0\left(\lambda \right),q^0\left(\lambda \right))$.

The constrained optimization problems (37)–(39) are equivalent to the unconstrained optimization problems

$$\begin{array}{cc}\hfill {{\sigma }_{\alpha }}_{\mathcal{D}}\left(f\right)& =\widetilde{{\sigma }_{\alpha }}\left(f\right)+\delta \left(f\right|{\mathcal{D}}_f)\to \inf ,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {{\sigma }_{\alpha }}_{\mathcal{D}}(f,g)& =\widetilde{{\sigma }_{\alpha }}(f,g)+\delta (f,g|{\mathcal{D}}_f\times {\mathcal{D}}_g)\to \inf ,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {{\sigma }_{\alpha }}_{\mathcal{D}}(f,q)& =\widetilde{{\sigma }_{\alpha }}(f,q)+\delta (f,q|{\mathcal{D}}_f\times {\mathcal{D}}_q)\to \inf ,\hfill \end{array}$$

(42)

where $\delta \left(f\right|{\mathcal{D}}_f)$, $\delta (f,g|{\mathcal{D}}_f\times {\mathcal{D}}_g)$, and $\delta (f,q|{\mathcal{D}}_f\times {\mathcal{D}}_q)$ are the indicator functions of the sets ${\mathcal{D}}_f$, ${\mathcal{D}}_f\times {\mathcal{D}}_g$, and ${\mathcal{D}}_f\times {\mathcal{D}}_q$. The solution $f^0$, $(f^0,g^0)$, and $(f^0,q^0)$ to these unconstrained optimization problems are characterized by the condition $0\in \partial {{\sigma }_{\alpha }}_{\mathcal{D}}\left(f^0\right)$, $0\in \partial {{\sigma }_{\alpha }}_{\mathcal{D}}(f^0,g^0)$, and $0\in \partial {{\sigma }_{\alpha }}_{\mathcal{D}}(f^0,q^0)$, where $\partial {{\sigma }_{\alpha }}_{\mathcal{D}}\left(f^0\right)$, $\partial {{\sigma }_{\alpha }}_{\mathcal{D}}(f^0,g^0)$, and $\partial {{\sigma }_{\alpha }}_{\mathcal{D}}(f^0,q^0)$ are the subdifferentials of the functional ${{\sigma }_{\alpha }}_{\mathcal{D}}\left(f\right)$ at point $f^0\in {\mathcal{D}}_f$, the functional ${{\sigma }_{\alpha }}_{\mathcal{D}}(f,g)$ at point $(f^0,g^0)\in {\mathcal{D}}_f\times {\mathcal{D}}_g$, and the functional ${{\sigma }_{\alpha }}_{\mathcal{D}}(f,q)$ at point $(f^0,q^0)\in {\mathcal{D}}_f\times {\mathcal{D}}_q$ [30,31]. These conditions make it possible to find the least favourable spectral densities in some special classes of spectral densities ${\mathcal{D}}_f$, ${\mathcal{D}}_f\times {\mathcal{D}}_g$, and ${\mathcal{D}}_f\times {\mathcal{D}}_q$.

5. Examples of Minimax Estimation for Specific Classes of Admissible Spectral Densities

5.1. Least Favorable Spectral Density in the Class ${\mathcal{D}}_f$

Consider the estimation problem without noise for the functional $A_N\xi ={\sum }_{k=0}^{N}a\left(k\right)\xi \left(k\right)$ which depends on unobserved values of the sequence $\xi \left(m\right)$ with $HS\alpha S$ increments based on its observations $\left\{\xi \right(m):m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\left\}\right\}$, where the set of admissible spectral densities is defined as follows:

$${\mathcal{D}}_f=\left\{f\left(\lambda \right)|{\int }_{-\pi }^{\pi }{\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda =P\right\}.$$

