# Explicit formulas for the inverses of Toeplitz matrices, with applications

AKIHIKO INOUE  A. Inoue
Department of Mathematics
Hiroshima University
Higashi-Hiroshima 739-8526
Japan<EMAIL_ADDRESS>
###### Abstract.

We derive novel explicit formulas for the inverses of truncated block Toeplitz
matrices that correspond to a multivariate minimal stationary process. The
main ingredients of the formulas are the Fourier coefficients of the phase
function attached to the spectral density of the process. The derivation of
the formulas is based on a recently developed finite prediction theory applied
to the dual process of the stationary process. We illustrate the usefulness of
the formulas by two applications. The first one is a strong convergence result
for solutions of general block Toeplitz systems for a multivariate short-
memory process. The second application is closed-form formulas for the
inverses of truncated block Toeplitz matrices corresponding to a multivariate
ARMA process. The significance of the latter is that they provide us with a
linear-time algorithm to compute the solutions of corresponding block Toeplitz
systems.

###### Key words and phrases:

Toeplitz matrix, Toeplitz system, explicit formula, closed-form formula,
linear-time algorithm.

###### 2010 Mathematics Subject Classification:

Primary 60G10; secondary 65F05, 60G25, 15B05.

##  1\. Introduction

Let $\mathbb{T}:=\\{z\in\mathbb{C}:|z|=1\\}$ be the unit circle in
$\mathbb{C}$. We write $\sigma$ for the normalized Lebesgue measure
$d\theta/(2\pi)$ on $([-\pi,\pi),\mathcal{B}([-\pi,\pi)))$, where
$\mathcal{B}([-\pi,\pi))$ is the Borel $\sigma$-algebra on $[-\pi,\pi)$; thus
we have $\sigma([-\pi,\pi))=1$. For $p\in[1,\infty)$, we write
$L_{p}(\mathbb{T})$ for the Lebesgue space of measurable functions
$f:\mathbb{T}\to\mathbb{C}$ such that $\|f\|_{p}<\infty$, where
$\|f\|_{p}:=\\{\int_{-\pi}^{\pi}|f(e^{i\theta})|^{p}\sigma(d\theta)\\}^{1/p}$.
Let $L_{p}^{m\times n}(\mathbb{T})$ be the space of $\mathbb{C}^{m\times
n}$-valued functions on $\mathbb{T}$ whose entries belong to
$L_{p}(\mathbb{T})$.

Let $d\in\mathbb{N}$. For $n\in\mathbb{N}$, we consider the block Toeplitz
matrix

$T_{n}(w):=\left(\begin{matrix}\gamma(0)&\gamma(-1)&\cdots&\gamma(-n+1)\cr\gamma(1)&\gamma(0)&\cdots&\gamma(-n+2)\cr\vdots&\vdots&\ddots&\vdots\cr\gamma(n-1)&\gamma(n-2)&\cdots&\gamma(0)\end{matrix}\right)\in\mathbb{C}^{dn\times
dn},$

where

(1.1)
$\gamma(k):=\int_{-\pi}^{\pi}e^{-ik\theta}w(e^{i\theta})\frac{d\theta}{2\pi}\in\mathbb{C}^{d\times
d},\qquad k\in\mathbb{Z},$

and the symbol $w$ satisfies the following two conditions:

(1.2) $w\in L^{d\times d}_{1}(\mathbb{T})$ and $w(e^{i\theta})$ is a positive
Hermitian matrix $\sigma$-a.e., (1.3) $\displaystyle w^{-1}\in L^{d\times
d}_{1}(\mathbb{T}).$

Let $\\{X_{k}:k\in\mathbb{Z}\\}$ be a $\mathbb{C}^{d}$-valued, centered,
weakly stationary process that has spectral density $w$, hence autocovariance
function $\gamma$. Then the conditions (1.2) and (1.3) imply that
$\\{X_{k}\\}$ is minimal (see Section 10 of [21, Chapter II]).

In this paper, we show novel explicit formulas for $T_{n}(w)^{-1}$ (Theorem
2.1), which are especially useful for large $n$ (see [2]). The formulas are
new even for $d=1$. The main ingredients of the formulas are the Fourier
coefficients of $h^{*}h_{\sharp}^{-1}=h^{-1}h_{\sharp}^{*}$, where $h$ and
$h_{\sharp}$ are $\mathbb{C}^{d\times d}$-valued outer functions on
$\mathbb{T}$ such that

(1.4)
$w(e^{i\theta})=h(e^{i\theta})h(e^{i\theta})^{*}=h_{\sharp}(e^{i\theta})^{*}h_{\sharp}(e^{i\theta}),\qquad\qquad\mbox{$\sigma$-a.e.}$

(see [10]; see also Section 2). We note that the unitary matrix valued
function $h^{*}h_{\sharp}^{-1}=h^{-1}h_{\sharp}^{*}$ on $\mathbb{T}$ attached
to $w$ is called the phase function of $w$ (see page 428 in [20]).

Let $\\{X_{k}\\}$ be as above, and let $\\{X^{\prime}_{k}:k\in\mathbb{Z}\\}$
be the dual process of $\\{X_{k}\\}$ (see [19]; see also Section 2 below). In
the proof of the above explicit formulas for $T_{n}(w)^{-1}$, the dual process
$\\{X^{\prime}_{k}\\}$ plays an important role. In fact, the key to the proof
of the explicit formulas for $T_{n}(w)^{-1}$ is the following equality
(Theorem 3.1):

(1.5) $\left(T_{n}(w)^{-1}\right)^{s,t}=\langle
X^{\prime}_{s},P_{[1,n]}X^{\prime}_{t}\rangle,\qquad s,t\in\\{1,\dots,n\\}.$

Here, $\langle\cdot,\cdot\rangle$ stands for the Gram matrix (see Section 3)
and $P_{[1,n]}X^{\prime}_{t}$ denotes the best linear predictor of
$X^{\prime}_{t}$ based on the observations $X_{1},\dots,X_{n}$ (see Section 2
for the precise definition). Moreover, for $n\in\mathbb{N}$,
$A\in\mathbb{C}^{dn\times dn}$ and $s,t\in\\{1,\dots,n\\}$, we write
$A^{s,t}\in\mathbb{C}^{d\times d}$ for the $(s,t)$ block of $A$; thus
$A=(A^{s,t})_{1\leq s,t\leq n}$. The equality (1.5) enables us to apply the
$P_{[1,n]}$-related methods developed in [11], [12], [14], [15], [16] and
others to derive the explicit formulas for $T_{n}(w)^{-1}$.

We illustrate the usefulness of the explicit formulas for $T_{n}(w)^{-1}$ by
two applications. The first one is a strong convergence result for solutions
of block Toeplitz systems. For this application, we assume (1.2) as well as
the following condition:

(1.6) $\sum_{k=-\infty}^{\infty}\|\gamma(k)\|<\infty$ and
$\min_{z\in\mathbb{T}}\det w(z)>0$.

Here, for $a\in\mathbb{C}^{d\times d}$, $\|a\|$ denotes the operator norm of
$a$. The condition (1.6) implies that $\\{X_{k}\\}$ with spectral density $w$
is a short-memory process. We note that (1.3) follows from (1.2) and (1.6)
(see Section 4). Under (1.2) and (1.6), for $n\in\mathbb{N}$ and a
$\mathbb{C}^{d\times d}$-valued sequence $\\{y_{k}\\}_{k=1}^{\infty}$ such
that $\sum_{k=1}^{\infty}\|y_{k}\|<\infty$, let

(1.7)
$Z_{n}=(z_{n,1}^{\top},\dots,z_{n,n}^{\top})^{\top}\in\mathbb{C}^{dn\times d}\
\ \mbox{with}\ \ z_{n,k}\in\mathbb{C}^{d\times d},\ k\in\\{1,\dots,n\\},$

be the solution to the block Toeplitz system

(1.8) $T_{n}(w)Z_{n}=Y_{n},$

where

(1.9) $Y_{n}:=(y_{1}^{\top},\dots,y_{n}^{\top})^{\top}\in\mathbb{C}^{dn\times
d}.$

Also, let

(1.10) $Z_{\infty}=(z_{1}^{\top},z_{2}^{\top},\dots)^{\top}\ \ \mbox{with}\ \
z_{k}\in\mathbb{C}^{d\times d},\ k\in\mathbb{N},$

be the solution to the corresponding infinite block Toeplitz system

(1.11) $T_{\infty}(w)Z_{\infty}=Y_{\infty},$

where

(1.12)
$T_{\infty}(w):=\left(\begin{matrix}\gamma(0)&\gamma(-1)&\gamma(-2)&\cdots\cr\gamma(1)&\gamma(0)&\gamma(-1)&\cdots\cr\gamma(2)&\gamma(1)&\gamma(0)&\cdots\cr\vdots&\vdots&\vdots&\ddots\end{matrix}\right)$

and

(1.13) $Y_{\infty}:=(y_{1}^{\top},y_{2}^{\top},\dots)^{\top}.$

Then, our result (Theorem 4.1) reads as follows:

(1.14) $\lim_{n\to\infty}\sum_{k=1}^{n}\|z_{n,k}-z_{k}\|=0.$

We explain the background of the result (1.14). As above, let
$\\{X_{k}:k\in\mathbb{Z}\\}$ be a $\mathbb{C}^{d}$-valued, centered, weakly
stationary process that has spectral density $w$. For $n\in\mathbb{N}$, the
finite and infinite predictor coefficients $\phi_{n,k}\in\mathbb{C}^{d\times
d}$, $k\in\\{1,\dots,n\\}$, and $\phi_{k}$, $k\in\mathbb{N}$, of $\\{X_{k}\\}$
are defined by

$P_{[1,n]}X_{n+1}=\sum_{k=1}^{n}\phi_{n,k}X_{n+1-k}\qquad\mbox{and}\qquad
P_{(-\infty,n]}X_{n+1}=\sum_{k=1}^{\infty}\phi_{k}X_{n+1-k},$

respectively; see Section 3 for the precise definitions of $P_{[1,n]}$ and
$P_{(-\infty,n]}$. We note that $\sum_{k=1}^{\infty}\|\phi_{k}\|<\infty$ holds
under (1.2) and (1.6) (see Section 4 below and (2.16) in [16]). Baxter’s
inequality in [1], [5], [9] states that, under (1.2) and (1.6), there exists
$K\in(0,\infty)$ such that

(1.15) $\sum_{k=1}^{n}\|\phi_{n,k}-\phi_{k}\|\leq
K\sum_{k=n+1}^{\infty}\|\phi_{k}\|,\qquad n\in\mathbb{N}.$

In particular, we have

(1.16) $\lim_{n\to\infty}\sum_{k=1}^{n}\|\phi_{n,k}-\phi_{k}\|=0.$

If we put $\tilde{w}(e^{i\theta}):=w(e^{-i\theta})$, then,
$(\phi_{n,1},\dots,\phi_{n,n})$ is the solution to the block Toeplitz system

$T_{n}(\tilde{w})(\phi_{n,1},\dots,\phi_{n,n})^{*}=(\gamma(1),\dots,\gamma(n))^{*},$

called the Yule–Walker equation, while $(\phi_{1},\phi_{2},\dots)$ is the
solution to the corresponding infinite block Toeplitz system

$T_{\infty}(\tilde{w})(\phi_{1},\phi_{2},\dots)^{*}=(\gamma(1),\gamma(2),\dots)^{*}.$

Clearly, $\tilde{w}$ satisfies (1.2) and (1.6) since so does $w$. Therefore,
our result (1.14) can be viewed as an extension to (1.16). It should be noted,
however, that we prove (1.14) directly, without proving an analogue of
Baxter’s inequality (1.15).

The convergence result (1.16) has various applications in time series
analysis, such as the autoregressive sieve bootstrap (see, e.g., [16] and the
references therein), whille Toeplitz systems of the form (1.8) appear in
various fields, such as filtering of signals. Therefore the extension (1.14),
as well as the other results explained below, may potentially be useful in
such fields. We note that Baxter’s inequality (1.15), hence (1.16), is also
proved for univariate and multivariate FARIMA (fractional autoregressive
integrated moving-average) processes, which are long-memory processes, in [14]
and [16], respectively. The FARIMA processes have singular spectral densities
$w$ but our explicit formulas for $T_{n}(w)^{-1}$ above also cover them since
we only assume minimality in the formulas. Applications of the explicit
formulas to univariate and multivariate FARIMA processes will be discussed
elsewhere. However, the problem of proving results of the type (1.14) for
FARIMA processes remains unsolved so far.

The second application of the explicit formulas for $T_{n}(w)^{-1}$ is closed-
form formulas for $T_{n}(w)^{-1}$ with rational $w$ that corresponds to a
univariate ($d=1$) or multivariate ($d\geq 2$) ARMA (autoregressive moving-
average) process (Theorem 5.2). More precisely, we assume that $w$ is of the
form

(1.17)
$w(e^{i\theta})=h(e^{i\theta})h(e^{i\theta})^{*},\qquad\theta\in[-\pi,\pi),$

where $h:\mathbb{T}\to\mathbb{C}^{d\times d}$ satisfies the following
condition:

(1.18) the entries of $h(z)$ are rational functions in $z$ that have no poles
in $\overline{\mathbb{D}}$, and $\det h(z)$ has no zeros in
$\overline{\mathbb{D}}$.

Here $\overline{\mathbb{D}}:=\\{z\in\mathbb{C}:|z|\leq 1\\}$ is the closed
unit disk in $\mathbb{C}$. The closed-form formulas for $T_{n}(w)^{-1}$
consist of several building block matrices that are of fixed sizes independent
of $n$. The significance of the formulas for $T_{n}(w)^{-1}$ is that they
provide us with a linear-time, or $O(n)$, algorithm to compute the solution
$Z\in\mathbb{C}^{dn\times d}$ to the block Toeplitz system

(1.19) $T_{n}(w)Z=Y$

for $Y\in\mathbb{C}^{dn\times d}$ (see Section 6). The famous Durbin–Levinson
algorithm solves the equation (1.19) for more general $w$ in $O(n^{2})$ time.
Algorithms for Toeplitz linear systems that run faster than $O(n^{2})$ are
called superfast. While our algorithm is restricted to the class of $w$
corresponding to ARMA processes, the class is important in applications, and
the linear-time algorithm is ideally superfast in the sense that there is no
algorithm faster than $O(n)$.

Toeplitz matrices appear in a variety of fields, including operator theory,
orthogonal polynomials on the unit circle, time series analysis, engineering,
and physics. Therefore, there is a vast amount of literature on Toeplitz
matrices. Here, we refer to [2], [3], [6], [8], [22], [23] and [24] for a
textbook treatment. For example, in [6, III], the Gohberg-Semencul formulas in
[7], which express the inverse of a Toeplitz matrix as a difference of
products of lower and upper triangular Toeplitz matrices, are explained.

After this work was completed, the author learned of [25] by Subba Rao and
Yang, where they also provide an explicit series expansion for $T_{n}(w)^{-1}$
that corresponds to a univariate stationary process satisfying some conditions
(see [25], Section 3.2). The main aim of [25] is to reconcile the Gaussian and
Whittle likelihood, and the series expansion in [25] is tailored to this
purpose, using the complete DFT (discrete Fourier transform) introduced in
[25]. It should be noticed that $T_{n}(w)^{-1}$ appears in the Gaussian
likelihood, while the Whittle likelihood is based on the ordinary DFT. Since
most results of the present paper directly concern $T_{n}(w)^{-1}$, some of
them may also be useful for studies related to the Gaussian likelihood.

This paper is organized as follows. We state the explicit formulas for
$T_{n}(w)^{-1}$ in Section 2. In Section 3, we first prove (1.5) and then use
it to prove the explicit formulas for $T_{n}(w)^{-1}$. In Section 4, we prove
(1.14) for $w$ satisfying (1.2) and (1.6), using the explicit formulas for
$T_{n}(w)^{-1}$. In Section 5, we prove the closed-form formulas for
$T_{n}(w)^{-1}$ with $w$ satisfying (1.18), using the explicit formulas for
$T_{n}(w)^{-1}$. In Section 6, we explain how the results in Section 5 give a
linear-time algorithm to compute the solution to (1.19). Finally, the Appendix
contains the omitted proofs of two lemmas.

##  2\. Explicit formulas

Let $\mathbb{C}^{m\times n}$ be the set of all complex $m\times n$ matrices;
we write $\mathbb{C}^{d}$ for $\mathbb{C}^{d\times 1}$. Let $I_{n}$ be the
$n\times n$ unit matrix. For $a\in\mathbb{C}^{m\times n}$, $a^{\top}$ denotes
the transpose of $a$, and $\overline{a}$ and $a^{*}$ the complex and Hermitian
conjugates of $a$, respectively; thus, in particular,
$a^{*}:=\overline{a}^{\top}$. For $a\in\mathbb{C}^{d\times d}$, we write
$\|a\|$ for the operator norm of $a$:

$\|a\|:=\sup_{u\in\mathbb{C}^{d},|u|\leq 1}|au|.$

Here $|u|:=(\sum_{i=1}^{d}|u^{i}|^{2})^{1/2}$ denotes the Euclidean norm of
$u=(u^{1},\dots,u^{d})^{\top}\in\mathbb{C}^{d}$. For $p\in[1,\infty)$ and
$K\subset\mathbb{Z}$, $\ell_{p}^{d\times d}(K)$ denotes the space of
$\mathbb{C}^{d\times d}$-valued sequences $\\{a_{k}\\}_{k\in K}$ such that
$\sum_{k\in K}\|a_{k}\|^{p}<\infty$. We write $\ell_{p+}^{d\times d}$ for
$\ell_{p}^{d\times d}(\mathbb{N}\cup\\{0\\})$ and $\ell_{p+}$ for
$\ell_{p+}^{1\times 1}=\ell_{p}^{1\times 1}(\mathbb{N}\cup\\{0\\})$.

Recall $\sigma$ from Section 1. The Hardy class $H_{2}(\mathbb{T})$ on
$\mathbb{T}$ is the closed subspace of $L_{2}(\mathbb{T})$ consisting of $f\in
L_{2}(\mathbb{T})$ such that
$\int_{-\pi}^{\pi}e^{im\theta}f(e^{i\theta})\sigma(d\theta)=0$ for
$m=1,2,\dots$. Let $H_{2}^{m\times n}(\mathbb{T})$ be the space of
$\mathbb{C}^{m\times n}$-valued functions on $\mathbb{T}$ whose entries belong
to $H_{2}(\mathbb{T})$. Let $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ be the
open unit disk in $\mathbb{C}$. We write $H_{2}(\mathbb{D})$ for the Hardy
class on $\mathbb{D}$, consisting of holomorphic functions $f$ on $\mathbb{D}$
such that
$\sup_{r\in[0,1)}\int_{-\pi}^{\pi}|f(re^{i\theta})|^{2}\sigma(d\theta)<\infty$.
As usual, we identify each function $f$ in $H_{2}(\mathbb{D})$ with its
boundary function $f(e^{i\theta}):=\lim_{r\uparrow 1}f(re^{i\theta})$,
$\sigma$-a.e., in $H_{2}(\mathbb{T})$. A function $h$ in $H_{2}^{d\times
d}(\mathbb{T})$ is called outer if $\det h$ is a $\mathbb{C}$-valued outer
function, that is, $\det h$ satisfies $\log|\det
h(0)|=\int_{-\pi}^{\pi}\log|\det h(e^{i\theta})|\sigma(d\theta)$ (see
Definition 3.1 in [18]).

