# Large deviations for quadratic functionals of stable Gauss-Markov chains and
entropy production

Marco Zamparo111Dipartimento di Fisica, Università degli Studi di Bari and
INFN, Sezione di Bari, via Amendola 173, 70126
aaz Bari, Italy. E-mail<EMAIL_ADDRESS>Massimiliano
Semeraro222Dipartimento di Fisica, Università degli Studi di Bari and INFN,
Sezione di Bari, via Amendola 173, 70126
aaz Bari, Italy. E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper we establish a large deviation principle for the entropy
production rate of possible non-stationary, centered stable Gauss-Markov
chains, verifying the Gallavotti-Cohen symmetry. We reach this goal by
developing a large deviation theory for quasi-Toeplitz quadratic functionals
of multivariate centered stable Gauss-Markov chains, which differ from a
perfect Toeplitz form by the addition of quadratic boundary terms.

##  1 Introduction

Let on a probability space $(\Omega,\mathscr{F},\mathbb{P})$ be given a
sequence $\\{X_{n}\\}_{n\geq 1}$ of random variables taking values in a Polish
space $\mathcal{X}$. For any integer $N\geq 1$, let
$\mu_{N}^{+}:=\mathbb{P}[(X_{1},\ldots,X_{N})\in\cdot\,]$ and
$\mu_{N}^{-}:=\mathbb{P}[(X_{N},\ldots,X_{1})\in\cdot\,]$ be the probability
measures on the Borel $\sigma$-field $\mathscr{B}(\mathcal{X}^{N})$ induced by
the direct process and the reverse process, respectively. The entropy
production rate up to time $N$ is the real random variable on
$(\Omega,\mathscr{F},\mathbb{P})$ defined by

$e_{N}:=\begin{cases}\frac{1}{N}\ln\Big{[}\frac{d\mu_{N}^{+}}{d\mu_{N}^{-}}(X_{1},\ldots,X_{N})\Big{]}&\mbox{if
}\mu_{N}^{+}\ll\mu_{N}^{-},\\\ +\infty&\mbox{otherwise}.\end{cases}$

The entropy production rate turns out to be a natural measure of
irreversibility since $e_{N}=0$ for all $N\geq 1$ if and only if the sequence
$\\{X_{n}\\}_{n\geq 1}$ is reversible, namely if and only if
$(X_{N},\ldots,X_{1})$ is distributed as $(X_{1},\ldots,X_{N})$ for every $N$.
The use of the entropy production rate to quantify the irreversibility of a
stochastic process was proposed by Kurchan [1] and in more generality by
Lebowitz, Spohn, and Maes [2, 3], who extended the seminal work by Gallavotti
and Cohen [4] in the context of deterministic dynamical systems. Since then,
the entropy production rate has become a basic topic in non-equilibrium
statistical physics [5, 6, 7, 8, 9, 10, 11, 12, 13]. The entropy production
rate came out with a supposed symmetry associated with its large fluctuations,
which in fact was discovered by Gallavotti and Cohen [4] prompted by results
of computer simulations [14]. They dubbed this symmetry a “fluctuation
theorem”. The appropriate formalism for describing the large fluctuations of
the entropy production rate is large deviation theory [15, 16]. The entropy
production rate $e_{N}$ is said to satisfy a large deviation principle with
the rate function $I$ if there exists a function $I$ with compact level sets
such that for each Borel set $\mathcal{B}\subseteq\mathbb{R}$

$-\inf_{w\in\mathcal{B}^{o}}\\{I(w)\\}\leq\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\big{[}e_{N}\in\mathcal{B}\big{]}\leq\limsup_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\big{[}e_{N}\in\mathcal{B}\big{]}\leq-\inf_{w\in\bar{\mathcal{B}}}\\{I(w)\\},$

where $\mathcal{B}^{o}$ and $\bar{\mathcal{B}}$ are the interior and the
closure of $\mathcal{B}$, respectively. The “fluctuation theorem” refers to a
property of the function $I$. The rate function $I$ is said to satisfy the
Gallavotti-Cohen symmetry if for all $w\in\mathbb{R}$

$I(-w)=I(w)+w.$

It has been pointed out that the Gallavotti-Cohen symmetry is an intrinsic
property of $I$, which is met whenever $e_{N}$ satisfies a large deviation
principle [10, 8, 9].

In this paper we investigate the large fluctuations of the entropy production
rate and the Gallavotti-Cohen symmetry for a possible non-stationary, centered
stable Gauss-Markov chain $\\{X_{n}\\}_{n\geq 1}$ valued in
$\mathcal{X}:=\mathbb{R}^{d}$ with any dimension $d\geq 1$. Thus, we assume
that there exists a drift matrix $S\in\mathbb{R}^{d\times d}$ with spectral
radius $\rho(S)<1$ such that

$X_{n+1}=SX_{n}+G_{n}$ (1.1)

for all $n\geq 1$, $\\{G_{n}\\}_{n\geq 1}$ being a sequence of i.i.d. standard
Gaussian random vectors valued in $\mathbb{R}^{d}$ and independent of $X_{1}$.
We suppose that $X_{1}$ is a Gaussian random vector with mean zero and general
positive-definite covariance matrix $\Sigma_{o}$. The process
$\\{X_{n}\\}_{n\geq 1}$ is stationary if and only if
$\Sigma_{o}=\Sigma_{s}:=\sum_{k\geq 0}S^{k}(S^{\top})^{k}$, and it is
reversible if and only if $S$ is symmetric and $\Sigma_{o}=\Sigma_{s}$ [17].
Stability corresponds to the hypothesis $\rho(S)<1$, which implies that
$\Sigma_{s}$ actually exists. Unstable Gauss-Markov chains for which
$\rho(S)\geq 1$ cannot achieve a stationary regime and obey a substantially
different mathematics, for which reason we leave them as a future project.

The large deviation principle for the entropy production rate and the
Gallavotti-Cohen symmetry have been rigorously established for finite Markov
chains with discrete and continuous time [8, 18]. The same has been done for
multivariate stationary Ornstein-Uhlenbeck processes with normal drift matrix
[19, 20, 21], i.e. with a drift matrix that commutes with its adjoint, and for
a model of heat conduction through a chain of anharmonic oscillators coupled
to two reservoirs at different temperatures [22]. The mathematical tool
underlying these works is the Gärtner-Ellis theorem [16, 15], and the lack of
a large deviation principle for the entropy production rate of more general
stochastic processes is due to non-satisfiability of the hypotheses of that
theorem. An attempt to overcome the limitations of the Gärtner-Ellis theorem
has been done for stationary diffusion processes, for which a large deviation
principle for the entropy production rate has been obtained in the limit of
vanishing noise by resorting to the classical Freidlin-Wentzell theory [23].
The autoregressive model (1.1) we consider basically is the discrete-time
version of a $d$-dimensional centered Ornstein-Uhlenbeck process. The main
contribution of our work stems from the fact that we do not assume that the
drift matrix $S$ is normal or that the chain is stationary. This generality
prevents the use of the Gärtner-Ellis theorem to get at a large deviation
principle for the entropy production rate $e_{N}$. The way we go around this
key point is to regard $e_{N}$ as a quadratic functional, and to establish a
large deviation principle for the class of quadratic functionals to which the
entropy production rate belongs via a time-dependent change of probability
measure. We need such a general principle to also tackle a problem of large
fluctuations in an active matter model [24]. The following lemma provides the
explicit expression of $e_{N}$ for the model (1.1) as a quadratic form. The
simple proof is reported in Appendix A. We denote by
$\langle\cdot,\cdot\rangle$ the standard inner product of $\mathbb{R}^{d}$.

###### Lemma 1.1.

Let $\\{X_{n}\\}_{n\geq 1}$ be a $d$-dimensional centered Gauss-Markov chain
with drift matrix $S$ and initial positive-definite covariance $\Sigma_{o}$.
For each $N\geq 1$

$\displaystyle Ne_{N}$
$\displaystyle=\frac{1}{2}\big{\langle}X_{1},(I-\Sigma_{o}^{-1}-S^{\top}S)X_{1}\big{\rangle}+\frac{1}{2}\big{\langle}X_{N},(\Sigma_{o}^{-1}+S^{\top}S-I)X_{N}\big{\rangle}$
$\displaystyle+\sum_{n=2}^{N}\big{\langle}X_{n},(S-S^{\top})X_{n-1}\big{\rangle}.$

Lemma 1.1 shows that the entropy production $Ne_{N}$ of the process
$\\{X_{n}\\}_{n\geq 1}$ is a particular instance of a quasi-Toeplitz quadratic
functional $W_{N}$ having the form

$W_{N}:=\frac{1}{2}\langle
X_{1},LX_{1}\rangle+\frac{1}{2}\sum_{n=2}^{N-1}\langle
X_{n},UX_{n}\rangle+\frac{1}{2}\langle
X_{N},RX_{N}\rangle+\sum_{n=2}^{N}\langle X_{n},VX_{n-1}\rangle,$ (1.2)

$L$, $U$, $R$, and $V$ being four matrices in $\mathbb{R}^{d\times d}$ with
$L$, $U$, and $R$ symmetric. In fact, $W_{N}$ turns out to be $Ne_{N}$ when
$L:=I-\Sigma_{o}^{-1}-S^{\top}S$, $U:=0$, $R:=\Sigma_{o}^{-1}+S^{\top}S-I$,
and $V:=S-S^{\top}$. The circumstance to be stressed is that the problem of
the entropy production rate leads to perturb a perfect Toeplitz structure by
quadratic boundary terms, in such a way that the coefficient matrix of $W_{N}$
differs from a block tridiagonal Toeplitz matrix by the first and last
diagonal blocks. This circumstance required new large deviation principles for
quadratic forms of Gauss-Markov chains to be developed, and to be added to the
literature on large deviations for Gaussian processes. Similarly to $e_{N}$,
we say that $W_{N}/N$ satisfies a large deviation principle with the rate
function $I$ if there exists a function $I$ with compact level sets such that
for each Borel set $\mathcal{B}\subseteq\mathbb{R}$

$-\inf_{w\in\mathcal{B}^{o}}\\{I(w)\\}\leq\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in\mathcal{B}\bigg{]}\leq\limsup_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in\mathcal{B}\bigg{]}\leq-\inf_{w\in\bar{\mathcal{B}}}\\{I(w)\\}.$

Large deviation principles for Gaussian processes have been an active field of
research since the pioneering works by Donsker and Varadhan [25] and Bryc and
Dembo [26] on the large fluctuations of empirical measures for stationary
Gaussian processes. The focus soon moved to large deviations of quadratic
functionals [27, 28], which in general cannot be tackled by a direct
application of the Gärtner-Ellis theorem since steepness of the asymptotic
cumulant generating function is not guaranteed. The asymptotic cumulant
generating function does not contain apparently the whole information on the
large deviation property of the process: there is a loss of information
passing to the limit. For Toeplitz quadratic forms of stationary centered
Gaussian sequences, large deviation principles are now well-established [29,
30, 31], as well as some moderate deviation principle [32]. These results have
been obtained by a sharp study of the spectrum of a product of two Toeplitz
matrices. For stationary centered Gaussian sequences, large deviations have
been also characterized for special Hermitian quadratic forms [29, 33] and a
sample path large deviation principle has been deduced for the squares of the
process [34]. Despite this progress, there are no general results to deal with
non-stationarity Gaussian sequences and perturbations of Toeplitz quadratic
functionals, which pose very specific problems.

The typical value of $W_{N}/N$ in the large $N$ limit is described by the law
of large numbers

$\lim_{N\uparrow\infty}\frac{W_{N}}{N}=\frac{1}{2}\mbox{tr}\Big{[}\big{(}U+V^{\top}S+S^{\top}V\big{)}\Sigma_{s}\Big{]}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{P}\mbox{-a.s.},$

which easily follows by specializing to our model standard results of the
theory of Markov chains [35]. It does not depend on the initial condition and
on the boundary terms. On the contrary, we shall see that non-stationarity and
quadratic boundary terms, which are intimately related for Gauss-Markov
chains, affect deviations of quadratic functionals from the mean and shape
rate functions. This is not surprising since squares of Gaussian random
variables have an exponential, rather than super-exponential, tail
probability. Researchers have already come across this issue. In fact, the
maximum likelihood estimator and the Yule-Walker estimator for the drift
parameter of a one-dimensional autoregressive stable process satisfy large
deviation principles with different rate functions [29]. These two estimators
are connected to quadratic functionals that differ exactly by a quadratic
boundary term. A similar phenomenology holds for the continuous-time
counterpart, i.e. the Ornstein-Uhlenbeck process [36]. Coming more close to
statistical physics, perturbations by quadratic boundary terms of the entropy
production rate for Ornstein-Uhlenbeck processes with normal drift matrix have
been considered to account for the heat dissipation rate of a network of
thermally driven harmonic oscillators [20].

The remainder of the paper is organized as follows. In Section 1.1 we present
the main results of this work: a large deviation principle for quadratic
functionals of type (1.2) in the context of multivariate centered stable
Gauss-Markov chains and the Gallavotti-Cohen symmetry of the entropy
production rate function. In Section 1.2 we apply the theory to the very
special class of stable Gauss-Markov chains with normal drift matrix, making
contact with previous results. Section 2 provides the proof of the large
deviation principle for the quadratic functionals. Section 3 reports the proof
of the Gallavotti-Cohen symmetry of the entropy production rate function.

###  1.1 Main results

From now on we regard $\mathbb{R}^{d}$ as a subset of $\mathbb{C}^{d}$ and we
denote by $\langle\cdot,\cdot\rangle$ the standard inner product of
$\mathbb{C}^{d}$. We write $A\succ 0$ to specify a positive-definite Hermitian
matrix $A\in\mathbb{C}^{d\times d}$.

Fix matrices $S$, $\Sigma_{o}$, $L$, $U$, $R$, and $V$ in $\mathbb{R}^{d\times
d}$ with $\rho(S)<1$, $\Sigma_{o}\succ 0$, and $L$, $U$, and $R$ symmetric.
According to (1.1) and (1.2), they define a centered stable Gauss-Markov chain
$\\{X_{n}\\}_{n\geq 1}$ and a quadratic functional $W_{N}$ for each $N\geq 1$.
For every $\lambda\in\mathbb{R}$ and $\theta\in[0,2\pi]$, we make use of $S$,
$U$, and $V$ to construct the Hermitian matrix

$F_{\lambda}(\theta):=\big{(}I-S^{\top}e^{\mathrm{i}\theta}\big{)}\big{(}I-Se^{-\mathrm{i}\theta}\big{)}-\lambda\big{(}U+Ve^{-\mathrm{i}\theta}+V^{\top}e^{\mathrm{i}\theta}\big{)}\in\mathbb{C}^{d\times
d},$ (1.3)

and we set

$f_{\lambda}:=\operatorname*{\inf\vphantom{p}}_{\theta\in[0,2\pi]}\inf_{\begin{subarray}{c}z\in\mathbb{C}^{d}\\\
z\neq 0\end{subarray}}\bigg{\\{}\frac{\langle
z,F_{\lambda}(\theta)z\rangle}{\langle z,z\rangle}\bigg{\\}}.$ (1.4)

As $f_{\lambda}$ bounds the spectrum of $F_{\lambda}(\theta)$ from below for
all $\theta$, if $f_{\lambda}>0$, then the functions that map
$\theta\in[0,2\pi]$ in $\ln\det F_{\lambda}(\theta)$ and
$F^{-1}_{\lambda}(\theta)$ are well-defined and continuous. Thus, for each
$\lambda\in\mathbb{R}$ such that $f_{\lambda}>0$ we can introduce the
integrals

$\varphi(\lambda):=-\frac{1}{4\pi}\int_{0}^{2\pi}\ln\det
F_{\lambda}(\theta)\,d\theta$ (1.5)

and

$\Phi_{\lambda}(n):=\frac{1}{2\pi}\int_{0}^{2\pi}F_{\lambda}^{-1}(\theta)e^{-\mathrm{i}n\theta}d\theta$
(1.6)

with $n\in\mathbb{Z}$. It will turn out that $\varphi(\lambda)$ is the value
at $\lambda$ of the cumulant generating function of $W_{N}$ in the large $N$
limit: $\lim_{N\uparrow\infty}(1/N)\ln\mathbb{E}[e^{\lambda
W_{N}}]=\varphi(\lambda)$. The boundary terms described by the matrices
$\Sigma_{o}$, $L$, and $R$ determine the effective domain of the asymptotic
cumulant generating function, and in order to specify this domain we need the
following technical lemma. The proof is reported in Section 2.

###### Lemma 1.2.

Let $\lambda\in\mathbb{R}$ be such that $f_{\lambda}>0$. The following
conclusions hold:

  1. 1.

$H_{\lambda}:=I+(S+\lambda V)\Phi_{\lambda}(1)\in\mathbb{C}^{d\times d}$ is
invertible, and the matrix

$\mathcal{L}_{\lambda}:=\displaystyle{\Sigma_{o}^{-1}+S^{\top}S-\lambda
L-(S^{\top}+\lambda V^{\top})\Phi_{\lambda}(0)H_{\lambda}^{-1}(S+\lambda
V)}\in\mathbb{C}^{d\times d}$

is Hermitian;

  2. 2.

$K_{\lambda}:=I+\Phi_{\lambda}(1)(S+\lambda V)\in\mathbb{C}^{d\times d}$ is
invertible, and the matrix

$\mathcal{R}_{\lambda}:=\displaystyle{I-\lambda R-(S+\lambda
V)K_{\lambda}^{-1}\Phi_{\lambda}(0)(S^{\top}+\lambda
V^{\top})}\in\mathbb{C}^{d\times d}$

is Hermitian.

