# Consistency and asymptotic normality in a class of nearly unstable processes

Marie Badreau  Laboratoire Manceau de Mathématiques, Le Mans Université,
Avenue Olivier Messiaen, 72085 LE MANS Cedex 09, France.
<EMAIL_ADDRESS>and  Frédéric Proïa  Univ Angers, CNRS,
LAREMA, SFR MATHSTIC, F-49000 Angers, France<EMAIL_ADDRESS>
###### Abstract.

This paper deals with inference in a class of stable but nearly-unstable
processes. Autoregressive processes are considered, in which the bridge
between stability and instability is expressed by a time-varying companion
matrix $A_{n}$ with spectral radius $\rho(A_{n})<1$ satisfying
$\rho(A_{n})\rightarrow 1$. This framework is particularly suitable to
understand unit root issues by focusing on the inner boundary of the unit
circle. Consistency is established for the empirical covariance and the OLS
estimation together with asymptotic normality under appropriate hypotheses
when $A$, the limit of $A_{n}$, has a real spectrum, and a particular case is
deduced when $A$ also contains complex eigenvalues. The asymptotic process is
integrated with either one unit root (located at 1 or $-1$), or even two unit
roots located at 1 and $-1$. Finally, a set of simulations illustrate the
asymptotic behavior of the OLS. The results are essentially proved by $L^{2}$
computations and the limit theory of triangular arrays of martingales.

###### Key words and phrases:

Nearly unstable autoregressive process, OLS estimation, Asymptotic behavior,
Unit root, Martingales.

##  1\. Introduction and motivation

This paper is dedicated to the boundary between stationarity and integration
in time series, which has long since proven to be crucial for practitioners,
e.g. in econometrics or finance. First of all, we refer the reader to
Brockwell and Davis (1991) for a large overview of linear time series. In the
context of autoregressive processes (AR), it is well-known that the least
squares (OLS) estimator is strongly consistent wherever its characteristic
roots lie, see Lai and Wei (1983), but with very different convergence rates
and limit distributions whether the process is stationary or integrated, or
even explosive. According to the terminology of Duflo (1997), a stationary
process will be described as stable and an integrated process as unstable in
what follows. In the first case, the OLS estimator is $\sqrt{n}$-consistent
with a Gaussian asymptotic behavior whereas in the other case, it is (at
least) $n$-consistent with an asymmetrical asymptotic behavior that can be
written as functionals of a standard Wiener process $(W(t),\,t\in[0,1])$.
Precisely, for example in the univariate case, in the stable (left-hand side)
and unstable (right-hand side, $\theta=\pm 1$) cases, either

$\sqrt{n}\,\big{(}\widehat{\theta}_{n}-\theta\big{)}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}(0,1-\theta^{2})\hskip
14.22636pt\text{or}\hskip
14.22636ptn\,\big{(}\widehat{\theta}_{n}-\theta\big{)}~{}\overset{d}{\longrightarrow}~{}\frac{\pm\int_{0}^{1}W(t)\,\mathrm{d}W(t)}{\int_{0}^{1}W^{2}(t)\,\mathrm{d}t}$

and these results are extended to the general case (see e.g. the substantial
work of Chan and Wei (1988)). This discontinuity has motivated numerous
studies devoted to intermediate AR models aimed at bridging these two
situations. Most of these studies involve the AR(1) process with time-varying
coefficients, which is an easy-to-use relevant case to understand the
underlying dynamics. Random coefficients have been suggested to bypass the
difficulty: as it is explained in Horváth and Trapani (2016), thanks to a
specific self-normalized WLS estimator of $\theta$, “there is no unit root
problem in case of the RCA model”. Indeed, the authors get the asymptotic
normality irrespective of the average value of the coefficient, only the rate
is affected. In a different context, the so-called volatility induced
stationarity detailed e.g. in Ling (2004) and Nielsen and Rahbek (2014),
enable to accommodate integration and stationarity in a double autoregression
through stochastic volatility despite a unit root in the conditional mean of
the process, see also Trapani (2021). We refer the reader to all the
references contained in those papers. An AR(1) with non-random but time-
varying coefficients may also be written in the triangular form111Double
indexing is customary to such representations: $X_{n,\,k}$ is the $k$-th
observation of a time series of size $n$ (apart from the initial value). The
triangular form of the process is
$\\{X_{1,\,0},X_{1,\,1}\\},\\{X_{2,\,0},X_{2,\,1},X_{2,\,2}\\},\ldots,\\{X_{n,\,0},\ldots,X_{n,\,n}\\}$.

$\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636ptX_{n,\,k}=\theta_{n}\,X_{n,\,k-1}+\varepsilon_{k}$

where $(\varepsilon_{k})$ is a sequence of zero-mean perturbations (usually
independent and identically distributed (i.i.d.) random variables, or
differences of martingale) and $X_{n,\,0}$ is an arbitrary initial value. In
that case, the OLS estimator is obviously given by

(1.1)
$\widehat{\theta}_{n}=\frac{\sum_{k=1}^{n}X_{n,\,k-1}\,X_{n,\,k}}{\sum_{k=1}^{n}X_{n,\,k-1}^{\,2}}.$

To focus on the inner neighborhood of the unit root, the idea is to consider a
sequence of coefficients that satisfies $|\theta_{n}|<1$ for all $n\geqslant
1$ but $|\theta_{n}|\rightarrow 1$, so that a new model corresponds to each
new observation, always stable but increasingly close to instability. Chan and
Wei (1987) show that, letting $\theta_{n}=1-c/n$ and under appropriate
assumptions, the OLS estimator (1.1) is $n$-consistent whether $c=0$ (which
corresponds to the standard integrated AR(1) process) or $c>0$. Once self-
normalized, it is shown in Thm. 1 that the estimation error has a non-Gaussian
limit distribution $\mathcal{L}(c)$ which is asymptotically $\mathcal{N}(0,1)$
as $c\rightarrow+\infty$ (see also Thm. 2 of Phillips (1987)). In this
context, it is natural to conjecture that any rate faster than $n$ shall lead
to the same conclusion. But slowing down the convergence of $\theta_{n}$ to 1
in order to make the aforementioned bridge, Giraitis and Phillips (2006) and
later Phillips and Magdalinos (2007) suggest to fix $\theta_{n}=1-c/v_{n}$ for
$1\ll v_{n}\ll n$. Among other results, they establish that

$\sqrt{n\,v_{n}}\,\big{(}\widehat{\theta}_{n}-\theta_{n}\big{)}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}(0,\,2c)$

as soon as $c>0$, that is in the stable but nearly unstable case, and that the
result also holds, under suitable assumptions, around the negative unit root
with $\theta_{n}=-1+c/v_{n}$. In the special case where $v_{n}=n^{\alpha}$
($0<\alpha<1$), the rate is $n^{(1+\alpha)/2}$ and the authors note that the
interval $\sqrt{n}\ll n^{(1+\alpha)/2}\ll n$ is covered but they emphasize
that boundaries do not match. The asymptotic variance is overestimated for
$\alpha\rightarrow 0^{+}$ ($2c$ instead of $2c-c^{2}$ when $\alpha=0$) whereas
the Gaussian limit distribution for $\alpha\rightarrow 1^{-}$ is no longer
Gaussian for $\alpha=1$. Note that the authors also consider the explosive
case $|\theta_{n}|>1$, whereas we will only focus here on the stable
alternative to the unstable process. In the same vein, Phillips and Lee (2015)
develop a limit theory for nonstationary vector autoregressions with mixed
roots in the vicinity of unity involving persistent and explosive components.

Now for any $p\geqslant 1$, consider the process generated according to the
triangular form

(1.2) $\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636ptX_{n,\,k}=\sum_{i=1}^{p}\theta_{n,\,i}\,X_{n,\,k-i}+\varepsilon_{k}$

where $(\varepsilon_{k})$ is a sequence of zero-mean i.i.d. random variables
with variance $\sigma^{2}>0$. In an equivalent way, the vector expression of
this process is

(1.3) $\Phi_{n,\,k}=A_{n}\,\Phi_{n,\,k-1}+E_{k}$

where $E_{k}=(\varepsilon_{k},0,\ldots,0)^{\,T}$ is a $p$-vectorial noise,
$\Phi_{n,\,k}=(X_{n,\,k},\ldots,X_{n,\,k-p+1})^{\,T}$ and

$A_{n}=\begin{pmatrix}\theta_{n,\,1}&\theta_{n,\,2}&\ldots&\theta_{n,\,p}\\\
&I_{p-1}&&0\end{pmatrix}$

is the associated $p\times p$ companion matrix. The initial value
$\Phi_{n,\,0}$ is supposed to have a finite moment of order 2 and to be
independent of $(E_{k})$. The OLS estimator of
$\theta_{n}=(\theta_{n,\,1},\ldots,\theta_{n,\,p})^{\,T}$ is then given by

(1.4)
$\widehat{\theta}_{n}=S_{n,\,n-1}^{\,-1}\sum_{k=1}^{n}\Phi_{n,\,k-1}\,X_{n,\,k}\hskip
14.22636pt\text{where}\hskip
14.22636ptS_{n,\,n-1}=\sum_{k=0}^{n-1}\Phi_{n,\,k}\,\Phi_{n,\,k}^{\,T}.$