We assume that $w$ satisfies (1.2) and (1.3). Then $\log\det w$ is in
$L_{1}(\mathbb{T})$ (see Section 3 in [16]). Therefore $w$ has the
decompositions (1.4) for two outer functions $h$ and $h_{\sharp}$ belonging to
$H_{2}^{d\times d}(\mathbb{T})$, and $h$ and $h_{\sharp}$ are unique up to
constant unitary factors (see Chapter II in [21] and Theorem 11 in [10]; see
also Section 3 in [16]). We may take $h_{\sharp}=h$ for the case $d=1$ but
there is no such simple relation between $h$ and $h_{\sharp}$ for $d\geq 2$.
We define the outer function $\tilde{h}$ in $H_{2}^{d\times d}(\mathbb{T})$ by

(2.1) $\tilde{h}(z):=\\{h_{\sharp}(\overline{z})\\}^{*}.$

All of $h^{-1}$, $h_{\sharp}^{-1}$ and $\tilde{h}^{-1}$ also belong to
$H_{2}^{d\times d}(\mathbb{T})$ since we have assumed (1.3).

We define four $\mathbb{C}^{d\times d}$-valued sequences $\\{c_{k}\\}$,
$\\{a_{k}\\}$, $\\{\tilde{c}_{k}\\}$ and $\\{\tilde{a}_{k}\\}$ by

(2.2) $\displaystyle h(z)$ $\displaystyle=\sum_{k=0}^{\infty}z^{k}c_{k},\qquad
z\in\mathbb{D},$ (2.3) $\displaystyle-h(z)^{-1}$
$\displaystyle=\sum_{k=0}^{\infty}z^{k}a_{k},\qquad z\in\mathbb{D},$ (2.4)
$\displaystyle\tilde{h}(z)$
$\displaystyle=\sum_{k=0}^{\infty}z^{k}\tilde{c}_{k},\qquad z\in\mathbb{D},$

and

(2.5) $-\tilde{h}(z)^{-1}=\sum_{k=0}^{\infty}z^{k}\tilde{a}_{k},\qquad
z\in\mathbb{D},$

respectively. By (1.3), all of $\\{c_{k}\\}$, $\\{a_{k}\\}$,
$\\{\tilde{c}_{k}\\}$ and $\\{\tilde{a}_{k}\\}$ belong to $\ell_{2+}^{d\times
d}$.

We define a $\mathbb{C}^{d\times d}$-valued sequence
$\\{\beta_{k}\\}_{k=-\infty}^{\infty}$ as the (minus of the) Fourier
coefficients of the phase function
$h^{*}h_{\sharp}^{-1}=h^{-1}h_{\sharp}^{*}$:

(2.6) $\displaystyle\beta_{k}$
$\displaystyle=-\int_{-\pi}^{\pi}e^{-ik\theta}h(e^{i\theta})^{*}h_{\sharp}(e^{i\theta})^{-1}\frac{d\theta}{2\pi}$
$\displaystyle=-\int_{-\pi}^{\pi}e^{-ik\theta}h(e^{i\theta})^{-1}h_{\sharp}(e^{i\theta})^{*}\frac{d\theta}{2\pi},\qquad
k\in\mathbb{Z}.$

For $n\in\mathbb{N}$, $u\in\\{1,\dots,n\\}$ and $k\in\mathbb{N}$, we can
define the sequences
$\\{b_{n,u,\ell}^{k}\\}_{\ell=0}^{\infty}\in\ell_{2+}^{d\times d}$ by the
recursion

(2.7) $\left\\{\begin{aligned} b_{n,u,\ell}^{1}&=\beta_{u+\ell},\\\
b_{n,u,\ell}^{2k}&=\sum_{m=0}^{\infty}b_{n,u,m}^{2k-1}\beta_{n+1+m+\ell}^{*},\qquad
b_{n,u,\ell}^{2k+1}=\sum_{m=0}^{\infty}b_{n,u,m}^{2k}\beta_{n+1+m+\ell}\end{aligned}\right.$

(see Section 3 below). Similarly, for $n\in\mathbb{N}$, $u\in\\{1,\dots,n\\}$
and $k\in\mathbb{N}$, we can define the sequences
$\\{\tilde{b}_{n,u,\ell}^{k}\\}_{\ell=0}^{\infty}\in\ell_{2+}^{d\times d}$ by
the recursion

(2.8) $\left\\{\begin{aligned}
\tilde{b}_{n,u,\ell}^{1}&=\beta_{n+1-u+\ell}^{*},\\\
\tilde{b}_{n,u,\ell}^{2k}&=\sum_{m=0}^{\infty}\tilde{b}_{n,u,m}^{2k-1}\beta_{n+1+m+\ell},\qquad\tilde{b}_{n,u,\ell}^{2k+1}=\sum_{m=0}^{\infty}\tilde{b}_{n,u,m}^{2k}\beta_{n+1+m+\ell}^{*}.\end{aligned}\right.$

Recall from Section 1 that $(T_{n}(w)^{-1})^{s,t}$ denotes the $(s,t)$ block
of $T_{n}(w)^{-1}$. Since $T_{n}(w)$, hence $T_{n}(w)^{-1}$, is self-adjoint,
we have

(2.9) $(T_{n}(w)^{-1})^{s,t}=((T_{n}(w)^{-1})^{t,s})^{*},\qquad
s,t\in\\{1,\dots,n\\}.$

We use the following notation:

$s\vee t:=\max(s,t),\qquad s\wedge t:=\min(s,t).$

We are ready to state the explicit formulas for $(T_{n}(w))^{-1}$.

######  Theorem 2.1.

We assume (1.2) and (1.3). Then the following two assertions hold.

  1. (i)

For $n\in\mathbb{N}$ and $s,t\in\\{1,\dots,n\\}$, we have

(2.10) $\displaystyle\left(T_{n}(w)^{-1}\right)^{s,t}=\sum_{\ell=1}^{s\wedge
t}\tilde{a}_{s-\ell}^{*}\tilde{a}_{t-\ell}$
$\displaystyle\qquad+\sum_{u=1}^{t}\sum_{k=1}^{\infty}\left\\{\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k-1}a_{n+1-s+\ell}+\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k}\tilde{a}_{s+\ell}\right\\}^{*}\tilde{a}_{t-u}.$

  2. (ii)

For $n\in\mathbb{N}$ and $s,t\in\\{1,\dots,n\\}$, we have

(2.11) $\displaystyle\left(T_{n}(w)^{-1}\right)^{s,t}=\sum_{\ell=s\vee
t}^{n}a_{\ell-s}^{*}a_{\ell-t}$
$\displaystyle\qquad+\sum_{u=t}^{n}\sum_{k=1}^{\infty}\left\\{\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k-1}\tilde{a}_{s+\ell}+\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k}a_{n+1-s+\ell}\right\\}^{*}a_{u-t}.$

The proof of Theorem 2.1 will be given in Section 3.

######  Corollary 2.1.

We assume (1.2) and (1.3). Then the following two assertions hold.

  1. (i)

For $n\in\mathbb{N}$ and $s,t\in\\{1,\dots,n\\}$, we have

(2.12) $\displaystyle\left(T_{n}(w)^{-1}\right)^{s,t}=\sum_{\ell=1}^{s\wedge
t}\tilde{a}_{s-\ell}^{*}\tilde{a}_{t-\ell}$
$\displaystyle\qquad+\sum_{u=1}^{s}\tilde{a}_{s-u}^{*}\sum_{k=1}^{\infty}\left\\{\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k-1}a_{n+1-t+\ell}+\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k}\tilde{a}_{t+\ell}\right\\}.$

  2. (ii)

For $n\in\mathbb{N}$ and $s,t\in\\{1,\dots,n\\}$, we have

(2.13) $\displaystyle\left(T_{n}(w)^{-1}\right)^{s,t}=\sum_{\ell=s\vee
t}^{n}a_{\ell-s}^{*}a_{\ell-t}$
$\displaystyle\qquad+\sum_{u=s}^{n}a_{u-s}^{*}\sum_{k=1}^{\infty}\left\\{\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k-1}\tilde{a}_{t+\ell}+\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k}a_{n+1-t+\ell}\right\\}.$

###### Proof.

Thanks to (2.9), we obtain (2.12) and (2.13) from (2.10) and (2.11),
respectively. ∎

######  Remark 2.1.

Recall $T_{\infty}(w)$ from (1.12). For $n\in\mathbb{N}\cup\\{0\\}$, we have
$\gamma(n)=\sum_{k=0}^{\infty}\tilde{c}_{k}\tilde{c}_{n+k}^{*}$ and
$\gamma(-n)=\sum_{k=0}^{\infty}\tilde{c}_{n+k}\tilde{c}_{k}^{*}$ (see (2.13)
in [16]), hence $T_{\infty}(w)=\tilde{C}_{\infty}(\tilde{C}_{\infty})^{*}$,
where

$\tilde{C}_{\infty}:=\left(\begin{matrix}\tilde{c}_{0}&\tilde{c}_{1}&\tilde{c}_{2}&\cdots\cr&\tilde{c}_{0}&\tilde{c}_{1}&\cdots\cr&&\tilde{c}_{0}&\cdots\cr\mbox{\huge
0}&&&\ddots\end{matrix}\right).$

On the other hand, it follows from $\tilde{h}(z)\tilde{h}(z)^{-1}=I_{d}$ that
$\sum_{k=0}^{n}\tilde{c}_{k}\tilde{a}_{n-k}=-\delta_{n0}I_{d}$ for
$n\in\mathbb{N}\cup\\{0\\}$, hence
$\tilde{C}_{\infty}\tilde{A}_{\infty}=-I_{\infty}$, where

$\tilde{A}_{\infty}:=\left(\begin{matrix}\tilde{a}_{0}&\tilde{a}_{1}&\tilde{a}_{2}&\cdots\cr&\tilde{a}_{0}&\tilde{a}_{1}&\cdots\cr&&\tilde{a}_{0}&\cdots\cr\mbox{\huge
0}&&&\ddots\end{matrix}\right),\qquad
I_{\infty}:=\left(\begin{matrix}1&0&0&\cdots\cr 0&1&0&\cdots\cr
0&0&1&\cdots\cr\vdots&\vdots&\vdots&\ddots\end{matrix}\right).$

Combining, we have
$T_{\infty}(w)^{-1}=(\tilde{A}_{\infty})^{*}\tilde{A}_{\infty}$. Thus, we find
that the first term $\sum_{\ell=1}^{s\wedge
t}\tilde{a}_{s-\ell}^{*}\tilde{a}_{t-\ell}$ in (2.10) or (2.12) coincides with
the $(s,t)$ block of $T_{\infty}(w)^{-1}$.

For $n\in\mathbb{N}$, we define

(2.14)
$\tilde{A}_{n}:=\left(\begin{matrix}\tilde{a}_{0}&\tilde{a}_{1}&\tilde{a}_{2}&\cdots&\tilde{a}_{n-1}\cr&\tilde{a}_{0}&\tilde{a}_{1}&\cdots&\tilde{a}_{n-2}\cr&&\ddots&\ddots&\vdots\cr&&&\ddots&\tilde{a}_{1}\cr\mbox{\huge
0}&&&&\tilde{a}_{0}\end{matrix}\right)\in\mathbb{C}^{dn\times dn}$

and

(2.15) $A_{n}:=\left(\begin{matrix}a_{0}&&&&\mbox{\huge 0}\cr
a_{1}&a_{0}&&&\cr a_{2}&a_{1}&\ddots&&\cr\vdots&\vdots&\ddots&\ddots&\cr
a_{n-1}&a_{n-2}&\cdots&a_{1}&a_{0}\end{matrix}\right)\in\mathbb{C}^{dn\times
dn}.$

The next lemma will turn out to be useful in Section 6.

######  Lemma 2.1.

For $n\in\mathbb{N}$ and $s,t\in\\{1,\dots,n\\}$, we have the following two
equalities:

$\displaystyle\left(\tilde{A}_{n}^{*}\tilde{A}_{n}\right)^{s,t}$
$\displaystyle=\sum_{\ell=1}^{s\wedge
t}\tilde{a}_{s-\ell}^{*}\tilde{a}_{t-\ell},$
$\displaystyle\left(A_{n}^{*}A_{n}\right)^{s,t}$
$\displaystyle=\sum_{\ell=s\vee t}^{n}a_{\ell-s}^{*}a_{\ell-t}.$

The proof of Lemma 2.1 is straightforward and will be omitted.

##  3\. Proof of Theorem 2.1

In this section, we prove Theorem 2.1. We assume (1.2) and (1.3). Let
$\\{X_{k}\\}=\\{X_{k}:k\in\mathbb{Z}\\}$ be a $\mathbb{C}^{d}$-valued,
centered, weakly stationary process, defined on a probability space
$(\Omega,\mathcal{F},P)$, that has spectral density $w$, hence autocovariance
function $\gamma$. Thus we have
$E[X_{k}X_{0}^{*}]=\gamma(k)=\int_{-\pi}^{\pi}e^{-ik\theta}w(e^{i\theta})(d\theta/(2\pi))$
for $k\in\mathbb{Z}$.

Write $X_{k}=(X^{1}_{k},\dots,X^{d}_{k})^{\top}$, and let $V$ be the complex
Hilbert space spanned by all the entries $\\{X^{j}_{k}:k\in\mathbb{Z},\
j=1,\dots,d\\}$ in $L^{2}(\Omega,\mathcal{F},P)$, which has inner product
$(x,y)_{V}:=E[x\overline{y}]$ and norm $\|x\|_{V}:=(x,x)_{V}^{1/2}$. For
$J\subset\mathbb{Z}$ such as $\\{n\\}$, $(-\infty,n]:=\\{n,n-1,\dots\\}$,
$[n,\infty):=\\{n,n+1,\dots\\}$, and $[m,n]:=\\{m,\dots,n\\}$ with $m\leq n$,
we define the closed subspace $V_{J}^{X}$ of $V$ by

$V_{J}^{X}:=\operatorname{\overline{\mathrm{sp}}}\\{X^{j}_{k}:j=1,\dots,d,\
k\in J\\}.$

Let $P_{J}$ and $P_{J}^{\perp}$ be the orthogonal projection operators of $V$
onto $V_{J}^{X}$ and $(V_{J}^{X})^{\perp}$, respectively, where
$(V_{J}^{X})^{\bot}$ denotes the orthogonal complement of $V_{J}^{X}$ in $V$.

By Theorem 3.1 in [11] for $d=1$ and Corollary 3.6 in [15] for general $d\geq
1$, the conditions (1.2) and (1.3) imply the following intersection of past
and future property:

(3.1) $V_{(-\infty,n]}^{X}\cap V_{[1,\infty)}^{X}=V_{[1,n]}^{X},\qquad
n\in\mathbb{N}.$

Let $V^{d}$ be the space of $\mathbb{C}^{d}$-valued random variables on
$(\Omega,\mathcal{F},P)$ whose entries belong to $V$. The norm $\|x\|_{V^{d}}$
of $x=(x^{1},\dots,x^{d})^{\top}\in V^{d}$ is given by
$\|x\|_{V^{d}}:=(\sum_{i=1}^{d}\|x^{i}\|_{V}^{2})^{1/2}$. For
$J\subset\mathbb{Z}$ and $x=(x^{1},\dots,x^{d})^{\top}\in V^{d}$, we write
$P_{J}x$ for $(P_{J}x^{1},\dots,P_{J}x^{d})^{\top}$. We define
$P_{J}^{\perp}x$ in a similar way. For $x=(x^{1},\dots,x^{d})^{\top}$ and
$y=(y^{1},\dots,y^{d})^{\top}$ in $V^{d}$,

$\langle x,y\rangle:=E[xy^{*}]=((x^{k},y^{\ell})_{V})_{1\leq k,\ell\leq
d}\in\mathbb{C}^{d\times d}$

stands for the Gram matrix of $x$ and $y$.

Let

$X_{k}=\int_{-\pi}^{\pi}e^{-ik\theta}\eta(d\theta),\qquad k\in\mathbb{Z},$

be the spectral representation of $\\{X_{k}\\}$, where $\eta$ is a
$\mathbb{C}^{d}$-valued random spectral measure. We define a $d$-variate
stationary process $\\{\varepsilon_{k}:k\in\mathbb{Z}\\}$, called the forward
innovation process of $\\{X_{k}\\}$, by

$\varepsilon_{k}:=\int_{-\pi}^{\pi}e^{-ik\theta}h(e^{i\theta})^{-1}\eta(d\theta),\qquad
k\in\mathbb{Z}.$

Then, $\\{\varepsilon_{k}\\}$ satisfies
$\langle\varepsilon_{n},\varepsilon_{m}\rangle=\delta_{nm}I_{d}$ and
$V_{(-\infty,n]}^{X}=V_{(-\infty,n]}^{\varepsilon}$ for $n\in\mathbb{Z}$,
hence

$(V_{(-\infty,n]}^{X})^{\bot}=V_{[n+1,\infty)}^{\varepsilon},\qquad
n\in\mathbb{Z}.$

Recall the outer function $h_{\sharp}$ in $H_{2}^{d\times d}(\mathbb{T})$ from
(1.4). We define the backward innovation process
$\\{\tilde{\varepsilon}_{k}:k\in\mathbb{Z}\\}$ of $\\{X_{k}\\}$ by

$\tilde{\varepsilon}_{k}:=\int_{-\pi}^{\pi}e^{ik\theta}\\{h_{\sharp}(e^{i\theta})^{*}\\}^{-1}\eta(d\theta),\qquad
k\in\mathbb{Z}.$

Then, $\\{\tilde{\varepsilon}_{k}\\}$ satisfies
$\langle\tilde{\varepsilon}_{n},\tilde{\varepsilon}_{m}\rangle=\delta_{nm}I_{d}$
and $V_{[-n,\infty)}^{X}=V_{(-\infty,n]}^{\tilde{\varepsilon}}$ for
$n\in\mathbb{Z}$, hence

$(V_{[-n,\infty)}^{X})^{\bot}=V_{[n+1,\infty)}^{\tilde{\varepsilon}},\qquad
n\in\mathbb{Z}$

(see Section 2 in [16]). Moreover, by Lemma 4.1 in [16], we have

(3.2)
$\langle\varepsilon_{\ell},\tilde{\varepsilon}_{m}\rangle=-\beta_{\ell+m},\qquad\langle\tilde{\varepsilon}_{m},\varepsilon_{\ell}\rangle=-\beta_{\ell+m}^{*},\qquad\ell,m\in\mathbb{Z}.$

By (3.2), for $\\{s_{\ell}\\}\in\ell_{2+}^{d\times d}$ and $n\in\mathbb{N}$,

(3.3) $\displaystyle
P_{[1,\infty)}^{\perp}\left(\sum_{\ell=0}^{\infty}s_{\ell}\varepsilon_{n+1+\ell}\right)$
$\displaystyle=-\sum_{\ell=0}^{\infty}\left(\sum_{m=0}^{\infty}s_{m}\beta_{n+1+\ell+m}\right)\tilde{\varepsilon}_{\ell},$
(3.4) $\displaystyle
P_{(-\infty,n]}^{\perp}\left(\sum_{\ell=0}^{\infty}s_{\ell}\tilde{\varepsilon}_{\ell}\right)$
$\displaystyle=-\sum_{\ell=0}^{\infty}\left(\sum_{m=0}^{\infty}s_{m}\beta_{n+1+\ell+m}^{*}\right)\varepsilon_{n+1+\ell}.$

Therefore,

$\left\\{\sum_{m=0}^{\infty}s_{m}\beta_{n+1+\ell+m}\right\\}_{\ell=0}^{\infty},\
\left\\{\sum_{m=0}^{\infty}s_{m}\beta_{n+1+\ell+m}^{*}\right\\}_{\ell=0}^{\infty}\
\in\ \ell_{2+}^{d\times d}.$

See Lemma 4.2 in [16]. In particular, for $n\in\mathbb{N}$,
$u\in\\{1,\dots,n\\}$ and $k\in\mathbb{N}$, we can define the sequences
$\\{b_{n,u,\ell}^{k}\\}_{\ell=0}^{\infty}\in\ell_{2+}^{d\times d}$ and
$\\{\tilde{b}_{n,u,\ell}^{k}\\}_{\ell=0}^{\infty}\in\ell_{2+}^{d\times d}$ by
the recursions (2.7) and (2.8), respectively.