Lemma 1.2 states that the matrices $\mathcal{L}_{\lambda}$ and
$\mathcal{R}_{\lambda}$ are well-defined and Hermitian when
$\lambda\in\mathbb{R}$ satisfies $f_{\lambda}>0$. It makes then sense to
consider the extended real numbers

$\lambda_{-}:=\inf\Big{\\{}\lambda\in\mathbb{R}\leavevmode\nobreak\
:\leavevmode\nobreak\ f_{\lambda}>0,\leavevmode\nobreak\
\mathcal{L}_{\lambda}\succ 0,\mbox{ and }\mathcal{R}_{\lambda}\succ
0\Big{\\}}$ (1.7)

and

$\lambda_{+}:=\sup\Big{\\{}\lambda\in\mathbb{R}\leavevmode\nobreak\
:\leavevmode\nobreak\ f_{\lambda}>0,\leavevmode\nobreak\
\mathcal{L}_{\lambda}\succ 0,\mbox{ and }\mathcal{R}_{\lambda}\succ
0\Big{\\}}.$ (1.8)

We are now in the position to present the first main result of the paper,
which establishes a large deviation principle for $W_{N}/N$ and is proved in
Section 2 via a time-dependent change of measure.

###### Theorem 1.1.

The following conclusions hold:

  1. 1.

$\lambda_{-}<0<\lambda_{+}$ and the convex function $I$ that maps
$w\in\mathbb{R}$ in
$I(w):=\sup_{\lambda\in(\lambda_{-},\lambda_{+})}\\{w\lambda-\varphi(\lambda)\\}$
has compact level sets;

  2. 2.

the quadratic functional $W_{N}/N$ associated with the stable Gauss-Markov
chain $\\{X_{n}\\}_{n\geq 1}$ satisfies a large deviation principle with the
rate function $I$.

Theorem 1.1 outperforms the Gärtner-Ellis theorem, which requires that the
asymptotic cumulant generating function exists and defines an essentially
smooth, lower semicontinuous function [16, 15]. In Section 2 we shall prove
that $\lim_{N\uparrow\infty}(1/N)\ln\mathbb{E}[e^{\lambda
W_{N}}]=\varphi(\lambda)$ if $\lambda\in(\lambda_{-},\lambda_{+})$ and
$\lim_{N\uparrow\infty}(1/N)\ln\mathbb{E}[e^{\lambda W_{N}}]=+\infty$ if
$\lambda\notin\overline{(\lambda_{-},\lambda_{+})}$. We shall also verify that
the function $\varphi$ that maps $\lambda\in(\lambda_{-},\lambda_{+})$ in
$\varphi(\lambda)$ is convex and differentiable, so that the limits
$\lim_{\lambda\downarrow\lambda_{-}}\varphi(\lambda)=:\varphi_{-}$,
$\lim_{\lambda\uparrow\lambda_{+}}\varphi(\lambda)=:\varphi_{+}$,
$\lim_{\lambda\downarrow\lambda_{-}}\varphi^{\prime}(\lambda)=:d_{-}$, and
$\lim_{\lambda\uparrow\lambda_{+}}\varphi^{\prime}(\lambda)=:d_{+}$ exist. If
even the limit $\lim_{N\uparrow\infty}(1/N)\ln\mathbb{E}[e^{\lambda W_{N}}]$
existed for all $\lambda\in\mathbb{R}$ and defined a lower semicontinuous
function as demanded by the Gärtner-Ellis theorem, what is generally missing
to guarantee essentially smoothness of the asymptotic cumulant generating
function is the steepness of $\varphi$, i.e. the property that $d_{-}=-\infty$
if $\lambda_{-}>-\infty$ and $d_{+}=+\infty$ if $\lambda_{+}<+\infty$. The
lack of steepness produces affine stretches in the graph of the rate function.
In fact, if $\lambda_{-}>-\infty$ and $d_{-}>-\infty$, then
$I(w)=w\lambda_{-}-\varphi_{-}$ for all $w<d_{-}$. Notice that $\varphi_{-}$
is finite in this case since
$\varphi(\lambda)\leq\varphi(0)+\varphi^{\prime}(\lambda)\lambda=\varphi^{\prime}(\lambda)\lambda$
for all $\lambda\in(\lambda_{-},\lambda_{+})$ by convexity, which gives
$\varphi_{-}\leq d_{-}\lambda_{-}$ by sending $\lambda$ to $\lambda_{-}$.
Similarly, $I(w)=w\lambda_{+}-\varphi_{+}$ for all $w>d_{+}$ with
$\varphi_{+}$ finite if $\lambda_{+}<+\infty$ and $d_{+}<+\infty$. The
following example involving a quadratic functional of a one-dimensional stable
Gauss-Markov chain demonstrates the presence of affine stretches.

###### Example 1.1.

Fix $s\in\mathbb{R}$ such that $|s|<1$ and consider the one-dimensional
autoregressive model $X_{n+1}=sX_{n}+G_{n}$ for $n\geq 1$. The large
fluctuations of the quadratic functional $W_{N}:=\sum_{n=1}^{N}X_{n}^{2}$ have
been already characterized for the non-stationary case $X_{1}:=0$ [37] and for
the stationary centered case corresponding to $\Sigma_{o}:=(1-s^{2})^{-1}$
[28]. We can use our theory to investigate centered non-stationary situations
with general initial variance $\Sigma_{o}>0$. In this example $S:=s$, $L:=2$,
$U:=2$, $R:=2$, and $V:=0$. For all $\lambda$ and $\theta$ we find

$F_{\lambda}(\theta)=1+s^{2}-2\lambda-2s\cos(\theta),$

so that $f_{\lambda}=1+s^{2}-2\lambda-2|s|$. If $f_{\lambda}>0$, i.e.
$2\lambda<(1-|s|)^{2}$, then easy calculations yield

$\varphi(\lambda)=-\frac{1}{2}\ln\frac{1+s^{2}-2\lambda+\sqrt{(1+s^{2}-2\lambda)^{2}-4s^{2}}}{2},$
$\mathcal{L}_{\lambda}=\Sigma_{o}^{-1}+\frac{s^{2}-1-2\lambda+\sqrt{(1+s^{2}-2\lambda)^{2}-4s^{2}}}{2},$

and

$\mathcal{R}_{\lambda}=\frac{1-s^{2}-2\lambda+\sqrt{(1+s^{2}-2\lambda)^{2}-4s^{2}}}{2}>0.$

The quantities $\mathcal{L}_{\lambda}$ and $\mathcal{R}_{\lambda}$ are defined
by Lemma 1.2. According to (1.7) and (1.8), we have $\lambda_{-}=-\infty$ and
$2\lambda_{+}=(1-|s|)^{2}$ if $\Sigma_{o}^{-1}\geq 1-|s|$, and
$\lambda_{-}=-\infty$ and
$2\lambda_{+}=(\Sigma_{o}^{-1}-1+s^{2})/(1-\Sigma_{o})$ if
$\Sigma_{o}^{-1}<1-|s|$. In the former case $\varphi$ is steep, whereas
steepness is missing in the latter case where
$d_{+}=1/\sqrt{(1+s^{2}-2\lambda_{+})^{2}-4s^{2}}<+\infty$. If
$\Sigma_{o}^{-1}\geq 1-|s|$, then the rate function is

$I(w)=J(w):=\begin{cases}+\infty&\mbox{if }w\leq 0,\\\
\frac{1}{2}(1+s^{2})w-\frac{1}{2}\ln(2w)-\frac{1}{2}\sqrt{1+(2sw)^{2}}+\frac{1}{2}\ln[1+\sqrt{1+(2sw)^{2}}]&\mbox{if
}w>0.\end{cases}$

If $\Sigma_{o}^{-1}<1-|s|$, then the rate function reads

$I(w)=\begin{cases}J(w)&\mbox{if }w<d_{+},\\\
w\lambda_{+}-\varphi_{+}&\mbox{if }w\geq d_{+}.\end{cases}$

As $W_{N}/N=e_{N}$ for all $N\geq 1$ when $L:=I-\Sigma_{o}^{-1}-S^{\top}S$,
$U:=0$, $R:=\Sigma_{o}^{-1}+S^{\top}S-I$, and $V:=S-S^{\top}$, Theorem 1.1
immediately shows that the entropy production rate $e_{N}$ satisfies a large
deviation principle. The Hermitian matrix $F_{\lambda}(\theta)$ corresponding
to $e_{N}$ reads for each $\lambda\in\mathbb{R}$ and $\theta\in[0,2\pi]$

$\displaystyle F_{\lambda}(\theta)$
$\displaystyle=\big{(}I-S^{\top}e^{\mathrm{i}\theta}\big{)}\big{(}I-Se^{-\mathrm{i}\theta}\big{)}+2\mathrm{i}\lambda\big{(}S-S^{\top}\big{)}\sin\theta$
$\displaystyle=I+S^{\top}S-(S+S^{\top})\cos\theta+\mathrm{i}(2\lambda+1)\big{(}S-S^{\top}\big{)}\sin\theta.$
(1.9)

The second main result of the paper, whose proof is reported in Section 3,
confirms the Gallavotti-Cohen symmetry. This symmetry comes from the manifest
relationship $F_{-\lambda-1}(\theta)=F_{\lambda}(2\pi-\theta)$.

###### Theorem 1.2.

The following conclusions hold:

  1. 1.

the entropy production rate $e_{N}$ of the stable Gauss-Markov chain
$\\{X_{n}\\}_{n\geq 1}$ satisfies a large deviation principle with the convex
rate function $I$;

  2. 2.

$\lambda_{-}=-\lambda_{+}-1$ and $I(-w)=I(w)+w$ for all $w\in\mathbb{R}$.

If the drift matrix $S$ is symmetric and $\Sigma_{o}=\Sigma_{s}$, then the
process $\\{X_{n}\\}_{n\geq 1}$ is reversible and $e_{N}=0$ for all $N\geq 1$.
The following example shows that there is entropy production when $S$ is
symmetric but $\\{X_{n}\\}_{n\geq 1}$ is not stationary.

###### Example 1.2.

Assume that the drift matrix $S$ is symmetric. We have
$\Sigma_{s}=(I-S^{2})^{-1}$ and formula (1.9) gives
$F_{\lambda}(\theta)=\big{(}I-Se^{\mathrm{i}\theta}\big{)}\big{(}I-Se^{-\mathrm{i}\theta}\big{)}$
for every $\lambda$ and $\theta$. One can easily verify that
$f_{\lambda}=[1-\rho(S)]^{2}>0$ and $\varphi(\lambda)=0$ for all
$\lambda\in\mathbb{R}$. Starting from the identity
$(I-Se^{\pm\mathrm{i}\theta})^{-1}=\sum_{k\geq
0}S^{k}e^{\pm\mathrm{i}k\theta}$ as $\rho(S)<1$, one can then deduce that for
all $\lambda\in\mathbb{R}$

$\mathcal{L}_{\lambda}=\mathcal{R}_{-\lambda-1}=(\lambda+1)\Sigma_{o}^{-1}-\lambda\Sigma_{s}^{-1}.$

Fix $\Sigma_{o}\succ 0$ different from $\Sigma_{s}$ and set
$\Delta:=(\Sigma_{s}-\Sigma_{o})(\Sigma_{s}+\Sigma_{o})^{-1}$. We claim that
the spectral radius $\rho(\Delta)$ of $\Delta$ is strictly positive and that

$\lambda_{\pm}=\frac{1}{2}\bigg{[}-1\pm\frac{1}{\rho(\Delta)}\bigg{]}.$ (1.10)

The entropy production rate satisfies a large deviation principle with the
rate function

$I(w)=\begin{cases}w\lambda_{-}&\mbox{if }w<0,\\\ w\lambda_{+}&\mbox{if }w\geq
0.\end{cases}$

To prove (1.10), let $A\in\mathbb{R}^{d\times d}$ be an invertible matrix such
that $(1/2)(\Sigma_{o}^{-1}+\Sigma_{s}^{-1})=AA^{\top}$ and set
$B:=(1/2)A^{-1}(\Sigma_{o}^{-1}-\Sigma_{s}^{-1})(A^{\top})^{-1}$. The matrix
$A$ exists since $\Sigma_{s}\succ 0$ and $\Sigma_{o}\succ 0$, and the spectral
radius $\rho(B)$ of the symmetric matrix $B$ is strictly positive since
$\Sigma_{o}\neq\Sigma_{s}$. Similarity transformations show that
$\rho(B)=\rho(\Delta)$. We have
$\mathcal{L}_{\lambda}=A[I+(2\lambda+1)B]A^{\top}\succ 0$ and
$\mathcal{R}_{\lambda}=A[I-(2\lambda+1)B]A^{\top}\succ 0$ if and only if
$|2\lambda+1|\rho(B)<1$. Thus, $(2\lambda_{\pm}+1)\rho(B)=\pm 1$.

###  1.2 Entropy production with a normal drift matrix

Analyzing the role of the conditions $\mathcal{L}_{\lambda}\succ 0$ and
$\mathcal{R}_{\lambda}\succ 0$ in determining those $\lambda\in\mathbb{R}$ for
which $\lim_{N\uparrow\infty}(1/N)\ln\mathbb{E}[e^{\lambda
W_{N}}]=\varphi(\lambda)$ is a difficult task. We stress that the
satisfiability of these conditions shapes the effective domain
$(\lambda_{-},\lambda_{+})$ of the asymptotic cumulant generating function of
$W_{N}$. Now our interest is in the entropy production $W_{N}:=Ne_{N}$.
Computer simulations suggest that, in the stationary case
$\Sigma_{o}=\Sigma_{s}$, the Hermitian matrices $\mathcal{L}_{\lambda}$ and
$\mathcal{R}_{\lambda}$ associated with $Ne_{N}$ are automatically positive-
definite for the values of $\lambda$ that satisfy the primary constraint
$f_{\lambda}>0$. If this is true in general, then we will conclude that
$\lambda_{-}=\inf\\{\lambda\in\mathbb{R}:f_{\lambda}>0\\}$ and
$\lambda_{+}=\sup\\{\lambda\in\mathbb{R}:f_{\lambda}>0\\}$ when
$\Sigma_{o}=\Sigma_{s}$. While we leave this general problem as an open
question, we verify the conjecture
$\lambda_{-}=\inf\\{\lambda\in\mathbb{R}:f_{\lambda}>0\\}$ and
$\lambda_{+}=\sup\\{\lambda\in\mathbb{R}:f_{\lambda}>0\\}$ for a stationary
stable Gauss-Markov chain $\\{X_{n}\\}_{n\geq 1}$ with normal drift matrix
$S$. Then, here we assume that $S^{\top}S=SS^{\top}$. This case is very
special because it allows for explicit results. We point out that large
deviation principles have been recently established for the entropy production
rate of stationary stable Ornstein-Uhlenbeck processes with normal drift
matrix [19, 20, 21]. In particular, Budhiraja, Chen, and Xu [21] have
exhibited explicitly the rate function, posing the question of whether the
same could have been done for the discrete-time autoregressive model. Our work
gives an affirmative answer to their question, and indeed we provide a large
deviation principle for any drift matrix.

Dealing with a normal drift matrix in the problem of entropy production
basically means dealing with a diagonal drift matrix. In fact, normality of
$S$ implies that there exists a unitary matrix $\Gamma\in\mathbb{C}^{d\times
d}$ such that $\Gamma S\Gamma^{-1}$ and $\Gamma S^{\top}\Gamma^{-1}=(\Gamma
S\Gamma^{-1})^{\dagger}$ are both diagonal. Let
$\alpha_{k}+\mathrm{i}\beta_{k}$ be the $k$th element of the diagonal of
$\Gamma S\Gamma^{-1}$, with $\alpha_{k}$ and $\beta_{k}$ real numbers, and
notice that the stability hypothesis $\rho(S)<1$ requires that
$\alpha_{k}^{2}+\beta_{k}^{2}<1$ as $\alpha_{k}+\mathrm{i}\beta_{k}$ obviously
is an eigenvalue of $S$. We suppose that $\beta_{k}\neq 0$ for some $k$ in
order to not to fall again in the class of symmetric drift matrices. According
to (1.9), $\Gamma F_{\lambda}(\theta)\Gamma^{-1}$ is diagonal for all
$\lambda\in\mathbb{R}$ and $\theta\in[0,2\pi]$, and the $k$th element of the
diagonal of $\Gamma F_{\lambda}(\theta)\Gamma^{-1}$ reads

$1+\alpha_{k}^{2}+\beta_{k}^{2}-2\alpha_{k}\cos\theta-2\beta_{k}(2\lambda+1)\sin\theta=\big{(}1+\alpha_{k}^{2}+\beta_{k}^{2}\big{)}\big{[}1-\varrho_{k}\cos(\theta-\vartheta_{k})\big{]}$

with

$\varrho_{k}:=2\frac{\sqrt{\alpha_{k}^{2}+(2\lambda+1)^{2}\beta_{k}^{2}}}{1+\alpha_{k}^{2}+\beta_{k}^{2}}\geq
0$

and

$\vartheta_{k}:=\arctan\bigg{(}\frac{\beta_{k}+2\lambda\beta_{k}}{\alpha_{k}}\bigg{)}.$

We omit to indicate the dependence of $\varrho_{k}$ and $\vartheta_{k}$ on
$\lambda$ for simplicity. We have

$f_{\lambda}=\inf_{\theta\in[0,2\pi]}\min_{1\leq k\leq
d}\Big{\\{}\big{(}1+\alpha_{k}^{2}+\beta_{k}^{2}\big{)}\big{[}1-\varrho_{k}\cos(\theta-\vartheta_{k})\big{]}\Big{\\}}=\min_{1\leq
k\leq
d}\Big{\\{}\big{(}1+\alpha_{k}^{2}+\beta_{k}^{2}\big{)}\big{(}1-\varrho_{k}\big{)}\Big{\\}},$

so that the condition $f_{\lambda}>0$ on $\lambda$ becomes $\max_{1\leq k\leq
d}\\{\varrho_{k}\\}<1$. If $\max_{1\leq k\leq d}\\{\varrho_{k}\\}<1$, then we
find from (1.5)