Note that we may add a small $\epsilon>0$ to $S_{n,\,n-1}$ to avoid a useless
invertibility assumption, without disturbing the asymptotic behavior.222 To be
rigorous, one should write $\widehat{\theta}_{n,\,n}$ instead of
$\widehat{\theta}_{n}$ to emphasize that the OLS is a function of
$X_{n,\,0},\ldots,X_{n,\,n}$. Similarly, $S_{n}$ will be used for $S_{n,\,n}$
(and $T_{n}$ for $T_{n,\,n}$, etc.) to lighten the notation when no confusion
can arise. The $p$-dimensional process (1.3) is stable when $\rho(A_{n})<1$,
that is, when the largest modulus of its eigenvalues is less than 1, see Def.
2.3.17 of Duflo (1997). Since the eigenvalues of $A_{n}$ are the inverses of
the roots of the complex polynomial
$\Theta_{n}(z)=1-\theta_{n,\,1}\,z-\ldots-\theta_{n,\,p}\,z^{\,p}$, this is
equivalent to say that $\Theta_{n}(z)\neq 0$ for all $|z|\leqslant 1$. On the
contrary, it is unstable when $\rho(A_{n})=1$. Along the same lines, we will
thus consider that $\rho(A_{n})<1$ for all $n\geqslant 1$ but
$\rho(A_{n})\rightarrow 1$, which corresponds to a stable but nearly unstable
AR$(p)$ process with time-varying coefficients. In this context and under
suitable assumptions, Proïa (2020) has established some moderate deviation
principles for the empirical covariance and the OLS estimator that hold for
any $p\geqslant 1$ (extending those of Miao et al. (2015) valid for $p=1$). In
particular, the sequence

$\left(\frac{\sqrt{n}}{b_{n}\,(1-\rho(A_{n}))^{1/2}}\,\big{(}\widehat{\theta}_{n}-\theta_{n}\big{)}\right)_{\\!n\,\geqslant\,1}$

satisfies a large deviation principle with a speed $(b_{n}^{\,2})$ and a rate
function depending on the renormalized limit covariance of the process. Let us
also mention the weak unit roots of Park (2003) containing more applications
than ours, including faster rates of convergence for the nearly unit root and
non-linear models, but in a more restricted setting (we will come back to this
in due time), or the recent work of Buchmann and Chan (2013) who introduce a
perturbation in the Jordan canonical form of the AR$(p)$ model (see Thm. 2.1)
and get a set of convergences in a context close to ours (although more
general). However, by directly dealing with the spectral radius of the
companion matrix, the approach of this paper seems easier to interpret and we
will see that in fine different kind of results are obtained via different
technical tools. Now, to complete these deviations and to generalize the
results of Phillips and Magdalinos (2007), we aim at proving the consistency
and asymptotic normality of the OLS estimator (1.4). The strategy remains the
same, but the calculation steps will prove to be much trickier. In the second
section, the assumptions and main results are provided together with comments
and examples. The third section is dedicated to the technical proofs whereas
the fourth section is the empirical part of the paper, containing simulations.
A quick conclusion with considerations about further improvements ends the
paper.

##  2\. Main results

First, let us start by describing three technical hypotheses that will be
needed to achieve our goals. In particular, the first one is a matter of
simplification of the reasonings since $A_{n}$ turns out to be diagonalizable
for a sufficiently large $n$, and that specificity will prove to be very
useful. The second hypothesis is related to the number of unit roots in the
asymptotic process (either $\pm 1$, or both 1 and $-1$). The third assumption
characterizes our stable but nearly-unstable setting. The complex eigenvalues
are sorted according to their modulus (in descending order), with ties broken
by lexicographic order (also in descending order).

  1. (H1)

Convergence of the companion matrix. There exists a $p\times p$ matrix $A$
such that

$\lim_{n\,\rightarrow\,+\infty}A_{n}=A$

with distinct eigenvalues
$0\,<\,|\lambda_{p}|\,\leqslant\,\ldots\,\leqslant\,|\lambda_{2}|\,\leqslant\,|\lambda_{1}|\,=\,\rho(A)=1$,
and the top-right element of $A$ is non-zero ($\theta_{p}\neq 0$).

  2. (H2)

Number of unit roots. Either (H2) (H21) or (H2) (H22) is true.

    1. (H21)

There is exactly one unit root in $A$ ($\lambda_{1}=\pm 1$ but
$|\lambda_{2}|<1$ if $p\geqslant 2$).

    2. (H22)

There is exactly two unit roots in $A$ ($\lambda_{1}=1$, $\lambda_{2}=-1$ but
$|\lambda_{3}|<1$ if $p\geqslant 3$).

  3. (H3)

Spectral radius of the companion matrix. The spectral radius of $A_{n}$ is
given by

$\rho(A_{n})=1-\frac{c}{v_{n}}$

for some $c>0$ and $1\ll v_{n}\ll n$.

Most of our results will be stated under (H2) (H21) but we will explain at the
end of the section that, in fact, they are still valid under (H2) (H22) with
only slight adjustments of the rates and limit behaviors. For readability
purposes, the calculations will not be developed in that case but left to the
reader since they follow exactly the same lines. A decisive argument in the
technical part of the paper rests on the diagonalization of $A_{n}$, on which
we will give more details in due course. But to summarize, we will explain
that there exists $n_{0}\geqslant 1$ such that, for all $n>n_{0}$,
$A_{n}=P_{n}\,D_{n}\,P_{n}^{-1}$ where
$D_{n}=\textnormal{diag}(\lambda_{n,\,1},\,\ldots,\,\lambda_{n,\,p})$ contains
the ordered distinct eigenvalues
$1\,>\,|\lambda_{n,\,1}|\,\geqslant\,\ldots\,\geqslant\,|\lambda_{n,\,p}|\,>\,0$
of $A_{n}$ which converge to those of $A$ (see (H1) above), and the basis of
eigenvectors of $A_{n}$ can be written in the standardized form

(2.1) $P_{n}=\begin{pmatrix}1&1&\ldots&1\\\
\frac{1}{\lambda_{n,\,1}}&\frac{1}{\lambda_{n,\,2}}&\ldots&\frac{1}{\lambda_{n,\,p}}\\\
\vdots&\vdots&&\vdots\\\
\frac{1}{\lambda_{n,\,1}^{p-1}}&\frac{1}{\lambda_{n,\,2}^{p-1}}&\ldots&\frac{1}{\lambda_{n,\,p}^{p-1}}\end{pmatrix}.$

In addition, $P_{n}\rightarrow P$ and $P_{n}^{-1}\rightarrow P^{-1}$ whose
entries of the first column are denoted by $\pi_{11},\ldots,\pi_{p1}$. The
$p\times p$ symmetric matrix $H$ and its standardized version $H_{0}$, that
will play the role of the precision matrix in the asymptotic normality of the
OLS estimate, are then defined as

(2.2) $H=\sigma^{2}\,\begin{pmatrix}\frac{\pi_{11}^{2}}{2}&0&\ldots&0\\\
0&\frac{\pi_{21}^{2}}{1-\lambda_{2}^{2}}&\ldots&\frac{\pi_{21}\pi_{p1}}{1-\lambda_{2}\lambda_{p}}\\\
\vdots&\vdots&\ddots&\vdots\\\
0&\frac{\pi_{p1}\pi_{21}}{1-\lambda_{p}\lambda_{2}}&\ldots&\frac{\pi_{p1}^{2}}{1-\lambda_{p}^{2}}\end{pmatrix}\,\overset{\Delta}{=}\,\sigma^{2}\,H_{0}.$

From now on, to lighten the expressions we will rather use
$\rho_{n}\,\overset{\Delta}{=}\,\rho(A_{n})$. Also, for any $d\geqslant 1$,
the usual $d\times d$ matrices

(2.3) $I_{d}=\begin{pmatrix}1&0&\ldots\\\ 0&\ddots&\\\
\vdots&&1\end{pmatrix}\hskip 14.22636pt\text{and}\hskip
14.22636ptK_{d}=\begin{pmatrix}1&0&\ldots\\\ 0&0&\\\
\vdots&&\ddots\end{pmatrix}$

will be frequently encountered, both in statements and proofs. Similarly, the
first vector of the canonical basis of $\mathbb{R}^{d}$ will be denoted by
$e_{d}=(1,0,\ldots,0)^{\,T}$. Finally, $f_{n}\asymp g_{n}$ will have the
meaning that both $f_{n}=O(g_{n})$ and $g_{n}=O(f_{n})$.

###  2.1. Causal representation and memory

Under (H3), there exists a causal representation of $(\Phi_{n,\,k})$ given by

$\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636pt\Phi_{n,\,k}=\sum_{\ell\,\geqslant\,0}A_{n}^{\ell}\,E_{k-\ell}$

that directly leads to the autocovariance function

(2.4) $\forall\,h\geqslant 0,\hskip
14.22636pt\Gamma_{n}(h)=\sigma^{2}A_{n}^{h}\sum_{\ell\,\geqslant\,0}A_{n}^{\ell}\,K_{p}\,(A_{n}^{\,T})^{\ell}\hskip
14.22636pt\text{and}\hskip 14.22636pt\Gamma_{n}(-h)=\Gamma_{n}^{\,T}(h)$

provided that the noise has a finite moment of order 2. In particular, we have
the following result related to the memory of the process.