Suppose that the linear functional ${\sigma }_{\alpha }(h_{\mu }\left(f^0\right);f)$ with respect to the variable f is bounded in the space $L_1$. Applying the Lagrange method of indefinite multipliers to optimization problem (40), we determine that the spectral density $f^0\in {\mathcal{D}}_f$ satisfies the equation

$${\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}{f}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{\alpha }{\alpha -1}}}{\left|C_{\mu ,N}^0\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{\alpha }{\alpha -1}}}=\gamma .$$

From this equation we find that the least favourable spectral density is of the form

$$f^0\left(\lambda \right)={\gamma }^{{\displaystyle \frac{1-\alpha }{\alpha }}}{\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}}}\left|C_{\mu ,N}^0\left(e^{i\lambda }\right)\right|.$$

(43)

Thus, we have the following statement.

Theorem 4.

Let $f^0\left(\lambda \right)\in {\mathcal{D}}_f$ be determined by Equation (43), where $C_{\mu ,N}^0\left(e^{i\lambda }\right)$ is derived from system of Equations (19). Suppose that it satisfies minimality condition (11), the restriction on the densities in the class ${\mathcal{D}}_f$, and yields a solution to optimization problem (37). Then, $f^0\left(\lambda \right)$ is the LFSD in the class ${\mathcal{D}}_f$ for the optimal linear estimation of $A_N\xi$ based on observations $\xi \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$. The MSC $h_{\mu }\left(f^0\right)$ is determined by Formula (18).

5.2. Least Favorable Spectral Densities in the Class ${\mathcal{D}}_f^{\beta }\times {\mathcal{D}}_g^{\beta }$

Consider the estimation problem with noise for the functional $A_N\xi ={\sum }_{k=0}^{N}a\left(k\right)\xi \left(k\right)$ which depends on unobserved values of the sequence $\xi \left(m\right)$ with $HS\alpha S$ increments based on observations $\left\{\xi \right(m)+\eta (m):m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\left\}\right\}$, where $\eta \left(m\right)$ is a $HS\alpha S$ sequence independent of $\xi \left(m\right)$. Let the set $\mathcal{D}$ of admissible spectral densities be defined as follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_f^{\beta }& =\left\{f\left(\lambda \right)|{\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)\right)}^{{\beta }_1}d\lambda =P_1\right\},\hfill \\ \hfill {\mathcal{D}}_g^{\beta }& =\left\{g\left(\lambda \right)|{\int }_{-\pi }^{\pi }{\left(g\left(\lambda \right)\right)}^{{\beta }_2}d\lambda =P_2\right\},\hfill \end{array}$$

where ${\beta }_1\ne 1$ and ${\beta }_2\ne 1$.

Remark 2.

The definition of the set ${\mathcal{D}}_g^{\beta }$ doesn’t contain the multiplier ${\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}$, in contrast to the set ${\mathcal{D}}_f^{\beta }$. The difference comes from the fact that the noise sequence $\eta \left(m\right)$ is an $HS\alpha S$ process, while $\xi \left(m\right)$ is a sequence with $HS\alpha S$ increments. In the next subsection dedicated to the semi-noise problem, when $\xi \left(m\right)$ and $\theta \left(m\right)$ are sequences with $HS\alpha S$ increments, the definitions of both sets ${\mathcal{D}}_f^{\beta }$ and ${\mathcal{D}}_q^{\beta }$ contain such a multiplier.