$\displaystyle\varphi(\lambda)$
$\displaystyle=-\frac{1}{4\pi}\sum_{k=1}^{d}\int_{0}^{2\pi}\ln\Big{\\{}\big{(}1+\alpha_{k}^{2}+\beta_{k}^{2}\big{)}\big{[}1-\varrho_{k}\cos(\theta-\vartheta_{k})\big{]}\Big{\\}}d\theta$
$\displaystyle=-\frac{1}{4\pi}\sum_{k=1}^{d}\int_{0}^{2\pi}\ln\big{(}1-\varrho_{k}\cos\theta\big{)}d\theta-\frac{1}{2}\sum_{k=1}^{d}\ln\big{(}1+\alpha_{k}^{2}+\beta_{k}^{2}\big{)}$
$\displaystyle=-\frac{1}{2}\sum_{k=1}^{d}\ln\frac{1+\sqrt{1-\varrho_{k}^{2}}}{2}-\frac{1}{2}\sum_{k=1}^{d}\ln\big{(}1+\alpha_{k}^{2}+\beta_{k}^{2}\big{)}.$

For each $n\in\mathbb{Z}$, the matrix $\Gamma\Phi_{\lambda}(n)\Gamma^{-1}$
defined by (1.6) is diagonal with $k$th diagonal element equal to

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\frac{e^{-\mathrm{i}n\theta}d\theta}{(1+\alpha_{k}^{2}+\beta_{k}^{2})[1-\varrho_{k}\cos(\theta-\vartheta_{k})]}$
$\displaystyle=\frac{e^{-\mathrm{i}n\vartheta_{k}}}{1+\alpha_{k}^{2}+\beta_{k}^{2}}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\cos(n\theta)d\theta}{1-\varrho_{k}\cos\theta}$
$\displaystyle=\frac{1}{1+\alpha_{k}^{2}+\beta_{k}^{2}}\frac{e^{-\mathrm{i}n\vartheta_{k}}}{\sqrt{1-\varrho_{k}^{2}}}\bigg{(}\frac{1-\sqrt{1-\varrho_{k}^{2}}}{\varrho_{k}}\bigg{)}^{|n|}.$

Under the constraint $\max_{1\leq k\leq d}\\{\varrho_{k}\\}<1$, the matrices
$\mathcal{L}_{\lambda}$ and $\mathcal{R}_{\lambda}$ associated by Lemma 1.2
with $L:=I-\Sigma_{o}^{-1}-S^{\top}S$, $U:=0$,
$R:=\Sigma_{o}^{-1}+S^{\top}S-I$, and $V:=S-S^{\top}$ can be written as

$\mathcal{L}_{\lambda}=(\lambda+1)(\Sigma_{o}^{-1}-\Sigma_{s}^{-1})+\mathcal{M}_{\lambda}$
(1.11)

and

$\mathcal{R}_{\lambda}=\lambda(\Sigma_{s}^{-1}-\Sigma_{o}^{-1})+\mathcal{M}_{\lambda},$
(1.12)

where $\Gamma\mathcal{M}_{\lambda}\Gamma^{-1}\in\mathbb{C}^{d\times d}$ is
diagonal with $k$th diagonal element given by

$\frac{1-\alpha_{k}^{2}-\beta_{k}^{2}}{2}+\frac{1+\alpha_{k}^{2}+\beta_{k}^{2}}{2}\sqrt{1-\varrho_{k}^{2}}>0.$
(1.13)

To obtain (1.11) and (1.12) we have used the facts that
$\Sigma_{s}=(I-SS^{\top})^{-1}$ and that $\Gamma\Sigma_{s}^{-1}\Gamma^{-1}$ is
diagonal with $k$th diagonal entry equal to $1-\alpha_{k}^{2}-\beta_{k}^{2}$.
Importantly, the Hermitian matrix $\mathcal{M}_{\lambda}$ is positive-definite
as demonstrated by (1.13).

If the chain $\\{X_{n}\\}_{n\geq 1}$ is stationary, i.e. if
$\Sigma_{o}=\Sigma_{s}$, then $\mathcal{L}_{\lambda}=\mathcal{M}_{\lambda}$
and $\mathcal{R}_{\lambda}=\mathcal{M}_{\lambda}$ are automatically positive-
definite when $\max_{1\leq k\leq d}\\{\varrho_{k}\\}<1$, namely when
$f_{\lambda}>0$. Thus, the conjecture
$\lambda_{-}=\inf\\{\lambda\in\mathbb{R}:f_{\lambda}>0\\}$ and
$\lambda_{+}=\sup\\{\lambda\in\mathbb{R}:f_{\lambda}>0\\}$ for a stationary
stable Gauss-Markov chain is true if the drift matrix is normal. Furthermore,
in this case $\lambda_{-}$ and $\lambda_{+}$ are the smallest and the largest
values of $\lambda$ for which $\max_{1\leq k\leq d}\\{\varrho_{k}\\}=1$, which
are explicitly given by the formulas

$\lambda_{+}=\lambda_{o}:=-\frac{1}{2}+\min_{1\leq k\leq
d}\Bigg{\\{}\sqrt{\frac{(1+\alpha_{k}^{2}+\beta_{k}^{2})^{2}-4\alpha_{k}^{2}}{16\beta_{k}^{2}}}\Bigg{\\}}$

and

$\lambda_{-}=-\lambda_{o}-1.$

Notice that $\lambda_{o}$ is finite since we are supposing that $\beta_{k}\neq
0$ for some $k$. With such $\lambda_{-}$ and $\lambda_{+}$, the function
$\varphi$ turns out to be steep in $(\lambda_{-},\lambda_{+})$. Thus, for each
$w\in\mathbb{R}$ there exists a unique $\lambda\in(\lambda_{-},\lambda_{+})$
such that $w=\varphi^{\prime}(\lambda)$ and, as a consequence,

$I(w)=w\lambda-\varphi(\lambda).$

Basically, this is the result found by Budhiraja, Chen, and Xu [21] for the
continuous-time model.

To conclude, let us briefly discuss what happens when the chain
$\\{X_{n}\\}_{n\geq 1}$ is not stationary, i.e. when
$\Sigma_{o}\neq\Sigma_{s}$. If $\mathcal{L}_{\lambda}\succ 0$ and
$\mathcal{R}_{\lambda}\succ 0$ for all
$\lambda\in(-\lambda_{o}-1,\lambda_{o})$, then $\lambda_{+}=\lambda_{o}$ and
$\lambda_{-}=-\lambda_{o}-1$, as before, and the function $\varphi$ is steep
in $(\lambda_{-},\lambda_{+})$. We have $\mathcal{L}_{\lambda}\succ 0$ and
$\mathcal{R}_{\lambda}\succ 0$ for all
$\lambda\in(-\lambda_{o}-1,\lambda_{o})$ if
$\mathcal{L}_{\lambda_{o}}=\mathcal{R}_{-\lambda_{o}-1}\succ 0$ and
$\mathcal{R}_{\lambda_{o}}=\mathcal{L}_{-\lambda_{o}-1}\succ 0$ as formulas
(1.11) and (1.12) show that the functions that map $\lambda$ in $\langle
z,\mathcal{L}_{\lambda}z\rangle$ and $\langle z,\mathcal{R}_{\lambda}z\rangle$
are concave for any given $z\in\mathbb{C}^{d}$. If, on the contrary, there
exists $\lambda\in(-\lambda_{o}-1,\lambda_{o})$ such that
$\mathcal{L}_{\lambda}\nsucc 0$ or $\mathcal{R}_{\lambda}\nsucc 0$, then
$\lambda_{+}<\lambda_{o}$ and $\lambda_{-}=-\lambda_{+}-1>-\lambda_{o}-1$. For
example, this occurs for $\Sigma_{o}=\sigma I$ with a sufficiently small
$\sigma>0$. In this case $\varphi$ is not steep in $(\lambda_{-},\lambda_{+})$
and the rate function at $w\in\mathbb{R}$ has the value

$I(w)=\begin{cases}w\lambda_{-}-\varphi_{-}&\mbox{if }w\leq d_{-},\\\
w\lambda-\varphi(\lambda)&\mbox{if }d_{-}<w<d_{+},\\\
w\lambda_{+}-\varphi_{+}&\mbox{if }w\geq d_{+},\end{cases}$

where, regarding the case $d_{-}<w<d_{+}$, $\lambda$ is the unique real number
in $(\lambda_{-},\lambda_{+})$ that satisfies $w=\varphi^{\prime}(\lambda)$.
Breaking stationarity can then involve affine stretches in the graph of the
entropy production rate function.

##  2 Proof of Lemma 1.2 and Theorem 1.1

In this section we prove Theorem 1.1, which states the large deviation
principle for the quadratic functional $W_{N}$ defined by (1.2). The proof of
Theorem 1.1 is based on a time-dependent change of measure and requires at
first to study the asymptotics of the cumulant generating function of $W_{N}$
as $N$ goes to infinity. In turn, this asks for investigation of Hermitian
block tridiagonal quasi-Toeplitz matrices that differ from Hermitian block
tridiagonal Toeplitz matrices by the first and last diagonal blocks. In
Section 2.1 we introduce these matrices and characterize their positive
definiteness property and their determinant. Section 2.2 uses the theory of
Section 2.1 to compute the scaled cumulant generating function of $W_{N}$ in
the large $N$ limit. The upper large deviation bound for closed sets is proved
in Section 2.3. Finally, the lower large deviation bound for open sets is
established in Section 2.4. Along the way we shall also verify Lemma 1.2.

As we have already said, we regard $\mathbb{R}^{d}$ as a subset of
$\mathbb{C}^{d}$. We denote by $\langle\cdot,\cdot\rangle$ the standard inner
product of $\mathbb{C}^{d}$ and by $\|\cdot\|$ the induced norm. If
$\zeta=(\zeta_{1},\ldots,\zeta_{N})$ and $z=(z_{1},\ldots,z_{N})$ are two
vectors in $(\mathbb{C}^{d})^{N}$, $N$ being a positive integer, we understand
that $\langle\zeta,z\rangle:=\sum_{n=1}^{N}\langle\zeta_{n},z_{n}\rangle$ and
$\|z\|^{2}:=\sum_{n=1}^{N}\langle
z_{n},z_{n}\rangle=\sum_{n=1}^{N}\|z_{n}\|^{2}$. For positive integers $M$ and
$N$, $\mathsf{BL}_{M,N}$ is the set of complex block matrices with $M\times N$
square blocks of size $d$. For any $\mathsf{A}\in\mathsf{BL}_{N,N}$,
$\|\mathsf{A}\|$ is the operator norm of $\mathsf{A}$ induced by the norm of
$(\mathbb{C}^{d})^{N}$:

$\|\mathsf{A}\|:=\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\|\mathsf{A}z\|}{\|z\|}\bigg{\\}}.$

Given a Hermitian matrix $\mathsf{A}\in\mathsf{BL}_{N,N}$, we denote by
$r(\mathsf{A})$ the infimum of the Rayleigh quotient of a $\mathsf{A}$, that
is the smallest eigenvalue of $\mathsf{A}$:

$r(\mathsf{A}):=\inf_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle z,\mathsf{A}z\rangle}{\langle
z,z\rangle}\bigg{\\}}.$

If a Hermitian matrix $\mathsf{A}\in\mathsf{BL}_{N,N}$ is positive-definite we
write $\mathsf{A}\succ 0$. We have $\mathsf{A}\succ 0$ if and only if
$r(\mathsf{A})>0$.

###  2.1 On Hermitian block tridiagonal quasi-Toeplitz matrices

The coefficient matrix of the quadratic functional $W_{N}$ is an element from
a sequence of Hermitian block tridiagonal matrices in the following class.

###### Definition 2.1.

A sequence of matrices $\\{\mathsf{Q}_{N}\\}_{N\geq 1}$, with
$\mathsf{Q}_{N}\in\mathsf{BL}_{N+2,N+2}$ for each $N$, is a Hermitian block
tridiagonal quasi-Toeplitz (HQT) matrix sequence if there exist four square
matrices $A$, $D$, $B$, and $E$ of size $d$, with $A$, $D$, and $B$ Hermitian,
such that for all $N\geq 1$

$\mathsf{Q}_{N}=\begin{pmatrix}A&E^{\dagger}&&&\\\ E&D&\ddots&&\\\
&\ddots&\ddots&\ddots&\\\ &&\ddots&D&E^{\dagger}\\\ &&&E&B\end{pmatrix}.$

In this section we characterize asymptotic positive definiteness and
asymptotic determinants of matrices from a HQT matrix sequence, postponing the
most technical proofs in the appendices. We stress that a HQT matrix sequence
is bounded in the following sense, which is demonstrated in Appendix B.

###### Lemma 2.1.

Let $\\{\mathsf{Q}_{N}\\}_{N\geq 1}$ be a HQT matrix sequence with $A$, $D$,
$B$, and $E$ as in the above definition. Then, for each $N\geq 1$

$\|\mathsf{Q}_{N}\|\leq\sqrt{2\|A\|^{2}+3\|D\|^{2}+2\|B\|^{2}+6\|E\|^{2}}.$

In order to deal with a HQT matrix sequence $\\{\mathsf{Q}_{N}\\}_{N\geq 1}$,
it is convenient to isolate the bulk matrix
$\mathsf{T}_{N}\in\mathsf{BL}_{N,N}$ of $\mathsf{Q}_{N}$ defined by

$\mathsf{T}_{N}:=\begin{pmatrix}D&E^{\dagger}&&\\\ E&\ddots&\ddots&\\\
&\ddots&\ddots&E^{\dagger}\\\ &&E&D\end{pmatrix}.$ (2.1)

The bulk matrix $\mathsf{T}_{N}$ is a Hermitian block tridiagonal Toeplitz
matrix, which allows $\mathsf{Q}_{N}$ to be written as

$\mathsf{Q}_{N}=\begin{pmatrix}A&E^{\dagger}\mathsf{C}^{\dagger}&0\\\
\mathsf{C}E&\mathsf{T}_{N}&\mathsf{R}^{\dagger}E^{\dagger}\\\
0&E\mathsf{R}&B\end{pmatrix},$ (2.2)

where

$\mathsf{C}:=\begin{pmatrix}I\\\ 0\\\ \vdots\\\
0\end{pmatrix}\in\mathsf{BL}_{N,1}$ (2.3)

and

$\mathsf{R}:=\begin{pmatrix}0&\cdots&0&I\end{pmatrix}\in\mathsf{BL}_{1,N}.$
(2.4)

When $\mathsf{T}_{N}$ is invertible we introduce the boundary matrix
$\mathsf{S}_{N}\in\mathsf{BL}_{2,2}$ of $\mathsf{Q}_{N}$ defined by

$\mathsf{S}_{N}:=\begin{pmatrix}A-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{C}E&-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\\\\[3.00003pt]
-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{C}E&B-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\end{pmatrix}.$
(2.5)

Manifestly, $\mathsf{S}_{N}$ is a Hermitian matrix. The following lemma
relates the positive definiteness and the determinant of $\mathsf{Q}_{N}$ to
those of the bulk matrix $\mathsf{T}_{N}$ and the boundary matrix
$\mathsf{S}_{N}$. The proof is reported in Appendix C.

###### Lemma 2.2.

Let $\\{\mathsf{Q}_{N}\\}_{N\geq 1}$ be a HQT matrix sequence with bulk
matrices $\mathsf{T}_{N}$ and boundary matrices $\mathsf{S}_{N}$. The
following conclusions hold for any $N\geq 1$:

  1. 1.

if $r(\mathsf{Q}_{N})\geq q$ for some real number $q>0$, then
$r(\mathsf{T}_{N})\geq q$ (which implies that $\mathsf{T}_{N}$ is invertible)
and $r(\mathsf{S}_{N})\geq q$;

  2. 2.

if $\mathsf{T}_{N}\succ\mathsf{0}$ (which implies that $\mathsf{T}_{N}$ is
invertible) and $\mathsf{S}_{N}\succ\mathsf{0}$, then
$\mathsf{Q}_{N}\succ\mathsf{0}$ and

$\ln\det\mathsf{Q}_{N}=\ln\det\mathsf{T}_{N}+\ln\det\mathsf{S}_{N}.$

We now examine the bulk matrices. For each $\theta\in[0,2\pi]$, let
$F(\theta)\in\mathbb{C}^{d\times d}$ be a Hermitian matrix defined by

$F(\theta):=Ee^{-\mathrm{i}\theta}+D+E^{\dagger}e^{\mathrm{i}\theta},$

$D$ and $E$ being the matrices that identify the bulk matrix $\mathsf{T}_{N}$
of $\mathsf{Q}_{N}$. In the theory of block Toeplitz matrices [38], the
function $F$ that maps $\theta$ in $F(\theta)$ is called the symbol of the
matrices $\mathsf{T}_{N}$. We shall equally call $F$ the symbol of
$\mathsf{T}_{N}$ or the symbol of $\mathsf{Q}_{N}$. The blocks of
$\mathsf{T}_{N}$ are related to the Fourier coefficients of the symbol $F$. In
fact, for all $N\geq 1$,
$\zeta=(\zeta_{1},\ldots,\zeta_{N})\in(\mathbb{C}^{d})^{N}$, and
$z=(z_{1},\ldots,z_{N})\in(\mathbb{C}^{d})^{N}$ we have

$\displaystyle\langle\zeta,\mathsf{T}_{N}z\rangle$
$\displaystyle=\sum_{m=1}^{N}\sum_{n=1}^{N}\Bigg{\langle}\zeta_{m},\frac{1}{2\pi}\int_{0}^{2\pi}F(\theta)e^{\mathrm{i}(m-n)\theta}d\theta\,z_{n}\Bigg{\rangle}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\Bigg{\langle}\sum_{n=1}^{N}\zeta_{n}e^{-\mathrm{i}n\theta},F(\theta)\sum_{n=1}^{N}z_{n}e^{-\mathrm{i}n\theta}\Bigg{\rangle}\,d\theta.$
(2.6)

The following lemma describes the positive definiteness and the determinant of
the bulk matrices $\mathsf{T}_{N}$. The proof is provided in Appendix D. We
stress that if $\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}>0$, then the
function that associates $\theta\in[0,2\pi]$ with $\ln\det F(\theta)$ is well-
defined and continuous.