By (1.2) and (1.3), $\\{X_{k}\\}$ has the dual process
$\\{X^{\prime}_{k}:k\in\mathbb{Z}\\}$, which is a $\mathbb{C}^{d}$-valued,
centered, weakly stationary process characterized by the biorthogonality
relation

$\langle X_{s},X^{\prime}_{t}\rangle=\delta_{st}I_{d},\qquad s,t\in\mathbb{Z}$

(see [19]). Recall $\\{a_{k}\\}\in\ell_{2+}^{d\times d}$ and
$\\{\tilde{a}_{k}\\}\in\ell_{2+}^{d\times d}$ from (2.3) and (2.5),
respectively. The dual process $\\{X^{\prime}_{k}\\}$ admits the following two
MA representations (see Section 5 in [16]):

(3.5) $\displaystyle X^{\prime}_{n}$
$\displaystyle=-\sum_{\ell=n}^{\infty}a_{\ell-n}^{*}\varepsilon_{\ell},\qquad
n\in\mathbb{Z},$ (3.6) $\displaystyle X^{\prime}_{n}$
$\displaystyle=-\sum_{\ell=-n}^{\infty}\tilde{a}_{\ell+n}^{*}\tilde{\varepsilon}_{\ell},\qquad
n\in\mathbb{Z}.$

The next theorem is the key to the proof of Theorem 2.1.

######  Theorem 3.1.

Assume (1.2) and (1.3). Then, for $n\in\mathbb{N}$ and
$s,t\in\\{1,\dots,n\\}$, we have (1.5).

###### Proof.

Fix $n\in\mathbb{N}$. For $s\in\\{1,\dots,n\\}$, we can write
$P_{[1,n]}X^{\prime}_{s}=\sum_{k=1}^{n}q_{s,k}X_{k}$ for some
$q_{s,k}\in\mathbb{C}^{d\times d}$, $k\in\\{1,\dots,n\\}$. For
$s,t\in\\{1,\dots,n\\}$, we have

$\displaystyle\delta_{st}I_{d}$ $\displaystyle=\langle
X^{\prime}_{s},X_{t}\rangle=\langle
X^{\prime}_{s},P_{[1,n]}X_{t}\rangle=\langle
P_{[1,n]}X^{\prime}_{s},X_{t}\rangle=\left\langle\sum_{k=1}^{n}q_{s,k}X_{k},X_{t}\right\rangle$
$\displaystyle=\sum_{k=1}^{n}q_{s,k}\left\langle
X_{k},X_{t}\right\rangle=\sum_{k=1}^{n}q_{s,k}\gamma(k-t),$

or $Q_{n}T_{n}(w)=I_{dn}$, where $Q_{n}:=(q_{s,k})_{1\leq s,k\leq
n}\in\mathbb{C}^{dn\times dn}$. Therefore, we have $Q_{n}=T_{n}(w)^{-1}$.
However,

$\langle X^{\prime}_{s},P_{[1,n]}X^{\prime}_{t}\rangle=\langle
P_{[1,n]}X^{\prime}_{s},X^{\prime}_{t}\rangle=\left\langle\sum_{k=1}^{n}q_{s,k}X_{k},X^{\prime}_{t}\right\rangle=\sum_{k=1}^{n}q_{s,k}\langle
X_{k},X^{\prime}_{t}\rangle=q_{s,t}.$

Thus, the theorem follows. ∎

######  Lemma 3.1.

Assume (1.2) and (1.3). Then, for $n\in\mathbb{N}$ and
$s,t\in\\{1,\dots,n\\}$, the following two equalities hold:

(3.7) $\displaystyle\langle X^{\prime}_{s},P_{[1,n]}X^{\prime}_{t}\rangle$
$\displaystyle=\sum_{\ell=s\vee
t}^{n}a_{\ell-s}^{*}a_{\ell-t}+\sum_{u=t}^{n}\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}\varepsilon_{u}\rangle a_{u-t},$ (3.8)
$\displaystyle\langle X^{\prime}_{s},P_{[1,n]}X^{\prime}_{t}\rangle$
$\displaystyle=\sum_{\ell=1}^{s\wedge
t}\tilde{a}_{s-\ell}^{*}\tilde{a}_{t-\ell}+\sum_{u=1}^{t}\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}\tilde{\varepsilon}_{-u}\rangle\tilde{a}_{t-u}.$

###### Proof.

First, we prove (3.7). Since $V_{[1,n]}^{X}\subset V_{(-\infty,n]}^{X}$, we
have

$\displaystyle\langle X^{\prime}_{s},P_{[1,n]}X^{\prime}_{t}\rangle$
$\displaystyle=\langle
X^{\prime}_{s},P_{[1,n]}P_{(-\infty,n]}X^{\prime}_{t}\rangle$
$\displaystyle=\langle
X^{\prime}_{s},P_{(-\infty,n]}X^{\prime}_{t}\rangle-\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}P_{(-\infty,n]}X^{\prime}_{t}\rangle.$

On the other hand, from (3.5), we have
$P_{(-\infty,n]}X^{\prime}_{t}=-\sum_{m=t}^{n}a_{m-t}^{*}\varepsilon_{m}$,
hence

$\langle
X^{\prime}_{s},P_{(-\infty,n]}X^{\prime}_{t}\rangle=\left\langle\sum_{\ell=s}^{\infty}a_{\ell-s}^{*}\varepsilon_{\ell},\sum_{m=t}^{n}a_{m-t}^{*}\varepsilon_{m}\right\rangle=\sum_{\ell=s\vee
t}^{n}a_{\ell-s}^{*}a_{\ell-t},$

and $\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}P_{(-\infty,n]}X^{\prime}_{t}\rangle$ is equal
to

$-\left\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}\left(\sum_{u=t}^{n}a_{u-t}^{*}\varepsilon_{u}\right)\right\rangle=-\sum_{u=t}^{n}\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}\varepsilon_{u}\rangle a_{u-t}.$

Combining, we obtain (3.7).

Next, we prove (3.8). Since $V_{[1,n]}^{X}\subset V_{[1,\infty)}^{X}$, we have

$\displaystyle\langle X^{\prime}_{s},P_{[1,n]}X^{\prime}_{t}\rangle$
$\displaystyle=\langle
X^{\prime}_{s},P_{[1,n]}P_{[1,\infty)}X^{\prime}_{t}\rangle$
$\displaystyle=\langle
X^{\prime}_{s},P_{[1,\infty)}X^{\prime}_{t}\rangle-\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}P_{[1,\infty)}X^{\prime}_{t}\rangle.$

On the other hand, from (3.6), we have
$P_{[1,\infty)}X^{\prime}_{t}=-\sum_{m=1}^{t}\tilde{a}_{t-m}^{*}\tilde{\varepsilon}_{-m}$,
hence

$\langle
X^{\prime}_{s},P_{[1,\infty)}X^{\prime}_{t}\rangle=\left\langle\sum_{\ell=-\infty}^{s}\tilde{a}_{s-\ell}^{*}\tilde{\varepsilon}_{-\ell},\sum_{m=1}^{t}\tilde{a}_{t-m}^{*}\tilde{\varepsilon}_{-m}\right\rangle=\sum_{\ell=1}^{s\wedge
t}\tilde{a}_{s-\ell}^{*}\tilde{a}_{t-\ell},$

and $\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}P_{[1,\infty)}X^{\prime}_{t}\rangle$ is equal
to

$-\left\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}\left(\sum_{u=1}^{t}\tilde{a}_{t-u}^{*}\tilde{\varepsilon}_{-u}\right)\right\rangle=-\sum_{u=1}^{t}\langle
X^{\prime}_{s},P_{[1,n]}^{\bot}\tilde{\varepsilon}_{-u}\rangle\tilde{a}_{t-u}.$

Combining, we obtain (3.8). ∎

For $n\in\mathbb{N}$ and $u\in\\{1,\dots,n\\}$, we define the sequence
$\\{W_{n,u}^{k}\\}_{k=1}^{\infty}$ in $V^{d}$ by

$\displaystyle W_{n,u}^{2k-1}$
$\displaystyle=-P_{[1,\infty)}^{\perp}(P_{(-\infty,n]}^{\perp}P_{[1,\infty)}^{\perp})^{k-1}\varepsilon_{u},\qquad
k\in\mathbb{N},$ $\displaystyle W_{n,u}^{2k}$
$\displaystyle=(P_{(-\infty,n]}^{\perp}P_{[1,\infty)}^{\perp})^{k}\varepsilon_{u},\qquad
k\in\mathbb{N}.$

######  Lemma 3.2.

We assume (1.2) and (1.3). Then, for $n\in\mathbb{N}$ and
$u\in\\{1,\dots,n\\}$, we have

(3.9) $P_{[1,n]}^{\perp}\varepsilon_{u}=-\sum_{k=1}^{\infty}W_{n,u}^{k},$

the sum converging strongly in $V^{d}$.

###### Proof.

Since $\varepsilon_{u}$ is in $V_{(-\infty,n]}^{X}$, (3.9) follows from (3.1)
and Theorem 3.2 in [16]. ∎

######  Proposition 3.1.

We assume (1.2) and (1.3). Then, for $n\in\mathbb{N}$, $u\in\\{1,\dots,n\\}$
and $k\in\mathbb{N}$, we have

(3.10) $\displaystyle W_{n,u}^{2k-1}$
$\displaystyle=\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k-1}\tilde{\varepsilon}_{\ell},$
(3.11) $\displaystyle W_{n,u}^{2k}$
$\displaystyle=\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k}\varepsilon_{n+1+\ell}.$

###### Proof.

Note that, from the definition of $W_{n,u}^{k}$,

$W_{n,u}^{2k+1}=-P_{[1,\infty)}^{\perp}W_{n,u}^{2k},\qquad
W_{n,u}^{2k+2}=-P_{(-\infty,n]}^{\perp}W_{n,u}^{2k+1}.$

We prove (3.10) and (3.11) by induction. First, by (3.2), we have

$W_{n,u}^{1}=-P_{[1,\infty)}^{\perp}\varepsilon_{u}=-\sum_{\ell=0}^{\infty}\langle\varepsilon_{u},\tilde{\varepsilon}_{\ell}\rangle\tilde{\varepsilon}_{\ell}=\sum_{\ell=0}^{\infty}\beta_{u+\ell}\tilde{\varepsilon}_{\ell}=\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{1}\tilde{\varepsilon}_{\ell}.$

For $k\in\mathbb{N}$, assume that
$W_{n,u}^{2k-1}=\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k-1}\tilde{\varepsilon}_{\ell}$.
Then, by (3.4),

$\displaystyle W_{n,u}^{2k}$
$\displaystyle=-P_{(-\infty,n]}^{\perp}\left(\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k-1}\tilde{\varepsilon}_{\ell}\right)=\sum_{\ell=0}^{\infty}\left(\sum_{m=0}^{\infty}b_{n,u,m}^{2k-1}\beta_{n+1+m+\ell}^{*}\right)\varepsilon_{n+1+\ell}$
$\displaystyle=\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k}\varepsilon_{n+1+\ell},$

and, by (3.3),

$\displaystyle W_{n,u}^{2k+1}$
$\displaystyle=-P_{[1,\infty)}^{\perp}\left(\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k}\varepsilon_{n+1+\ell}\right)=\sum_{\ell=0}^{\infty}\left(\sum_{m=0}^{\infty}b_{n,u,m}^{2k}\beta_{n+1+m+\ell}\right)\tilde{\varepsilon}_{\ell}$
$\displaystyle=\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k+1}\tilde{\varepsilon}_{\ell}.$

Thus (3.10) and (3.11) follow. ∎

For $n\in\mathbb{N}$ and $u\in\\{1,\dots,n\\}$, we define the sequence
$\\{\tilde{W}_{n,u}^{k}\\}_{k=1}^{\infty}$ in $V^{d}$ by

$\displaystyle\tilde{W}_{n,u}^{2k-1}$
$\displaystyle=-P_{(-\infty,n]}^{\perp}(P_{[1,\infty)}^{\perp}P_{(-\infty,n]}^{\perp})^{k-1}\tilde{\varepsilon}_{-u},\qquad
k\in\mathbb{N},$ $\displaystyle\tilde{W}_{n,u}^{2k}$
$\displaystyle=(P_{[1,\infty)}^{\perp}P_{(-\infty,n]}^{\perp})^{k}\tilde{\varepsilon}_{-u},\qquad
k\in\mathbb{N}.$

######  Lemma 3.3.

We assume (1.2) and (1.3). Then, for $n\in\mathbb{N}$ and
$u\in\\{1,\dots,n\\}$, we have

(3.12)
$P_{[1,n]}^{\perp}\tilde{\varepsilon}_{-u}=-\sum_{k=1}^{\infty}\tilde{W}_{n,u}^{k},$

the sum converging strongly in $V^{d}$.

###### Proof.

Since $\tilde{\varepsilon}_{-u}$ is in $V_{[1,\infty)}^{X}$, (3.12) follows
from (3.1) and Theorem 3.2 in [16]. ∎

######  Proposition 3.2.

We assume (1.2) and (1.3). Then, for $n\in\mathbb{N}$, $u\in\\{1,\dots,n\\}$
and $k\in\mathbb{N}$, we have

(3.13) $\displaystyle\tilde{W}_{n,u}^{2k-1}$
$\displaystyle=\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k-1}\varepsilon_{n+1+\ell},$
(3.14) $\displaystyle\tilde{W}_{n,u}^{2k}$
$\displaystyle=\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k}\tilde{\varepsilon}_{\ell}.$

###### Proof.

Note that, from the definition of $\tilde{W}_{n,u}^{k}$,

$\tilde{W}_{n,u}^{2k+1}=-P_{(-\infty,n]}^{\perp}\tilde{W}_{n,u}^{2k},\qquad\tilde{W}_{n,u}^{2k+2}=-P_{[1,\infty)}^{\perp}\tilde{W}_{n,u}^{2k+1}.$

We prove (3.13) and (3.14) by induction. First, by (3.2), we have

$\displaystyle\tilde{W}_{n,u}^{1}$
$\displaystyle=-P_{(-\infty,n]}^{\perp}\tilde{\varepsilon}_{-u}=-\sum_{\ell=0}^{\infty}\langle\tilde{\varepsilon}_{-u},\varepsilon_{n+1+\ell}\rangle\varepsilon_{n+1+\ell}$
$\displaystyle=\sum_{\ell=0}^{\infty}\beta_{n+1-u+\ell}^{*}\varepsilon_{n+1+\ell}=\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{1}\varepsilon_{n+1+\ell}.$

For $k\in\mathbb{N}$, assume that
$\tilde{W}_{n,u}^{2k-1}=\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k-1}\varepsilon_{n+1+\ell}$.
Then, by (3.3),

$\displaystyle\tilde{W}_{n,u}^{2k}$
$\displaystyle=-P_{[1,\infty)}^{\perp}\left(\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k-1}\varepsilon_{n+1+\ell}\right)=\sum_{\ell=0}^{\infty}\left(\sum_{m=0}^{\infty}\tilde{b}_{n,u,m}^{2k-1}\beta_{n+1+m+\ell}\right)\tilde{\varepsilon}_{l}$
$\displaystyle=\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k}\tilde{\varepsilon}_{\ell},$

and, by (3.4),

$\displaystyle\tilde{W}_{n,u}^{2k+1}$
$\displaystyle=-P_{(\infty,n]}^{\perp}\left(\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k}\tilde{\varepsilon}_{\ell}\right)=\sum_{\ell=0}^{\infty}\left(\sum_{m=0}^{\infty}\tilde{b}_{n,u,m}^{2k}\beta_{n+1+m+\ell}^{*}\right)\varepsilon_{n+1+\ell}$
$\displaystyle=\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k+1}\varepsilon_{n+1+\ell}.$

Thus (3.13) and (3.14) follow. ∎

We are ready to prove Theorem 2.1.

###### Proof.

(i) For $n\in\mathbb{N}$, $s,u\in\\{1,\dots,n\\}$ and $k\in\mathbb{N}$, we see
from (3.5) and (3.13) that

$\langle
X^{\prime}_{s},\tilde{W}_{n,u}^{2k-1}\rangle=-\sum_{\ell=0}^{\infty}a_{n+1-s+\ell}^{*}(\tilde{b}_{n,u,\ell}^{2k-1})^{*},$

and from (3.6) and (3.14) that

$\langle
X^{\prime}_{s},\tilde{W}_{n,u}^{2k}\rangle=-\sum_{\ell=0}^{\infty}\tilde{a}_{s+\ell}^{*}(\tilde{b}_{n,u,\ell}^{2k})^{*}.$

Therefore, by Lemma 3.3, $\langle
X^{\prime}_{s},P_{[1,n]}^{\perp}\tilde{\varepsilon}_{-u}\rangle$ is equal to

$-\sum_{k=1}^{\infty}\langle
X^{\prime}_{s},\tilde{W}_{n,u}^{k}\rangle=\sum_{k=1}^{\infty}\left\\{\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k-1}a_{n+1-s+\ell}+\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k}\tilde{a}_{s+\ell}\right\\}^{*}.$

The assertion (i) follows from this, Theorem 3.1 and Lemma 3.1.