######  Proposition 2.1.

Assume that (H1), (H2) and (H3) hold, and that
$\mathbb{E}[\varepsilon_{1}^{\,2}]=\sigma^{2}<+\infty$. Then,

$\forall\,n>n_{0},\hskip
14.22636pt\left\|\sum_{h\,\in\,\mathbb{Z}}\Gamma_{n}(h)\right\|<+\infty.$

Besides, as $n\rightarrow+\infty$,

$\sum_{h\,\in\,\mathbb{Z}}\Gamma_{n}(h)\asymp\left\\{\begin{array}[]{ll}(1-\rho_{n})^{-2}&\textnormal{under
\ref{HypUR1} with $\lambda_{1}=1$ or \ref{HypUR2}},\\\ 1&\textnormal{under
\ref{HypUR1} with $\lambda_{1}=-1$.}\end{array}\right.$

###### Proof.

See Section 3.2. ∎

In other terms, using the terminology of Sec. 1.3.1 of Beran et al. (2013),
$(\Phi_{n,\,k})$ has a short memory at fixed $n$ whereas, as $n$ tends to
infinity, it turns to a long memory process when $\lambda_{1}=1$ but keeps a
short memory when $\lambda_{1}=-1$. This is especially clear in the univariate
case for which

$\sum_{h\,\in\,\mathbb{Z}}\gamma_{n}(h)=\frac{\sigma^{2}}{(1-\theta_{n})^{2}}$

is an increasing function of $\theta_{n}$, minimal for
$\theta_{n}\rightarrow-1^{+}$ and diverging to infinity for
$\theta_{n}\rightarrow 1^{-}$. We are now ready to state the following result
dedicated to the asymptotic behavior of the covariance matrix of the process.

######  Proposition 2.2.

Assume that (H1), (H2) (H21) and (H3) hold, and that
$\mathbb{E}[\varepsilon_{1}^{\,2}]=\sigma^{2}<+\infty$. Then, the empirical
covariance matrix $S_{n,\,n}=S_{n}$ given in (1.4) satisfies

$(1-\rho_{n})\,\frac{S_{n}}{n}~{}\overset{p}{\longrightarrow}~{}\frac{\sigma^{2}}{2}\,\textnormal{vec}^{-1}\big{(}(A^{*}\otimes
A^{*})\,e_{p^{2}}\big{)}\,\overset{\Delta}{=}\,\Gamma\hskip
14.22636pt\text{with}\hskip 14.22636ptA^{*}=P\,K_{p}\,P^{-1}$

where $\textnormal{vec}^{-1}:\mathbb{R}^{p^{2}}\rightarrow\mathbb{R}^{p\times
p}$ is the vectorization inverse operator and the invertible matrix $P$ is the
limit of $P_{n}$ in (2.1). In addition,

$(1-\rho_{n})\left\|\frac{S_{n}}{n}-\Gamma_{n}\right\|~{}\overset{p}{\longrightarrow}~{}0\hskip
14.22636pt\text{where}\hskip
14.22636pt\Gamma_{n}=\Gamma_{n}(0)=\sigma^{2}\sum_{\ell\,\geqslant\,0}A_{n}^{\ell}\,K_{p}\,(A_{n}^{\,T})^{\ell}$

is the covariance of the stationary process (at fixed $n$).

###### Proof.

See Section 3.2. ∎

######  Remark 2.1.

For the empirical covariance of the process to be consistent, it must be
renormalized by $1-\rho_{n}$. In the stable case (stationary and ergodic), it
is well-known that $S_{n}$ converges at rate $n$ whereas in the unstable case
(integrated with one unit root, either positive or negative), $S_{n}$
converges at rate $n^{2}$. Let $p=1$ to simplify. Then, under suitable
hypotheses, in the stable (left-hand side) and unstable (right-hand side)
cases, either

$\frac{S_{n}}{n}~{}\overset{a.s.}{\longrightarrow}~{}\gamma_{0}\hskip
14.22636pt\text{or}\hskip
14.22636pt\frac{S_{n}}{n^{2}}~{}\overset{d}{\longrightarrow}~{}\sigma^{2}\\!\int_{0}^{1}W^{2}(t)\,\mathrm{d}t$

where $\gamma_{0}>0$ is the variance of the stationary process and
$(W(t),\,t\in[0,1])$ is a standard Wiener process. Proposition 2.2 establishes
the convergence of $S_{n,\,n}$ at rate $n/(1-\rho_{n})\propto n^{1+\alpha}$
for $v_{n}=n^{\alpha}$ and $0<\alpha<1$. One can see that our model bridges
the stable and unstable cases in terms of empirical covariance even if the
limit behaviors as $\alpha\rightarrow 0^{+}$ and $\alpha\rightarrow 1^{-}$ do
not make a connection with $\alpha=0$ and $\alpha=1$.

Even if it is instructive to control the behavior of the covariance matrix in
a stationary process, that result shall not help to prove the following
theorems, dedicated to the OLS estimator, since the limit matrix is not
invertible as soon as $p>1$. Indeed a direct calculation shows that
$(A^{*}\otimes A^{*})\,e_{p^{2}}$ is a column vector of size $p^{2}$ with
entries $\pm\pi^{\,2}_{11}$ and different rates of convergence must be used to
achieve our objectives.

###  2.2. OLS estimation

In view of the above, consider the $p\times p$ diagonal matrix defined as

(2.5) $V_{n}=\textnormal{diag}(1-\rho_{n},\,1,\,\ldots,\,1)$

together with the $p^{2}\times p^{2}$ matrix given, for $n>n_{0}$, by

(2.6) $W_{n}=(V_{n}^{\,1/2}P_{n}^{-1})\otimes(V_{n}^{\,1/2}P_{n}^{-1})$

where $P_{n}$ is the (invertible) basis of eigenvectors of $A_{n}$.

######  Theorem 2.3.

Assume that (H1), (H2) (H21) and (H3) hold, and that
$\mathbb{E}[\varepsilon_{1}^{\,2}]=\sigma^{2}<+\infty$. Then we have the
consistency

$\big{\|}\widehat{\theta}_{n}-\theta_{n}\big{\|}~{}\overset{p}{\longrightarrow}~{}0$

where $\widehat{\theta}_{n}$ is the OLS estimator (1.4) in the nearly unstable
AR$(p)$ process (1.2).

###### Proof.

See Section 3.3. ∎

For the asymptotic normality, we need a slightly stronger hypothesis on the
noise $(\varepsilon_{k})$. Let us assume from now on that there exists a
moment of order strictly greater than 2.

######  Theorem 2.4.

Assume that (H1), (H2) (H21) and (H3) hold, and that
$\mathbb{E}[|\varepsilon_{1}|^{\,2+\nu}]=\eta_{\nu}<+\infty$ for some $\nu>0$.
Then, if the eigenvalues of $A$ are real, we have the asymptotic normality

$\sqrt{n}\,V_{n}^{-1/2}P_{n}^{T}\,\big{(}\widehat{\theta}_{n}-\theta_{n}\big{)}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}_{p}(0,\,H_{0}^{-1})$

where $\widehat{\theta}_{n}$ is the OLS estimator (1.4) in the nearly unstable
AR$(p)$ process (1.2), the matrix of rates $V_{n}$ is given in (2.5), $P_{n}$
is given in (2.1), and the standardized positive definite precision matrix
$H_{0}$ is given in (2.2). If some eigenvalues of $A$ are complex, we have the
asymptotic normality

$\sqrt{n\,v_{n}}\,\big{\langle}L_{n},\,\widehat{\theta}_{n}-\theta_{n}\big{\rangle}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}(0,\,h_{0}^{2})$

where

$L_{n}=\left(1,\frac{1}{\lambda_{1}\rho_{n}},\ldots,\frac{1}{(\lambda_{1}\rho_{n})^{p-1}}\right)^{\,T}\hskip
14.22636pt\text{and}\hskip 14.22636pth_{0}^{2}=\frac{2c}{\pi_{11}^{2}}>0.$

###### Proof.

See Section 3.4. ∎

######  Corollary 2.5.

Under the same assumptions as in Theorem 2.4 with $p=1$, we have the
asymptotic normality

$\sqrt{n\,v_{n}}\,\big{(}\widehat{\theta}_{n}-\theta_{n}\big{)}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}(0,\,2c).$

###### Proof.

Just take $V_{n}=1-\rho_{n}=c/v_{n}$, $P_{n}=1$ and $H_{0}=1/2$ in Theorem
2.4. ∎

One can note that this is precisely the statement of Thm. 3.2(c) of Phillips
and Magdalinos (2007) when the unit root is positive ($\lambda_{1}=1$).
Nevertheless, our result also holds for a negative unit root
($\lambda_{1}=-1$) whereas in the reference just mentioned, an extra symmetry
argument is needed on the distribution of $(\varepsilon_{k})$. We may
therefore say that Theorem 2.4 reinforces and extends the result to the
general near-stationary setting.

######  Remark 2.2.

On the basis of Theorem 2.3, the triangle inequality directly implies that,
under the same hypotheses,

(2.7) $\widehat{\theta}_{n}~{}\overset{p}{\longrightarrow}~{}\theta.$

But it is important to note that Theorem 2.4 cannot lead to the corresponding
asymptotic normality. To understand this, consider for example the univariate
setting of Corollary 2.5 where $\theta_{n}-\theta\propto 1/v_{n}$. Then,

$\sqrt{n\,v_{n}}\,\big{|}\theta_{n}-\theta\big{|}\propto\sqrt{\frac{n}{v_{n}}}~{}\longrightarrow~{}+\infty.$

However, this is not surprising. Indeed, the limit AR process generated by
$\theta$ is unstable. In that case, the OLS estimate is still consistent but
not asymptotically normal.