###### Lemma 2.3.

Let $\mathsf{T}_{N}$ be the bulk matrices of a HQT matrix sequence with symbol
$F$. The following conclusions hold:

  1. 1.

if there exists a diverging sequence $\\{N_{k}\\}_{k\geq 0}$ of positive
integers such that $r(\mathsf{T}_{N_{k}})\geq t$ for all $k\geq 0$ with some
$t\in\mathbb{R}$, then $r(\mathsf{T}_{N})\geq t$ for all $N\geq 1$;

  2. 2.

$r(\mathsf{T}_{N})\geq t$ for all $N\geq 1$ with some $t\in\mathbb{R}$ if and
only if $\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}\geq t$;

  3. 3.

if $\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}>0$, then

$\lim_{N\uparrow\infty}\frac{1}{N}\ln\det\mathsf{T}_{N}=\frac{1}{2\pi}\int_{0}^{2\pi}\ln\det
F(\theta)\,d\theta.$

The analysis of the boundary matrices $\mathsf{S}_{N}$ is based on the
possibility to determine a limit boundary matrix when $N$ is sent to infinity.
This is done by the following lemma, which is proved in Appendix E. Let $A$,
$D$, $B$, and $E$ as in Definition 2.1. Set for each $n\in\mathbb{Z}$

$\Phi(n):=\frac{1}{2\pi}\int_{0}^{2\pi}F^{-1}(\theta)e^{-\mathrm{i}n\theta}d\theta,$

which is a well-defined matrix under the hypothesis
$\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}>0$.

###### Lemma 2.4.

Let $\mathsf{S}_{N}$ be the boundary matrices of a HQT matrix sequence with
symbol $F$. Assume that $\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}>0$. The
following conclusions hold:

  1. 1.

$H:=I-E\Phi(1)\in\mathbb{C}^{d\times d}$ and
$K:=I-\Phi(1)E\in\mathbb{C}^{d\times d}$ are invertible;

  2. 2.

the limit $\lim_{N\uparrow\infty}\mathsf{S}_{N}=:\mathsf{S}_{\infty}$ exists
and $\mathsf{S}_{\infty}=\begin{pmatrix}\mathcal{L}&0\\\
0&\mathcal{R}\end{pmatrix}$ with Hermitian matrices $\mathcal{L}$ and
$\mathcal{R}$ defined, respectively, by

$\mathcal{L}:=A-E^{\dagger}\Phi(0)H^{-1}E$

and

$\mathcal{R}:=B-EK^{-1}\Phi(0)E^{\dagger}.$

We call $\mathsf{S}_{\infty}$ the limit boundary matrix of the HQT matrix
sequence $\\{\mathsf{Q}_{N}\\}_{N\geq 1}$. Putting the pieces together in the
following proposition, we finally solve the positive definiteness and the
determinants of the matrices $\mathsf{Q}_{N}$ in the large $N$ limit.

###### Proposition 2.1.

Let $\\{\mathsf{Q}_{N}\\}_{N\geq 1}$ be a HQT matrix sequence with symbol $F$
and limit boundary matrix $\mathsf{S}_{\infty}=\begin{pmatrix}\mathcal{L}&0\\\
0&\mathcal{R}\end{pmatrix}$. Assume that
$\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}>0$, $\mathcal{L}\succ 0$, and
$\mathcal{R}\succ 0$. Then, $\mathsf{Q}_{N}\succ\mathsf{0}$ for all
sufficiently large $N$ and

$\lim_{N\uparrow\infty}\frac{1}{N}\ln\det\mathsf{Q}_{N}=\frac{1}{2\pi}\int_{0}^{2\pi}\ln\det
F(\theta)\,d\theta.$

###### Proof.

According to Lemma 2.3, the hypothesis
$\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}>0$ gives
$\mathsf{T}_{N}\succ\mathsf{0}$ for all $N\geq 1$ and

$\lim_{N\uparrow\infty}\frac{1}{N}\ln\det\mathsf{T}_{N}=\frac{1}{2\pi}\int_{0}^{2\pi}\ln\det
F(\theta)\,d\theta.$

It also shows that $\lim_{N\uparrow\infty}\mathsf{S}_{N}=\mathsf{S}_{\infty}$
exists by Lemma 2.4. Since $\mathsf{S}_{\infty}\succ\mathsf{0}$ by hypothesis,
the boundary matrices $\mathsf{S}_{N}$ are positive definite for all
sufficiently large $N$. It follows by Lemma 2.2 that
$\mathsf{Q}_{N}\succ\mathsf{0}$ for all sufficiently large $N$ and that

$\lim_{N\uparrow\infty}\frac{1}{N}\ln\det\mathsf{Q}_{N}=\lim_{N\uparrow\infty}\frac{1}{N}\ln\det\mathsf{T}_{N}+\lim_{N\uparrow\infty}\frac{1}{N}\ln\det\mathsf{S}_{N}=\frac{1}{2\pi}\int_{0}^{2\pi}\ln\det
F(\theta)\,d\theta.\qed$

###  2.2 The cumulant generating function of $W_{N}$

Let us move to the stable Gauss-Markov chain $\\{X_{n}\\}_{n\geq 1}$ and the
quadratic functional $W_{N}$. According to (1.1), for each $N\geq 1$ the law
of $(X_{1},\ldots,X_{N+2})$ is the multivariate Gaussian distribution that at
$x=(x_{1},\ldots,x_{N+2})\in(\mathbb{R}^{d})^{N+2}$ has probability density

$\frac{e^{-\frac{1}{2}\langle
x,\Sigma_{N}^{-1}x\rangle}}{\sqrt{(2\pi)^{(N+2)d}\det\Sigma_{N}}}=\frac{e^{-\frac{1}{2}\langle
x_{1},\Sigma_{o}^{-1}x_{1}\rangle}}{\sqrt{(2\pi)^{d}\det\Sigma_{o}}}\prod_{n=2}^{N+2}\frac{1}{\sqrt{(2\pi)^{d}}}\,e^{-\frac{1}{2}\|x_{n}-Sx_{n-1}\|^{2}}.$

We see that the inverse $\Sigma_{N}^{-1}$ of the covariance matrix
$\Sigma_{N}$ of $(X_{1},\ldots,X_{N+2})$ is the real symmetric block
tridiagonal matrix in $\mathsf{BL}_{N+2,N+2}$ given by

$\Sigma_{N}^{-1}:=\begin{pmatrix}\Sigma_{o}^{-1}+S^{\top}S&-S^{\top}&&&\\\
-S&I+S^{\top}S&\ddots&&\\\ &\ddots&\ddots&\ddots&\\\
&&\ddots&I+S^{\top}S&-S^{\top}\\\ &&&-S&I\end{pmatrix},$

and we have $\det\Sigma_{N}=\det\Sigma_{o}$. Together with $\Sigma_{N}^{-1}$,
we introduce the real symmetric block tridiagonal matrix
$\mathsf{M}_{N}\in\mathsf{BL}_{N+2,N+2}$ defined by

$\mathsf{M}_{N}:=\begin{pmatrix}L&V^{\top}&&&\\\ V&U&\ddots&&\\\
&\ddots&\ddots&\ddots&\\\ &&\ddots&U&V^{\top}\\\ &&&V&R\end{pmatrix}.$

The matrix $\mathsf{M}_{N}$ allows us to express the quadratic functional
$W_{N+2}$ as $(1/2)\langle X,\mathsf{M}_{N}X\rangle$ with
$X:=(X_{1},\ldots,X_{N+2})$. The cumulant generating function of $W_{N}$ is
the function that maps $\lambda\in\mathbb{R}$ in
$(1/N)\ln\mathbb{E}[e^{\lambda W_{N}}]$. We start with the following
elementary result involving Gaussian integrals.

###### Lemma 2.5.

For each $N\geq 1$ and $\lambda\in\mathbb{R}$

$\ln\mathbb{E}\big{[}e^{\lambda
W_{N+2}}\big{]}=\begin{cases}-\frac{1}{2}\ln\det\Sigma_{o}-\frac{1}{2}\ln\det\big{(}\Sigma_{N}^{-1}-\lambda\mathsf{M}_{N}\big{)}&\mbox{if}\leavevmode\nobreak\
\Sigma_{N}^{-1}-\lambda\mathsf{M}_{N}\succ\mathsf{0},\\\
+\infty&\mbox{otherwise}.\end{cases}$

We aim to investigate the asymptotics of the cumulant generating function.
According to Definition 2.1, the matrices
$\mathsf{Q}_{N}:=\Sigma_{N}^{-1}-\lambda\mathsf{M}_{N}\in\mathsf{BL}_{N+2,N+2}$
with some $\lambda\in\mathbb{R}$ form a HQT matrix sequence. Explicitly, we
have $A:=\Sigma_{o}^{-1}+S^{\top}S-\lambda L$, $D:=I+S^{\top}S-\lambda U$,
$B:=I-\lambda R$, and $E:=-S-\lambda V$. The symbol $F_{\lambda}$ of this HQT
matrix sequence reads

$\displaystyle F_{\lambda}(\theta)$ $\displaystyle:=-\big{(}S+\lambda
V\big{)}e^{-\mathrm{i}\theta}+I+S^{\top}S-\lambda U-\big{(}S^{\top}+\lambda
V^{\top}\big{)}e^{\mathrm{i}\theta}$
$\displaystyle=\big{(}I-S^{\top}e^{\mathrm{i}\theta}\big{)}\big{(}I-Se^{-\mathrm{i}\theta}\big{)}-\lambda\big{(}U+Ve^{-\mathrm{i}\theta}+V^{\top}e^{\mathrm{i}\theta}\big{)}$

for every $\theta\in[0,2\pi]$. It is exactly the matrix (1.3). According to
(1.4), the real number $f_{\lambda}$ is related to Rayleigh quotients of the
symbol $F_{\lambda}$ by
$\inf_{\theta\in[0,2\pi]}\\{r(F_{\lambda}(\theta))\\}=f_{\lambda}$. Lemma 2.4
proves the technical Lemma 1.2, and the matrices $\mathcal{L}_{\lambda}$ and
$\mathcal{R}_{\lambda}$ defined by Lemma 1.2 enter the limit boundary matrix
of $\\{\mathsf{Q}_{N}\\}_{N\geq 1}$:
$\mathsf{S}_{\infty}=\begin{pmatrix}\mathcal{L}_{\lambda}&0\\\
0&\mathcal{R}_{\lambda}\end{pmatrix}$. By combining Lemma 2.5 with Proposition
2.1 we get that if $f_{\lambda}>0$, $\mathcal{L}_{\lambda}\succ 0$, and
$\mathcal{R}_{\lambda}\succ 0$, then

$\lim_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{E}\big{[}e^{\lambda
W_{N}}\big{]}=-\frac{1}{4\pi}\int_{0}^{2\pi}\ln\det
F_{\lambda}(\theta)\,d\theta=:\varphi(\lambda),$ (2.7)

$\varphi(\lambda)$ being the integral already defined in (1.5). We want to
prove here that the set

$\Lambda:=\Big{\\{}\lambda\in\mathbb{R}\leavevmode\nobreak\
:\leavevmode\nobreak\ f_{\lambda}>0,\leavevmode\nobreak\
\mathcal{L}_{\lambda}\succ 0,\mbox{ and }\mathcal{R}_{\lambda}\succ
0\Big{\\}}$ (2.8)

is an interval. Formulas (1.7) and (1.8) states that
$\lambda_{-}=\inf\\{\Lambda\\}$ and $\lambda_{+}=\sup\\{\Lambda\\}$. To begin
with, we need the following bound for $r(\Sigma_{N}^{-1})$, which is based on
the hypothesis that the spectral radius $\rho(S)$ of $S$ is smaller than 1 and
is proved in Appendix F.

###### Lemma 2.6.

There exists a real number $\sigma>0$ such that $r(\Sigma_{N}^{-1})\geq\sigma$
for all $N\geq 1$.

The following lemma shows that $\Lambda$ is a convex set, and hence it is an
interval.

###### Lemma 2.7.

The following limits exist and are finite:

$\adjustlimits{\lim}_{\begin{subarray}{c}N\uparrow\infty\\\
\phantom{x}\end{subarray}}{\inf}_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}=:\xi_{-}.$

and

$\adjustlimits{\lim}_{\begin{subarray}{c}N\uparrow\infty\\\
\phantom{x}\end{subarray}}{\sup}_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}=:\xi_{+}.$

If $\lambda\in\Lambda$, then $\lambda\xi_{-}\leq 1$ and $\lambda\xi_{+}\leq
1$. If $\lambda\in\mathbb{R}$ is such that $\lambda\xi_{-}<1$ and
$\lambda\xi_{+}<1$, then $\lambda\in\Lambda$.

###### Proof.

Fix a real number $\lambda_{o}$ and set

$\adjustlimits{\liminf}_{\begin{subarray}{c}N\uparrow\infty\\\
\phantom{x}\end{subarray}}{\sup}_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\lambda_{o}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}=:\xi_{o}.$

The limit $\xi_{o}$ is finite since $\|\mathsf{M}_{N}\|\leq C$ for every
$N\geq 1$ with some constant $C<+\infty$ by Lemma 2.1 and $\langle
z,\Sigma_{N}^{-1}z\rangle\geq\sigma\langle z,z\rangle$ and for all $N\geq 1$
and $z\in(\mathbb{C}^{d})^{N+2}$ by Lemma 2.6. Let us show that

$\adjustlimits{\limsup}_{\begin{subarray}{c}N\uparrow\infty\\\
\phantom{x}\end{subarray}}{\sup}_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\lambda_{o}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}\leq\xi_{o}.$ (2.9)

By choosing $\lambda_{o}=-1$ and $\lambda_{o}=1$, this proves that the
following limits exist and are finite:

$\adjustlimits{\lim}_{\begin{subarray}{c}N\uparrow\infty\\\
\phantom{x}\end{subarray}}{\inf}_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}=:\xi_{-}.$

and

$\adjustlimits{\lim}_{\begin{subarray}{c}N\uparrow\infty\\\
\phantom{x}\end{subarray}}{\sup}_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}=:\xi_{+}.$

Pick an arbitrary real number $\xi>\xi_{o}$ and $\epsilon>0$ such that
$\xi_{o}+2\epsilon\leq\xi$. Consider the HQT matrix sequence
$\\{\mathsf{Q}_{N}\\}_{N\geq 1}$ with
$\mathsf{Q}_{N}:=\xi\Sigma_{N}^{-1}-\lambda_{o}\mathsf{M}_{N}\in\mathsf{BL}_{N+2,N+2}$
for all $N\geq 1$. By definition of $\xi_{o}$, there exists a diverging
sequence $\\{N_{k}\\}_{k\geq 0}$ of positive integers with the property that
for all $k$ and $z\in(\mathbb{C}^{d})^{N_{k}+2}$

$\lambda_{o}\langle z,\mathsf{M}_{N_{k}}z\rangle\leq(\xi-\epsilon)\langle
z,\Sigma_{N_{k}}^{-1}z\rangle.$

It follows that $r(\mathsf{Q}_{N_{k}})\geq\epsilon
r(\Sigma_{N_{k}}^{-1})\geq\epsilon\sigma>0$ for any $k\geq 0$. Then, part 1 of
Lemma 2.2 tells us that $r(\mathsf{T}_{N_{k}})\geq\epsilon\sigma$ and
$r(\mathsf{S}_{N_{k}})\geq\epsilon\sigma$ for all $k$, $\mathsf{T}_{N}$ being
the bulk matrix of $\mathsf{Q}_{N}$ and $\mathsf{S}_{N}$ being its boundary
matrix. As a consequence, parts 1 and 2 of Lemma 2.3 give
$r(\mathsf{T}_{N})\geq\epsilon\sigma$ for every $N\geq 1$ and
$r(F(\theta))\geq\epsilon\sigma$ for all $\theta\in[0,2\pi]$, $F$ being the
symbol of the Hermitian block Toeplitz matrices $\mathsf{T}_{N}$. This way,
Lemma 2.4 shows that
$\lim_{N\uparrow\infty}\mathsf{S}_{N}=\mathsf{S}_{\infty}$ exists and is well-
defined. Since $r(\mathsf{S}_{N_{k}})\geq\epsilon\sigma$ for all $k$ we have
$r(\mathsf{S}_{\infty})\geq\epsilon\sigma$, so that $r(\mathsf{S}_{N})>0$ for
all sufficiently large $N$. In conclusion, we find that both
$\mathsf{T}_{N}\succ\mathsf{0}$ and $\mathsf{S}_{N}\succ\mathsf{0}$ for all
sufficiently large $N$, and part 2 of Lemma 2.2 ensures us that
$\mathsf{Q}_{N}\succ\mathsf{0}$ for all such $N$. This means that

$\langle z,\mathsf{Q}_{N}z\rangle=\langle
z,\big{(}\xi\Sigma_{N}^{-1}-\lambda_{o}\mathsf{M}_{N}\big{)}z\rangle>0$ (2.10)

for all sufficiently large $N$ and $z\in(\mathbb{C}^{d})^{N+2}$. It follows
that

$\adjustlimits{\limsup}_{\begin{subarray}{c}N\uparrow\infty\\\
\phantom{x}\end{subarray}}{\sup}_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\lambda_{o}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}\leq\xi,$

which demonstrates (2.9) thanks to the arbitrariness of $\xi>\xi_{o}$.

Let us demonstrate now the connection between the set $\Lambda$ and the number
$\xi_{-}$ and $\xi_{+}$. Fix $\lambda\in\Lambda$ and consider the HQT matrix
sequence $\\{\mathsf{Q}_{N}\\}_{N\geq 1}$ with matrices
$\mathsf{Q}_{N}:=\Sigma_{N}^{-1}-\lambda\mathsf{M}_{N}$. We already know that
this HQT matrix sequence has symbol $F_{\lambda}$ and limit boundary matrix
$\mathsf{S}_{\infty}=\begin{pmatrix}\mathcal{L}_{\lambda}&0\\\
0&\mathcal{R}_{\lambda}\end{pmatrix}$. Since $f_{\lambda}>0$,
$\mathcal{L}_{\lambda}\succ 0$, and $\mathcal{R}_{\lambda}\succ 0$ by
hypothesis, we have $\mathsf{Q}_{N}\succ\mathsf{0}$ for all sufficiently large
$N$ according to Proposition 2.1. This shows that

$1-\lambda\frac{\langle z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}>0$

for all sufficiently large $N$ and $z\in(\mathbb{C}^{d})^{N+2}$. Thus, by
taking the infimum over $z$ we have

$1-\lambda\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}>0$

if $\lambda\geq 0$ and

$1-\lambda\inf_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle z,\mathsf{M}_{N}z\rangle}{\langle
z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}>0$

if $\lambda<0$. By sending $N$ to infinity we realize that $\lambda\xi_{+}\leq
1$ if $\lambda\geq 0$, which also gives $\lambda\xi_{-}\leq 1$ as
$\xi_{-}\leq\xi_{+}$, and that $\lambda\xi_{-}\leq 1$ if $\lambda<0$, which
also gives $\lambda\xi_{+}\leq 1$.