(ii) For $n\in\mathbb{N}$, $s,u\in\\{1,\dots,n\\}$ and $k\in\mathbb{N}$, we
see from (3.6) and (3.10) that

$\langle
X^{\prime}_{s},W_{n,u}^{2k-1}\rangle=-\sum_{\ell=0}^{\infty}\tilde{a}_{s+\ell}^{*}(b_{n,u,\ell}^{2k-1})^{*},$

and from (3.5) and (3.11) that

$\langle
X^{\prime}_{s},W_{n,u}^{2k}\rangle=-\sum_{\ell=0}^{\infty}a_{n+1-s+\ell}^{*}(b_{n,u,\ell}^{2k})^{*}.$

Therefore, by Lemma 3.2, $\langle
X^{\prime}_{s},P_{[1,n]}^{\perp}\varepsilon_{u}\rangle$ is equal to

$-\sum_{k=1}^{\infty}\langle
X^{\prime}_{s},W_{n,u}^{k}\rangle=\sum_{k=1}^{\infty}\left\\{\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k-1}\tilde{a}_{s+\ell}+\sum_{\ell=0}^{\infty}b_{n,u,\ell}^{2k}a_{n+1-s+\ell}\right\\}^{*}.$

The assertion (ii) follows from this, Theorem 3.1 and Lemma 3.1. ∎

##  4\. Strong convergence result for Toeplitz systems

In this section, we use Theorem 2.1 to show a strong convergence result for
solutions of block Toeplitz systems. We assume (1.2) and (1.6). Then $w$ is
continuous on $\mathbb{T}$ since
$w(e^{i\theta})=(2\pi)^{-1}\sum_{k\in\mathbb{Z}}e^{ik\theta}\gamma(k)$. In
particular, (1.3) is also satisfied. The conditions (1.2) and (1.6) also imply
that all of $\\{a_{k}\\}$, $\\{c_{k}\\}$, $\\{\tilde{a}_{k}\\}$ and
$\\{\tilde{c}_{k}\\}$ belong to $\ell_{1+}^{d\times d}$. See Theorem 3.3 and
(3.3) in [17]; see also Theorem 4.1 in [12]. In particular, we have
$h(e^{i\theta})^{-1}=-\sum_{k=0}^{\infty}e^{ik\theta}a_{k}$ and
$h_{\sharp}(e^{i\theta})=\tilde{h}(e^{-i\theta})^{*}=\sum_{k=0}^{\infty}e^{ik\theta}\tilde{c}_{k}^{*}$,
hence, by (2.6),

(4.1) $\beta_{k}=\sum_{j=0}^{\infty}a_{j+k}\tilde{c}_{j},\qquad
k\in\mathbb{N}\cup\\{0\\}.$

Under (1.2) and (1.6), we define

$F(n):=\left(\sum_{j=0}^{\infty}\|\tilde{c}_{j}\|\right)\sum_{\ell=n}^{\infty}\|a_{\ell}\|,\qquad
n\in\mathbb{N}\cup\\{0\\}.$

Then $F(n)$ decreases to zero as $n\to\infty$.

We need the next lemma in the proof of Theorem 4.1 below.

######  Lemma 4.1.

Assume (1.2) and (1.6). Then, for $n,k\in\mathbb{N}$ and
$u\in\\{1,\dots,n\\}$, we have

(4.2) $\sum_{\ell=0}^{\infty}\|\tilde{b}^{k}_{n,u,\ell}\|\leq
F(n+1)^{k-1}F(n+1-u).$

###### Proof.

For $m\in\mathbb{N}$, we see from (4.1) that

$\sum_{\ell=0}^{\infty}\|\beta_{m+\ell}\|\leq\sum_{j=0}^{\infty}\|\tilde{c}_{j}\|\sum_{\ell=0}^{\infty}\|a_{m+j+\ell}\|\leq\sum_{j=0}^{\infty}\|\tilde{c}_{j}\|\sum_{\ell=m}^{\infty}\|a_{\ell}\|,$

hence

(4.3) $\sum_{\ell=0}^{\infty}\|\beta_{m+\ell}\|\leq F(m).$

Let $n\in\mathbb{N}$ and $u\in\\{1,\dots,n\\}$. We use induction on $k$ to
prove (4.2). Since $\tilde{b}^{1}_{n,u,\ell}=\beta_{n+1-u+\ell}^{*}$, we see
from (4.3) that

$\sum_{\ell=0}^{\infty}\|\tilde{b}^{1}_{n,u,\ell}\|=\sum_{\ell=0}^{\infty}\|\beta_{n+1-u+\ell}\|\leq
F(n+1-u).$

We assume (4.2) for $k\in\mathbb{N}$. Then, again by (4.3),

$\displaystyle\sum_{\ell=0}^{\infty}\|\tilde{b}^{k+1}_{n,u,\ell}\|$
$\displaystyle\leq\sum_{m=0}^{\infty}\|\tilde{b}^{k}_{n,u,m}\|\sum_{\ell=0}^{\infty}\|\beta_{n+1+m+\ell}\|$
$\displaystyle\leq F(n+1)\sum_{m=0}^{\infty}\|\tilde{b}^{k}_{n,u,m}\|\leq
F(n+1)^{k}F(n+1-u).$

Thus (4.2) with $k$ replaced by $k+1$ also holds. ∎

For $\\{y_{k}\\}_{k=1}^{\infty}\in\ell_{1}^{d\times d}(\mathbb{N})$, the
solution $Z_{\infty}$ to (1.11) with (1.12) and (1.13) is given by (1.10) with

(4.4) $z_{s}=\sum_{t=1}^{\infty}\sum_{\ell=1}^{s\wedge
t}\tilde{a}_{s-\ell}^{*}\tilde{a}_{t-\ell}y_{t}\in\mathbb{C}^{d\times
d},\qquad s\in\mathbb{N}$

(see Remark 2.1 in Section 2). Notice that the sum in (4.4) converges
absolutely.

######  Theorem 4.1.

We assume (1.2) and (1.6). Let $\\{y_{k}\\}_{k=1}^{\infty}\in\ell_{1}^{d\times
d}(\mathbb{N})$. Then, for $Z_{n}$ in (1.7)–(1.9) and $Z_{\infty}$ in
(1.10)–(1.13), we have (1.14).

###### Proof.

By Theorem 2.1 (i), we have

$\displaystyle z_{n,s}$ $\displaystyle=\sum_{t=1}^{n}\sum_{\ell=1}^{s\wedge
t}\tilde{a}_{s-\ell}^{*}\tilde{a}_{t-\ell}y_{t}+\sum_{t=1}^{n}\sum_{u=1}^{t}\sum_{\ell=0}^{\infty}a_{n+1-s+\ell}^{*}\beta_{n+1-u+\ell}\tilde{a}_{t-u}y_{t}$
$\displaystyle\quad\quad+\sum_{t=1}^{n}\sum_{u=1}^{t}\sum_{k=1}^{\infty}\left\\{\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k+1}a_{n+1-s+\ell}+\sum_{\ell=0}^{\infty}\tilde{b}_{n,u,\ell}^{2k}\tilde{a}_{s+\ell}\right\\}^{*}\tilde{a}_{t-u}y_{t},$

hence, by (4.4), $\sum_{s=1}^{n}\|z_{n,s}-z_{s}\|\leq
S_{1}(n)+S_{2}(n)+S_{3}(n)+S_{4}(n)$, where

$\displaystyle S_{1}(n)$
$\displaystyle:=\sum_{t=n+1}^{\infty}\sum_{s=1}^{n}\sum_{\ell=1}^{s}\|\tilde{a}_{s-\ell}\|\|\tilde{a}_{t-\ell}\|\|y_{t}\|,$
$\displaystyle S_{2}(n)$
$\displaystyle:=\sum_{s=1}^{n}\sum_{t=1}^{n}\sum_{u=1}^{t}\sum_{\ell=0}^{\infty}\|a_{n+1-s+\ell}\|\|\beta_{n+1-u+\ell}\|\|\tilde{a}_{t-u}\|\|y_{t}\|,$
$\displaystyle S_{3}(n)$
$\displaystyle:=\sum_{s=1}^{n}\sum_{t=1}^{n}\sum_{u=1}^{t}\sum_{k=1}^{\infty}\sum_{\ell=0}^{\infty}\|\tilde{b}_{n,u,\ell}^{2k+1}\|\|a_{n+1-s+\ell}\|\|\tilde{a}_{t-u}\|\|y_{t}\|$

and

$S_{4}(n)=\sum_{s=1}^{n}\sum_{t=1}^{n}\sum_{u=1}^{t}\sum_{k=1}^{\infty}\sum_{\ell=0}^{\infty}\|\tilde{b}_{n,u,\ell}^{2k}\|\|\tilde{a}_{s+\ell}\|\|\tilde{a}_{t-u}\|\|y_{t}\|.$

By the change of variables $m=s-\ell+1$, we have

$\displaystyle S_{1}(n)$
$\displaystyle=\sum_{t=n+1}^{\infty}\sum_{s=1}^{n}\sum_{m=1}^{s}\|\tilde{a}_{m-1}\|\|\tilde{a}_{t+m-s-1}\|\|y_{t}\|$
$\displaystyle=\sum_{t=n+1}^{\infty}\|y_{t}\|\sum_{m=1}^{n}\|\tilde{a}_{m-1}\|\sum_{s=m}^{n}\|\tilde{a}_{t+m-s-1}\|$
$\displaystyle\leq\left(\sum_{k=0}^{\infty}\|\tilde{a}_{k}\|\right)^{2}\sum_{t=n+1}^{\infty}\|y_{t}\|\quad\to\quad
0,\qquad n\to\infty.$

By (4.2) with $k=1$ or (4.3), we have

$\displaystyle S_{2}(n)$
$\displaystyle=\sum_{t=1}^{n}\sum_{u=1}^{t}\|\tilde{a}_{t-u}\|\|y_{t}\|\sum_{\ell=0}^{\infty}\|\beta_{n+1-u+\ell}\|\sum_{s=1}^{n}\|a_{n+1-s+\ell}\|$
$\displaystyle\leq\left(\sum_{s=1}^{\infty}\|a_{s}\|\right)\sum_{t=1}^{n}\sum_{u=1}^{t}\|\tilde{a}_{t-u}\|\|y_{t}\|F(n+1-u).$

Furthermore, by the change of variables $v=t-u+1$, we obtain

$\displaystyle\sum_{t=1}^{n}\sum_{u=1}^{t}\|\tilde{a}_{t-u}\|\|y_{t}\|F(n+1-u)$
$\displaystyle=\sum_{t=1}^{\infty}\sum_{u=1}^{t}\|\tilde{a}_{t-u}\|\|y_{t}\|1_{[0,n]}(t)F(n+1-u)$
$\displaystyle=\sum_{t=1}^{\infty}\sum_{v=1}^{t}\|\tilde{a}_{v-1}\|\|y_{t}\|1_{[0,n]}(t)F(n-t+v)$
$\displaystyle\leq\sum_{t=1}^{\infty}\sum_{v=1}^{\infty}\|\tilde{a}_{v-1}\|\|y_{t}\|1_{[0,n]}(t)F(n-t+v).$

Since

$\displaystyle\lim_{n\to\infty}\|\tilde{a}_{v-1}\|\|y_{t}\|1_{[0,n]}(t)F(n-t+v)=0,\qquad
t,v\in\mathbb{N},$
$\displaystyle\|\tilde{a}_{v-1}\|\|y_{t}\|1_{[0,n]}(t)F(n-t+v)\leq
F(1)\|\tilde{a}_{v-1}\|\|y_{t}\|,\qquad t,v\in\mathbb{N},$
$\displaystyle\sum_{t=1}^{\infty}\sum_{v=1}^{\infty}\|\tilde{a}_{v-1}\|\|y_{t}\|<\infty,$

the dominated convergence theorem yields

$\lim_{n\to\infty}\sum_{t=1}^{\infty}\sum_{v=1}^{\infty}\|\tilde{a}_{v-1}\|\|y_{t}\|1_{[0,n]}(t)F(n-t+v)=0,$

hence $\lim_{n\to\infty}S_{2}(n)=0$.

Choose $N\in\mathbb{N}$ such that $F(N+1)<1$. Then, by Lemma 4.1, we have, for
$n\geq N$,

$\displaystyle S_{3}(n)$
$\displaystyle=\sum_{t=1}^{n}\sum_{u=1}^{t}\|\tilde{a}_{t-u}\|\|y_{t}\|\sum_{k=1}^{\infty}\sum_{\ell=0}^{\infty}\|\tilde{b}_{n,u,\ell}^{2k+1}\|\sum_{s=1}^{n}\|a_{n+1-s+\ell}\|$
$\displaystyle\leq
F(1)\left(\sum_{s=1}^{\infty}\|a_{s}\|\right)\sum_{t=1}^{n}\|y_{t}\|\sum_{u=1}^{t}\|\tilde{a}_{t-u}\|\sum_{k=1}^{\infty}F(n+1)^{2k}$
$\displaystyle\leq
F(1)\left(\sum_{s=1}^{\infty}\|a_{s}\|\right)\left(\sum_{u=0}^{\infty}\|\tilde{a}_{u}\|\right)\left(\sum_{t=1}^{\infty}\|y_{t}\|\right)\frac{F(n+1)^{2}}{1-F(n+1)^{2}}.$

Thus $\lim_{n\to\infty}S_{3}(n)=0$. Similarly, we have, for $n\geq N$,

$S_{4}(n)\leq
F(1)\left(\sum_{s=0}^{\infty}\|\tilde{a}_{s}\|\right)^{2}\left(\sum_{t=1}^{\infty}\|y_{t}\|\right)\frac{F(n+1)}{1-F(n+1)^{2}},$

hence $\lim_{n\to\infty}S_{4}(n)=0$.

Combining, we obtain (1.14). ∎

##  5\. Closed-form formulas

In this section, we use Theorem 2.1 to derive closed-form formulas for
$T_{n}(w)^{-1}$ with rational symbol $w$ that corresponds to a $d$-variate
ARMA process. We assume that the symbol $w$ of $T_{n}(w)$ is of the form
(1.17) with $h:\mathbb{T}\to\mathbb{C}^{d\times d}$ satisfying (1.18). Then
$h$ is an outer function in $H_{2}^{d\times d}(\mathbb{T})$, and another outer
function $h_{\sharp}\in H_{2}^{d\times d}(\mathbb{T})$ that appears in (1.4)
also satisfies (1.18); see Section 6.2 in [16]. Notice that (1.17) with (1.18)
implies (1.2) and (1.3).

We can write $h(z)^{-1}$ in the form

(5.1)
$h(z)^{-1}=-\rho_{0,0}-\sum_{{\mu}=1}^{K}\sum_{j=1}^{m_{\mu}}\frac{1}{(1-\overline{p}_{\mu}z)^{j}}\rho_{{\mu},j}-\sum_{j=1}^{m_{0}}z^{j}\rho_{0,j},$

where

(5.2) $\left\\{\begin{aligned} &K\in\mathbb{N}\cup\\{0\\},\qquad
m_{\mu}\in\mathbb{N},\quad\mu\in\\{1,\dots,K\\},\qquad
m_{0}\in\mathbb{N}\cup\\{0\\},\\\
&p_{\mu}\in\mathbb{D}\setminus\\{0\\},\quad\mu\in\\{1,\dots,K\\},\qquad
p_{\mu}\neq p_{\nu},\quad\mu\neq\nu,\\\ &\rho_{{\mu},j}\in\mathbb{C}^{d\times
d},\quad\mu\in\\{0,\dots,K\\},\
j\in\\{1,\dots,m_{\mu}\\},\qquad\rho_{0,0}\in\mathbb{C}^{d\times d},\\\
&\rho_{{\mu},m_{\mu}}\neq 0,\quad\mu\in\\{1,\dots,K\\},\\\ &\rho_{0,m_{0}}\neq
0\quad\mbox{if}\ \ m_{0}\geq 1.\end{aligned}\right.$

Here the convention $\sum_{k=1}^{0}=0$ is adopted in the sums on the right-
hand side of (5.1). For example, if $m_{0}=0$, then

$h(z)^{-1}=-\rho_{0,0}-\sum_{{\mu}=1}^{K}\sum_{j=1}^{m_{\mu}}\frac{1}{(1-\overline{p}_{\mu}z)^{j}}\rho_{{\mu},j},$

while, if $K=0$, then

(5.3) $h(z)^{-1}=-\rho_{0,0}-\sum_{j=1}^{m_{0}}z^{j}\rho_{0,j}$

and the corresponding stationary process $\\{X_{k}\\}$ is a $d$-variate
AR$(m_{0})$ process.

######  Remark 5.1.

It should be noticed that the expression (5.1) with (5.2) is uniquely
determined, up to a constant unitary factor, from $\\{X_{k}\\}$ satisfying
(1.17) with (1.18) since so is $h$ in the factorization (1.17) with (1.18)
(see Section 2). Suppose that we start with a $d$-variate, causal and
invertible ARMA process $\\{X_{k}\\}$ in the sense of [4], that is, a
$\mathbb{C}^{d}$-valued, centered, weakly stationary process described by the
ARMA equation

$\Phi(B)X_{n}=\Psi(B)\xi_{n},\qquad n\in\mathbb{Z},$

where, for $r,s\in\mathbb{N}\cup\\{0\\}$ and
$\Phi_{i},\Psi_{j}\in\mathbb{C}^{d\times d}\ (i=1,\dots,r,\ j=1,\dots,s)$,

$\Phi(z)=I_{d}-z\Phi_{1}-\cdots-z^{r}\Phi_{r}\quad\mbox{and}\quad\Psi(z)=I_{d}-z\Psi_{1}-\cdots-z^{s}\Psi_{s}$

are $\mathbb{C}^{d\times d}$-valued polynomials satisfying $\det\Phi(z)\neq 0$
and $\det\Psi(z)\neq 0$ on $\overline{\mathbb{D}}$, $B$ is the backward shift
operator defined by $BX_{m}=X_{m-1}$, and $\\{\xi_{k}:k\in\mathbb{Z}\\}$ is a
$d$-variate white noise, that is, a $d$-variate, centered process such that
$E[\xi_{n}\xi_{m}^{*}]=\delta_{nm}V$ for some positive-definite
$V\in\mathbb{C}^{d\times d}$. Notice that the pair $(\Phi(z),\Psi(z))$ is not
uniquely determined from $\\{X_{k}\\}$; for example, we can replace
$(\Phi(z),\Psi(z))$ by $((2-z)\Phi(z),(2-z)\Psi(z))$. However, if we put
$h(z)=\Phi(z)^{-1}\Psi(z)V^{1/2}$, then $h$ is an outer function belonging to
$H_{2}^{d\times d}(\mathbb{T})$ and satisfies (1.17) for the spectral density
$w$ of $\\{X_{k}\\}$. Therefore, $h$ is uniquely determined, up to a constant
unitary factor, from $\\{X_{k}\\}$. In particular, the expression (5.1) with
(5.2) for $h$ is also uniquely determined, up to a constant unitary factor,
from $\\{X_{k}\\}$. From these observations and the results in [13] and this
paper, we are led to the idea of parameterizing the ARMA processes by the
expression (5.1) with (5.2) (see Remark 8 in [13]). This point will be
discussed in future work.