In terms of estimation, the mixing induced by $P_{n}^{T}$ is somewhat
troublesome because it relies on the true and unknown eigenvalues of $A_{n}$.
Since the eigenvalues of a square matrix depend continuously on its entries,
see e.g. Thm. 2.4.9.2 of Horn and Johnson (2012), the consistency of
$\widehat{\theta}_{n}$ would justify replacing $A_{n}$ by $\widehat{A}_{n}$
before calculating its spectrum, so as to get consistent estimates of all
quantities appearing in Theorem 2.4 and build hypotheses tests. For
simulations (Section 4), the following corollary will be used to illustrate
the latter result.

######  Corollary 2.6.

Assume that the assumptions of Theorem 2.4 hold. Then, for $\lambda_{1}=\pm
1$, we have the asymptotic distribution

$\frac{\pi_{11}^{\,2}\,n\,v_{n}}{2\,c}\,\left[\sum_{i=1}^{p}\frac{1}{(\lambda_{1}\rho_{n})^{\,i-1}}\,\big{(}\widehat{\theta}_{n,\,i}-\theta_{n,\,i}\big{)}\right]^{\,2}~{}\overset{d}{\longrightarrow}~{}\chi^{2}_{1}.$

###### Proof.

This is in fact the first marginal convergence of the asymptotic normality
combined with the continuous mapping theorem, (2.2) and the fact that the
first column of $P_{n}$ is
$(1,1/\lambda_{n,\,1},\ldots,1/\lambda_{n,\,1}^{p-1})$. ∎

Before switching to the proofs of the results, some examples are provided. The
goal is to get a clear insight into the structure of the fundamental matrix
$H_{0}$ in simple cases.

######  Examples.

Let us consider the examples $p\in\\{2,3,4\\}$ with $\lambda_{1}=\pm 1$ (see
Corollary 2.5 for $p=1$). In fact, $H_{0}$ has a simple form deduced from the
expressions

$\forall\,1\leqslant i\leqslant p,\hskip 14.22636ptf_{i}=\prod_{j\neq
i}(\lambda_{j}-\lambda_{i}).$

  * •

For $p=2$ and $\textnormal{sp}(A)=\\{\pm 1,\lambda_{2}\\}$ with
$0<|\lambda_{2}|<1$,

$H_{0}=\begin{pmatrix}\frac{\lambda_{1}^{2}}{2\,f_{1}^{\,2}}&0\\\
0&\frac{\lambda_{2}^{2}}{f_{2}^{\,2}\,(1-\lambda_{2}^{2})}\end{pmatrix}.$

  * •

For $p=3$ and $\textnormal{sp}(A)=\\{\pm 1,\lambda_{2},\lambda_{3}\\}$ with
$0<|\lambda_{3}|\leqslant|\lambda_{2}|<1$ and $\lambda_{2}\neq\lambda_{3}$,

$H_{0}=\begin{pmatrix}\frac{\lambda_{1}^{4}}{2\,f_{1}^{\,2}}&0&0\\\
0&\frac{\lambda_{2}^{4}}{f_{2}^{\,2}\,(1-\lambda_{2}^{2})}&\frac{\lambda_{2}^{2}\,\lambda_{3}^{2}}{f_{2}\,f_{3}\,(1-\lambda_{2}\lambda_{3})}\\\
0&\frac{\lambda_{3}^{2}\,\lambda_{2}^{2}}{f_{3}\,f_{2}\,(1-\lambda_{3}\lambda_{2})}&\frac{\lambda_{3}^{4}}{f_{3}^{\,2}\,(1-\lambda_{3}^{2})}\end{pmatrix}.$

  * •

For $p=4$ and $\textnormal{sp}(A)=\\{\pm
1,\lambda_{2},\lambda_{3},\lambda_{4}\\}$ with
$0<|\lambda_{4}|\leqslant|\lambda_{3}|\leqslant|\lambda_{2}|<1$ and
$\lambda_{2}\neq\lambda_{3}\neq\lambda_{4}$,

$H_{0}=\begin{pmatrix}\frac{\lambda_{1}^{6}}{2\,f_{1}^{\,2}}&0&0&0\\\
0&\frac{\lambda_{2}^{6}}{f_{2}^{\,2}\,(1-\lambda_{2}^{2})}&\frac{\lambda_{2}^{3}\,\lambda_{3}^{3}}{f_{2}\,f_{3}\,(1-\lambda_{2}\lambda_{3})}&\frac{\lambda_{2}^{3}\,\lambda_{4}^{3}}{f_{2}\,f_{4}\,(1-\lambda_{2}\lambda_{4})}\\\
0&\frac{\lambda_{3}^{3}\,\lambda_{2}^{3}}{f_{3}\,f_{2}\,(1-\lambda_{3}\lambda_{2})}&\frac{\lambda_{3}^{6}}{f_{3}^{\,2}\,(1-\lambda_{3}^{2})}&\frac{\lambda_{3}^{3}\,\lambda_{4}^{3}}{f_{3}\,f_{4}\,(1-\lambda_{3}\lambda_{4})}\\\
0&\frac{\lambda_{4}^{3}\,\lambda_{2}^{3}}{f_{4}\,f_{2}\,(1-\lambda_{4}\lambda_{2})}&\frac{\lambda_{4}^{3}\,\lambda_{3}^{3}}{f_{4}\,f_{3}\,(1-\lambda_{4}\lambda_{3})}&\frac{\lambda_{4}^{6}}{f_{4}^{\,2}\,(1-\lambda_{4}^{2})}\end{pmatrix}.$

As mentioned at the beginning of the section, Proposition 2.2 and Theorems 2.3
and 2.4 are true under (H2) (H22), i.e. when $A$ contains the eigenvalues $1$
and $-1$ ordered this way (according to lexicographic descending order). That
could correspond to a situation in which $\lambda_{n,\,1}=1-c/v_{n}$ and
$\lambda_{n,\,2}=-1+d/w_{n}$ with $c,\,d>0$, $1\ll v_{n},w_{n}\ll n$ and
$c/v_{n}\leqslant d/w_{n}$ (otherwise switching $\lambda_{n,\,1}$ and
$\lambda_{n,\,2}$ in $P_{n}$ and $D_{n}$). More details will be given in due
course (Remarks 3.1–3.4). Let us now prove our results.

##  3\. Technical tools

In this section, $C_{st}$ denotes a generic positive constant that is not
necessarily identical from one line to another and we use the conventions
$\sum_{\varnothing}=0$ and $\Pi_{\varnothing}=1$. Let us also define a
fundamental matrix for the reasoning to come,

(3.1) $B_{n}=I_{p^{2}}-A_{n}\otimes A_{n}.$

In all the sequel, $\|\cdot\|$ will refer to the Frobenius norm
$\|\cdot\|_{F}=\|\textnormal{vec}(\cdot)\|_{2}$ induced by the inner product
$\llangle\cdot,\cdot\rrangle=\llangle\cdot,\cdot\rrangle_{F}=\langle\textnormal{vec}(\cdot),\textnormal{vec}(\cdot)\rangle$.
The distinction is not made between matrices and vectors for which we simply
have $\|\cdot\|_{F}=\|\cdot\|_{2}$ and
$\llangle\cdot,\cdot\rrangle_{F}=\langle\cdot,\cdot\rangle$.

###  3.1. Linear Algebra

This section gathers all the linear algebra tools that we shall need in the
proofs of our results.

######  Lemma 3.1.

Assume that (H1) holds. Then, there exists $n_{0}\geqslant 1$ such that, for
all $n>n_{0}$, $A_{n}$ is diagonalizable in the form
$A_{n}=P_{n}\,D_{n}\,P_{n}^{-1}$ with
$D_{n}=\textnormal{diag}(\lambda_{n,\,1},\,\ldots,\,\lambda_{n,\,p})$
containing ordered distinct eigenvalues
$1\,>\,|\lambda_{n,\,1}|\,\geqslant\,\ldots\,\geqslant\,|\lambda_{n,\,p}|\,>\,0$.
In addition, $\|P_{n}\|\leqslant C_{st}$ and $\|P_{n}^{-1}\|\leqslant C_{st}$.

###### Proof.

See Lem. 3.1 in Proïa (2020). ∎

######  Lemma 3.2.

Assume that (H1) holds. Then in the context of Lemma 3.1, for all $n>n_{0}$,
the first row of $P_{n}^{-1}$ is real.

###### Proof.

This is obvious for $p=1$, so let $p>1$. Suppose that, for $j\geqslant 2$,
$\lambda_{n,\,j}\in\textnormal{sp}(A_{n})$ is a complex eigenvalue so that,
since $A_{n}$ is real, we must also have
$\bar{\lambda}_{n,\,j}\in\textnormal{sp}(A_{n})$. Taking $i=1$ in formula
(3.4) of Proïa (2020), the first row of $P_{n}^{-1}$ is written

$\left(\frac{T_{n,\,1,\,k}\,\lambda_{n,\,1}^{p-1}\,\prod_{j=2}^{p}\lambda_{n,\,j}}{\prod_{j=2}^{p}(\lambda_{n,\,j}-\lambda_{n,\,1})}\right)_{1\,\leqslant\,k\,\leqslant\,p}$

where $T_{n,\,1,\,k}$ can be retrieved from the relation

$\prod_{j=2}^{p}\left(X-\frac{1}{\lambda_{n,\,j}}\right)=T_{n,\,1,\,1}+T_{n,\,1,\,2}\,X+\ldots+T_{n,\,1,\,p}\,X^{p-1}$

for real values of $X$. Combining such terms together with the fact that, for
any $x\in\mathbb{R}$ and for any $z\in\mathbb{C}$,

$(x-z)(x-\bar{z})=x^{2}+|z|^{2}-2\,\textnormal{Re}(z)x\,\in\mathbb{R}\hskip
14.22636pt\text{and}\hskip
14.22636pt\lambda_{n,\,1}=\pm\rho_{n}\in\mathbb{R},$

each entry of the first row of $P_{n}^{-1}$ is real. ∎

######  Lemma 3.3.

Assume that (H1) holds. Then, each element of the first column of $P^{-1}$ is
non-zero.