Conversely, if $\lambda\in\mathbb{R}$ is such that $\lambda\xi_{-}<1$ and
$\lambda\xi_{+}<1$, then there exists $\epsilon>0$ such that
$1-\lambda\xi_{-}\geq 2\epsilon$ and $1-\lambda\xi_{+}\geq 2\epsilon$. This
yields that for all sufficiently large $N$ and $z\in(\mathbb{C}^{d})^{N+2}$

$\langle z,\Sigma_{N}^{-1}z\rangle-\lambda\langle
z,\mathsf{M}_{N}z\rangle\geq\epsilon\langle z,\Sigma_{N}^{-1}z\rangle.$

This way, $r(\mathsf{Q}_{N})\geq\epsilon r(\Sigma_{N}^{-1})\geq\epsilon\sigma$
for all sufficiently large $N$, where
$\mathsf{Q}_{N}:=\Sigma_{N}^{-1}-\lambda\mathsf{M}_{N}$ and Lemma 2.1 has been
invoked. It follows from Lemma 2.2 that $r(\mathsf{T}_{N})\geq\epsilon\sigma$
and $r(\mathsf{S}_{N})\geq\epsilon\sigma$ for all sufficiently large $N$, so
that $\inf_{\theta\in[0,2\pi]}\\{r(F_{\lambda}(\theta))\\}\geq\epsilon\sigma$
by Lemma 2.3 and $r(S_{\infty})\geq\epsilon\sigma$ by Lemma 2.4 with
$\mathsf{S}_{\infty}=\begin{pmatrix}\mathcal{L}_{\lambda}&0\\\
0&\mathcal{R}_{\lambda}\end{pmatrix}$. Thus, $f_{\lambda}>0$,
$\mathcal{L}_{\lambda}\succ 0$, and $\mathcal{R}_{\lambda}\succ 0$, so that
$\lambda\in\Lambda$. ∎

The limit (2.7) together with the fact that $\Lambda$ is an interval finally
give the following important result. We stress that $\Lambda$ contains an open
neighborhood of the origin, as it is manifest by Lemma 2.7, so that
$\lambda_{-}<0<\lambda_{+}$.

###### Proposition 2.2.

For all $\lambda\in(\lambda_{-},\lambda_{+})$

$\lim_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{E}\big{[}e^{\lambda
W_{N}}\big{]}=-\frac{1}{4\pi}\int_{0}^{2\pi}\ln\det
F_{\lambda}(\theta)\,d\theta.$

The function $\varphi$ that maps any $\lambda\in(\lambda_{-},\lambda_{+})$ in

$\varphi(\lambda):=-\frac{1}{4\pi}\int_{0}^{2\pi}\ln\det
F_{\lambda}(\theta)\,d\theta$

is convex. A rapid way to demonstrate this fact is to observe that $\varphi$
is the limit of a sequence of convex functions by Proposition 2.2. The
function $\varphi$ is also differentiable since $F_{\lambda}(\theta)$ is
differentiable with respect to $\lambda$ for each $\theta\in[0,2\pi]$. The
asymptotic theory of sequences of convex functions (see [39], Theorem 24.5)
gives for all $\lambda\in(\lambda_{-},\lambda_{+})$

$\lim_{N\uparrow\infty}\frac{1}{N}\frac{d}{d\lambda}\ln\mathbb{E}\big{[}e^{\lambda
W_{N}}\big{]}=\varphi^{\prime}(\lambda).$ (2.11)

Limit (2.11) will serve us to verify the lower large deviation bound. Other
notable consequences of convexity are that the limits
$\lim_{\lambda\downarrow\lambda_{-}}\varphi(\lambda)=:\varphi_{-}$ and
$\lim_{\lambda\uparrow\lambda_{+}}\varphi(\lambda)=:\varphi_{+}$ exist (see
[39], Theorem 7.5) and that $\varphi^{\prime}$ is non-decreasing, in such a
way that also the limits
$\lim_{\lambda\downarrow\lambda_{-}}\varphi^{\prime}(\lambda)=:d_{-}$ and
$\lim_{\lambda\uparrow\lambda_{+}}\varphi^{\prime}(\lambda)=:d_{+}$ exist.

###### Remark 2.1.

In Section 1.1 we have claimed that
$\lim_{N\uparrow\infty}(1/N)\ln\mathbb{E}[e^{\lambda W_{N}}]=+\infty$ if
$\lambda\notin\overline{(\lambda_{-},\lambda_{+})}=\bar{\Lambda}$. Although we
do not need this limit to prove a large deviation principle, we can verify it
as follows. Assume for instance that $\lambda_{+}<+\infty$ and pick
$\lambda>\lambda_{+}>0$. It must be $\lambda\xi_{+}>1$ since, on the contrary,
$\lambda\xi_{-}\leq\lambda\xi_{+}\leq 1$ and $\lambda\in\bar{\Lambda}$ as a
consequence according to Lemma 2.7. Fix $N\geq 1$ and let
$\mathsf{A}\in\mathsf{BL}_{N+2,N+2}$ be a real invertible matrix such that
$\Sigma_{N}=\mathsf{A}\mathsf{A}^{\top}$, which exists because
$\Sigma_{N}\succ 0$. Denoting by $m_{1},\ldots,m_{(N+2)d}$ the eigenvalues of
the real symmetric matrix
$\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}\in\mathsf{BL}_{N+2,N+2}$, we have

$\displaystyle\xi_{N}:=\max\big{\\{}m_{1},\ldots,m_{(N+2)d}\big{\\}}$
$\displaystyle=\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}z\rangle}{\langle
z,z\rangle}\bigg{\\}}=\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle\mathsf{A}z,\mathsf{M}_{N}\mathsf{A}z\rangle}{\langle
z,z\rangle}\bigg{\\}}$
$\displaystyle=\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle\mathsf{A}^{-1}z,\mathsf{A}^{-1}z\rangle}\bigg{\\}}=\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}.$

Thus, Lemma 2.7 tells us that the number $\lambda\xi_{N}$ approaches
$\lambda\xi_{+}>1$ at large $N$, so that $1-\lambda\xi_{N}\leq 0$ if $N$
exceeds a threshold value $N_{o}$. This shows that the matrix
$\Sigma_{N}^{-1}-\lambda\mathsf{M}_{N}=(\mathsf{A}^{-1})^{\top}(\mathsf{I}-\lambda\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A})\mathsf{A}^{-1}$
is not positive-definite for $N>N_{o}$. Lemma 2.5 concludes the proof.

###  2.3 The upper large deviation bound

In this section we prove the upper large deviation bound for closed sets. We
start with some standard results from the theory of large deviations that we
shall use to prove both the upper large deviation bound and the lower large
deviation bound. For each $\eta\in(\lambda_{-},\lambda_{+})$ and $N\geq 1$,
let $\mathbb{P}_{\eta,N}$ be the probability measure on $(\Omega,\mathscr{F})$
defined by the exponential change of measure

$\frac{d\mathbb{P}_{\eta,N}}{d\mathbb{P}}:=\frac{e^{\eta
W_{N}}}{\mathbb{E}[e^{\eta W_{N}}]}.$ (2.12)

Let $\varphi_{\eta}$ be the function that maps any $\lambda\in\mathbb{R}$ in

$\varphi_{\eta}(\lambda):=\limsup_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{E}_{\eta,N}\big{[}e^{\lambda
W_{N}}\big{]}.$ (2.13)

Since $0\in(\lambda_{-},\lambda_{+})$, we have $\mathbb{P}_{0,N}=\mathbb{P}$
for all $N\geq 1$ and $\varphi_{0}(\lambda)=\varphi(\lambda)$ for all
$\lambda\in(\lambda_{-},\lambda_{+})$, $\varphi$ being the convex
differentiable function introduced at the end of the last section. Moreover,
if $\lambda\in\mathbb{R}$ is such that
$\lambda+\eta\in(\lambda_{-},\lambda_{+})$, then

$\varphi_{\eta}(\lambda)=\limsup_{N\uparrow\infty}\frac{1}{N}\ln\frac{\mathbb{E}[e^{(\lambda+\eta)W_{N}}]}{\mathbb{E}[e^{\eta
W_{N}}]}=\varphi(\lambda+\eta)-\varphi(\eta)<+\infty.$

It follows that the function $\varphi_{\eta}$ is finite and differentiable in
an open neighborhood of the origin with
$\varphi_{\eta}^{\prime}(0)=\varphi^{\prime}(\eta)$. The following lemma
states an upper large deviation bound with respect to the measure
$\mathbb{P}_{\eta,N}$. We recall that the Fenchel-Legendre transform
$I_{\eta}$ of $\varphi_{\eta}$ is the convex function that associates
$w\in\mathbb{R}$ with

$I_{\eta}(w):=\sup_{\lambda\in\mathbb{R}}\big{\\{}w\lambda-\varphi_{\eta}(\lambda)\big{\\}}.$

###### Lemma 2.8.

Fix $\eta\in(\lambda_{-},\lambda_{+})$. The following conclusions hold:

  1. 1.

the Fenchel-Legendre transform $I_{\eta}$ of $\varphi_{\eta}$ has compact
level sets;

  2. 2.

for each closed set $\mathcal{F}\subseteq\mathbb{R}$

$\limsup_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}_{\eta,N}\bigg{[}\frac{W_{N}}{N}\in\mathcal{F}\bigg{]}\leq-\inf_{w\in\mathcal{F}}\\{I_{\eta}(w)\\};$

  3. 3.

for each $\epsilon>0$ there exists $\kappa>0$ such that for all sufficiently
large $N$

$\mathbb{P}_{\eta,N}\bigg{[}\bigg{|}\frac{W_{N}}{N}-\varphi^{\prime}(\eta)\bigg{|}\geq\epsilon\bigg{]}\leq
e^{-\kappa N}.$

###### Proof.

$I_{\eta}$ is lower semicontinuous as any Fenchel-Legendre transform (see
[39], Theorem 12.2). Due to lower semicontinuity, the level sets of $I_{\eta}$
are closed. In order to prove part 1, it remains to verify that they are
bounded. As the function $\varphi_{\eta}$ is finite in an open neighborhood of
the origin, there exists $\delta>0$ such that $\varphi_{\eta}(\delta)<+\infty$
and $\varphi_{\eta}(-\delta)<+\infty$. If $I_{\eta}(w)\leq a$ for given real
numbers $w$ and $a$, then $w\delta-\varphi_{\eta}(\delta)\leq I_{\eta}(w)\leq
a$ and $-w\delta-\varphi_{\eta}(-\delta)\leq I_{\eta}(w)\leq a$ by definition,
that is $-[a+\varphi_{\eta}(-\delta)]/\delta\leq
w\leq[a+\varphi_{\eta}(\delta)]/\delta$.

Part 2 is a standard result from large deviation theory (see [15], Theorem
2.3.6 and Exercise 2.3.25). In a nutshell, the upper large deviation bound for
compact sets is a manipulation of the Chernoff bound and holds without any
assumption on the function $\varphi_{\eta}$. Extension to all closed sets is
made possible by finiteness of $\varphi_{\eta}$ in an open neighborhood of the
origin, which entails exponential tightness.

As far as part 3 is concerned, in the light of part 2 it suffices to
demonstrate that

$\inf_{v\notin(w-\epsilon,w+\epsilon)}\\{I_{\eta}(v)\\}>0$ (2.14)

for each $\epsilon>0$ with $w:=\varphi^{\prime}(\eta)$. To begin with, let us
observe that $I_{\eta}(v)>0$ if $v\neq w$. On the contrary, if
$I_{\eta}(v)=0$, then for all $\lambda$ in a neighborhood of the origin we
would have $\varphi(\lambda+\eta)-\varphi(\eta)=\varphi_{\eta}(\lambda)\geq
v\lambda$ by definition of $I_{\eta}(v)$. This would imply
$v=\varphi^{\prime}(\eta)=:w$, which contradicts the hypothesis $v\neq w$. We
can now verify (2.14). Pick $\epsilon>0$ and notice that the set
$\mathcal{A}:=\\{v\in\mathbb{R}:I_{\eta}(v)\leq 1\\}$ is compact by part 1. If
$(w-\epsilon,w+\epsilon)^{c}\,\cap\mathcal{A}=\emptyset$, then
$\inf_{v\notin(w-\epsilon,w+\epsilon)}\\{I_{\eta}(v)\\}\geq 1$. If
$(w-\epsilon,w+\epsilon)^{c}\,\cap\mathcal{A}\neq\emptyset$, then there exists
$v_{\star}\in(w-\epsilon,w+\epsilon)^{c}\,\cap\mathcal{A}$ such that
$I_{\eta}(v)\geq I_{\eta}(v_{\star})$ for all
$v\in(w-\epsilon,w+\epsilon)^{c}\,\cap\mathcal{A}$, and hence for all
$v\notin(w-\epsilon,w+\epsilon)$, as $I_{\eta}$ is a lower semicontinuous
function and $(w-\epsilon,w+\epsilon)^{c}\,\cap\mathcal{A}$ is a compact set.
On the other hand, we have $I_{\eta}(v_{\star})>0$ since $v_{\star}\neq w$. ∎

Lemma 2.8 gives the following upper large deviation bound for the quadratic
functionals $W_{N}$ of the stable Gauss-Markov processes $\\{X_{n}\\}_{n\geq
1}$.

###### Proposition 2.3.

The following conclusions hold:

  1. 1.

the convex function $I$ that maps $w\in\mathbb{R}$ in
$I(w):=\sup_{\lambda\in(\lambda_{-},\lambda_{+})}\\{w\lambda-\varphi(\lambda)\\}$
has compact level sets;

  2. 2.

for each closed set $\mathcal{F}\subseteq\mathbb{R}$

$\limsup_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in\mathcal{F}\bigg{]}\leq-\inf_{w\in\mathcal{F}}\\{I(w)\\}.$

###### Proof.

$I$ is the Fenchel-Legendre transform of the function that associates
$\lambda\in(\lambda_{-},\lambda_{+})$ with $\varphi(\lambda)$ and
$\lambda\notin(\lambda_{-},\lambda_{+})$ with $+\infty$. Then, part 1 is
proved in the same way of part 1 of Lemma 2.8. Part 2 follows from part 2 of
Lemma 2.8 with $\eta=0$ as
$I_{0}(w):=\sup_{\lambda\in\mathbb{R}}\\{w\lambda-\varphi_{0}(\lambda)\\}\geq\sup_{\lambda\in(\lambda_{-},\lambda_{+})}\\{w\lambda-\varphi(\lambda)\\}=:I(w)$
for every $w\in\mathbb{R}$. ∎

###  2.4 The lower large deviation bound

In this section we prove the lower large deviation bound for open sets, namely
that for each open set $\mathcal{G}\subseteq\mathbb{R}$

$\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in\mathcal{G}\bigg{]}\geq-\inf_{w\in\mathcal{G}}\\{I(w)\\},$

where $I$ is the function that maps $w\in\mathbb{R}$ in
$I(w):=\sup_{\lambda\in(\lambda_{-},\lambda_{+})}\\{w\lambda-\varphi(\lambda)\\}$.
This is tantamount to state that for all $w\in\mathbb{R}$ and $\delta>0$

$\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in(w-\delta,w+\delta)\bigg{]}\geq-I(w).$
(2.15)

We start with the following lower bound based on Lemma 2.8.