By Theorem 2 in [13], $h_{\sharp}^{-1}$ has the same $m_{0}$ and the same
poles with the same multiplicities as $h^{-1}$, that is, for $m_{0}$, $K$ and
$(p_{1},m_{1}),\dots,(p_{K},m_{K})$ in $(\ref{eq:hinverse162})$ with
$(\ref{eq:hinverse163})$, $h_{\sharp}^{-1}$ has the form

(5.4)
$h_{\sharp}(z)^{-1}=-\rho_{0,0}^{\sharp}-\sum_{{\mu}=1}^{K}\sum_{j=1}^{m_{\mu}}\frac{1}{(1-\overline{p}_{\mu}z)^{j}}\rho_{{\mu},j}^{\sharp}-\sum_{j=1}^{m_{0}}z^{j}\rho_{0,j}^{\sharp},$

where

$\left\\{\begin{aligned} &\rho_{{\mu},j}^{\sharp}\in\mathbb{C}^{d\times
d},\quad\mu\in\\{0,\dots,K\\},\
j\in\\{1,\dots,m_{\mu}\\},\qquad\rho_{0,0}^{\sharp}\in\mathbb{C}^{d\times
d},\\\ &\rho_{{\mu},m_{\mu}}^{\sharp}\neq 0,\quad\mu\in\\{1,\dots,K\\},\\\
&\rho_{0,m_{0}}^{\sharp}\neq 0\quad\mbox{if}\ \ m_{0}\geq
1.\end{aligned}\right.$

Notice that if $d=1$, then we can take $h_{\sharp}=h$, hence
$\rho_{0,0}=\rho_{0,0}^{\sharp}$ and $\rho_{{\mu},j}=\rho_{{\mu},j}^{\sharp}$
for $\mu\in\\{1,\dots,K\\}$ and $j\in\\{1,\dots,m_{\mu}\\}$.

Recall $\tilde{h}$ from (2.1). From (5.4), we have

$\tilde{h}(z)^{-1}=-\tilde{\rho}_{0,0}-\sum_{{\mu}=1}^{K}\sum_{j=1}^{m_{\mu}}\frac{1}{(1-p_{\mu}z)^{j}}\tilde{\rho}_{{\mu},j}-\sum_{j=1}^{m_{0}}z^{j}\tilde{\rho}_{0,j},$

where

$\tilde{\rho}_{0,0}:=(\rho_{0,0}^{\sharp})^{*},\qquad\tilde{\rho}_{\mu,j}:=(\rho_{\mu,j}^{\sharp})^{*},\quad\mu\in\\{0,\dots,K\\},\
j\in\\{1,\dots,m_{\mu}\\}.$

Recall the sequences $\\{a_{k}\\}$ and $\\{\tilde{a}_{k}\\}$ from (2.3) and
(2.5), respectively. We have

(5.5) $\displaystyle a_{n}$
$\displaystyle=\sum_{{\mu}=1}^{K}\sum_{j=1}^{m_{\mu}}\binom{n+j-1}{j-1}\overline{p}_{\mu}^{n}\rho_{{\mu},j},\qquad
n\geq m_{0}+1,$ (5.6) $\displaystyle\tilde{a}_{n}$
$\displaystyle=\sum_{{\mu}=1}^{K}\sum_{j=1}^{m_{\mu}}\binom{n+j-1}{j-1}p_{\mu}^{n}\tilde{\rho}_{{\mu},j},\qquad
n\geq m_{0}+1$

and

(5.7) $\displaystyle a_{n}$
$\displaystyle=\rho_{0,n}+\sum_{{\mu}=1}^{K}\sum_{j=1}^{m_{\mu}}\binom{n+j-1}{j-1}\overline{p}_{\mu}^{n}\rho_{{\mu},j},\qquad
n\in\\{0,\dots,m_{0}\\},$ (5.8) $\displaystyle\tilde{a}_{n}$
$\displaystyle=\tilde{\rho}_{0,n}+\sum_{{\mu}=1}^{K}\sum_{j=1}^{m_{\mu}}\binom{n+j-1}{j-1}p_{\mu}^{n}\tilde{\rho}_{{\mu},j},\qquad
n\in\\{0,\dots,m_{0}\\},$

where the convention $\binom{0}{0}=1$ is adopted; see Proposition 4 in [13].

We first consider the case of $K=0$ that corresponds to a $d$-variate
AR$(m_{0})$ process. As can be seen from the following theorem, in this case,
we have simple closed-form formulas for $T_{n}(w)^{-1}$.

######  Theorem 5.1.

We assume (1.17), (1.18) and $K=0$ for $K$ in (5.1). Thus we assume (5.3).
Then the following four assertions hold.

  1. (i)

For $n\geq m_{0}+1$, $s\in\\{1,\dots,n\\}$ and $t\in\\{1,\dots,n-m_{0}\\}$, we
have

(5.9) $\left(T_{n}(w)^{-1}\right)^{s,t}=\sum_{\lambda=1}^{s\wedge
t}\tilde{a}_{s-\lambda}^{*}\tilde{a}_{t-\lambda}.$

  2. (ii)

For $n\geq m_{0}+1$, $s\in\\{1,\dots,n-m_{0}\\}$ and $t\in\\{1,\dots,n\\}$, we
have

(5.10) $\left(T_{n}(w)^{-1}\right)^{s,t}=\sum_{\lambda=1}^{s\wedge
t}\tilde{a}_{s-\lambda}^{*}\tilde{a}_{t-\lambda}.$

  3. (iii)

For $n\geq m_{0}+1$, $s\in\\{1,\dots,n\\}$ and $t\in\\{m_{0}+1,\dots,n\\}$, we
have

(5.11) $\left(T_{n}(w)^{-1}\right)^{s,t}=\sum_{\lambda=s\vee
t}^{n}a_{\lambda-s}^{*}a_{\lambda-t}.$

  4. (iv)

For $n\geq m_{0}+1$, $s\in\\{m_{0}+1,\dots,n\\}$ and $t\in\\{1,\dots,n\\}$, we
have

(5.12) $\left(T_{n}(w)^{-1}\right)^{s,t}=\sum_{\lambda=s\vee
t}^{n}a_{\lambda-s}^{*}a_{\lambda-t}.$

###### Proof.

For $w$ satisfying (1.17), (1.18) and $K=0$, let $\\{X_{k}\\}$,
$\\{X^{\prime}_{k}\\}$, $\\{\varepsilon_{k}\\}$ and
$\\{\tilde{\varepsilon}_{k}\\}$ be as in Section 3.

(i) By (5.4) with $K=0$, we have $\tilde{a}_{0}=\tilde{\rho}_{0}$,
$\tilde{a}_{k}=\tilde{\rho}_{0,k}$ for $k\in\\{1,\dots,m_{0}\\}$ and
$\tilde{a}_{k}=0$ for $k\geq m_{0}+1$. In particular, we have
$\sum_{k=0}^{m_{0}}\tilde{a}_{k}X_{u+k}+\tilde{\varepsilon}_{-u}=0$ for
$u\in\mathbb{Z}$; see (2.15) in [16]. This implies
$\tilde{\varepsilon}_{-u}\in V_{[1,n]}^{X}$, or
$P_{[1,n]}^{\bot}\tilde{\varepsilon}_{-u}=0$, for $u\in\\{1,\dots,n-m_{0}\\}$.
Therefore, (5.9) follows from Theorem 3.1 and (3.8).

(iii) By (5.3), we have $a_{0}=\rho_{0,0}$, $a_{k}=\rho_{0,k}$ for
$k\in\\{1,\dots,m_{0}\\}$ and $a_{k}=0$ for $k\geq m_{0}+1$. In particular,
$\sum_{k=0}^{m_{0}}a_{k}X_{u-k}+\varepsilon_{u}=0$ for $u\in\mathbb{Z}$; see
(2.15) in [16]. This implies $\varepsilon_{u}\in V_{[1,n]}^{X}$, or
$P_{[1,n]}^{\bot}\varepsilon_{u}=0$, for $u\in\\{m_{0}+1,\dots,n\\}$.
Therefore, (5.11) follows from Theorem 3.1 and (3.7).

(ii), (iv) By (2.9), (ii) and (iv) follow from (i) and (iii), respectively. ∎

We turn to the case of $K\geq 1$. In what follows in this section, for $K$ in
(5.1), we assume

$K\geq 1.$

For $m_{1},\dots,m_{K}$ in (5.1), we define $M\in\mathbb{N}$ by

(5.13) $M:=\sum_{\mu=1}^{K}m_{\mu}.$

For $\mu\in\\{1,\dots,K\\}$, $p_{\mu}$ in (5.1) and $i\in\mathbb{N}$, we
define $p_{{\mu},i}:\mathbb{Z}\to\mathbb{C}^{d\times d}$ by

(5.14) $p_{{\mu},i}(k):=\binom{k}{i-1}p_{\mu}^{k-i+1}I_{d},\qquad
k\in\mathbb{Z}.$

Notice that

$p_{\mu,i}(0)=\binom{0}{i-1}p_{\mu}^{-i+1}I_{d}=\delta_{i,1}I_{d}.$

For $n\in\mathbb{Z}$, we also define $\mathbf{p}_{n}\in\mathbb{C}^{dM\times
d}$ by the following block representation:

$\displaystyle\mathbf{p}_{n}$
$\displaystyle:=(p_{1,1}(n),\dots,p_{1,m_{1}}(n)\ |\
p_{2,1}(n),\dots,p_{2,m_{2}}(n)\ |$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdots|\
p_{K,1}(n),\dots,p_{K,m_{K}}(n))^{\top}.$

Notice that

$\mathbf{p}_{0}=(I_{d},0,\dots,0\ |\ I_{d},0,\dots,0|\cdots|\
I_{d},0,\dots,0)^{\top}\in\mathbb{C}^{dM\times d}.$

We define $\Lambda\in\mathbb{C}^{dM\times dM}$ by

$\Lambda:=\sum_{\ell=0}^{\infty}\mathbf{p}_{\ell}\mathbf{p}_{\ell}^{*}.$

For ${\mu},{\nu}\in\\{1,2,\dots,K\\}$, we define
$\Lambda^{{\mu},{\nu}}\in\mathbb{C}^{dm_{\mu}\times dm_{\nu}}$ by the block
representation

$\Lambda^{{\mu},{\nu}}:=\left(\begin{matrix}\lambda^{{\mu},{\nu}}(1,1)&\lambda^{{\mu},{\nu}}(1,2)&\cdots&\lambda^{{\mu},{\nu}}(1,m_{\nu})\cr\lambda^{{\mu},{\nu}}(2,1)&\lambda^{{\mu},{\nu}}(2,2)&\cdots&\lambda^{{\mu},{\nu}}(2,m_{\nu})\cr\vdots&\vdots&&\vdots\cr\lambda^{\mu,\nu}(m_{\mu},1)&\lambda^{\mu,\nu}(m_{\mu},2)&\cdots&\lambda^{\mu,\nu}(m_{\mu},m_{\nu})\cr\end{matrix}\right),$

where, for $i\in\\{1,\dots,m_{\mu}\\}$ and $j\in\\{1,\dots,m_{\nu}\\}$,

$\lambda^{{\mu},{\nu}}(i,j):=\sum_{r=0}^{j-1}\binom{i-1}{r}\binom{i+j-r-2}{i-1}\frac{p_{\mu}^{j-r-1}\overline{p}_{\nu}^{i-r-1}}{(1-p_{\mu}\overline{p}_{\nu})^{i+j-r-1}}I_{d}\in\mathbb{C}^{d\times
d}.$

Then, by Lemma 3 in [13], the matrix $\Lambda$ has the following block
representation:

$\Lambda=\left(\begin{matrix}\Lambda^{1,1}&\Lambda^{1,2}&\cdots&\Lambda^{1,K}\cr\Lambda^{2,1}&\Lambda^{2,2}&\cdots&\Lambda^{2,K}\cr\vdots&\vdots&\ddots&\vdots\cr\Lambda^{K,1}&\Lambda^{K,2}&\cdots&\Lambda^{K,K}\cr\end{matrix}\right).$

We define, for $\mu\in\\{1,\dots,K\\}$ and $j\in\\{1,\dots,m_{\mu}\\}$,

(5.15) $\theta_{{\mu},j}:=-\lim_{z\to
p_{\mu}}\frac{1}{(m_{\mu}-j)!}\frac{d^{m_{\mu}-j}}{dz^{m_{\mu}-j}}\left\\{(z-p_{\mu})^{m_{\mu}}h_{\sharp}(z)h^{\dagger}(z)^{-1}\right\\}\in\mathbb{C}^{d\times
d},$

where

(5.16) $h^{\dagger}(z):=h(1/\overline{z})^{*}.$

We define $\Theta\in\mathbb{C}^{dM\times dM}$ by the block representation

$\Theta:=\left(\begin{matrix}\Theta_{1}&0&\cdots&0\cr
0&\Theta_{2}&\cdots&0\cr\vdots&\vdots&\ddots&\vdots\cr
0&0&\cdots&\Theta_{K}\end{matrix}\right),$

where, for $\mu\in\\{1,\dots,K\\}$, $\Theta_{\mu}\in\mathbb{C}^{dm_{\mu}\times
dm_{\mu}}$ is defined by

$\Theta_{\mu}:=\left(\begin{matrix}\theta_{{\mu},1}&\theta_{{\mu},2}&\cdots&\theta_{{\mu},m_{\mu}-1}&\theta_{{\mu},m_{\mu}}\cr\theta_{{\mu},2}&\theta_{{\mu},3}&\cdots&\theta_{{\mu},m_{\mu}}&\cr\vdots&\vdots&&&\cr\theta_{{\mu},m_{\mu}-1}&\theta_{{\mu},m_{\mu}}&&&\cr\theta_{{\mu},m_{\mu}}&&&&\mbox{\huge
0}\end{matrix}\right)$

using $\theta_{\mu,j}$ in (5.15) with (5.16).

For $n\in\mathbb{Z}$, we define $\Pi_{n}\in\mathbb{C}^{dM\times dM}$ by the
block representation

$\Pi_{n}:=\left(\begin{matrix}\Pi_{1,n}&0&\cdots&0\cr
0&\Pi_{2,n}&\cdots&0\cr\vdots&\vdots&\ddots&\vdots\cr
0&0&\cdots&\Pi_{K,n}\end{matrix}\right),$

where, for $\mu\in\\{1,\dots,K\\}$ and $n\in\mathbb{Z}$,
$\Pi_{\mu,n}\in\mathbb{C}^{dm_{\mu}\times dm_{\mu}}$ is defined by

$\Pi_{{\mu},n}:=\left(\begin{matrix}p_{{\mu},1}(n)&p_{{\mu},2}(n)&p_{{\mu},3}(n)&\cdots&p_{{\mu},m_{\mu}}(n)\cr&p_{{\mu},1}(n)&p_{{\mu},2}(n)&\cdots&p_{{\mu},m_{\mu}-1}(n)\cr&&\ddots&\ddots&\vdots\cr&&&\ddots&p_{{\mu},2}(n)\cr\mbox{\huge
0}&&&&p_{{\mu},1}(n)\end{matrix}\right)$

using $p_{{\mu},i}(n)$ in (5.14).

The next lemma slightly extends Lemma 17 in [13].

######  Lemma 5.1.

We assume (1.17), (1.18) and $K\geq 1$ for $K$ in (5.1). Then, for
$n,k,\ell\in\mathbb{Z}$ such that $n+k+\ell\geq m_{0}$, we have

$\beta_{n+k+\ell+1}^{*}=\mathbf{p}_{\ell}^{\top}\Pi_{n}\Theta\mathbf{p}_{k},$

hence

$\beta_{n+k+\ell+1}=\mathbf{p}_{k}^{*}(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}.$

The proof of Lemma 5.1 is almost the same as that of Lemma 17 in [13], hence
we omit it.

For $n\in\mathbb{Z}$, we define $G_{n},\tilde{G}_{n}\in\mathbb{C}^{dM\times
dM}$ by

$G_{n}:=\Pi_{n}\Theta\Lambda,\qquad\tilde{G}_{n}:=(\Pi_{n}\Theta)^{*}\Lambda^{\top}.$

######  Lemma 5.2.

We assume (1.17), (1.18) and $K\geq 1$ for $K$ in (5.1). Then the following
two assertions hold.

  1. (i)

We assume $n\geq u\geq m_{0}+1$. Then, for $k\in\mathbb{N}$ and
$\ell\in\mathbb{N}\cup\\{0\\}$, we have

(5.17) $\displaystyle b_{n,u,\ell}^{2k-1}$
$\displaystyle=\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell},$
(5.18) $\displaystyle b_{n,u,\ell}^{2k}$
$\displaystyle=\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}\tilde{G}_{n}\Pi_{n}\Theta\mathbf{p}_{\ell}.$

  2. (ii)

We assume $1\leq u\leq n-m_{0}$. Then, for $k\in\mathbb{N}$ and
$\ell\in\mathbb{N}\cup\\{0\\}$, we have

(5.19) $\displaystyle\tilde{b}_{n,u,\ell}^{2k-1}$
$\displaystyle=\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}\Pi_{n}\Theta\mathbf{p}_{\ell},$
(5.20) $\displaystyle\tilde{b}_{n,u,\ell}^{2k}$
$\displaystyle=\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}G_{n}(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}.$

The proof of Lemma 5.2 will be given in the Appendix.