Suppose that the linear functional ${\sigma }_{\alpha }(h_{\mu }(f^0,g^0);f,g)$ with respect to the variable $(f,g)$ is bounded in the space $L_1$. Applying the Lagrange method of indefinite multipliers to optimization problem (41), we determine that the spectral densities $f^0\in {\mathcal{D}}_f^{\beta }$ and $g^0\in {\mathcal{D}}_g^{\beta }$ satisfy the equations

$${\left|B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{\alpha }={\gamma }_1{\left({\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}{f}^0\left(\lambda \right)\right)}^{{\beta }_1-1}$$

and

$${\left|B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{\alpha }{|1-e^{-i\lambda \mu }|}^{\alpha n}={\gamma }_2{\left(g^0\left(\lambda \right)\right)}^{{\beta }_2-1}.$$

From these equations we find that the least favourable spectral densities are of the form

$$\begin{array}{cc}\hfill f^0\left(\lambda \right)& ={\gamma }_1^{{\displaystyle \frac{-1}{{\beta }_1-1}}}{\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}}}{\left|B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{\alpha }{{\beta }_1-1}}},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill g^0\left(\lambda \right)& ={\gamma }_2^{{\displaystyle \frac{-1}{{\beta }_2-1}}}{|1-e^{-i\lambda \mu }|}^{{\displaystyle \frac{\alpha n}{{\beta }_2-1}}}{\left|B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{\alpha }{{\beta }_2-1}}}.\hfill \end{array}$$

(45)

Thus, we have the following statement.

Theorem 5.

Let $f^0\left(\lambda \right)\in {\mathcal{D}}_f^{\beta }$ and $g^0\left(\lambda \right)\in {\mathcal{D}}_g^{\beta }$ be determined by Equations (44) and (45), where $H_{\mu }^0\left(e^{i\lambda }\right)$ is derived from system of Equations (24). Suppose that they satisfy minimality condition (12), the restrictions on the densities in the classes ${\mathcal{D}}_f^{\beta }$ and ${\mathcal{D}}_g^{\beta }$, and yield a solution to optimization problem (38). Then, $f^0\left(\lambda \right)$ and $g^0\left(\lambda \right)$ are the LFSDs in the class ${\mathcal{D}}_f^{\beta }\times {\mathcal{D}}_g^{\beta }$ for the optimal linear estimation of $A_N\xi$ based on observations $\xi \left(m\right)+\eta \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$. The MSC $h_{\mu }(f^0,g^0)$ is determined by Formula (23).

5.3. Least Favorable Spectral Densities in the Class ${\mathcal{D}}_f^{\beta }\times {\mathcal{D}}_q^{\beta }$

Consider the semi-noise estimation problem for the functional $A_N\xi ={\sum }_{k=0}^{N}a\left(k\right)\xi \left(k\right)$ which depends on unobserved values of the sequence $\xi \left(m\right)$ with $HS\alpha S$ increments based on observations $\left\{\xi \right(m):m\le -1\}\cup \left\{\xi \right(m)+\theta (m):m\ge N+1\}$, where $\theta \left(m\right)$ is a sequence with $HS\alpha S$ increments independent of $\xi \left(m\right)$. Let the set $\mathcal{D}$ of admissible spectral densities be defined as follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_f^{\beta }& =\left\{f\left(\lambda \right)|{\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)\right)}^{{\beta }_1}d\lambda =P_1\right\},\hfill \\ \hfill {\mathcal{D}}_q^{\beta }& =\left\{q\left(\lambda \right)|{\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}q\left(\lambda \right)\right)}^{{\beta }_2}d\lambda =P_2\right\},\hfill \end{array}$$

where ${\beta }_1\ne -1/(\alpha -1)$ and ${\beta }_2\ne -1/(\alpha -1)$.

Suppose that the linear functional ${\sigma }_{\alpha }(h_{\mu }(f^0,q^0);f,q)$ with respect to the variable $(f,q)$ is bounded in the space $L_1$. Applying the Lagrange method of indefinite multipliers to optimization problem (42), we obtain that the spectral densities $f^0\in {\mathcal{D}}_f^{\beta }$, $q^0\in {\mathcal{D}}_q^{\beta }$ satisfy the equations

$${\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}{f}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{\alpha }{\alpha -1}}}{\left|C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{\alpha }{\alpha -1}}}={\gamma }_1{\left({\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}{f}^0\left(\lambda \right)\right)}^{{\beta }_1-1}$$

and

$${\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}{q}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{\alpha }{\alpha -1}}}{|C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)-C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right)|}^{{\displaystyle \frac{\alpha }{\alpha -1}}}={\gamma }_2{\left({\displaystyle \frac{{|1-e^{-i\lambda \mu }|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}{q}^0\left(\lambda \right)\right)}^{{\beta }_2-1}.$$