###### Lemma 2.9.

Fix $w\in\mathbb{R}$ and assume that there exists
$\eta\in(\lambda_{-},\lambda_{+})$ such that $w=\varphi^{\prime}(\eta)$. Then,
for every $\delta>0$

$\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in(w-\delta,w+\delta)\bigg{]}\geq\varphi(\eta)-w\eta.$

###### Proof.

Let $\mathbb{P}_{\eta,N}$ and $\varphi_{\eta}$ be the probability measure
(2.12) and the function (2.13), respectively. Fix $\delta>0$ and pick
$\epsilon\in(0,\delta)$. The fact that $\eta W_{N}-Nw\eta-N\epsilon|\eta|\leq
0$ if $W_{N}/N\in(w-\epsilon,w+\epsilon)$ gives for each $N\geq 1$

$\displaystyle\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in(w-\delta,w+\delta)\bigg{]}$
$\displaystyle\geq e^{-Nw\eta-N\epsilon|\eta|}\,\mathbb{E}\bigg{[}e^{\eta
W_{N}}\mathds{1}_{\big{\\{}\frac{W_{N}}{N}\in(w-\epsilon,w+\epsilon)\big{\\}}}\bigg{]}$
$\displaystyle=e^{-Nw\eta-N\epsilon|\eta|}\,\mathbb{E}\big{[}e^{\eta
W_{N}}\big{]}\,\mathbb{P}_{\eta,N}\bigg{[}\bigg{|}\frac{W_{N}}{N}-w\bigg{|}<\epsilon\bigg{]},$

and part 3 of Lemma 2.8 shows that

$\lim_{N\uparrow\infty}\mathbb{P}_{\eta,N}\bigg{[}\bigg{|}\frac{W_{N}}{N}-w\bigg{|}<\epsilon\bigg{]}=1.$

Thus, by invoking Proposition 2.2 we obtain

$\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in(w-\delta,w+\delta)\bigg{]}\geq\varphi(\eta)-w\eta-\epsilon|\eta|.$

The lemma follows from here by sending $\epsilon$ to 0. ∎

Lemma 2.9 allows us to demonstrate the lower large deviation bound (2.15) for
$w$ in the closure $\overline{(d_{-},d_{+})}$ of $(d_{-},d_{+})$, where
$d_{-}:=\lim_{\lambda\downarrow\lambda_{-}}\varphi^{\prime}(\lambda)$ and
$d_{+}:=\lim_{\lambda\uparrow\lambda_{+}}\varphi^{\prime}(\lambda)$ as in
Section 2.2. Notice that convexity and differentiability of $\varphi$ yield
$\varphi(\lambda)\geq\varphi(\eta)+\varphi^{\prime}(\eta)(\lambda-\eta)$ for
every $\lambda$ and $\eta$ in $(\lambda_{-},\lambda_{+})$, so that
$I(w)=w\eta-\varphi(\eta)$ if $w=\varphi^{\prime}(\eta)$ for some
$\eta\in(\lambda_{-},\lambda_{+})$. Since $d_{-}\leq\varphi^{\prime}(0)\leq
d_{+}$ as $\varphi^{\prime}$ is non-decreasing, if $d_{-}=d_{+}$, then
$\overline{(d_{-},d_{+})}$ contains only $\varphi^{\prime}(0)$ and bound
(2.15) directly follows from Lemma 2.9 with $\eta=0$. If $d_{-}<d_{+}$ and
$w\in(d_{-},d_{+})$, then there exists $\eta\in(\lambda_{-},\lambda_{+})$ such
that $w=\varphi^{\prime}(\eta)$ and bound (2.15) follows again from Lemma 2.9
with such $\eta$. If $d_{-}<d_{+}<+\infty$, then $\overline{(d_{-},d_{+})}$
contains $d_{+}$, and we tackle the case $w=d_{+}$ as follows. Fix $\delta>0$.
There exist $v\in(d_{-},d_{+})$ arbitrarily close to $w$ and $\epsilon>0$ such
that $(v-\epsilon,v+\epsilon)\subseteq(w-\delta,w+\delta)$. This way, since
(2.15) holds for $v$ we find

$\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in(w-\delta,w+\delta)\bigg{]}\geq\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in(v-\epsilon,v+\epsilon)\bigg{]}\geq-I(v).$

From here, we get bound (2.15) for $w=d_{+}$ by sending $v$ to $w$ and by
observing that $\lim_{v\uparrow w}I(v)=I(w)$ by convexity and lower
semicontinuity of $I$ (see [39], Corollary 7.5.1). Similar arguments can be
used to solve the case $-\infty<d_{-}<d_{+}$ and $w=d_{-}$.

In order to complete the proof of the lower large deviation bound (2.15), it
remains to address the case $d_{+}<+\infty$ and $w>d_{+}$, as well as the case
$d_{-}>-\infty$ and $w<d_{-}$. They are similar, so that we discuss in detail
the former only, omitting the proof of the latter. Assume that $d_{+}<+\infty$
and fix $w>d_{+}$. We claim that the case $\lambda_{+}=+\infty$ is trivial, so
that we also suppose $\lambda_{+}<+\infty$. In fact, convexity and
differentiability of $\varphi$ combined with $\varphi(0)=0$ give
$\varphi(\lambda)\leq\lambda\varphi^{\prime}(\lambda)$ for all
$\lambda\in(\lambda_{-},\lambda_{+})$. It follows that $I(w)\geq
w\lambda-\varphi(\lambda)\geq\lambda[w-\varphi^{\prime}(\lambda)]$ for all
$\lambda\in(\lambda_{-},\lambda_{+})$. Thus, if $\lambda_{+}=+\infty$, then we
realize that $I(w)=+\infty$ by sending $\lambda$ to $\lambda_{+}$, as
$\lim_{\lambda\uparrow\lambda_{+}}\varphi^{\prime}(\lambda)=d_{+}<w$, and the
lower bound (2.15) is trivial. Observe that if $\lambda_{+}<+\infty$ and
$d_{+}<+\infty$, then
$\lim_{\lambda\uparrow\lambda_{+}}\varphi(\lambda)=:\varphi_{+}<+\infty$ as
$\varphi(\lambda)\leq\lambda\varphi^{\prime}(\lambda)$ for all
$\lambda\in(\lambda_{-},\lambda_{+})$. Since the function that associates
$\lambda\in(\lambda_{-},\lambda_{+})$ with $w\lambda-\varphi(\lambda)$ is
increasing under the hypothesis $w>d_{+}$, we have
$I(w):=\sup_{\lambda\in(\lambda_{-},\lambda_{+})}\\{w\lambda-\varphi(\lambda)\\}=w\lambda_{+}-\varphi_{+}$.

The idea to prove (2.15) for $w>d_{+}$ and $\lambda_{+}<+\infty$ is to make a
change of measure like in Lemma 2.9, but this time the parameter $\eta$ must
depend on the time $N$. Let us introduce such parameter. Pick $N\geq 1$. Since
the covariance matrix $\Sigma_{N}\in\mathsf{BL}_{N+2,N+2}$ is symmetric
positive-definite, there exists a real invertible matrix
$\mathsf{A}\in\mathsf{BL}_{N+2,N+2}$ such that
$\Sigma_{N}=\mathsf{A}\mathsf{A}^{\top}$. Like in Remark 2.1, let
$m_{1},\ldots,m_{(N+2)d}$ be the eigenvalues of the real symmetric matrix
$\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}\in\mathsf{BL}_{N+2,N+2}$ and
observe that

$\displaystyle\xi_{N}:=\max\big{\\{}m_{1},\ldots,m_{(N+2)d}\big{\\}}$
$\displaystyle=\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}z\rangle}{\langle
z,z\rangle}\bigg{\\}}$
$\displaystyle=\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle z,\mathsf{M}_{N}z\rangle}{\langle
z,(\mathsf{A}^{-1})^{\top}\mathsf{A}^{-1}z\rangle}\bigg{\\}}=\sup_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}.$

Similarly

$\min\big{\\{}m_{1},\ldots,m_{(N+2)d}\big{\\}}=\inf_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\
z\neq 0\end{subarray}}\bigg{\\{}\frac{\langle
z,\mathsf{M}_{N}z\rangle}{\langle z,\Sigma_{N}^{-1}z\rangle}\bigg{\\}}.$

We have $\lim_{N\uparrow\infty}\xi_{N}=\xi_{+}$ by Lemma 2.7, whereas
$\min\\{m_{1},\ldots,m_{(N+2)d}\\}$ approaches $\xi_{-}$ at large $N$. Lemma
2.7 also gives $\xi_{+}>0$ since $\lambda_{+}<+\infty$ by hypothesis. Indeed,
$\xi_{+}\leq 0$ would entail that the set $\Lambda$ contains all positive real
numbers. Lemma 2.5 and the fact that $\det\Sigma_{N}=\det\Sigma_{o}$ show that
if $\mathsf{I}-\lambda\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}\succ 0$,
namely if $1-\lambda m_{l}>0$ for $l=1,\ldots,(N+2)d$, then

$\ln\mathbb{E}\big{[}e^{\lambda
W_{N+2}}\big{]}=-\frac{1}{2}\ln\det\big{(}\mathsf{I}-\lambda\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}\big{)}=-\frac{1}{2}\sum_{l=1}^{(N+2)d}\ln(1-\lambda
m_{l}).$ (2.16)

We claim that for all sufficiently large $N$ there exists
$\eta_{N}\in(0,\xi_{N})$ such that

$\frac{1}{2(N+2)}\sum_{l=1}^{(N+2)d}\frac{m_{l}}{1-\eta_{N}m_{l}}=w.$ (2.17)

Notice that $\xi_{N}>0$ for all sufficiently large $N$ as
$\lim_{N\uparrow\infty}\xi_{N}=\xi_{+}>0$. In fact, identity (2.16) in
combination with (2.11) yields that $[2(N+2)]^{-1}\sum_{l=1}^{(N+2)d}m_{l}$
approaches $\varphi^{\prime}(0)\leq d_{+}<w$ when $N$ is sent to infinity.
Thus, for all sufficiently large $N$, the continuous function that maps
$\lambda\in[0,\xi_{N})$ in $[2(N+2)]^{-1}\sum_{l=1}^{(N+2)d}m_{l}(1-\lambda
m_{l})^{-1}$ increases from a value smaller than $w$ at $\lambda=0$ to
$+\infty$ at $\lambda=\xi_{N}$, so that there exists a unique $\eta_{N}$
satisfying (2.17). We must have $\lim_{N\uparrow\infty}\eta_{N}=\lambda_{+}$.
On the contrary, there would exist $\epsilon>0$ and a diverging sequence
$\\{N_{k}\\}_{k\geq 0}$ of positive integers such that
$\eta_{N_{k}}<\lambda_{+}-\epsilon$ for all $k\geq 0$. Then, for every $k$

$w=\frac{1}{2(N_{k}+2)}\sum_{l=1}^{(N_{k}+2)d}\frac{m_{l}}{1-\eta_{N_{k}}m_{l}}\leq\frac{1}{2(N_{k}+2)}\sum_{l=1}^{(N_{k}+2)d}\frac{m_{l}}{1-(\lambda_{+}-\epsilon)m_{l}}.$

By sending $k$ to infinity and by combining (2.16) with (2.11), from here we
would get $w\leq\varphi^{\prime}(\lambda_{+}-\epsilon)\leq d_{+}$, which
contradicts the assumption $w>d_{+}$. Another property of $\eta_{N}$ is that

$\liminf_{N\uparrow\infty}\frac{1}{N+2}\ln\mathbb{E}\big{[}e^{\eta_{N}W_{N+2}}\big{]}\geq\varphi_{+}.$
(2.18)

In order to verify this bound, fix $\lambda\in(0,\lambda_{+})$ and bear in
mind that $\eta_{N}\geq\lambda$ for all sufficiently large $N$ as
$\lim_{N\uparrow\infty}\eta_{N}=\lambda_{+}$, so that

$-\ln(1-\eta_{N}m_{l})\geq-\ln(1-\lambda
m_{l})+(\eta_{N}-\lambda)\min\\{0,m_{1},\ldots,m_{(N+2)d}\\}$

for $l$ and sufficiently large $N$. Then, for all sufficiently large $N$ we
have

$\displaystyle\frac{1}{N+2}\ln\mathbb{E}\big{[}e^{\eta_{N}W_{N+2}}\big{]}$
$\displaystyle=-\frac{1}{2(N+2)}\sum_{l=1}^{(N+2)d}\ln(1-\eta_{N}m_{l})$
$\displaystyle\geq-\frac{1}{2(N+2)}\sum_{l=1}^{(N+2)d}\ln(1-\lambda
m_{l})+(\eta_{N}-\lambda)d\min\\{0,m_{1},\ldots,m_{(N+2)d}\\}$
$\displaystyle=\frac{1}{N+2}\ln\mathbb{E}\big{[}e^{\lambda
W_{N+2}}\big{]}+(\eta_{N}-\lambda)d\min\\{0,m_{1},\ldots,m_{(N+2)d}\\}.$

By sending $N$ to infinity and by recalling that
$\min\\{m_{1},\ldots,m_{(N+2)d}\\}$ approaches $\xi_{-}$ in this limit,
Proposition 2.2 shows that

$\liminf_{N\uparrow\infty}\frac{1}{N+2}\ln\mathbb{E}\big{[}e^{\eta_{N}W_{N+2}}\big{]}\geq\varphi(\lambda)+(\lambda_{+}-\lambda)d\min\\{0,\xi_{-}\\},$

which demonstrates (2.18) once $\lambda$ is sent $\lambda_{+}$.

We now move to bound (2.15) and put $\eta_{N}$ into context. Fix $\delta>0$
and pick $\epsilon\in(0,\delta)$. For all sufficiently large $N$, $\eta_{N}$
is positive as $\lim_{N\uparrow\infty}\eta_{N}=\lambda_{+}$ and we have

$\displaystyle\mathbb{P}\bigg{[}\frac{W_{N+2}}{N+2}\in(w-\delta,w+\delta)\bigg{]}$
$\displaystyle\geq
e^{-(N+2)(w+\epsilon)\eta_{N}}\,\mathbb{E}\bigg{[}e^{\eta_{N}W_{N+2}}\mathds{1}_{\big{\\{}\frac{W_{N+2}}{N+2}\in(w-\epsilon,w+\epsilon)\big{\\}}}\bigg{]}$
$\displaystyle=e^{-(N+2)(w+\epsilon)\eta_{N}}\,\mathbb{E}\big{[}e^{\eta_{N}W_{N+2}}\big{]}\,\mathbb{P}_{\eta_{N},N+2}\bigg{[}\bigg{|}\frac{W_{N+2}}{N+2}-w\bigg{|}<\epsilon\bigg{]},$

where $\mathbb{P}_{\eta_{N},N+2}$ is the probability measure (2.12) associated
with $\eta_{N}$. This bound, together with (2.18), yields

$\displaystyle\liminf_{N\uparrow\infty}\frac{1}{N}\ln\mathbb{P}\bigg{[}\frac{W_{N}}{N}\in(w-\delta,w+\delta)\bigg{]}$
$\displaystyle\geq\varphi_{+}-w\lambda_{+}-\epsilon\lambda_{+}$
$\displaystyle+\liminf_{N\uparrow\infty}\frac{1}{N+2}\ln\mathbb{P}_{\eta_{N},N+2}\bigg{[}\bigg{|}\frac{W_{N+2}}{N+2}-w\bigg{|}<\epsilon\bigg{]}.$

This way, as $w\lambda_{+}-\varphi_{+}=I(w)$, we get at the lower large
deviation bound (2.15) from here if we can prove that

$\lim_{\epsilon\downarrow
0}\,\liminf_{N\uparrow\infty}\frac{1}{N+2}\ln\mathbb{P}_{\eta_{N},N+2}\bigg{[}\bigg{|}\frac{W_{N+2}}{N+2}-w\bigg{|}<\epsilon\bigg{]}=0.$
(2.19)

Verifying (2.19) is our last task. To this aim, we resort to the following
result, which was introduced by Bryc and Dembo (see [28], Lemma 2) to deal
with a similar problem.

###### Lemma 2.10.

If $\\{Z_{l}\\}_{l\geq 1}$ is a sequence of i.i.d. random variables with mean
zero, finite second moment, and positive probability density function at 0
with respect to a probability measure $P$, then for each $\epsilon>0$ there
exists $p>0$ such that the following property holds:

$P\Bigg{[}\Bigg{|}\sum_{l\geq 1}a_{l}Z_{l}\Bigg{|}<\epsilon\Bigg{]}\geq p$

for any numerical sequence $\\{a_{l}\\}_{l\geq 1}$ such that $\sum_{l\geq
1}|a_{l}|\leq 1$.

Let $\\{Y_{l}\\}_{l\geq 1}$ be a sequence of independent standard Gaussian
random variables with respect to a probability measure $P$. Lemma 2.10 ensures
that for each $\epsilon>0$ there exists $p>0$ with the property that

$P\Bigg{[}\Bigg{|}\sum_{l\geq
1}a_{l}\big{(}Y_{l}^{2}-1\big{)}\Bigg{|}<\frac{\epsilon}{1+|w|+3d|\xi_{-}|}\Bigg{]}\geq
p$ (2.20)

for any numerical sequence $\\{a_{l}\\}_{l\geq 1}$ such that $\sum_{l\geq
1}|a_{l}|\leq 1$, $\xi_{-}$ being the number introduced by Lemma 2.7. We make
use of property (2.20) to prove (2.19). Since the real symmetric matrix
$\mathsf{I}-\eta_{N}\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}\in\mathsf{BL}_{N+2,N+2}$
has positive eigenvalues $1-\eta_{N}m_{1},\ldots,1-\eta_{N}m_{(N+2)d}$, if we
build a diagonal matrix $\mathsf{D}$ with
$\sqrt{1-\eta_{N}m_{1}},\ldots,\sqrt{1-\eta_{N}m_{(N+2)d}}$ on the diagonal,
then
$\mathsf{I}-\eta_{N}\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}=\mathsf{O}^{\top}\mathsf{D}^{2}\mathsf{O}$
with an orthogonal matrix $\mathsf{O}\in\mathsf{BL}_{N+2,N+2}$. This way, if
we write $W_{N+2}=(1/2)\langle X,\mathsf{M}_{N}X\rangle$ with
$X:=(X_{1},\ldots,X_{N+2})$, then standard manipulations of Gaussian integrals
yield for all $k\in(\mathbb{R}^{d})^{N+2}$

$\displaystyle\mathbb{E}_{\eta_{N},N+2}\big{[}e^{\mathrm{i}\langle
k,\mathsf{D}\mathsf{O}\mathsf{A}^{-1}X\rangle}\big{]}$
$\displaystyle=\frac{\mathbb{E}[e^{\mathrm{i}\langle(\mathsf{A}^{\top})^{-1}\mathsf{O}^{\top}\mathsf{D}k,X\rangle+\frac{1}{2}\eta_{N}\langle
X,\mathsf{M}_{N}X\rangle}]}{\mathbb{E}[e^{\frac{1}{2}\eta_{N}\langle
X,\mathsf{M}_{N}X\rangle}]}$
$\displaystyle=e^{-\frac{1}{2}\langle(\mathsf{A}^{\top})^{-1}\mathsf{O}^{\top}\mathsf{D}k,(\Sigma_{N}^{-1}-\eta_{N}\mathsf{M}_{N})^{-1}(\mathsf{A}^{\top})^{-1}\mathsf{O}^{\top}\mathsf{D}k\rangle}$
$\displaystyle=e^{-\frac{1}{2}\langle\mathsf{O}^{\top}\mathsf{D}k,(\mathsf{I}-\eta_{N}\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A})^{-1}\mathsf{O}^{\top}\mathsf{D}k\rangle}=e^{-\frac{1}{2}\langle
k,k\rangle}.$