###### Proof.

We remind that the entries of the first column of $P^{-1}$ are denoted by
$\pi_{k1}$. Then, taking $j=1$ in the same formula as in the previous proof,

(3.2)
$\pi_{k1}=\frac{(-\lambda_{k})^{p-1}}{\prod_{\ell\,\neq\,k}(\lambda_{\ell}-\lambda_{k})}.$

Under (H1), all eigenvalues are distinct and non-zero, which concludes the
proof. ∎

######  Lemma 3.4.

Assume that (H3) holds. Then, for all $n\geqslant 1$, the matrix $B_{n}$ given
in (3.1) is invertible.

###### Proof.

Indeed, since $\rho(A_{n}\otimes A_{n})=\rho_{n}^{\,2}<1$ under (H3), the
conclusion follows e.g. from Cor. 5.6.16 of Horn and Johnson (2012). ∎

######  Lemma 3.5.

Assume that (H1), (H2) (H21) and (H3) hold. Then,

$\lim_{n\,\rightarrow\,+\infty}(1-\rho_{n})\,B_{n}^{-1}=\frac{1}{2}\,(A^{*}\otimes
A^{*})$

where $A^{*}$ is defined in Proposition 2.2.

###### Proof.

Once again the result is obvious for $p=1$, so let $p>1$. Because
$\rho(A_{n}\otimes A_{n})<1$ (see the proof of Lemma 3.4),

$B_{n}^{-1}=\sum_{k=0}^{+\infty}A_{n}^{k}\otimes A_{n}^{k}.$

For $n>n_{0}$, it follows from simple linear algebra that

(3.3) $B_{n}^{-1}=(P_{n}\otimes
P_{n})\,\left[\sum_{k=0}^{+\infty}D_{n}^{k}\otimes
D_{n}^{k}\right](P_{n}^{-1}\otimes P_{n}^{-1}).$

From Lem. 3.1 of Proïa (2020), we know that $P_{n}\rightarrow P$ where the
limit matrix $P$ is invertible under (H1), as a Vandermonde matrix with
distinct entries. Thus we also have $P_{n}^{-1}\rightarrow P^{-1}$. It remains
to observe that $D_{n}^{k}\otimes D_{n}^{k}$ is a $p^{2}\times p^{2}$ diagonal
matrix with top-left element $(\pm\rho_{n})^{\,2k}$ and all other elements
given either by $(\pm\rho_{n})^{\,k}\,\lambda^{k}_{n,\,i}$, depending on
whether $\lambda_{1}=1$ or $\lambda_{1}=-1$, or by
$\lambda^{k}_{n,\,i}\,\lambda^{k}_{n,\,j}$ for $2\leqslant i,j\leqslant p$.
Under (H3),

(3.4)
$\lim_{n\,\rightarrow\,+\infty}(1-\rho_{n})\,\sum_{k=0}^{+\infty}(\pm\rho_{n})^{\,2k}=\frac{1}{2}$

and, for $2\leqslant i,j\leqslant p$,

(3.5)
$\lim_{n\,\rightarrow\,+\infty}\sum_{k=0}^{+\infty}(\pm\rho_{n})^{\,k}\,\lambda^{k}_{n,\,i}=\frac{1}{1\mp\lambda_{i}}\hskip
14.22636pt\text{and}\hskip
14.22636pt\lim_{n\,\rightarrow\,+\infty}\sum_{k=0}^{+\infty}\lambda^{k}_{n,\,i}\,\lambda^{k}_{n,\,j}=\frac{1}{1-\lambda_{i}\lambda_{j}}.$

Under (H2) (H21), we can conclude that

(3.6)
$\lim_{n\,\rightarrow\,+\infty}(1-\rho_{n})\,B_{n}^{-1}=\frac{1}{2}\,(P\otimes
P)\,K_{p^{2}}\,(P^{-1}\otimes P^{-1})$

where $K_{p^{2}}$ is given in (2.3). In a more ‘elegant’ way,

$\frac{1}{2}\,(P\otimes P)\,K_{p^{2}}\,(P^{-1}\otimes
P^{-1})=\frac{1}{2}\,(P\,K_{p}\,P^{-1})\otimes(P\,K_{p}\,P^{-1})$

using the fact that $K_{p^{2}}=K_{p}\otimes K_{p}$, which concludes the proof.
∎

######  Remark 3.1.

If (H2) (H22) is true instead of (H2) (H21), the lemma still holds and the
reasoning is similar but $K_{p^{2}}$ in (3.6) must be replaced by the matrix
of size $p^{2}\times p^{2}$ having the diagonal block structure

$K^{*}_{p^{2}}=\begin{pmatrix}K_{p}&&\\\ &\bar{K}_{p}&&\\\
&&0&\end{pmatrix}\hskip 14.22636pt\text{with}\hskip
14.22636pt\bar{K}_{p}=\textnormal{diag}(0,1,0,\ldots,0).$

######  Lemma 3.6.

Assume that (H1) and (H2) (H21) hold, and that $p>1$. Then, the
$(p-1)\times(p-1)$ bottom-right block of $H_{0}$, which is the symmetric
matrix given by

(3.7)
$\Lambda=\begin{pmatrix}\frac{\pi_{21}^{2}}{1-\lambda_{2}^{2}}&\ldots&\frac{\pi_{21}\pi_{p1}}{1-\lambda_{2}\lambda_{p}}\\\
\vdots&\ddots&\vdots\\\
\frac{\pi_{p1}\pi_{21}}{1-\lambda_{p}\lambda_{2}}&\ldots&\frac{\pi_{p1}^{2}}{1-\lambda_{p}^{2}}\end{pmatrix},$

is invertible.

###### Proof.

First under (H1), Lemma 3.3 implies $\pi_{k1}\neq 0$ for all $k$, so the
result is obvious for $p=2$. Now let $p>2$ and denote by $\Lambda_{k}$ the
$(k-1)\times(k-1)$ top-left submatrix of $\Lambda$ (so that
$\Lambda_{p}=\Lambda$) and note that, although it is not a Cauchy matrix, it
is closely related to it (see Sec. 0.9.12 of Horn and Johnson (2012)). Let us
adopt the usual reasoning for calculating the determinant of such matrices.
Multiplying the $(j-1)$-th column by
$(1-\lambda_{j}\lambda_{k})/(\pi_{j1}\pi_{k1})$ for $2\leqslant j\leqslant k$,
and subtracting the last column from each other column, it follows that

$\displaystyle d_{k}$ $\displaystyle=$
$\displaystyle\prod_{j=2}^{k}\frac{\pi_{j1}\pi_{k1}}{(1-\lambda_{j}\lambda_{k})}$
$\displaystyle\hskip 14.22636pt\hskip
14.22636pt\times\begin{vmatrix}\frac{b_{2}(\lambda_{2}-\lambda_{k})^{2}}{(1-\lambda_{2}^{2})(1-\lambda_{2}\lambda_{k})}&\frac{b_{2}(\lambda_{2}-\lambda_{k})(\lambda_{3}-\lambda_{k})}{(1-\lambda_{2}\lambda_{3})(1-\lambda_{2}\lambda_{k})}&\ldots&\frac{b_{2}(\lambda_{2}-\lambda_{k})(\lambda_{k-1}-\lambda_{k})}{(1-\lambda_{2}\lambda_{k-1})(1-\lambda_{2}\lambda_{k})}&*\\\
\frac{b_{3}(\lambda_{3}-\lambda_{k})(\lambda_{2}-\lambda_{k})}{(1-\lambda_{3}\lambda_{2})(1-\lambda_{3}\lambda_{k})}&\frac{b_{3}(\lambda_{3}-\lambda_{k})^{2}}{(1-\lambda_{3}^{2})(1-\lambda_{3}\lambda_{k})}&\ldots&\frac{b_{3}(\lambda_{3}-\lambda_{k})(\lambda_{k-1}-\lambda_{k})}{(1-\lambda_{3}\lambda_{k-1})(1-\lambda_{3}\lambda_{k})}&*\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\frac{b_{k-1}(\lambda_{k-1}-\lambda_{k})(\lambda_{2}-\lambda_{k})}{(1-\lambda_{k-1}\lambda_{2})(1-\lambda_{k-1}\lambda_{k})}&\frac{b_{k-1}(\lambda_{k-1}-\lambda_{k})(\lambda_{3}-\lambda_{k})}{(1-\lambda_{k-1}\lambda_{3})(1-\lambda_{k-1}\lambda_{k})}&\ldots&\frac{b_{k-1}(\lambda_{k-1}-\lambda_{k})^{2}}{(1-\lambda_{k-1}^{2})(1-\lambda_{k-1}\lambda_{k})}&*\\\
0&0&\ldots&0&1\end{vmatrix}$

where $d_{k}=\det(\Lambda_{k})$,
$b_{i}=(\pi_{i1}\pi_{j1})/(\pi_{j1}\pi_{k1})=\pi_{i1}/\pi_{k1}$ ($j$ stands
for the column), and $*$ symbolizes useless entries for $d_{k}$. Thus,
factorizing everything that can be factorized yields, for every $3\leqslant
k\leqslant p$,