For $n\in\mathbb{N}$ and ${\mu},{\nu}\in\\{1,2,\dots,K\\}$, we define
$\Xi_{n}^{{\mu},{\nu}}\in\mathbb{C}^{dm_{\mu}\times dm_{\nu}}$ by the block
representation

$\Xi_{n}^{{\mu},{\nu}}:=\left(\begin{matrix}\xi_{n}^{{\mu},{\nu}}(1,1)&\xi_{n}^{{\mu},{\nu}}(1,2)&\cdots&\xi_{n}^{{\mu},{\nu}}(1,m_{\nu})\cr\xi_{n}^{{\mu},{\nu}}(2,1)&\xi_{n}^{{\mu},{\nu}}(2,2)&\cdots&\xi_{n}^{{\mu},{\nu}}(2,m_{\nu})\cr\vdots&\vdots&&\vdots\cr\xi_{n}^{\mu,\nu}(m_{\mu},1)&\xi_{n}^{\mu,\nu}(m_{\mu},2)&\cdots&\xi_{n}^{\mu,\nu}(m_{\mu},m_{\nu})\cr\end{matrix}\right),$

where, for $n\in\mathbb{N}$, $i\in\\{1,\dots,m_{\mu}\\}$ and
$j\in\\{1,\dots,m_{\nu}\\}$, $\xi_{n}^{{\mu},{\nu}}(i,j)\in\mathbb{C}^{d\times
d}$ is defined by

$\xi_{n}^{{\mu},{\nu}}(i,j):=\sum_{r=0}^{j-1}\binom{n+i+j-2}{r}\binom{i+j-r-2}{i-1}\frac{p_{\mu}^{j-r-1}\overline{p}_{\nu}^{n+i+j-r-2}}{(1-p_{\mu}\overline{p}_{\nu})^{i+j-r-1}}I_{d}.$

For $n\in\mathbb{N}$, we define $\Xi_{n}\in\mathbb{C}^{dM\times dM}$ by

$\Xi_{n}:=\left(\begin{matrix}\Xi_{n}^{1,1}&\Xi_{n}^{1,2}&\cdots&\Xi_{n}^{1,K}\cr\Xi_{n}^{2,1}&\Xi_{n}^{2,2}&\cdots&\Xi_{n}^{2,K}\cr\vdots&\vdots&\ddots&\vdots\cr\Xi_{n}^{K,1}&\Xi_{n}^{K,2}&\cdots&\Xi_{n}^{K,K}\cr\end{matrix}\right).$

We also define $\rho\in\mathbb{C}^{dM\times d}$ and
$\tilde{\rho}\in\mathbb{C}^{dM\times d}$ by the block representations

$\rho:=(\rho_{1,1}^{\top},\dots,\rho_{1,m_{1}}^{\top}\ |\
\rho_{2,1}^{\top},\dots,\rho_{2,m_{2}}^{\top}\ |\ \cdots\ |\
\rho_{K,1}^{\top},\dots,\rho_{K,m_{K}}^{\top})^{\top}$

and

$\displaystyle\tilde{\rho}$
$\displaystyle:=(\tilde{\rho}_{1,1}^{\top},\dots,\tilde{\rho}_{1,m_{1}}^{\top}\
|\ \tilde{\rho}_{2,1}^{\top},\dots,\tilde{\rho}_{2,m_{2}}^{\top}\ |\ \cdots\
|\ \tilde{\rho}_{K,1}^{\top},\dots,\tilde{\rho}_{K,m_{K}}^{\top})^{\top}$
$\displaystyle=\left(\overline{\rho_{1,1}^{\sharp}},\dots,\overline{\rho_{1,m_{1}}^{\sharp}}\
|\ \overline{\rho_{2,1}^{\sharp}},\dots,\overline{\rho_{2,m_{2}}^{\sharp}}\ |\
\cdots\ |\
\overline{\rho_{K,1}^{\sharp}},\dots,\overline{\rho_{K,m_{K}}^{\sharp}}\right)^{\top},$

respectively. For $n\in\mathbb{N}$, we define
$v_{n},\tilde{v}_{n}\in\mathbb{C}^{dM\times d}$ by

$v_{n}:=\sum_{\ell=0}^{\infty}\mathbf{p}_{\ell}a_{n+\ell},\qquad\tilde{v}_{n}:=\sum_{\ell=0}^{\infty}\overline{\mathbf{p}}_{\ell}\tilde{a}_{n+\ell}.$

Then, by Lemma 5 in [13], we have

$v_{n}=\Xi_{n}\rho,\qquad\tilde{v}_{n}=\overline{\Xi}_{n}\tilde{\rho},\qquad
n\geq m_{0}+1.$

Moreover, if $m_{0}\geq 1$, then we have

$v_{n}=\Xi_{n}\rho+\sum_{\ell=0}^{m_{0}-n}\mathbf{p}_{\ell}\rho_{0,n+\ell},\quad\tilde{v}_{n}=\overline{\Xi}_{n}\tilde{\rho}+\sum_{\ell=0}^{m_{0}-n}\overline{\mathbf{p}}_{\ell}\tilde{\rho}_{0,n+\ell},\quad
n\in\\{1,\dots,m_{0}\\}.$

For $n\in\mathbb{Z}$, we define $w_{n},\tilde{w}_{n}\in\mathbb{C}^{dM\times
d}$ by

$w_{n}:=\sum_{\ell=0}^{\infty}\mathbf{p}_{\ell-n}a_{\ell},\qquad\tilde{w}_{n}:=\sum_{\ell=0}^{\infty}\overline{\mathbf{p}}_{\ell-n}\tilde{a}_{\ell}.$

To give closed-form expressions for $w_{n}$ and $\tilde{w}_{n}$, we introduce
some matrices. For $n\in\mathbb{Z}$ and ${\mu},{\nu}\in\\{1,2,\dots,K\\}$, we
define $\Phi_{n}^{{\mu},{\nu}}\in\mathbb{C}^{dm_{\mu}\times dm_{\nu}}$ by the
block representation

$\Phi_{n}^{{\mu},{\nu}}:=\left(\begin{matrix}\varphi_{n}^{{\mu},{\nu}}(1,1)&\varphi_{n}^{{\mu},{\nu}}(1,2)&\cdots&\varphi_{n}^{{\mu},{\nu}}(1,m_{\nu})\cr\varphi_{n}^{{\mu},{\nu}}(2,1)&\varphi_{n}^{{\mu},{\nu}}(2,2)&\cdots&\varphi_{n}^{{\mu},{\nu}}(2,m_{\nu})\cr\vdots&\vdots&&\vdots\cr\varphi_{n}^{\mu,\nu}(m_{\mu},1)&\varphi_{n}^{\mu,\nu}(m_{\mu},2)&\cdots&\varphi_{n}^{\mu,\nu}(m_{\mu},m_{\nu})\cr\end{matrix}\right),$

where, for $n\in\mathbb{Z}$, $i=1,\dots,m_{\mu}$ and $j=1,\dots,m_{\nu}$,
$\varphi_{n}^{{\mu},{\nu}}(i,j)\in\mathbb{C}^{d\times d}$ is defined by

$\varphi_{n}^{{\mu},{\nu}}(i,j):=\sum_{q=0}^{i-1}\sum_{r=0}^{j-1}\binom{j-1}{r}\binom{r+q}{q}\binom{r-n}{i-q-1}\frac{p_{\mu}^{r+q+1-i-n}\overline{p}_{\nu}^{r+q}}{(1-p_{\mu}\overline{p}_{\nu})^{r+q+1}}I_{d}.$

For $n\in\mathbb{Z}$, we define $\Phi_{n}\in\mathbb{C}^{dM\times dM}$ by

$\Phi_{n}:=\left(\begin{matrix}\Phi_{n}^{1,1}&\Phi_{n}^{1,2}&\cdots&\Phi_{n}^{1,K}\cr\Phi_{n}^{2,1}&\Phi_{n}^{2,2}&\cdots&\Phi_{n}^{2,K}\cr\vdots&\vdots&\ddots&\vdots\cr\Phi_{n}^{K,1}&\Phi_{n}^{K,2}&\cdots&\Phi_{n}^{K,K}\cr\end{matrix}\right).$

Here are closed-form expressions for $w_{n}$ and $\tilde{w}_{n}$.

######  Lemma 5.3.

We have

$\displaystyle w_{n}$
$\displaystyle=\Phi_{n}\rho+\sum_{\ell=0}^{m_{0}}\mathbf{p}_{\ell-n}\rho_{0,\ell},\qquad
n\in\mathbb{Z},$ $\displaystyle\tilde{w}_{n}$
$\displaystyle=\overline{\Phi}_{n}\tilde{\rho}+\sum_{\ell=0}^{m_{0}}\overline{\mathbf{p}}_{\ell-n}\tilde{\rho}_{0,\ell},\qquad
n\in\mathbb{Z}.$

The proof of Lemma 5.3 will be given in the Appendix.

Recall $M$ from (5.13). For $n\in\mathbb{N}$ and $s\in\\{1,\dots,n\\}$, we
define

$\displaystyle\ell_{n,s}$
$\displaystyle:=\\{w_{n+1-s}-v_{n+1-s}\\}^{*}(I_{dM}-\tilde{G}_{n}G_{n})^{-1}\in\mathbb{C}^{d\times
dM},$ $\displaystyle\tilde{\ell}_{n,s}$
$\displaystyle:=\\{\tilde{w}_{s}-\tilde{v}_{s}\\}^{*}(I_{dM}-G_{n}\tilde{G}_{n})^{-1}\in\mathbb{C}^{d\times
dM},$ $\displaystyle\ r_{n,s}$
$\displaystyle:=(\Pi_{n}\Theta)^{*}\tilde{v}_{s}+\tilde{G}_{n}\Pi_{n}\Theta
v_{n+1-s}\in\mathbb{C}^{dM\times d}$

and

$\tilde{r}_{n,s}:=\Pi_{n}\Theta
v_{n+1-s}+G_{n}(\Pi_{n}\Theta)^{*}\tilde{v}_{s}\in\mathbb{C}^{dM\times d}.$

Here are closed-form formulas for $(T_{n}(w))^{-1}$ with $w$ satisfying (1.18)
and $K\geq 1$.

######  Theorem 5.2.

We assume (1.17), (1.18) and $K\geq 1$ for $K$ in (5.1). Then the following
four assertions hold.

  1. (i)

For $n\geq m_{0}+1$, $s\in\\{1,\dots,n\\}$ and $t\in\\{1,\dots,n-m_{0}\\}$, we
have

$\left(T_{n}(w)^{-1}\right)^{s,t}=\tilde{r}_{n,s}^{*}\tilde{\ell}_{n,t}^{*}+\sum_{\lambda=1}^{s\wedge
t}\tilde{a}_{s-\lambda}^{*}\tilde{a}_{t-\lambda}.$

  2. (ii)

For $n\geq m_{0}+1$, $s\in\\{1,\dots,n-m_{0}\\}$ and $t\in\\{1,\dots,n\\}$, we
have

$\left(T_{n}(w)^{-1}\right)^{s,t}=\tilde{\ell}_{n,s}\tilde{r}_{n,t}+\sum_{\lambda=1}^{s\wedge
t}\tilde{a}_{s-\lambda}^{*}\tilde{a}_{t-\lambda}.$

  3. (iii)

For $n\geq m_{0}+1$, $s\in\\{1,\dots,n\\}$ and $t\in\\{m_{0}+1,\dots,n\\}$, we
have

$\left(T_{n}(w)^{-1}\right)^{s,t}=r_{n,s}^{*}\ell_{n,t}^{*}+\sum_{\lambda=s\vee
t}^{n}a_{\lambda-s}^{*}a_{\lambda-t}.$

  4. (iv)

For $n\geq m_{0}+1$, $s\in\\{m_{0}+1,\dots,n\\}$ and $t\in\\{1,\dots,n\\}$, we
have

$\left(T_{n}(w)^{-1}\right)^{s,t}=\ell_{n,s}r_{n,t}+\sum_{\lambda=s\vee
t}^{n}a_{\lambda-s}^{*}a_{\lambda-t}.$

###### Proof.

(i) We assume $n\geq m_{0}+1$, $s\in\\{1,\dots,n\\}$ and
$t\in\\{1,\dots,n-m_{0}\\}$. Then, by Lemma 5.2 (ii) above and Lemma 19 in
[13], we have

$\displaystyle\sum_{u=1}^{t}\sum_{k=1}^{\infty}\left\\{\sum_{\lambda=0}^{\infty}\tilde{b}_{n,u,\lambda}^{2k-1}a_{n+1-s+\lambda}\right\\}^{*}\tilde{a}_{t-u}$
$\displaystyle\qquad=\sum_{u=1}^{t}\sum_{k=1}^{\infty}\left\\{\sum_{\lambda=0}^{\infty}\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}\Pi_{n}\Theta\mathbf{p}_{\lambda}a_{n+1-s+\lambda}\right\\}^{*}\tilde{a}_{t-u}$
$\displaystyle\qquad=\sum_{u=1}^{t}\sum_{k=1}^{\infty}\left\\{\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}\Pi_{n}\Theta
v_{n+1-s}\right\\}^{*}\tilde{a}_{t-u}$
$\displaystyle\qquad=\sum_{u=1}^{t}\left\\{\mathbf{p}_{-u}^{\top}(I_{dM}-G_{n}\tilde{G}_{n})^{-1}\Pi_{n}\Theta
v_{n+1-s}\right\\}^{*}\tilde{a}_{t-u}$
$\displaystyle\qquad=v_{n+1-s}^{*}(\Pi_{n}\Theta)^{*}(I_{dM}-\tilde{G}_{n}^{*}G_{n}^{*})^{-1}\sum_{u=1}^{t}\overline{\mathbf{p}}_{-u}\tilde{a}_{t-u}.$

Similarly, by Lemma 5.2 (ii) above and Lemma 19 in [13],

$\sum_{u=1}^{t}\sum_{k=1}^{\infty}\left\\{\sum_{\lambda=0}^{\infty}\tilde{b}_{n,u,\lambda}^{2k}\tilde{a}_{s+\lambda}\right\\}^{*}\tilde{a}_{t-u}=\tilde{v}_{s}^{*}\Pi_{n}\Theta
G_{n}^{*}(I_{dM}-\tilde{G}_{n}^{*}G_{n}^{*})^{-1}\sum_{u=1}^{t}\overline{\mathbf{p}}_{-u}\tilde{a}_{t-u}.$

However,
$\sum_{u=1}^{t}\overline{\mathbf{p}}_{-u}\tilde{a}_{t-u}=\sum_{\lambda=0}^{\infty}\overline{\mathbf{p}}_{\lambda-t}\tilde{a}_{\lambda}-\sum_{\lambda=0}^{\infty}\overline{\mathbf{p}}_{\lambda}\tilde{a}_{t+\lambda}=\tilde{w}_{t}-\tilde{v}_{t}$.
Therefore, the assertion (i) follows from Theorem 2.1 (i).

(iii) We assume $n\geq m_{0}+1$, $s\in\\{1,\dots,n\\}$ and
$t\in\\{m_{0}+1,\dots,n\\}$. Then, by Lemma 5.2 (i) above and Lemma 19 in
[13], we have

$\displaystyle\sum_{u=t}^{n}\sum_{k=1}^{\infty}\left\\{\sum_{\lambda=0}^{\infty}b_{n,u,\lambda}^{2k-1}\tilde{a}_{s+\lambda}\right\\}^{*}a_{u-t}$
$\displaystyle\qquad=\sum_{u=t}^{n}\sum_{k=1}^{\infty}\left\\{\sum_{\lambda=0}^{\infty}\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\lambda}\tilde{a}_{s+\lambda}\right\\}^{*}a_{u-t}$
$\displaystyle\qquad=\sum_{u=t}^{n}\sum_{k=1}^{\infty}\left\\{\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}(\Pi_{n}\Theta)^{*}\tilde{v}_{s}\right\\}^{*}a_{u-t}$
$\displaystyle\qquad=\sum_{u=t}^{n}\left\\{\mathbf{p}_{u-n-1}^{*}(I_{dM}-\tilde{G}_{n}G_{n})^{-1}(\Pi_{n}\Theta)^{*}\tilde{v}_{s}\right\\}^{*}a_{u-t}$
$\displaystyle\qquad=\tilde{v}_{s}^{*}\Pi_{n}\Theta(I_{dM}-G_{n}^{*}\tilde{G}_{n}^{*})^{-1}\sum_{u=t}^{n}\mathbf{p}_{u-n-1}a_{u-t}.$

Similarly, by Lemma 5.2 (i) above and Lemma 19 in [13], we have

$\displaystyle\sum_{u=t}^{n}\sum_{k=1}^{\infty}\left\\{\sum_{\lambda=0}^{\infty}b_{n,u,\lambda}^{2k}a_{n+1-s+\lambda}\right\\}^{*}a_{u-t}$
$\displaystyle\qquad=v_{n+1-s}^{*}(\Pi_{n}\Theta)^{*}\tilde{G}_{n}^{*}(I_{dM}-G_{n}^{*}\tilde{G}_{n}^{*})^{-1}\sum_{u=t}^{n}\mathbf{p}_{u-n-1}a_{u-t}.$

However, $\sum_{u=t}^{n}\mathbf{p}_{u-n-1}a_{u-t}=w_{n+1-t}-v_{n+1-t}$.
Therefore, the assertion (ii) follows from Theorem 2.1 (ii).