This formula states that the characteristic function of the random vector
$Y:=\mathsf{D}\mathsf{O}\mathsf{A}^{-1}X$ with respect to the probability
measure $\mathbb{P}_{\eta_{N},N+2}$ is the characteristic function of $(N+2)d$
independent standard Gaussian random variables. Thus, the components
$Y_{1},\ldots,Y_{(N+2)d}$ of $Y$ are independent standard Gaussian random
variables with respect to the probability measure $\mathbb{P}_{\eta_{N},N+2}$.
It follows from (2.20) that for each $\epsilon>0$ there exists $p>0$ with the
property that

$\mathbb{P}_{\eta_{N},N+2}\Bigg{[}\Bigg{|}\sum_{l=1}^{(N+2)d}a_{l}\big{(}Y_{l}^{2}-1\big{)}\Bigg{|}<\frac{\epsilon}{1+|w|+3d|\xi_{-}|}\Bigg{]}\geq
p$ (2.21)

for all $N\geq 1$ and real numbers $a_{1},\ldots,a_{(N+2)d}$ such that
$\sum_{l=1}^{(N+2)d}|a_{l}|\leq 1$. Let us observe now that

$W_{N+2}=\frac{1}{2}\langle
X,\mathsf{M}_{N}X\rangle=\frac{1}{2}\langle\mathsf{O}^{\top}\mathsf{D}^{-1}Y,\mathsf{A}^{\top}\mathsf{M}_{N}\mathsf{A}\mathsf{O}^{\top}\mathsf{D}^{-1}Y\rangle=\frac{1}{2}\sum_{l=1}^{(N+2)d}\frac{m_{l}}{1-\eta_{N}m_{l}}Y_{l}^{2}.$

This identity combined with (2.17) shows that for all $\epsilon>0$ and
sufficiently large $N$

$\mathbb{P}_{\eta_{N},N+2}\bigg{[}\bigg{|}\frac{W_{N+2}}{N+2}-w\bigg{|}<\epsilon\bigg{]}=\mathbb{P}_{\eta_{N},N+2}\Bigg{[}\Bigg{|}\sum_{l=1}^{(N+2)d}a_{l}\big{(}Y_{l}^{2}-1\big{)}\Bigg{|}<\frac{\epsilon}{1+|w|+3d|\xi_{-}|}\Bigg{]},$
(2.22)

where, for $l=1,\ldots,(N+2)d$, we have set

$a_{l}:=\frac{1}{2(N+2)(1+|w|+3d|\xi_{-}|)}\frac{m_{l}}{1-\eta_{N}m_{l}}.$

We have $\sum_{l=1}^{(N+2)d}|a_{l}|\leq 1$ for all sufficiently large $N$. In
fact, since

$\frac{|m_{l}|}{1-\eta_{N}m_{l}}\leq\frac{m_{l}}{1-\eta_{N}m_{l}}-2\min\big{\\{}0,m_{1},\ldots,m_{(N+2)d}\big{\\}}$

for every $l$, by invoking (2.17) and by recalling that
$\min\\{m_{1},\ldots,m_{(N+2)d}\\}$ approaches $\xi_{-}$ at large $N$, for all
sufficiently large $N$ we find

$\displaystyle(1+|w|+3d|\xi_{-}|)\sum_{l=1}^{(N+2)d}|a_{l}|$
$\displaystyle=\frac{1}{2(N+2)}\sum_{l=1}^{(N+2)d}\frac{|m_{l}|}{1-\eta_{N}m_{l}}$
$\displaystyle\leq\frac{1}{2(N+2)}\sum_{l=1}^{(N+2)d}\frac{m_{l}}{1-\eta_{N}m_{l}}-2d\min\big{\\{}0,m_{1},\ldots,m_{(N+2)d}\big{\\}}$
$\displaystyle\leq
w+2d\Big{|}\min\big{\\{}m_{1},\ldots,m_{(N+2)d}\big{\\}}\Big{|}\leq|w|+3d|\xi_{-}|.$

In conclusion, by comparing (2.22) with (2.21) we realize that for each
$\epsilon>0$ there exists $p>0$ such that

$\mathbb{P}_{\eta_{N},N+2}\bigg{[}\bigg{|}\frac{W_{N+2}}{N+2}-w\bigg{|}<\epsilon\bigg{]}\geq
p$

for all sufficiently large $N$. This bound proves (2.19).

##  3 Proof of Theorem 1.2

We know that the entropy production $Ne_{N}$ is the quadratic functional
$W_{N}$ corresponding to the matrices $L:=I-\Sigma_{o}^{-1}-S^{\top}S$,
$U:=0$, $R:=\Sigma_{o}^{-1}+S^{\top}S-I$, and $V:=S-S^{\top}$. Part 1 of the
theorem then follows from Theorem 1.1 with the rate function $I$ that maps
$w\in\mathbb{R}$ in
$\sup_{\lambda\in(\lambda_{-},\lambda_{+})}\\{w\lambda-\varphi(\lambda)\\}$.
It remains to verify the Gallavotti-Cohen symmetry stated by part 2.

Formula (1.9) shows that the Hermitian matrices $F_{\lambda}(\theta)$
associated with the entropy production satisfy
$F_{-\lambda-1}(\theta)=F_{\lambda}(2\pi-\theta)$ for all
$\lambda\in\mathbb{R}$ and $\theta\in[0,2\pi]$. According to (1.4) and (1.5),
this identity immediately gives $f_{-\lambda-1}=f_{\lambda}$ for any $\lambda$
and $\varphi(-\lambda-1)=\varphi(\lambda)$ for any $\lambda$ such that
$f_{-\lambda-1}=f_{\lambda}>0$. We shall show in a moment that
$\lambda_{-}=-\lambda_{+}-1$. It follows that for every $w\in\mathbb{R}$

$\displaystyle I(-w)-w$
$\displaystyle=\sup_{\lambda\in(\lambda_{-},\lambda_{+})}\big{\\{}w(-\lambda-1)-\varphi(\lambda)\big{\\}}$
$\displaystyle=\sup_{\lambda\in(-\lambda_{+}-1,-\lambda_{-}-1)}\big{\\{}w\lambda-\varphi(-\lambda-1)\big{\\}}=\sup_{\lambda\in(\lambda_{-},\lambda_{+})}\big{\\{}w\lambda-\varphi(\lambda)\big{\\}}=I(w),$

which demonstrates the Gallavotti-Cohen symmetry for $I$.

Let us verify that $\lambda_{-}=-\lambda_{+}-1$. Recalling that
$\lambda_{-}=\inf\\{\Lambda\\}$ and $\lambda_{+}=\sup\\{\Lambda\\}$, $\Lambda$
being the set defined by (2.8), it suffices to prove that
$-\lambda-1\in\Lambda$ whenever $\lambda\in\Lambda$. Fix $\lambda\in\Lambda$.
Then, $f_{-\lambda-1}=f_{\lambda}>0$, which implies that the matrices
$\Phi_{-\lambda-1}(n)$ given by (1.6) and $H_{-\lambda-1}$, $K_{-\lambda-1}$,
$\mathcal{L}_{-\lambda-1}$, and $\mathcal{R}_{-\lambda-1}$ introduced by Lemma
1.2 are well-defined. The identity
$F_{-\lambda-1}(\theta)=F_{\lambda}(2\pi-\theta)$ shows that
$\Phi_{-\lambda-1}(n)=\Phi_{\lambda}(-n)=\Phi_{\lambda}^{\dagger}(n)$ for all
$n\in\mathbb{Z}$. The latter entails that $H_{-\lambda-1}$ and $K_{\lambda}$
are related by the law

$H_{-\lambda-1}=I+\big{[}(\lambda+1)S^{\top}-\lambda
S\big{]}\Phi_{-\lambda-1}(1)=I+\big{[}(\lambda+1)S^{\top}-\lambda
S\big{]}\Phi_{\lambda}^{\dagger}(1)=K_{\lambda}^{\dagger}.$

This law induces a relationship between the matrices
$\mathcal{L}_{-\lambda-1}$ and $\mathcal{R}_{\lambda}$. In fact

$\displaystyle\mathcal{L}_{-\lambda-1}$
$\displaystyle=(\lambda+1)I-\lambda(\Sigma_{o}^{-1}+S^{\top}S)-\big{[}(\lambda+1)S-\lambda
S^{\top}\big{]}\Phi_{-\lambda-1}(0)H_{-\lambda-1}^{-1}\big{[}(\lambda+1)S^{\top}-\lambda
S\big{]}$
$\displaystyle=(\lambda+1)I-\lambda(\Sigma_{o}^{-1}+S^{\top}S)-\big{[}(\lambda+1)S-\lambda
S^{\top}\big{]}\Phi_{\lambda}(0)(K_{\lambda}^{-1})^{\dagger}\big{[}(\lambda+1)S^{\top}-\lambda
S\big{]},$

which, by taking adjoint on both the sides and by bearing in mind that
$\mathcal{L}_{\lambda}$ is Hermitian, yields

$\mathcal{L}_{-\lambda-1}=(\lambda+1)I-\lambda(\Sigma_{o}^{-1}+S^{\top}S)-\big{[}(\lambda+1)S-\lambda
S^{\top}\big{]}K_{\lambda}^{-1}\Phi_{\lambda}(0)\big{[}(\lambda+1)S^{\top}-\lambda
S\big{]}=\mathcal{R}_{\lambda}.$

Since $\mathcal{R}_{\lambda}\succ 0$ by hypothesis, we obtain
$\mathcal{L}_{-\lambda-1}\succ 0$. By similar arguments, we find that
$K_{\lambda-1}=H_{\lambda}^{\dagger}$ and
$\mathcal{R}_{-\lambda-1}=\mathcal{L}_{\lambda}\succ 0$. In conclusion,
$f_{-\lambda-1}>0$, $\mathcal{L}_{-\lambda-1}\succ 0$, and
$\mathcal{R}_{-\lambda-1}\succ 0$, so that $-\lambda-1\in\Lambda$.

##  Appendix A Proof of Lemma 1.1

Fix $N\geq 1$. Let $\mu_{N}^{+}:=\mathbb{P}[(X_{1},\ldots,X_{N})\in\cdot\,]$
and $\mu_{N}^{-}:=\mathbb{P}[(X_{N},\ldots,X_{1})\in\cdot\,]$ be the
probability measures on the Borel sets of $(\mathbb{R}^{d})^{N}$ induced by
the Gauss-Markov chain $\\{X_{n}\\}_{n\geq 1}$. According to (1.1),
$\mu_{N}^{+}$ and $\mu_{N}^{-}$ are the multivariate Gaussian distributions
that have densities

$\frac{d\mu_{N}^{+}}{d\ell}(x_{1},\ldots,x_{N}):=\frac{e^{-\frac{1}{2}\langle
x_{1},\Sigma_{o}^{-1}x_{1}\rangle}}{\sqrt{(2\pi)^{d}\det\Sigma_{o}}}\prod_{n=2}^{N}\frac{1}{\sqrt{(2\pi)^{d}}}\,e^{-\frac{1}{2}\|x_{n}-Sx_{n-1}\|^{2}}$

and

$\frac{d\mu_{N}^{-}}{d\ell}(x_{1},\ldots,x_{N}):=\frac{d\mu_{N}^{+}}{d\ell}(x_{N},\ldots,x_{1})=\frac{e^{-\frac{1}{2}\langle
x_{N},\Sigma_{o}^{-1}x_{N}\rangle}}{\sqrt{(2\pi)^{d}\det\Sigma_{o}}}\prod_{n=2}^{N}\frac{1}{\sqrt{(2\pi)^{d}}}\,e^{-\frac{1}{2}\|x_{n-1}-Sx_{n}\|^{2}}$

with respect to the Lebesgue measure $\ell$. Thus,
$\mu_{N}^{+}\ll\mu_{N}^{-}\ll\ell$ and standard results about measure theory
[40] give for all $(x_{1},\ldots,x_{N})\in(\mathbb{R}^{d})^{N}$

$\displaystyle\ln\bigg{[}\frac{d\mu_{N}^{+}}{d\mu_{N}^{-}}(x_{1},\ldots,x_{N})\bigg{]}$
$\displaystyle=\ln\bigg{[}\frac{d\mu_{N}^{+}}{d\ell}(x_{1},\ldots,x_{N})\bigg{/}\frac{d\mu_{N}^{-}}{d\ell}(x_{1},\ldots,x_{N})\bigg{]}$
$\displaystyle=\frac{1}{2}\langle
x_{1},(I-\Sigma_{o}^{-1}-S^{\top}S)x_{1}\rangle+\frac{1}{2}\langle
x_{N},(\Sigma_{o}^{-1}+S^{\top}S-I)x_{N}\rangle$
$\displaystyle+\sum_{n=2}^{N}\langle x_{n},(S-S^{\top})x_{n-1}\rangle.$

##  Appendix B Proof of Lemma 2.1

For all $N\geq 1$ and $z=(z_{1},\ldots,z_{N+2})\in(\mathbb{C}^{d})^{N+2}$ we
have

$\displaystyle\big{\|}\mathsf{Q}_{N}z\big{\|}^{2}$
$\displaystyle=\big{\|}Az_{1}+E^{\dagger}z_{2}\big{\|}^{2}+\sum_{n=2}^{N+1}\big{\|}Ez_{n-1}+Dz_{n}+E^{\dagger}z_{n+1}\big{\|}^{2}+\big{\|}Ez_{N+1}+Bz_{N+2}\big{\|}^{2}$
$\displaystyle\leq\Big{(}\|A\|\|z_{1}\|+\|E\|\|z_{2}\|\Big{)}^{2}+\sum_{n=2}^{N+1}\Big{(}\|E\|\|z_{n-1}\|+\|D\|\|z_{n}\|+\|E\|\|z_{n+1}\|\Big{)}^{2}$
$\displaystyle+\Big{(}\|E\|\|z_{N+1}\|+\|B\|\|z_{N+2}\|\Big{)}^{2}$
$\displaystyle\leq
2\|A\|^{2}\|z_{1}\|^{2}+2\|E\|^{2}\|z_{2}\|^{2}+3\sum_{n=2}^{N+1}\Big{(}\|E\|^{2}\|z_{n-1}\|^{2}+\|D\|^{2}\|z_{n}\|^{2}+\|E\|^{2}\|z_{n+1}\|^{2}\Big{)}$
$\displaystyle+2\|E\|^{2}\|z_{N+1}\|^{2}+2\|B\|^{2}\|z_{N+2}\|^{2}$
$\displaystyle\leq\Big{(}2\|A\|^{2}+3\|D\|^{2}+2\|B\|^{2}+6\|E\|^{2}\Big{)}\|z\|^{2}.$

##  Appendix C Proof of Lemma 2.2

Fix $N\geq 1$. Assume that there exists $q>0$ such that $\langle
z,\mathsf{Q}_{N}z\rangle\geq q\langle z,z\rangle$ for all
$z\in(\mathbb{C}^{d})^{N+2}$. Bearing in mind (2.2) and by writing $z$ as
$(a,t_{1},\ldots,t_{N},b)$ with $s:=(a,b)\in(\mathbb{C}^{d})^{2}$ and
$t:=(t_{1},\ldots,t_{N})\in(\mathbb{C}^{d})^{N}$, this condition reads

$\langle a,Aa\rangle+2\langle t,\mathsf{C}Ea\rangle+\langle
t,\mathsf{T}_{N}t\rangle+2\langle b,E\mathsf{R}t\rangle+\langle
b,Bb\rangle\geq q\langle a,a\rangle+q\langle t,t\rangle+q\langle b,b\rangle.$

This way, by setting $a:=0$ and $b:=0$ we find $\langle
t,\mathsf{T}_{N}t\rangle\geq q\langle t,t\rangle$ for any
$t\in(\mathbb{C}^{d})^{N}$. This shows in particular that $\mathsf{T}_{N}$ is
invertible. By setting
$t:=-\mathsf{T}_{N}^{-1}\mathsf{C}Ea-\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}b$
we obtain

$\langle
a,\big{(}A-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{C}E\big{)}a\rangle-2\langle
a,E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}b\rangle+\langle
b,\big{(}B-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\big{)}b\rangle\geq
q\langle a,a\rangle+q\langle b,b\rangle,$

that is $\langle s,\mathsf{S}_{N}s\rangle\geq q\langle s,s\rangle$ for all
$s\in(\mathbb{C}^{d})^{2}$. Part 1 is thus verified.

As far as part 2 is concerned, if $\mathsf{T}_{N}$ is invertible, then we can
write down the identity

$\mathsf{Q}_{N}=\mathsf{L}^{\dagger}\begin{pmatrix}A-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{C}E&\mathsf{0}&-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\\\
\mathsf{0}&\mathsf{T}_{N}&\mathsf{0}\\\
-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{C}E&\mathsf{0}&B-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\end{pmatrix}\mathsf{L}$
(C.1)

with

$\mathsf{L}:=\begin{pmatrix}I&\mathsf{0}&0\\\
\mathsf{T}_{N}^{-1}\mathsf{C}E&\mathsf{I}&\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\\\
0&\mathsf{0}&I\end{pmatrix}\in\mathsf{BL}_{N+2,N+2}.$

Since $\det\mathsf{L}=1$, it follows by permutations of rows and columns that

$\displaystyle\det\mathsf{Q}_{N}$
$\displaystyle=\det\begin{pmatrix}A-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{C}E&\mathsf{0}&-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\\\
\mathsf{0}&\mathsf{T}_{N}&\mathsf{0}\\\
-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{C}E&\mathsf{0}&B-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\end{pmatrix}$
$\displaystyle=\det\begin{pmatrix}\mathsf{T}_{N}&\mathsf{0}&\mathsf{0}\\\
\mathsf{0}&A-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{C}E&-E^{\dagger}\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\\\
\mathsf{0}&-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{C}E&B-E\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}E^{\dagger}\end{pmatrix}=\det\mathsf{T}_{N}\cdot\det\mathsf{S}_{N}.$

Moreover, given $z\in(\mathbb{C}^{d})^{N+2}$, by writing
$Lz=(a,t_{1},\ldots,t_{N},b)$ with $s:=(a,b)\in(\mathbb{C}^{d})^{2}$ and
$t:=(t_{1},\ldots,t_{N})\in(\mathbb{C}^{d})^{N}$, we realize from (C.1) that
$\langle z,\mathsf{Q}_{N}z\rangle=\langle t,\mathsf{T}_{N}t\rangle+\langle
s,\mathsf{S}_{N}s\rangle$. Thus, if $\mathsf{T}_{N}\succ\mathsf{0}$,
$\mathsf{S}_{N}\succ\mathsf{0}$, and $z\neq 0$, then we have $\langle
z,\mathsf{Q}_{N}z\rangle>0$ since $\mathsf{L}$ is invertible.