$d_{k}=\frac{\pi_{k1}^{2}}{1-\lambda_{k}^{2}}\left(\prod_{j=2}^{k-1}\frac{(\lambda_{j}-\lambda_{k})^{2}}{(1-\lambda_{j}\lambda_{k})^{2}}\right)d_{k-1}\hskip
14.22636pt\text{with}\hskip
14.22636ptd_{2}=\frac{\pi_{21}^{2}}{1-\lambda_{2}^{2}}.$

The solution of the recurrence is given by

(3.8) $\forall\,2\leqslant k\leqslant p,\hskip
14.22636ptd_{k}=\left(\prod_{j=2}^{k}\frac{\pi_{j1}^{2}}{1-\lambda_{j}^{2}}\right)\left(\prod_{j=2}^{k}\prod_{i\,<\,j}\frac{(\lambda_{i}-\lambda_{j})^{2}}{(1-\lambda_{i}\lambda_{j})^{2}}\right).$

If (H1) holds, then all eigenvalues are distinct and since $\pi_{j1}$ is non-
zero for all $j$, $d_{k}$ is non-zero for all $k$. Taking $k=p$, the proof is
now complete. If $A$ has only real eigenvalues, we can even show that, under
(H1), $d_{k}>0$ for all $k$ which implies the positive definiteness of
$\Lambda$ using Sylvester’s criterion, see e.g. Thm. 7.2.5 of Horn and Johnson
(2012). ∎

######  Remark 3.2.

Under (H2) (H22) and for $p>2$, we only have to consider the
$(p-2)\times(p-2)$ bottom-right submatrix of $\Lambda$, which is obviously
invertible (and positive definite when $A$ has only real eigenvalues), based
on the above.

###  3.2. Proof of Propositions 2.1 and 2.2

Let us start by noting that, under Lemma 3.1 and (H3), for a fixed $n>n_{0}$,

$\displaystyle\|\Gamma_{n}(0)\|$ $\displaystyle\leqslant$ $\displaystyle
C_{st}\sum_{\ell\,\geqslant\,0}\|A_{n}^{\ell}\,K_{p}\,(A_{n}^{\,T})^{\ell}\|$
$\displaystyle\leqslant$ $\displaystyle
C_{st}\sum_{\ell\,\geqslant\,0}\|D_{n}^{\ell}\|^{2}\,\leqslant\,\frac{C_{st}}{1-\rho_{n}^{2}}\,<\,+\infty$

where $\Gamma_{n}(\cdot)$ is the autocovariance function given in (2.4). Thus,
by the same reasoning,

$\displaystyle\left\|\sum_{h\,\in\,\mathbb{Z}}\Gamma_{n}(h)\right\|\,\leqslant\,\sum_{h\,\in\,\mathbb{Z}}\|\Gamma_{n}(h)\|$
$\displaystyle\leqslant$ $\displaystyle
C_{st}\,\|\Gamma_{n}(0)\|\sum_{h\,\in\,\mathbb{Z}}\|A_{n}^{|h|}\|$
$\displaystyle\leqslant$
$\displaystyle\frac{C_{st}}{1-\rho_{n}^{2}}\sum_{h\,\in\,\mathbb{Z}}\|D_{n}^{|h|}\|\,\leqslant\,\frac{C_{st}}{(1-\rho_{n})^{2}}\,<\,+\infty.$

Moreover, as we will see later in (3.22), there exists a positive semi-
definite matrix $\Gamma$ such that
$(1-\rho_{n})\Gamma_{n}(0)\rightarrow\Gamma$. Hence, since
$\Gamma_{n}(-h)=\Gamma_{n}^{\,T}(h)$,

(3.9)
$(1-\rho_{n})^{2}\sum_{h\,\in\,\mathbb{Z}}\Gamma_{n}(h)~{}\longrightarrow~{}A^{*}\Gamma+(A^{*}\Gamma)^{\,T}\,\overset{\Delta}{=}\,\Gamma^{+}$

under (H2) (H21) with $\lambda_{1}=1$ or (H2) (H22), where
$A^{*}=P\,K_{p}\,P^{-1}$ and $K_{p}$ is given in (2.3). The top-left element
of $\Gamma^{+}$ is $\sigma^{2}\pi_{11}^{2}\neq 0$ (take e.g. Lemma 3.3 with
$k=1$). Under (H2) (H21) with $\lambda_{1}=-1$, by the same calculations, it
turns out that

(3.10) $\sum_{h\,\in\,\mathbb{Z}}\Gamma_{n}(h)~{}\longrightarrow~{}\Gamma^{-}$

where $\Gamma^{-}$ is also a non-zero matrix. The result is established by
combining (3.9) and (3.10). ∎

As for the proof of Proposition 2.2, it will result from a chain of
intermediate lemmas. See footnote 2 on page 2 for notation.

######  Lemma 3.7 (Variance decomposition).

Assume that (H3) holds. Then, for all $n\geqslant 1$,

(3.11) $\displaystyle\textnormal{vec}\\!\left(\frac{S_{n,\,n-1}}{n}\right)$
$\displaystyle=$ $\displaystyle
B_{n}^{-1}\,\textnormal{vec}\\!\left(\frac{T_{n}}{n}\right)+B_{n}^{-1}\,(I_{p}\otimes
A_{n})\,\textnormal{vec}\\!\left(\frac{Z_{n}}{n}\right)$ $\displaystyle\hskip
14.22636pt\hskip 14.22636pt+~{}B_{n}^{-1}\,(A_{n}\otimes
I_{p})\,\textnormal{vec}\\!\left(\frac{Z_{n}^{\,T}}{n}\right)+B_{n}^{-1}\,\textnormal{vec}\\!\left(\frac{L_{n}}{n}\right)$

where $S_{n,\,n-1}$ is given in (1.4), $B_{n}$ is given in (3.1),

(3.12) $\hskip
14.22636ptZ_{n}=\sum_{k=1}^{n}\Phi_{n,\,k-1}\,E_{k}^{\,T},\hskip
14.22636ptL_{n}=\sum_{k=1}^{n}E_{k}\,E_{k}^{\,T}$

and $T_{n}=\Phi_{n,\,0}\,\Phi_{n,\,0}^{\,T}-\Phi_{n,\,n}\,\Phi_{n,\,n}^{\,T}$.

###### Proof.

By direct calculation, we first obtain that, for all $n\geqslant 1$,

$\Phi_{n,\,k}\,\Phi_{n,\,k}^{\,T}=A_{n}\,\Phi_{n,\,k-1}\,\Phi_{n,\,k-1}^{\,T}\,A_{n}^{\,T}+A_{n}\,\Phi_{n,\,k-1}\,E_{k}^{\,T}+E_{k}\,\Phi_{n,\,k-1}^{\,T}\,A_{n}^{\,T}+E_{k}\,E_{k}^{\,T}.$

Then, summing over $k$, it is not hard to see that

$S_{n,\,n-1}=T_{n}+A_{n}\,S_{n,\,n-1}\,A_{n}^{\,T}+A_{n}\,Z_{n}+Z_{n}^{\,T}A_{n}^{\,T}+L_{n}.$

This is a generalized Sylvester equation w.r.t. $S_{n,\,n-1}$ having a unique
solution since $\rho_{n}<1$, see e.g. Lem. 2.1(2) of Jiang and Wei (2003). We
easily deduce that

$B_{n}\,\textnormal{vec}(S_{n,\,n-1})=\textnormal{vec}(T_{n})+(I_{p}\otimes
A_{n})\,\textnormal{vec}(Z_{n})+(A_{n}\otimes
I_{p})\,\textnormal{vec}(Z_{n}^{\,T})+\textnormal{vec}(L_{n})$

which, via Lemma 3.4, gives the result. ∎

######  Lemma 3.8 (Isolated terms).

Assume that (H1) and (H3) hold. Then,

$\frac{T_{n}}{n}~{}\overset{p}{\longrightarrow}~{}0$

where $T_{n}$ is given in (3.12).