From these equations, under the conditions ${\beta }_1\ne -1/(\alpha -1)$ and ${\beta }_2\ne -1/(\alpha -1)$, we find that the least favourable spectral densities are of the form

$$\begin{array}{cc}\hfill f^0\left(\lambda \right)& ={\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}}}{\gamma }_1^{{\displaystyle \frac{\alpha -1}{-\alpha -(\alpha -1)({\beta }_1-1)}}}{\left|C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{-\alpha }{-\alpha -(\alpha -1)({\beta }_1-1)}}},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill q^0\left(\lambda \right)& ={\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{{|1-e^{-i\lambda \mu }|}^{\alpha n}}}{\gamma }_2^{{\displaystyle \frac{\alpha -1}{-\alpha -(\alpha -1)({\beta }_2-1)}}}{|C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)-C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right)|}^{{\displaystyle \frac{-\alpha }{-\alpha -(\alpha -1)({\beta }_2-1)}}}.\hfill \end{array}$$

(47)

The conditions ${\beta }_1\ne -1/(\alpha -1)$ and ${\beta }_2\ne -1/(\alpha -1)$ guarantee that the denominators of the fractions ${\displaystyle \frac{\alpha -1}{-\alpha -(\alpha -1)({\beta }_1-1)}}$ and ${\displaystyle \frac{\alpha -1}{-\alpha -(\alpha -1)({\beta }_2-1)}}$ are nonzero.

Thus, we have the following statement.

Theorem 6.

Let $f^0\left(\lambda \right)\in {\mathcal{D}}_f^{\beta }$ and $q^0\left(\lambda \right)\in {\mathcal{D}}_q^{\beta }$ be determined by Equations (46) and (47), where $C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)$ and $C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right)$ are derived from system of Equations (30) and (31). Suppose that they satisfy minimality conditions (11), the restrictions on the densities in the classes ${\mathcal{D}}_f^{\beta }$ and ${\mathcal{D}}_q^{\beta }$, and yield a solution to optimization problem (39). Then, $f^0\left(\lambda \right)$ and $q^0\left(\lambda \right)$ are the LFSDs in the class ${\mathcal{D}}_f^{\beta }\times {\mathcal{D}}_q^{\beta }$ for the optimal linear estimation of $A_N\xi$ based on observations $\xi \left(m\right)$ at points $m\le -1$ and $\xi \left(m\right)+\theta \left(m\right)$ at points $m\ge N+1$. The MSC $h_{\mu }(f^0,q^0)$ is determined by Formulas (28) and (29).

6. Discussion

In this paper, the results of an investigation of the classical and minimax estimation problems for a class of stochastic sequences with harmonizable symmetric $\alpha$-stable nth increments are presented.

Extrapolation, interpolation and filtering problems were previously investigated for the linear functionals from the harmonizable symmetric $\alpha$-stable stochastic sequence and from the stochastic sequence with stationary increments. Particularly, the interpolation problem for the $HS\alpha S$ has been studied in the paper by Moklyachuk and Ostapenko [17]. The interpolation problem for the stationary nth increment sequences with stationary noise has been studied in the paper by Luz and Moklyachuk [24]. Comprehensive results for the estimation problems for the stationary increment sequences and processes were published in our book [25]. Thus, the novelty current work consists of extending our previous results in the random processes estimation theory to a new class of stochastic sequences, which combines two stated patterns of non-stationarity.

Future research directions will include extrapolation and filtering problems for the considered sequences as well as estimation of functionals of continuous time random processes with harmonizable symmetric $\alpha$-stable nth increments.