##  Appendix D Proof of Lemma 2.3

Part 1 is immediate since $r(\mathsf{T}_{N})$ is non-increasing with respect
to $N$. In fact, given any $z=(z_{1},\ldots,z_{N})\in(\mathbb{C}^{d})^{N}$, by
setting $\zeta:=(z_{1},\ldots,z_{N},0)\in(\mathbb{C}^{d})^{N+1}$ we see that
$r(\mathsf{T}_{N+1})\langle
z,z\rangle=r(\mathsf{T}_{N+1})\langle\zeta,\zeta\rangle\leq\langle\zeta,\mathsf{T}_{N+1}\zeta\rangle=\langle
z,\mathsf{T}_{N}z\rangle$. Part 3 is nothing but the Szegö theorem for the
determinant of Hermitian block Toeplitz matrices (see [38], Theorem 7). Let us
focus on part 2. Assume that $r(\mathsf{T}_{N})\geq t$ for all $N\geq 1$ and
pick a positive continuous function $\varphi$ with period $2\pi$ and a vector
$u\in\mathbb{C}^{d}$. Due to the assumed properties of $\varphi$, there exists
a sequence $\\{p_{N}\\}_{N\geq 0}$ of trigonometric polynomials that converges
uniformly to $\sqrt{\varphi}$, $p_{N}$ having degree $N$ (see [40], Theorem
4.25). Write $p_{N}(\theta)$ as $\sum_{n=-N}^{N}c_{N,n}e^{-\mathrm{i}n\theta}$
for each $N$ and $\theta$. Since $r(\mathsf{T}_{2N+1})\geq t$, by setting
$\zeta_{n}=z_{n}:=c_{N,N-n+1}e^{\mathrm{i}(N+1)}u$ for $n=1,\ldots,2N+1$ in
(2.6) we obtain

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\big{\langle}u,F(\theta)u\big{\rangle}\,p_{N}^{2}(\theta)\,d\theta$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\Bigg{\langle}\sum_{n=1}^{2N+1}z_{n}e^{-\mathrm{i}n\theta},F(\theta)\sum_{n=1}^{2N+1}z_{n}e^{-\mathrm{i}n\theta}\Bigg{\rangle}\,d\theta$
$\displaystyle=\langle z,\mathsf{T}_{2N+1}z\rangle\geq
t\sum_{n=1}^{2N+1}\langle z_{n},z_{n}\rangle=t\langle
u,u\rangle\,\frac{1}{2\pi}\int_{0}^{2\pi}p_{N}^{2}(\theta)\,d\theta$

for all $N\geq 0$. By sending $N$ to infinity we get

$\frac{1}{2\pi}\int_{0}^{2\pi}\big{\langle}u,F(\theta)u\big{\rangle}\,\varphi(\theta)\,d\theta\geq
t\langle u,u\rangle\,\frac{1}{2\pi}\int_{0}^{2\pi}\varphi(\theta)\,d\theta.$

The arbitrariness of $\varphi$ and $u$ shows that $\langle
z,F(\theta)z\rangle\geq t\langle z,z\rangle$ for all $\theta\in[0,2\pi]$ and
$z\in\mathbb{C}^{d}$.

Conversely, if $r(F(\theta))\geq t$ for every $\theta\in[0,2\pi]$, then by
invoking (2.6) again we can write for all $N\geq 1$ and
$z=(z_{1},\ldots,z_{N})\in(\mathbb{C}^{d})^{N}$

$\langle
z,\mathsf{T}_{N}z\rangle\geq\frac{t}{2\pi}\int_{0}^{2\pi}\Bigg{\langle}\sum_{n=1}^{N}z_{n}e^{-\mathrm{i}n\theta},\sum_{n=1}^{N}z_{n}e^{-\mathrm{i}n\theta}\Bigg{\rangle}\,d\theta=t\sum_{n=1}^{N}\langle
z_{n},z_{n}\rangle.$

##  Appendix E Proof of Lemma 2.4

Suppose for a moment that the matrix $H$ is invertible. Then, the matrix $K$
is proved to be invertible by contradiction. In fact, if $K$ is not
invertible, then there exists a vector $u\in\mathbb{C}^{d}$ different from $0$
such that $Ku=[I-\Phi(1)E]u=0$. We must have $Eu\neq 0$, otherwise $u=0$.
Since $HE=EK$, we get $HEu=0$ with $Eu\neq 0$, which contradicts the
assumption that $H$ is invertible.

Let us demonstrate now that the matrix $H$ is invertible. This will prove part
1 of the lemma. We proceed by contradiction. Suppose that there exists a
vector $u\in\mathbb{C}^{d}$ different from $0$ such that $Hu=0$. Pick an
arbitrary integer $N\geq 3$ and for $n=1,\ldots,N$ consider the vectors

$z_{n}:=\Phi(1-n)u=\frac{1}{2\pi}\int_{0}^{2\pi}F^{-1}(\theta)e^{\mathrm{i}(n-1)\theta}d\theta\,u.$

We have $z_{1}\neq 0$ since $\Phi(0)$ is invertible. We claim that

$\mathsf{T}_{N}\begin{pmatrix}z_{1}\\\ z_{2}\\\ \vdots\\\ z_{N-1}\\\
z_{N}\end{pmatrix}=\begin{pmatrix}Dz_{1}+E^{\dagger}z_{2}\\\
Ez_{1}+Dz_{2}+E^{\dagger}z_{3}\\\ \vdots\\\
Ez_{N-2}+Dz_{N-1}+E^{\dagger}z_{N}\\\
Ez_{N-1}+Dz_{N}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ \vdots\\\ 0\\\
-E^{\dagger}\Phi(-N)u\end{pmatrix}.$ (E.1)

Indeed, for $n=2,\ldots,N-1$ we have

$\displaystyle Ez_{n-1}+Dz_{n}+E^{\dagger}z_{n+1}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\big{[}Ee^{-\mathrm{i}\theta}+D+E^{\dagger}e^{\mathrm{i}\theta}\big{]}F^{-1}(\theta)e^{\mathrm{i}(n-1)\theta}d\theta\,u$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{\mathrm{i}(n-1)\theta}d\theta\,u=0$

and

$\displaystyle Dz_{1}+E^{\dagger}z_{2}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\big{[}D+E^{\dagger}e^{\mathrm{i}\theta}\big{]}F^{-1}(\theta)\,d\theta\,u$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\big{[}F(\theta)-Ee^{-\mathrm{i}\theta}\big{]}F^{-1}(\theta)\,d\theta\,u=\big{[}I-E\Phi(1)\big{]}u=Hu=0.$

Finally, we see that

$\displaystyle Ez_{N-1}+Dz_{N}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\big{[}Ee^{-\mathrm{i}\theta}+D\big{]}F^{-1}(\theta)e^{\mathrm{i}(N-1)\theta}d\theta\,u$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\big{[}F(\theta)-E^{\dagger}e^{\mathrm{i}\theta}\big{]}F^{-1}(\theta)e^{\mathrm{i}(N-1)\theta}d\theta\,u=-E^{\dagger}\Phi(-N)u.$

Due to (E.1), it follows from (2.6) with
$\zeta=z:=(z_{1},\ldots,z_{N})\in(\mathbb{C}^{d})^{N}$ and the hypothesis
$t:=\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}>0$ that

$-\Big{\langle}E^{\dagger}\Phi(1-N)u,E^{\dagger}\Phi(-N)u\Big{\rangle}=\langle
z,\mathsf{T}_{N}z\rangle\geq t\sum_{n=1}^{N}\langle z_{n},z_{n}\rangle\geq
t\langle z_{1},z_{1}\rangle.$

This bound is absurd since $z_{1}\neq 0$ and
$\lim_{N\uparrow\infty}\Phi(-N)=0$ by the Riemann-Lebesgue lemma.

Let us move to part 2. As $\inf_{\theta\in[0,2\pi]}\\{r(F(\theta))\\}>0$,
$\mathsf{T}_{N}$ is invertible by Lemma 2.3 and we can set

$\mathsf{T}_{N}^{-1}\mathsf{C}=:\begin{pmatrix}C_{1}\\\ \vdots\\\ C_{N}\\\
\end{pmatrix}$ (E.2)

and

$\mathsf{R}\mathsf{T}_{N}^{-1}=:\begin{pmatrix}R_{1}&\cdots&R_{N}\end{pmatrix}.$
(E.3)

The matrices $\mathsf{C}$ and $\mathsf{R}$ were defined in (2.3) and (2.4),
respectively. We have
$\mathsf{C}^{\dagger}\mathsf{T}_{N}^{-1}\mathsf{C}=C_{1}$ and
$\mathsf{R}\mathsf{T}_{N}^{-1}\mathsf{R}^{\dagger}=R_{N}$, which on the one
hand show that $C_{1}$ and $R_{N}$ are Hermitian and on the other hand allow
us to write

$\mathsf{S}_{N}=\begin{pmatrix}A-E^{\dagger}C_{1}E&-E^{\dagger}R_{1}^{\dagger}E^{\dagger}\\\
-EC_{N}E&B-ER_{N}E^{\dagger}\end{pmatrix}.$

Let us verify that $C_{1}$ approaches the matrix $\Phi(0)H^{-1}$ and $R_{N}$
approaches the matrix $K^{-1}\Phi(0)$ when $N$ is sent to infinity, whereas
$C_{N}$ and $R_{1}$ approach 0. These facts prove part 2 of the lemma.

To begin with, we observe that since $H^{\dagger}$ and $K$ are non-singular
and $\lim_{N\uparrow\infty}\Phi(\pm N)=0$ by the Riemann-Lebesgue lemma, the
matrix

$\mathsf{Z}:=\begin{pmatrix}H^{\dagger}&-\Phi(N)E\\\
-\Phi(-N)E^{\dagger}&K\end{pmatrix}\in\mathsf{BL}_{2,2}$ (E.4)

is invertible if $N>N_{o}$, $N_{o}\geq 2$ being a sufficiently large integer.
Pick $N>N_{o}$. By multiplying (E.2) by $\mathsf{T}_{N}$ on the left and (E.3)
by $\mathsf{T}_{N}$ on the right we explicitly have

$\begin{cases}DC_{1}+E^{\dagger}C_{2}=I&\mbox{for }n=1,\\\
EC_{n-1}+DC_{n}+E^{\dagger}C_{n+1}=0&\mbox{for }n=2,\ldots,N-1,\\\
EC_{N-1}+DC_{N}=0&\mbox{for }n=N\end{cases}$ (E.5)

and

$\begin{cases}DR_{1}^{\dagger}+E^{\dagger}R_{2}^{\dagger}=0&\mbox{for }n=1,\\\
ER_{n-1}^{\dagger}+DR_{n}^{\dagger}+E^{\dagger}R_{n+1}^{\dagger}=0&\mbox{for
}n=2,\ldots,N-1,\\\ ER_{N-1}^{\dagger}+DR_{N}^{\dagger}=I&\mbox{for
}n=N.\end{cases}$ (E.6)

By multiplying the $n$th equation in (E.5) by $e^{-\mathrm{i}n\theta}$ and
then by carrying out the sum over $n$ we get

$F(\theta)\sum_{n=1}^{N}C_{n}e^{-\mathrm{i}n\theta}=Ie^{-\mathrm{i}\theta}+E^{\dagger}C_{1}+EC_{N}e^{-\mathrm{i}(N+1)\theta},$

which gives for $n=1,\ldots,N$

$\displaystyle C_{n}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}F^{-1}(\theta)\Big{[}Ie^{\mathrm{i}(n-1)\theta}+E^{\dagger}C_{1}e^{\mathrm{i}n\theta}+EC_{N}e^{\mathrm{i}(n-N-1)\theta}\Big{]}d\theta$
$\displaystyle=\Phi(1-n)+\Phi(-n)E^{\dagger}C_{1}+\Phi(N-n+1)EC_{N}.$ (E.7)

Similarly, (E.6) shows that for $n=1,\ldots,N$

$R_{n}^{\dagger}=\Phi(N-n)+\Phi(-n)E^{\dagger}R_{1}^{\dagger}+\Phi(N-n+1)ER_{N}^{\dagger}.$
(E.8)

At this point, by setting $n:=1$ and $n:=N$ in (E.7) and by recalling that
$C_{1}=C_{1}^{\dagger}$ we realize that

$\mathsf{Z}\begin{pmatrix}C_{1}^{\dagger}\\\
C_{N}\end{pmatrix}=\begin{pmatrix}\Phi(0)\\\ \Phi(1-N)\end{pmatrix},$

$\mathsf{Z}$ being the matrix defined in (E.4). It follows that

$\begin{pmatrix}C_{1}^{\dagger}\\\
C_{N}\end{pmatrix}=\mathsf{Z}^{-1}\begin{pmatrix}\Phi(0)\\\
\phi(1-N)\end{pmatrix}$

as $\mathsf{Z}$ is invertible for $N>N_{o}$. Similarly, (E.8) for $n:=1$ and
$n:=N$ and the fact that $R_{N}^{\dagger}=R_{N}$ yield

$\begin{pmatrix}R_{1}^{\dagger}\\\
R_{N}\end{pmatrix}=\mathsf{Z}^{-1}\begin{pmatrix}\Phi(N-1)\\\
\Phi(0)\end{pmatrix}.$

This way, the Riemann-Lebesgue lemma entails that $C_{1}$ approaches
$\Phi(0)H^{-1}$ and $R_{N}$ approaches $K^{-1}\Phi(0)$ when $N$ is sent to
infinity, whereas $C_{N}$ and $R_{1}$ approach 0.

##  Appendix F Proof of Lemma 2.6

As the spectral radius $\rho(S)$ of $S$ is smaller than $1$ by hypothesis,
Gelfand’s formula for spectral radii gives
$\lim_{n\uparrow\infty}\|S^{n}\|^{\frac{1}{n}}=\rho(S)<1$. Then, there exist
$s\in(0,1)$ and a positive constant $c$ such that $\|S^{n}\|\leq cs^{n}$ for
all $n\geq 0$. Let us show that the lemma holds with $\sigma:=[1\wedge
r(\Sigma_{o}^{-1})](1-s)^{2}c^{-2}$, which is positive since, obviously,
$r(\Sigma_{o}^{-1})>0$. Fix $N\geq 1$ and, to begin with, observe that

$\displaystyle r(\Sigma_{N}^{-1})$
$\displaystyle=\inf_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\langle
z_{1},\Sigma_{o}^{-1}z_{1}\rangle+\sum_{n=2}^{N+2}\|z_{n}-Sz_{n-1}\|^{2}}{\sum_{n=1}^{N+2}\langle
z_{n},z_{n}\rangle}\bigg{\\}}$ $\displaystyle\geq 1\wedge
r(\Sigma_{o}^{-1})\inf_{\begin{subarray}{c}z\in(\mathbb{C}^{d})^{N+2}\\\ z\neq
0\end{subarray}}\bigg{\\{}\frac{\|z_{1}\|^{2}+\sum_{n=2}^{N+2}\|z_{n}-Sz_{n-1}\|^{2}}{\sum_{n=1}^{N+2}\langle
z_{n},z_{n}\rangle}\bigg{\\}}.$

Since for any $z=(z_{1},\ldots,z_{N+2})\in(\mathbb{C}^{d})^{N+2}$ there exists
$\zeta=(\zeta_{1},\ldots,\zeta_{N+2})\in(\mathbb{C}^{d})^{N+2}$ such that
$z_{n}=\sum_{k=1}^{n}S^{n-k}\zeta_{k}$ for each $n$, this bound yields

$r(\Sigma_{N}^{-1})\geq 1\wedge
r(\Sigma_{o}^{-1})\inf_{\begin{subarray}{c}\zeta\in(\mathbb{C}^{d})^{N+2}\\\
\zeta\neq
0\end{subarray}}\bigg{\\{}\frac{\sum_{n=1}^{N+2}\|\zeta_{n}\|^{2}}{\sum_{n=1}^{N+2}\sum_{h=1}^{n}\sum_{k=1}^{n}\langle
S^{n-h}\zeta_{h},S^{n-k}\zeta_{k}\rangle}\bigg{\\}}.$

At this point, it suffices to invoke the Cauchy-Schwarz inequality to conclude
that for every $\zeta=(\zeta_{1},\ldots,\zeta_{N+2})\in(\mathbb{C}^{d})^{N+2}$

$\displaystyle\sum_{n=1}^{N+2}\sum_{h=1}^{n}\sum_{k=1}^{n}\langle
S^{n-h}\zeta_{h},S^{n-k}\zeta_{k}\rangle$
$\displaystyle\leq\sum_{n=1}^{N+2}\sum_{h=1}^{n}\sum_{k=1}^{n}\|S^{n-h}\zeta_{h}\|\|S^{n-k}\zeta_{k}\|$
$\displaystyle\leq
c^{2}\sum_{n=1}^{N+2}\sum_{h=1}^{n}\sum_{k=1}^{n}s^{2n-h-k}\|\zeta_{h}\|\|\zeta_{k}\|$
$\displaystyle\leq\frac{c^{2}}{2}\sum_{n=1}^{N+2}\sum_{h=1}^{n}\sum_{k=1}^{n}s^{2n-h-k}\Big{(}\|\zeta_{h}\|^{2}+\|\zeta_{k}\|^{2}\Big{)}$
$\displaystyle\leq\frac{c^{2}}{(1-s)^{2}}\sum_{k=1}^{N+2}\|\zeta_{k}\|^{2}.$

## Acknowledgements

The authors are grateful to Giuseppe Gonnella for suggesting the problem of
large deviations for the entropy production rate and quadratic functionals of
autoregressive models.

## Data availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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