###### Proof.

First, since $\Phi_{n,\,0}$ has a finite moment of order 2, Markov’s
inequality directly gives

(3.13)
$\frac{\|\Phi_{n,\,0}\,\Phi_{n,\,0}^{\,T}\|}{n}~{}\overset{p}{\longrightarrow}~{}0.$

Then, by the triangle inequality and exploiting Lemma 3.1, for all $n>n_{0}$,

$\displaystyle\|\Phi_{n,\,n}\,\Phi_{n,\,n}^{\,T}\|$ $\displaystyle\leqslant$
$\displaystyle
2\,\|A_{n}^{n}\|^{2}\,\|\Phi_{n,\,0}\|^{2}+2\,\left\|\sum_{k=0}^{n-1}A_{n}^{k}\,E_{n-k}\right\|^{2}$
$\displaystyle\leqslant$ $\displaystyle
C_{st}\,\rho_{n}^{\,2n}\,\|\Phi_{n,\,0}\|^{2}+2\,\left\|\sum_{k=0}^{n-1}A_{n}^{k}\,E_{n-k}\right\|^{2}~{}\,\overset{\Delta}{=}\,~{}T_{1,\,n}+T_{2,\,n}.$

Note that, according to (H3),

$\displaystyle\rho_{n}^{\,2n}$ $\displaystyle=$
$\displaystyle\left(1-\frac{c}{v_{n}}\right)^{2n}$ $\displaystyle=$
$\displaystyle\exp\left(\frac{-2c\,n}{v_{n}}+o\left(\frac{n}{v_{n}}\right)\right)~{}\longrightarrow~{}0.$

We can conclude that $\mathbb{E}[|T_{1,\,n}|]\rightarrow 0$, and thus

(3.14) $\frac{T_{1,\,n}}{n}~{}\overset{p}{\longrightarrow}~{}0.$

Moreover,

$\displaystyle\left\|\sum_{k=0}^{n-1}A_{n}^{k}\,E_{n-k}\right\|^{2}$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{n-1}\sum_{\ell=0}^{n-1}\llangle
P_{n}\,D_{n}^{k}\,P_{n}^{-1}\,E_{n-k},\,P_{n}\,D_{n}^{\ell}\,P_{n}^{-1}\,E_{n-\ell}\rrangle$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}\sum_{\ell=0}^{n-1}\textnormal{tr}\big{(}P_{n}\,D_{n}^{\ell}\,P_{n}^{-1}\,E_{n-\ell}\,E_{n-k}^{\,T}\,(P_{n}^{-1})^{\,T}\,D_{n}^{k}\,P_{n}^{\,T}\big{)}.$

Consequently,

$\displaystyle\mathbb{E}[|T_{2,\,n}|]$ $\displaystyle=$ $\displaystyle
2\,\sigma^{2}\,\sum_{k=0}^{n-1}\textnormal{tr}\big{(}P_{n}\,D_{n}^{k}\,P_{n}^{-1}\,K_{p}\,(P_{n}^{-1})^{\,T}\,D_{n}^{k}\,P_{n}^{\,T}\big{)}$
$\displaystyle=$ $\displaystyle
C_{st}\,\sum_{k=0}^{n-1}\|P_{n}\,D_{n}^{k}\,P_{n}^{-1}\,K_{p}\|^{2}$

where $K_{p}$ is given in (2.3), and obviously satisfies $K_{p}=K_{p}^{\,T}$
and $K_{p}^{\,2}=K_{p}$. Hence, using the same arguments as before,

(3.15) $\displaystyle\mathbb{E}[|T_{2,\,n}|]$ $\displaystyle\leqslant$
$\displaystyle C_{st}\,\sum_{k=0}^{n-1}\rho_{n}^{\,2k}$ $\displaystyle=$
$\displaystyle
C_{st}\,v_{n}\,\frac{1-\exp\left(\frac{-2c\,n}{v_{n}}+o\left(\frac{n}{v_{n}}\right)\right)}{2c-\frac{c^{2}}{v_{n}}}~{}=~{}O(v_{n}).$

Since $v_{n}=o(n)$, that leads to

(3.16) $\frac{T_{2,\,n}}{n}~{}\overset{p}{\longrightarrow}~{}0.$

The combination of (3.13), (3.14) and (3.16) concludes the proof. ∎

######  Lemma 3.9 (Martingale terms).

Assume that (H1) and (H3) hold. Then,

$\frac{Z_{n}}{n}~{}\overset{p}{\longrightarrow}~{}0$

where $Z_{n}$ is given in (3.12).

###### Proof.

By the triangle inequality,

$\displaystyle\|Z_{n}\|$ $\displaystyle\leqslant$
$\displaystyle\|\Phi_{n,\,0}\,E_{1}^{\,T}\|+\left\|\sum_{k=2}^{n}A_{n}^{k-1}\,\Phi_{n,\,0}\,E_{k}^{\,T}\right\|+\left\|\sum_{k=2}^{n}\sum_{\ell=0}^{k-2}A_{n}^{\ell}\,E_{k-1-\ell}\,E_{k}^{\,T}\right\|$
$\displaystyle\,\overset{\Delta}{=}\,$ $\displaystyle
T_{0,\,n}+T_{1,\,n}+T_{2,\,n}.$

Clearly, by independence and existence of moments of order 2,

(3.17) $\frac{T_{0,\,n}}{n}~{}\overset{p}{\longrightarrow}~{}0.$

Then, a direct calculation shows that

$\displaystyle T_{1,\,n}^{\,2}$ $\displaystyle=$
$\displaystyle\sum_{k=2}^{n}\sum_{\ell=2}^{n}\llangle
A_{n}^{k-1}\,\Phi_{n,\,0}\,E_{k}^{\,T},\,A_{n}^{\ell-1}\,\Phi_{n,\,0}\,E_{\ell}^{\,T}\rrangle$
$\displaystyle=$
$\displaystyle\sum_{k=2}^{n}\sum_{\ell=2}^{n}\textnormal{tr}\big{(}\Phi_{n,\,0}^{\,T}\,(A_{n}^{\,T})^{\ell-1}\,A_{n}^{k-1}\,\Phi_{n,\,0}\,E_{k}^{\,T}E_{\ell}\big{)}$

exploiting the cyclic property of the trace. Hence, for any $n>n_{0}$,

$\displaystyle\mathbb{E}[T_{1,\,n}^{\,2}]$ $\displaystyle=$
$\displaystyle\sum_{k=2}^{n}\sum_{\ell=2}^{n}\textnormal{tr}\big{(}\mathbb{E}\big{[}\Phi_{n,\,0}^{\,T}\,(A_{n}^{\,T})^{\ell-1}\,A_{n}^{k-1}\,\Phi_{n,\,0}\big{]}\,\mathbb{E}[E_{k}^{\,T}E_{\ell}]\big{)}$
$\displaystyle=$
$\displaystyle\sigma^{2}\,\sum_{k=2}^{n}\mathbb{E}\big{[}\textnormal{tr}\big{(}\Phi_{n,\,0}^{\,T}\,(A_{n}^{k-1})^{\,T}\,A_{n}^{k-1}\,\Phi_{n,\,0}\big{)}\big{]}$
$\displaystyle=$
$\displaystyle\sigma^{2}\,\sum_{k=2}^{n}\mathbb{E}\big{[}\|A_{n}^{k-1}\,\Phi_{n,\,0}\|^{2}\big{]}$
$\displaystyle\leqslant$ $\displaystyle
C_{st}\,\sum_{k=2}^{n}\rho_{n}^{\,2(k-1)}\,\mathbb{E}[\|\Phi_{n,\,0}\|^{2}]~{}=~{}O(v_{n})$

using (3.15), Lemma 3.1 and the second-order moments of $\Phi_{n,\,0}$. Since
$v_{n}=o(n)$,

(3.18) $\frac{T_{1,\,n}^{\,2}}{n}~{}\overset{p}{\longrightarrow}~{}0\hskip
14.22636pt\text{so that}\hskip
14.22636pt\frac{T_{1,\,n}}{n}~{}\overset{p}{\longrightarrow}~{}0.$

To handle the last term of the decomposition, let, for any $k\geqslant 1$,

$U_{k}=\sum_{j=1}^{k}A_{n}^{k-j}\,E_{j}$

and note that

(3.19) $\forall\,k\neq\ell,\hskip
14.22636pt\mathbb{E}\big{[}U_{\ell-1}^{\,T}\,U_{k-1}\,E_{k}^{\,T}E_{\ell}\big{]}=0.$

Now,

$\displaystyle T_{2,\,n}^{\,2}$ $\displaystyle=$
$\displaystyle\left\|\sum_{k=2}^{n}U_{k-1}\,E_{k}^{\,T}\right\|^{2}$
$\displaystyle=$ $\displaystyle\sum_{k=2}^{n}\sum_{\ell=2}^{n}\llangle
U_{k-1}\,E_{k}^{\,T},\,U_{\ell-1}\,E_{\ell}^{\,T}\rrangle~{}=~{}\sum_{k=2}^{n}\sum_{\ell=2}^{n}\textnormal{tr}\big{(}U_{\ell-1}^{\,T}\,U_{k-1}\,E_{k}^{\,T}E_{\ell}\big{)}$

exploiting again the cyclic property of the trace. Thanks to relations (3.19),
it follows that, for $n>n_{0}$,

$\displaystyle\mathbb{E}[T_{2,\,n}^{\,2}]$ $\displaystyle=$
$\displaystyle\sigma^{2}\,\sum_{k=2}^{n}\textnormal{tr}\big{(}\mathbb{E}\big{[}U_{k-1}^{\,T}\,U_{k-1}\big{]}\big{)}$
$\displaystyle=$
$\displaystyle\sigma^{2}\,\sum_{k=2}^{n}\sum_{j=1}^{k-1}\sum_{i=1}^{k-1}\textnormal{tr}\big{(}(A_{n}^{\,T})^{k-1-i}\,A_{n}^{k-1-j}\,\mathbb{E}[E_{j}\,E_{i}^{\,T}]\big{)}$
$\displaystyle=$
$\displaystyle\sigma^{4}\,\sum_{k=2}^{n}\sum_{j=1}^{k-1}\textnormal{tr}\big{(}(A_{n}^{\,T})^{k-1-j}\,A_{n}^{k-1-j}\,K_{p}\big{)}=\sigma^{4}\,\sum_{k=2}^{n}\sum_{\ell=0}^{k-2}\|A_{n}^{\ell}\,K_{p}\|^{2}~{}=~{}O(nv_{n})$

where $K_{p}$ is given in (2.3), and where the rate $v_{n}$ is obtained via
the same lines as (3.15). Consequently,

(3.20) $\frac{T_{2,\,n}^{\,2}}{n^{2}}~{}\overset{p}{\longrightarrow}~{}0\hskip
14.22636pt\text{so that}\hskip
14.22636pt\frac{T_{2,\,n}}{n}~{}\overset{p}{\longrightarrow}~{}0.$