(ii), (iv) By (2.9), (ii) and (iv) follow from (i) and (iii), respectively. ∎

######  Example 5.1.

Suppose that $K\geq 1$, $m_{\mu}=1$ for $\mu\in\\{1,\dots,K\\}$ and $m_{0}=0$.
Then,

$h(z)^{-1}=-\rho_{0,0}-\sum_{{\mu}=1}^{K}\frac{1}{1-\overline{p}_{\mu}z}\rho_{{\mu},1},\qquad
h_{\sharp}(z)^{-1}=-\rho^{\sharp}_{0,0}-\sum_{{\mu}=1}^{K}\frac{1}{1-\overline{p}_{\mu}z}\rho^{\sharp}_{{\mu},1}.$

We have

$\displaystyle\mathbf{p}_{n}^{\top}=(p_{1}^{n}I_{d},\dots,p_{K}^{n}I_{d})\in\mathbb{C}^{d\times
dK},\quad n\in\mathbb{Z},$
$\displaystyle\rho^{\top}=(\rho_{1,1}^{\top},\rho_{2,1}^{\top},\dots,\rho_{K,1}^{\top})\in\mathbb{C}^{dK\times
d},\quad\tilde{\rho}^{\top}=\left(\overline{\rho^{\sharp}_{1,1}},\overline{\rho^{\sharp}_{2,1}},\dots,\overline{\rho^{\sharp}_{K,1}}\right)\in\mathbb{C}^{dK\times
d}.$

We also have

$\displaystyle\Theta$
$\displaystyle=\left(\begin{matrix}p_{1}h_{\sharp}(p_{1})\rho_{1,1}^{*}&0&\cdots&0\cr
0&p_{2}h_{\sharp}(p_{2})\rho_{2,1}^{*}&\cdots&0\cr\vdots&\vdots&\ddots&\vdots\cr
0&0&\cdots&p_{K}h_{\sharp}(p_{K})\rho_{K,1}^{*}\end{matrix}\right)\in\mathbb{C}^{dK\times
dK},$ $\displaystyle\Lambda$
$\displaystyle=\left(\begin{matrix}\frac{1}{1-p_{1}\overline{p}_{1}}I_{d}&\frac{1}{1-p_{1}\overline{p}_{2}}I_{d}&\cdots&\frac{1}{1-p_{1}\overline{p}_{K}}I_{d}\cr\frac{1}{1-p_{2}\overline{p}_{1}}I_{d}&\frac{1}{1-p_{2}\overline{p}_{2}}I_{d}&\cdots&\frac{1}{1-p_{2}\overline{p}_{K}}I_{d}\cr\vdots&\vdots&\ddots&\vdots\cr\frac{1}{1-p_{K}\overline{p}_{1}}I_{d}&\frac{1}{1-p_{K}\overline{p}_{2}}I_{d}&\cdots&\frac{1}{1-p_{K}\overline{p}_{K}}I_{d}\cr\end{matrix}\right)\in\mathbb{C}^{dK\times
dK},$ $\displaystyle\Pi_{n}$
$\displaystyle=\left(\begin{matrix}p_{1}^{n}I_{d}&0&\cdots&0\cr
0&p_{2}^{n}I_{d}&\cdots&0\cr\vdots&\vdots&\ddots&\vdots\cr
0&0&\cdots&p_{K}^{n}I_{d}\end{matrix}\right)\in\mathbb{C}^{dK\times dK},\qquad
n\in\mathbb{Z},$ $\displaystyle\Xi_{n}$
$\displaystyle=\left(\begin{matrix}\frac{\overline{p}_{1}^{n}}{1-p_{1}\overline{p}_{1}}I_{d}&\frac{\overline{p}_{2}^{n}}{1-p_{1}\overline{p}_{2}}I_{d}&\cdots&\frac{\overline{p}_{K}^{n}}{1-p_{1}\overline{p}_{K}}I_{d}\cr\frac{\overline{p}_{1}^{n}}{1-p_{2}\overline{p}_{1}}I_{d}&\frac{\overline{p}_{2}^{n}}{1-p_{2}\overline{p}_{2}}I_{d}&\cdots&\frac{\overline{p}_{K}^{n}}{1-p_{2}\overline{p}_{K}}I_{d}\cr\vdots&\vdots&\ddots&\vdots\cr\frac{\overline{p}_{1}^{n}}{1-p_{K}\overline{p}_{1}}I_{d}&\frac{\overline{p}_{2}^{n}}{1-p_{K}\overline{p}_{2}}I_{d}&\cdots&\frac{\overline{p}_{K}^{n}}{1-p_{K}\overline{p}_{K}}I_{d}\cr\end{matrix}\right)\in\mathbb{C}^{dK\times
dK},\qquad n\in\mathbb{N},$ $\displaystyle\Phi_{n}$
$\displaystyle=\left(\begin{matrix}\frac{p_{1}^{-n}}{1-p_{1}\overline{p}_{1}}I_{d}&\frac{p_{1}^{-n}}{1-p_{1}\overline{p}_{2}}I_{d}&\cdots&\frac{p_{1}^{-n}}{1-p_{1}\overline{p}_{K}}I_{d}\cr\frac{p_{2}^{-n}}{1-p_{2}\overline{p}_{1}}I_{d}&\frac{p_{2}^{-n}}{1-p_{2}\overline{p}_{2}}I_{d}&\cdots&\frac{p_{2}^{-n}}{1-p_{2}\overline{p}_{K}}I_{d}\cr\vdots&\vdots&\ddots&\vdots\cr\frac{p_{K}^{-n}}{1-p_{K}\overline{p}_{1}}I_{d}&\frac{p_{K}^{-n}}{1-p_{K}\overline{p}_{2}}I_{d}&\cdots&\frac{p_{K}^{-n}}{1-p_{K}\overline{p}_{K}}I_{d}\cr\end{matrix}\right)\in\mathbb{C}^{dK\times
dK},\qquad n\in\mathbb{Z},$ $\displaystyle G_{n}$
$\displaystyle=\Pi_{n}\Theta\Lambda\in\mathbb{C}^{dK\times
dK},\quad\tilde{G}_{n}=(\Pi_{n}\Theta)^{*}\Lambda^{\top}\in\mathbb{C}^{dK\times
dK},\quad n\in\mathbb{Z},$ $\displaystyle v_{n}$
$\displaystyle=\Xi_{n}\rho\in\mathbb{C}^{dK\times
d},\quad\tilde{v}_{n}=\overline{\Xi}_{n}\tilde{\rho}\in\mathbb{C}^{dK\times
d},\quad n\in\mathbb{N},$ $\displaystyle w_{n}$
$\displaystyle=\Phi_{n}\rho+\mathbf{p}_{-n}\rho_{0,0}\in\mathbb{C}^{dK\times
d},\quad\tilde{w}_{n}=\overline{\Phi}_{n}\tilde{\rho}+\overline{\mathbf{p}}_{-n}\tilde{\rho}_{0,0}\in\mathbb{C}^{dK\times
d},\quad n\in\mathbb{Z}.$

######  Example 5.2.

In Example 5.1, we further assume $d=K=1$. Then, we can write
$h(z)=h_{\sharp}(z)=-(1-\overline{p}z)/\rho$, where
$\rho\in\mathbb{C}\setminus\\{0\\}$ and $p\in\mathbb{D}\setminus\\{0\\}$. It
follows that

$\displaystyle c_{0}=-1/\rho,\qquad c_{1}=\overline{p}/\rho,\qquad
c_{k}=0\quad(k\geq 2),$ $\displaystyle
a_{k}=\rho(\overline{p})^{k},\qquad\tilde{a}_{k}=\overline{a}_{k},\qquad
k\in\mathbb{N}\cup\\{0\\}.$

Since $\gamma(k)=\sum_{\ell=0}^{\infty}c_{k+\ell}\overline{c}_{\ell}$ and
$\gamma(-k)=\overline{\gamma(k)}$ for $k\in\mathbb{N}\cup\\{0\\}$, we have

$T_{2}(w)=\frac{1}{|\rho|^{2}}\left(\begin{matrix}1+|p|^{2}&-p\cr-\overline{p}&1+|p|^{2}\end{matrix}\right),$

hence

$T_{2}(w)^{-1}=\frac{|\rho|^{2}}{1+|p|^{2}+|p|^{4}}\left(\begin{matrix}1+|p|^{2}&p\cr\overline{p}&1+|p|^{2}\end{matrix}\right).$

We also have

$\tilde{A}_{2}=\overline{\rho}\left(\begin{matrix}1&p\cr
0&1\end{matrix}\right)\quad\mbox{and}\quad
A_{2}=\rho\left(\begin{matrix}1&0\cr\overline{p}&1\end{matrix}\right)$

for $\tilde{A}_{2}$ and $A_{2}$ in (2.14) and (2.15), respectively. By simple
calculations, we have

$\displaystyle(\ell_{2,1},\ell_{2,2})$
$\displaystyle=\frac{\overline{\rho}}{(\overline{p})^{2}(1-|p|^{6})}(1+|p|^{2},\overline{p}),\qquad\left(\tilde{\ell}_{2,1},\tilde{\ell}_{2,2}\right)=\left(\overline{\ell_{2,2}},\overline{\ell_{2,1}}\right),$
$\displaystyle(r_{2,1},r_{2,2})$
$\displaystyle=-\rho\overline{p}|p|^{2}(1-|p|^{2})(\overline{p}(1+|p|^{2}),|p|^{2}),\qquad\left(\tilde{r}_{2,1},\tilde{r}_{2,2}\right)=\left(\overline{r_{2,2}},\overline{r_{2,1}}\right)$

hence

$T_{2}(w)^{-1}=\tilde{A}_{2}^{*}\tilde{A}_{2}+\left(\begin{matrix}\tilde{\ell}_{2,1}\cr\tilde{\ell}_{2,2}\end{matrix}\right)\left(\tilde{r}_{2,1},\tilde{r}_{2,2}\right)=A_{2}^{*}A_{2}+\left(\begin{matrix}\ell_{2,1}\cr\ell_{2,2}\end{matrix}\right)\left(r_{2,1},r_{2,2}\right)$

which agrees with equalities in Theorem 5.2.

##  6\. Linear-time algorithm

As in Section 5, we assume (1.17) and (1.18). Let $K$ be as in (5.1) with
(5.2). In this section, we explain how Theorems 5.1 and 5.2 above provide us
with a linear-time algorithm to compute the solution $Z$ to the block Toeplitz
system (1.19).

For

(6.1) $Y=(y_{1}^{\top},\dots,y_{n}^{\top})^{\top}\in\mathbb{C}^{dn\times
d}\quad\mbox{with}\quad y_{s}\in\mathbb{C}^{d\times d},\quad
s\in\\{1,\dots,n\\},$

let

$Z=(z_{1}^{\top},\dots,z_{n}^{\top})^{\top}\in\mathbb{C}^{dn\times
d}\quad\mbox{with}\quad z_{s}\in\mathbb{C}^{d\times d},\quad
s\in\\{1,\dots,n\\},$

be the solution to (1.19), that is, $Z=T_{n}(w)^{-1}Y$. For $m_{0}$ in (5.1),
let $n\geq 2m_{0}+1$ so that $n-m_{0}\geq m_{0}+1$ holds.

Recall $\tilde{A}_{n}$ and $A_{n}$ from (2.14) and (2.15), respectively. If
$K=0$, then it follows from Lemma 2.1 and Theorem 5.1 (ii), (iv) that

$\displaystyle z_{s}$ $\displaystyle=\tilde{\alpha}_{n,s},\qquad
s\in\\{1,\dots,n-m_{0}\\},$ $\displaystyle z_{s}$
$\displaystyle=\alpha_{n,s},\qquad s\in\\{m_{0}+1,\dots,n\\},$

where

$\displaystyle(\tilde{\alpha}_{n,1}^{\top},\dots,\tilde{\alpha}_{n,n}^{\top})^{\top}$
$\displaystyle:=\tilde{A}_{n}^{*}\tilde{A}_{n}Y\quad\mbox{with}\quad\tilde{\alpha}_{n,s}\in\mathbb{C}^{d\times
d},\quad s\in\\{1,\dots,n\\},$
$\displaystyle(\alpha_{n,1}^{\top},\dots,\alpha_{n,n}^{\top})^{\top}$
$\displaystyle:=A_{n}^{*}A_{n}Y\quad\mbox{with}\quad\alpha_{n,s}\in\mathbb{C}^{d\times
d},\quad s\in\\{1,\dots,n\\}.$

On the other hand, if $K\geq 1$, then we see from Lemma 2.1 and Theorem 5.2
(ii), (iv) that

$\displaystyle z_{s}$
$\displaystyle=\tilde{\ell}_{n,s}\tilde{R}_{n}+\tilde{\alpha}_{n,s},\qquad
s\in\\{1,\dots,n-m_{0}\\},$ $\displaystyle z_{s}$
$\displaystyle=\ell_{n,s}R_{n}+\alpha_{n,s},\qquad s\in\\{m_{0}+1,\dots,n\\},$

where

$\tilde{R}_{n}:=\sum_{t=1}^{n}\tilde{r}_{n,t}y_{t}\in\mathbb{C}^{d\times
d},\qquad R_{n}:=\sum_{t=1}^{n}r_{n,t}y_{t}\in\mathbb{C}^{d\times d}.$

Therefore, algorithms to compute $\tilde{A}_{n}^{*}\tilde{A}_{n}Y$ and
$A_{n}^{*}A_{n}Y$ in $O(n)$ operations imply that of $Z$. We present the
former ones below.

For $n\in\mathbb{N}\cup\\{0\\}$, $\mu\in\\{1,\dots,K\\}$ and
$j\in\\{1,\dots,m_{\mu}\\}$, we define $q_{\mu,j}(n)\in\mathbb{C}^{d\times d}$
by $q_{\mu,j}(n):=p_{\mu,j}(n+j-1)$, that is,

(6.2) $q_{\mu,j}(n)=\binom{n+j-1}{j-1}p_{\mu}^{n}I_{d}.$

For $n\in\mathbb{N}$, $\mu\in\\{1,\dots,K\\}$ and $j\in\\{1,\dots,m_{\mu}\\}$,
we define the upper trianglular block Toeplitz matrix
$Q_{\mu,j,n}\in\mathbb{C}^{dn\times dn}$ by

$Q_{\mu,j,n}:=\left(\begin{matrix}q_{\mu,j}(0)&q_{\mu,j}(1)&q_{\mu,j}(2)&\cdots&q_{\mu,j}(n-1)\cr&q_{\mu,j}(0)&q_{\mu,j}(1)&\cdots&q_{\mu,j}(n-2)\cr&&\ddots&\ddots&\vdots\cr&&&\ddots&q_{\mu,j}(1)\cr\mbox{\huge
0}&&&&q_{\mu,j}(0)\end{matrix}\right).$

Notice that

$Q^{*}_{\mu,j,n}:=\left(\begin{matrix}q^{*}_{\mu,j}(0)&&&&\mbox{\huge 0}\cr
q^{*}_{\mu,j}(1)&q^{*}_{\mu,j}(0)&&&\cr
q^{*}_{\mu,j}(2)&q^{*}_{\mu,j}(1)&\ddots&&\cr\vdots&\vdots&\ddots&\ddots&\cr
q^{*}_{\mu,j}(n-1)&q^{*}_{\mu,j}(n-2)&\cdots&q^{*}_{\mu,j}(1)&q^{*}_{\mu,j}(0)\end{matrix}\right)$

with $q^{*}_{\mu,j}(n)=\binom{n+j-1}{j-1}\overline{p}_{\mu}^{n}I_{d}$. For
$n\in\mathbb{N}$, $\mu\in\\{1,\dots,K\\}$ and $j\in\\{1,\dots,m_{\mu}\\}$, we
define the block diagonal matrices $\tilde{D}_{\mu,j,n}\in\mathbb{C}^{dn\times
dn}$ and $D_{\mu,j,n}\in\mathbb{C}^{dn\times dn}$ by

$\tilde{D}_{\mu,j,n}:=\left(\begin{matrix}\tilde{\rho}_{{\mu},j}&0&\cdots&0\cr
0&\tilde{\rho}_{{\mu},j}&\cdots&0\cr\vdots&\vdots&\ddots&\vdots\cr
0&0&\cdots&\tilde{\rho}_{{\mu},j}\end{matrix}\right)\quad\mbox{and}\quad
D_{\mu,j,n}:=\left(\begin{matrix}\rho_{{\mu},j}&0&\cdots&0\cr
0&\rho_{{\mu},j}&\cdots&0\cr\vdots&\vdots&\ddots&\vdots\cr
0&0&\cdots&\rho_{{\mu},j}\end{matrix}\right),$

respectively. Moreover, for $n\geq m_{0}+1$, we define the upper and lower
triangular block Toeplitz matrices $\tilde{\Delta}_{n}\in\mathbb{C}^{dn\times
dn}$ and $\Delta_{n}\in\mathbb{C}^{dn\times dn}$ by

$\tilde{\Delta}_{n}:=\left(\begin{matrix}\tilde{\rho}_{0,0}&\tilde{\rho}_{0,1}&\cdots&\tilde{\rho}_{0,m_{0}}&&\mbox{\huge
0}\cr&\tilde{\rho}_{0,0}&\tilde{\rho}_{0,1}&&\ddots&\cr&&\ddots&\ddots&&\tilde{\rho}_{0,m_{0}}\cr&&&\ddots&\ddots&\vdots\cr&&&&\tilde{\rho}_{0,0}&\tilde{\rho}_{0,1}\cr\mbox{\huge
0}&&&&&\tilde{\rho}_{0,0}\end{matrix}\right)$

and

$\Delta_{n}:=\left(\begin{matrix}\rho_{0,0}&&&&&\mbox{\huge
0}\cr\rho_{0,1}&\rho_{0,0}&&&&\cr\vdots&\rho_{0,1}&\ddots&&&\cr\rho_{0,m_{0}}&&\ddots&\ddots&&\cr&\ddots&&\ddots&\rho_{0,0}&\cr\mbox{\huge
0}&&\rho_{0,m_{0}}&\cdots&\rho_{0,1}&\rho_{0,0}\end{matrix}\right),$

respectively. Note that both $\tilde{\Delta}_{n}$ and $\Delta_{n}$ are sparse
matrices in the sense that they have only $O(n)$ nonzero elements.

By (5.5)–(5.8), we have

$\displaystyle\tilde{A}_{n}$
$\displaystyle=\tilde{\Delta}_{n}+\sum_{\mu=1}^{K}\sum_{j=1}^{m_{\mu}}Q_{\mu,j,n}\tilde{D}_{\mu,j,n},\qquad
n\geq m_{0}+1,$ $\displaystyle A_{n}$
$\displaystyle=\Delta_{n}+\sum_{\mu=1}^{K}\sum_{j=1}^{m_{\mu}}Q^{*}_{\mu,j,n}D_{\mu,j,n},\qquad
n\geq m_{0}+1.$

Therefore, it is enough to give linear-time algorithms to compute
$Q_{\mu,i,n}Y$ and $Q^{*}_{\mu,i,n}Y$ for $Y\in\mathbb{C}^{dn\times d}$ in
$O(n)$ operations. The following two propositions provide such linear-time
algorithms.

######  Proposition 6.1.

Let $n\in\mathbb{N}$, $\mu\in\\{1,\dots,K\\}$ and $Y$ be as in (6.1). We put
$Z_{\mu,i}=Q_{\mu,i,n}Y$ for $i\in\\{1,\dots,m_{\mu}\\}$. Then the component
blocks $z_{\mu,i}(s)$ of
$Z_{\mu,i}=(z_{\mu,i}^{\top}(1),\dots,z_{\mu,i}^{\top}(n))^{\top}$ satisfy the
following equalities:

(6.3) $\displaystyle z_{\mu,i}(n)=q_{\mu,i}(0)y_{n},\qquad
i\in\\{1,\dots,m_{\mu}\\},$ (6.4) $\displaystyle
z_{\mu,1}(s)=p_{\mu}z_{\mu,1}(s+1)+q_{\mu,1}(0)y_{s},\qquad
s\in\\{1,\dots,n-1\\}$ (6.5) $\displaystyle\begin{aligned}
z_{\mu,i}(s)&=p_{\mu}z_{\mu,i}(s+1)+z_{\mu,i-1}(s)+\\{q_{\mu,i}(0)-q_{\mu,i-1}(0)\\}y_{s},\\\
&\qquad\qquad\qquad\qquad\qquad i\in\\{2,\dots,m_{\mu}\\},\
s\in\\{1,\dots,n-1\\}.\end{aligned}$

###### Proof.

From the definition of $Q_{\mu,i,n}$, (6.3) is trivial. For $q_{\mu,i}(k)$ in
(6.2), Pascal’s rule yields the following recursions:

(6.6) $\displaystyle q_{\mu,1}(k+1)=p_{\mu}q_{\mu,1}(k),\quad
k\in\mathbb{N}\cup\\{0\\},$ (6.7) $\displaystyle
q_{\mu,i}(k+1)=p_{\mu}q_{\mu,i}(k)+q_{\mu,i-1}(k+1),\quad
i\in\\{2,\dots,j\\},\ k\in\mathbb{N}\cup\\{0\\}.$

For $s\in\\{1,\dots,n-1\\}$, we see, from (6.6),

$\displaystyle z_{\mu,1}(s)$
$\displaystyle=q_{\mu,1}(0)y_{s}+\sum_{t=0}^{n-s-1}q_{\mu,1}(t+1)y_{s+t+1}$
$\displaystyle=q_{\mu,1}(0)y_{s}+p_{\mu}\sum_{t=0}^{n-s-1}q_{\mu,1}(t)y_{s+t+1}=q_{\mu,1}(0)y_{s}+p_{\mu}z_{\mu,1}(s+1),$

and, from (6.7),

$\displaystyle z_{\mu,i}(s)$
$\displaystyle=q_{\mu,i}(0)y_{s}+\sum_{t=0}^{n-s-1}q_{\mu,i}(t+1)y_{s+t+1}$
$\displaystyle=\\{q_{\mu,i}(0)-q_{\mu,i-1}(0)\\}y_{s}+p_{\mu}\sum_{t=0}^{n-s-1}q_{\mu,1}(t)y_{s+t+1}+\sum_{t=0}^{n-s}q_{\mu,i-1}(t)y_{s+t}$
$\displaystyle=\\{q_{\mu,i}(0)-q_{\mu,i-1}(0)\\}y_{s}+p_{\mu}z_{\mu,i}(s+1)+z_{\mu,i-1}(s)$

for $i\in\\{2,\dots,j\\}$. Thus, (6.4) and (6.5) follow. ∎

By Proposition 6.1, we can compute $z_{\mu,i}(s)$ in the following order in
$O(n)$ operations:

$\displaystyle z_{\mu,1}(n)\ \to\ \cdots\to\ z_{\mu,1}(1)\ \to\ z_{\mu,2}(n)\
\to\cdots\to\ z_{\mu,2}(1)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\to\cdots\to\
z_{\mu,m_{\mu}}(n)\ \to\cdots\to z_{\mu,m_{\mu}}(1).$

######  Proposition 6.2.

Let $n\in\mathbb{N}$, $\mu\in\\{1,\dots,K\\}$ and $Y$ be as in (6.1). We put
$W_{\mu,i}=Q^{*}_{\mu,i,n}Y$ for $i\in\\{1,\dots,m_{\mu}\\}$. Then the
component blocks $w_{\mu,i}(s)$ of
$W_{\mu,i}=(w_{\mu,i}^{\top}(1),\dots,w_{\mu,i}^{\top}(n))^{\top}$ satisfy the
following equalities:

$\displaystyle w_{\mu,i}(1)=q_{\mu,i}^{*}(0)y_{1},\qquad
i\in\\{1,\dots,m_{\mu}\\},$ $\displaystyle
w_{\mu,1}(s+1)=\overline{p}_{\mu}w_{\mu,1}(s)+q_{\mu,1}^{*}(0)y_{s+1},\qquad
s\in\\{1,\dots,n-1\\}$ $\displaystyle\begin{aligned}
w_{\mu,i}(s+1)&=\overline{p}_{\mu}w_{\mu,i}(s)+w_{\mu,i-1}(s+1)+\\{q_{\mu,i}^{*}(0)-q_{\mu,i-1}^{*}(0)\\}y_{s+1},\\\
&\qquad\qquad\qquad\qquad\qquad i\in\\{2,\dots,m_{\mu}\\},\
s\in\\{1,\dots,n-1\\}.\end{aligned}$

The proof of Proposition 6.2 is similar to that of Proposition 6.1; we omit
it.

By Proposition 6.2, we can compute $w_{\mu,i}(s)$ in the following order in
$O(n)$ operations:

$\displaystyle w_{\mu,1}(1)\ \to\ \cdots\to\ w_{\mu,1}(n)\ \to\ w_{\mu,2}(1)\
\to\cdots\to\ w_{\mu,2}(n)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\to\cdots\to\
w_{\mu,m_{\mu}}(1)\ \to\cdots\to w_{\mu,m_{\mu}}(n).$

##  Appendix A Proofs of Lemmas 5.2 and 5.3

As in Section 5, we assume (1.17) and (1.18). We use the same notation as in
Section 5. For $K$ in (5.1) with (5.2), we assume $K\geq 1$.

We prove Lemma 5.2.

###### Proof.

(i) We assume $n\geq u\geq m_{0}+1$, and prove (5.17) and (5.18) by induction.
First, from Lemma 5.1,

$b_{n,u,\ell}^{1}=\beta_{u+l}=\mathbf{p}_{u-n-1}^{*}(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}.$

Next, for $k=1,2,\dots$, we assume (5.17). Then, by Lemma 5.1,

$\displaystyle b_{n,u,\ell}^{2k}$
$\displaystyle=\sum_{m=0}^{\infty}b_{n,u,m}^{2k-1}\beta_{n+1+m+\ell}^{*}$
$\displaystyle=\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}(\Pi_{n}\Theta)^{*}\left(\sum_{m=0}^{\infty}\overline{\mathbf{p}}_{m}\mathbf{p}_{m}^{\top}\right)\Pi_{n}\Theta\mathbf{p}_{\ell}$
$\displaystyle=\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}(\Pi_{n}\Theta)^{*}\Lambda^{\top}\Pi_{n}\Theta\mathbf{p}_{\ell}=\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}\tilde{G}_{n}\Pi_{n}\Theta\mathbf{p}_{\ell}$

or (5.18). From this as well as Lemma 5.1,

$\displaystyle b_{n,u,\ell}^{2k+1}$
$\displaystyle=\sum_{m=0}^{\infty}b_{n,u,m}^{2k}\beta_{n+1+m+\ell}$
$\displaystyle=\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}\tilde{G}_{n}\Pi_{n}\Theta\left(\sum_{m=0}^{\infty}\mathbf{p}_{m}\mathbf{p}_{m}^{*}\right)(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}$
$\displaystyle=\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k-1}\tilde{G}_{n}\Pi_{n}\Theta\Lambda(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}=\mathbf{p}_{u-n-1}^{*}(\tilde{G}_{n}G_{n})^{k}(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}$

or (5.17) with $k$ replaced by $k+1$. Thus (5.17) and (5.18) follow.

(ii) We assume $1\leq u\leq n-m_{0}$, and prove (5.19) and (5.20) by
induction. First, from Lemma 5.1,

$\tilde{b}_{n,u,\ell}^{1}=\beta_{n+1-u+\ell}^{*}=\mathbf{p}_{-u}^{\top}\Pi_{n}\Theta\mathbf{p}_{\ell}.$

Next, for $k=1,2,\dots$, we assume (5.19). Then, by Lemma 5.1,

$\displaystyle\tilde{b}_{n,u,\ell}^{2k}$
$\displaystyle=\sum_{m=0}^{\infty}\tilde{b}_{n,u,m}^{2k-1}\beta_{n+1+m+\ell}$
$\displaystyle=\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}\Pi_{n}\Theta\left(\sum_{m=0}^{\infty}\mathbf{p}_{m}\mathbf{p}_{m}^{*}\right)(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}$
$\displaystyle=\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}\Pi_{n}\Theta\Lambda(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}=\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}G_{n}(\Pi_{n}\Theta)^{*}\overline{\mathbf{p}}_{\ell}$

or (5.20). From this as well as Lemma 5.1,

$\displaystyle\tilde{b}_{n,u,\ell}^{2k+1}$
$\displaystyle=\sum_{m=0}^{\infty}\tilde{b}_{n,u,m}^{2k}\beta_{n+1+m+\ell}^{*}=\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}G_{n}(\Pi_{n}\Theta)^{*}\left(\sum_{m=0}^{\infty}\overline{\mathbf{p}}_{m}\mathbf{p}_{m}^{\top}\right)\Pi_{n}\Theta\mathbf{p}_{\ell}$
$\displaystyle=\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k-1}G_{n}(\Pi_{n}\Theta)^{*}\Lambda^{\top}\Pi_{n}\Theta\mathbf{p}_{\ell}=\mathbf{p}_{-u}^{\top}(G_{n}\tilde{G}_{n})^{k}\Pi_{n}\Theta\mathbf{p}_{\ell}$

or (5.19) with $k$ replaced by $k+1$. Thus (5.19) and (5.20) follow. ∎

To prove Lemma 5.3, we need some propositions.

######  Proposition A.1.

For $m,n\in\mathbb{Z}$, $i,j\in\mathbb{N}\cup\\{0\\}$ and $x,y\in\mathbb{D}$,
we have

$\displaystyle\frac{1}{i!j!}\left(\frac{\partial}{\partial
x}\right)^{i}\left(\frac{\partial}{\partial
y}\right)^{j}\frac{x^{m}y^{n}}{1-xy}$
$\displaystyle\qquad\qquad\qquad=\sum_{q=0}^{i}\sum_{r=0}^{j}\binom{n}{j-r}\binom{q+r}{q}\binom{m+r}{i-q}\frac{x^{m+q+r-i}y^{n+q+r-j}}{(1-xy)^{q+r+1}}.$

###### Proof.

Let $m,n\in\mathbb{Z}$, $i,j\in\mathbb{N}\cup\\{0\\}$ and $x,y\in\mathbb{D}$.
Then, we have

$\displaystyle\frac{1}{j!}\left(\frac{\partial}{\partial
y}\right)^{j}\frac{y^{n}}{1-xy}$
$\displaystyle=\sum_{r=0}^{j}\left\\{\frac{1}{r!}\left(\frac{\partial}{\partial
y}\right)^{r}\frac{1}{1-xy}\right\\}\left\\{\frac{1}{(j-r)!}\left(\frac{\partial}{\partial
y}\right)^{j-r}y^{n}\right\\}$
$\displaystyle=\sum_{r=0}^{j}\binom{n}{j-r}\frac{x^{r}y^{n+r-j}}{(1-xy)^{r+1}},$

hence

$\displaystyle\frac{1}{i!j!}\left(\frac{\partial}{\partial
x}\right)^{i}\left(\frac{\partial}{\partial
y}\right)^{j}\frac{x^{m}y^{n}}{1-xy}$
$\displaystyle=\sum_{r=0}^{j}\binom{n}{j-r}y^{n+r-j}\sum_{q=0}^{i}\left\\{\frac{1}{q!}\left(\frac{\partial}{\partial
x}\right)^{q}\frac{1}{(1-xy)^{r+1}}\right\\}\left\\{\frac{1}{(i-q)!}\left(\frac{\partial}{\partial
x}\right)^{i-q}x^{m+r}\right\\}$
$\displaystyle=\sum_{r=0}^{j}\binom{n}{j-r}y^{n+r-j}\sum_{q=0}^{i}\left\\{\binom{q+r}{q}\frac{y^{q}}{(1-xy)^{q+r+1}}\right\\}\left\\{\binom{m+r}{i-q}x^{m+q+r-i}\right\\}$
$\displaystyle=\sum_{q=0}^{i}\sum_{r=0}^{j}\binom{n}{j-r}\binom{q+r}{q}\binom{m+r}{i-q}\frac{x^{m+q+r-i}y^{n+q+r-j}}{(1-xy)^{q+r+1}}.$

Thus, the proposition follows. ∎

######  Proposition A.2.

For $n\in\mathbb{Z}$, $i,j\in\mathbb{N}\cup\\{0\\}$ and $x,y\in\mathbb{D}$, we
have

$\sum_{\ell=0}^{\infty}\binom{n+\ell}{i}\binom{j+\ell}{j}x^{n+\ell-i}y^{\ell}=\sum_{q=0}^{i}\sum_{r=0}^{j}\binom{j}{r}\binom{r+q}{q}\binom{n+r}{i-q}\frac{x^{n+r+q-i}y^{r+q}}{(1-xy)^{r+q+1}}.$

###### Proof.

Let $n\in\mathbb{Z}$, $i,j\in\mathbb{N}\cup\\{0\\}$ and $x,y\in\mathbb{D}$.
Since $x^{n}y^{j}/(1-xy)=\sum_{\ell=0}^{\infty}x^{n+\ell}y^{j+\ell}$, we have

$\frac{1}{i!j!}\left(\frac{\partial}{\partial
x}\right)^{i}\left(\frac{\partial}{\partial
y}\right)^{j}\frac{x^{n}y^{j}}{1-xy}=\sum_{\ell=0}^{\infty}\binom{n+\ell}{i}\binom{j+\ell}{j}x^{n+\ell-i}y^{\ell}.$

On the other hand, by Proposition A.1, we have

$\frac{1}{i!j!}\left(\frac{\partial}{\partial
x}\right)^{i}\left(\frac{\partial}{\partial
y}\right)^{j}\frac{x^{n}y^{j}}{1-xy}=\sum_{q=0}^{i}\sum_{r=0}^{j}\binom{j}{r}\binom{r+q}{q}\binom{n+r}{i-q}\frac{x^{n+r+q-i}y^{r+q}}{(1-xy)^{r+q+1}}.$

Comparing, we obtain the proposition. ∎

We are ready to prove Lemma 5.3.

###### Proof.

By (5.5)–(5.8) and Proposition A.2, we have, for $n\in\mathbb{Z}$,
$\mu\in\\{1,\dots,K\\}$ and $i\in\\{1,\dots,m_{\mu}\\}$,

$\displaystyle\sum_{\ell=0}^{\infty}p_{\mu,i}(\ell-n)a_{\ell}$
$\displaystyle=\sum_{\ell=0}^{m_{0}}p_{\mu,i}(\ell-n)\rho_{0,\ell}$
$\displaystyle\qquad+\sum_{{\nu}=1}^{K}\sum_{j=1}^{m_{\nu}}\left\\{\sum_{\ell=0}^{\infty}\binom{\ell-n}{i-1}\binom{\ell+j-1}{j-1}p_{\mu}^{\ell-i+1-n}\overline{p}_{\nu}^{\ell}\right\\}\rho_{\nu,j}$
$\displaystyle=\sum_{\ell=0}^{m_{0}}p_{\mu,i}(\ell-n)\rho_{0,\ell}+\sum_{{\nu}=1}^{K}\sum_{j=1}^{m_{\nu}}\varphi_{n}^{{\mu},{\nu}}(i,j)\rho_{\nu,j}$

and

$\displaystyle\sum_{\ell=0}^{\infty}\overline{p}_{\mu,i}(\ell-n)\tilde{a}_{\ell}$
$\displaystyle=\sum_{\ell=0}^{m_{0}}\overline{p}_{\mu,i}(\ell-n)\tilde{\rho}_{0,\ell}$
$\displaystyle\qquad+\sum_{{\nu}=1}^{K}\sum_{j=1}^{m_{\nu}}\left\\{\sum_{\ell=0}^{\infty}\binom{\ell-n}{i-1}\binom{\ell+j-1}{j-1}\overline{p}_{\mu}^{\ell-i+1-n}p_{\nu}^{\ell}\right\\}\tilde{\rho}_{\nu,j}$
$\displaystyle=\sum_{\ell=0}^{m_{0}}\overline{p}_{\mu,i}(\ell-n)\tilde{\rho}_{0,\ell}+\sum_{{\nu}=1}^{K}\sum_{j=1}^{m_{\nu}}\overline{\varphi}_{n}^{{\mu},{\nu}}(i,j)\tilde{\rho}_{\nu,j}.$

Thus, the lemma follows. ∎

## Acknowledgments

The author is grateful to two referees for their valuable comments and helpful
suggestions.

## References

  * [1] Baxter, G.: An asymptotic result for the finite predictor. Math. Scand. 10, 137–144 (1962)
  * [2] Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer, New York (1999)
  * [3] Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2006)
  * [4] Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods, 2nd edn. Springer, New York (1991)
  * [5] Cheng, R., Pourahmadi, M.: Baxter’s inequality and convergence of finite predictors of multivariate stochastic processes. Probab. Theory Relat. Fields 95(1), 115–124 (1993)
  * [6] Gohberg, I.C., Fel’dman, I.A.: Convolution Equations and Projection Methods for Their Solution, volume 41 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1974)
  * [7] Gohberg, I.C., Semencul, A.A.: The inversion of finite Toeplitz matrices and their continual analogues. (Russian) Mat. Issled. 7(2), 201–223 (1972)
  * [8] Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications, 2nd edn. Chelsea Publishing Co., New York (1984)
  * [9] Hannan, E.J., Deistler, M.: The Statistical Theory of Linear Systems. Wiley, New York (1988)
  * [10] Helson, H., Lowdenslager, D.: Prediction theory and Fourier series in several variables. II. Acta Math. 106, 175–213 (1961)
  * [11] Inoue, A.: Asymptotics for the partial autocorrelation function of a stationary process. J. Anal. Math. 81, 65–109 (2000)
  * [12] Inoue, A.: AR and MA representation of partial autocorrelation functions, with applications. Probab. Theory Relat. Fields 140(3-4), 523–551 (2008)
  * [13] Inoue, A.: Closed-form expression for finite predictor coefficients of multivariate ARMA processes. J. Multivariate Anal. 176, 104578 (2020)
  * [14] Inoue, A., Kasahara, Y.: Explicit representation of finite predictor coefficients and its applications. Ann. Stat. 34(2), 973–993 (2006)
  * [15] Inoue, A., Kasahara, Y., Pourahmadi, M.: The intersection of past and future for multivariate stationary processes. Proc. Amer. Math. Soc. 144(4), 1779–1786 (2016)
  * [16] Inoue, A., Kasahara, Y., Pourahmadi, M.: Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes. Bernoulli 24(2), 1202–1232 (2018)
  * [17] Kasahara, Y., Bingham, N.H.: Matricial Baxter’s theorem with a Nehari sequence. Math. Nachr. 291(17-18), 2590–2598 (2018)
  * [18] Katsnelson, V.E., Kirstein, B.: On the theory of matrix-valued functions belonging to the Smirnov class. In: Topics in Interpolation Theory. (Leipzig, 1994). Oper. Theory Adv. Appl. vol. 95, pp. 299–350, Birkhäuser (1997)
  * [19] Masani, P.: The prediction theory of multivariate stochastic processes. III. Unbounded spectral densities. Acta Math. 104, 141–162 (1960)
  * [20] Peller, V.V.: Hankel Operators and Their Applications. Springer, New York (2003)
  * [21] Rozanov, Yu.A.: Stationary Random Processes. Holden-Day, San Francisco (1967)
  * [22] Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, American Mathematical Society, Providence (2005)
  * [23] Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory, American Mathematical Society, Providence (2005)
  * [24] Simon, B.: Szegö’s Theorem and Its Descendants. Spectral Theory for $L^{2}$ Perturbations of Orthogonal Polynomials, Princeton University Press, Princeton (2011)
  * [25] Subba Rao, S., Yang, J.: Reconciling the Gaussian and Whittle Likelihood with an application to estimation in the frequency domain. Ann. Stat. 49(5), 2774–2802 (2021)