The combination of (3.17), (3.18) and (3.20) concludes the proof. ∎

Let us now return to the proof of Proposition 2.2. By the law of large
numbers, the last term of (3.11) is such that

(3.21) $\frac{L_{n}}{n}~{}\overset{p}{\longrightarrow}~{}\sigma^{2}\,K_{p}$

where $K_{p}$ is given in (2.3). By combining Lemmas 3.5, 3.7, 3.8 and 3.9,
convergence (3.21) and hypotheses (H1), (H2) (H21) and (H3), the first result
is proved. It is important to note that $A^{*}=P\,K_{p}\,P^{-1}$ is a real
matrix (see Lemma 3.2), so that the limit of the renormalized empirical
covariance is obviously real itself. For the second result, we have

(3.22) $\displaystyle(1-\rho_{n})\,\textnormal{vec}(\Gamma_{n})$
$\displaystyle=$
$\displaystyle\sigma^{2}\,(1-\rho_{n})\left[\sum_{k=0}^{+\infty}A_{n}^{k}\otimes
A_{n}^{k}\right]\textnormal{vec}(K_{p})$ $\displaystyle=$
$\displaystyle\sigma^{2}\left((1-\rho_{n})\,B_{n}^{-1}\right)e_{p^{2}}$
$\displaystyle\longrightarrow$
$\displaystyle\frac{\sigma^{2}}{2}\,(A^{*}\otimes
A^{*})e_{p^{2}}~{}=~{}\textnormal{vec}(\Gamma)$

from Lemma 3.5, where $\Gamma$ is defined in Proposition 2.2 and noting that
$\textnormal{vec}(K_{p})=e_{p^{2}}$. Finally,

$(1-\rho_{n})\left\|\frac{S_{n}}{n}-\Gamma_{n}\right\|\,\leqslant\,\left\|(1-\rho_{n})\,\frac{S_{n}}{n}-\Gamma\right\|+\big{\|}(1-\rho_{n})\,\Gamma_{n}-\Gamma\big{\|}~{}\overset{p}{\longrightarrow}~{}0$

exploiting the first part of the proposition and (3.22). The proof is now
complete. ∎

######  Remark 3.3.

If (H2) (H22) is true instead of (H2) (H21), the results of the proposition
still hold provided that the limit is adjusted (see Remark 3.1).

###  3.3. Proof of Theorem 2.3

Note that for all $n>n_{0}$, under (H1) and like in (3.3),

$W_{n}\,B_{n}^{-1}=(V_{n}^{\,1/2}\otimes
V_{n}^{\,1/2})\,\left[\sum_{k=0}^{+\infty}D_{n}^{k}\otimes
D_{n}^{k}\right](P_{n}^{-1}\otimes P_{n}^{-1})$

where $W_{n}$ comes from (2.6). The matrix of rates is structured in $p$
blocks as follows,

$V_{n}^{\,1/2}\otimes
V_{n}^{\,1/2}=\textnormal{diag}\big{(}\sqrt{1-\rho_{n}}\,V_{n}^{\,1/2},\,V_{n}^{\,1/2},\,\ldots,\,V_{n}^{\,1/2}\big{)}.$

Thus, taking over the reasoning of (3.4) and (3.5), we can show that

(3.23)
$\lim_{n\,\rightarrow\,+\infty}W_{n}\,B_{n}^{-1}=\textnormal{diag}\left(\frac{K_{p}}{2},\,\Delta_{2},\,\ldots,\,\Delta_{p}\right)(P^{-1}\otimes
P^{-1})\,\overset{\Delta}{=}\,H^{*}$

using (H2) (H21) and (H3), where $K_{p}$ is given in (2.3) and

$\forall\,2\leqslant i\leqslant p,\hskip
14.22636pt\Delta_{i}=\textnormal{diag}\left(0,\,\frac{1}{1-\lambda_{i}\,\lambda_{2}},\,\ldots,\,\frac{1}{1-\lambda_{i}\,\lambda_{p}}\right).$

That leads, together with Lemmas 3.7, 3.8, 3.9 and formula (3.21), to the
convergence

$W_{n}\,\textnormal{vec}\\!\left(\frac{S_{n,\,n-1}}{n}\right)~{}\overset{p}{\longrightarrow}~{}\sigma^{2}H^{*}e_{p^{2}}.$

Since

$W_{n}\,\textnormal{vec}\\!\left(\frac{S_{n,\,n-1}}{n}\right)=\textnormal{vec}\\!\left(V_{n}^{\,1/2}\,P_{n}^{-1}\,\frac{S_{n,\,n-1}}{n}\,(P_{n}^{-1})^{\,T}\,V_{n}^{\,1/2}\right),$

we obtain that

(3.24)
$V_{n}^{\,1/2}\,P_{n}^{-1}\,\frac{S_{n,\,n-1}}{n}\,(P_{n}^{-1})^{\,T}\,V_{n}^{\,1/2}~{}\overset{p}{\longrightarrow}~{}\textnormal{vec}^{-1}(\sigma^{2}H^{*}e_{p^{2}})$

where $\textnormal{vec}^{-1}:\mathbb{C}^{p^{2}}\rightarrow\mathbb{C}^{p\times
p}$ is the vectorization inverse operator. A straightforward calculation shows
that $\textnormal{vec}^{-1}(\sigma^{2}H^{*}e_{p^{2}})=H$ where $H$ is given in
(2.2), and this limit is invertible. Indeed, on the one hand,

$\pi_{11}=\frac{(-\lambda_{1})^{p-1}}{\prod_{j=2}^{p}(\lambda_{j}-\lambda_{1})}$

from formula (3.2) with $k=1$, and we know from Lemma 3.2 that $\pi_{11}$ is
real and non-zero (positive). On the other hand, Lemma 3.6 implies that
$\Lambda$ is invertible so that $\det(H)$ is obviously non-zero. In addition,
the estimation error satisfies

(3.25) $\displaystyle\widehat{\theta}_{n}-\theta_{n}$ $\displaystyle=$
$\displaystyle(P_{n}^{\,T})^{-1}\,V_{n}^{\,1/2}\,\left(V_{n}^{\,1/2}\,P_{n}^{-1}\,\frac{S_{n,\,n-1}}{n}\,(P_{n}^{-1})^{\,T}\,V_{n}^{\,1/2}\right)^{-1}$
$\displaystyle\hskip 14.22636pt\hskip
14.22636pt\times\left(V_{n}^{\,1/2}\,P_{n}^{-1}\frac{Z_{n}}{n}\,e_{p}\right)$

where in this expression, $Z_{n}$ comes from (3.12). By the reasoning above
and Lemmas 3.1 and 3.9,

$\big{\|}\widehat{\theta}_{n}-\theta_{n}\big{\|}\,\leqslant\,C_{st}\left\|\left(V_{n}^{\,1/2}\,P_{n}^{-1}\,\frac{S_{n,\,n-1}}{n}\,(P_{n}^{-1})^{\,T}\,V_{n}^{\,1/2}\right)^{-1}\right\|\,\left\|\frac{Z_{n}}{n}\right\|~{}\overset{p}{\longrightarrow}~{}0.$

That establishes the consistency of the OLS estimator. ∎

###  3.4. Proof of Theorem 2.4

Suppose first that the eigenvalues of $A$ are real, and consider the
filtration

$\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636pt\mathcal{F}_{n,\,k}=\sigma(\Phi_{n,\,0},\,\varepsilon_{1},\,\ldots,\,\varepsilon_{k}).$

For all $a\in\mathbb{R}^{p}\backslash\\{0\\}$, let also

(3.26)
$m_{n,\,k}^{(a)}=a^{\,T}\,V_{n}^{\,1/2}\,P_{n}^{-1}\,\Phi_{n,\,k-1}\,\varepsilon_{k}\hskip
14.22636pt\text{and}\hskip
14.22636ptM_{n}^{(a)}=\sum_{k=1}^{n}m_{n,\,k}^{(a)}$

the way it appears in the right-hand side of (3.25). The sequence
$(m_{n,\,k}^{(a)})$ is clearly a scalar martingale difference array w.r.t.
$\mathcal{F}_{n,\,k}$ at fixed $n$ and for $1\leqslant k\leqslant n$. The
predictable quadratic variation of $M_{n}^{(a)}$ is

$\displaystyle\langle M^{(a)}\rangle_{n}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\mathbb{E}[(m_{n,\,k}^{(a)})^{2}\,|\,\mathcal{F}_{n,\,k-1}]$
$\displaystyle=$
$\displaystyle\sigma^{2}\,a^{\,T}\,V_{n}^{\,1/2}\,P_{n}^{-1}\left[\sum_{k=1}^{n}\Phi_{n,\,k-1}\,\Phi_{n,\,k-1}^{\,T}\right](P_{n}^{-1})^{\,T}\,V_{n}^{\,1/2}\,a$

since $(\varepsilon_{k})$ is a white noise and $\Phi_{n,\,k-1}$ is obviously
$\mathcal{F}_{n,\,k-1}$-measurable. Together with (3.24) and the definition of
$S_{n,\,n-1}$ in (1.4), that leads to the convergence

(3.27) $\frac{\langle
M^{(a)}\rangle_{n}}{n}~{}\overset{p}{\longrightarrow}~{}\sigma^{2}\,a^{\,T}Ha\,>\,0$

where $H$ is the covariance matrix given in (2.2), positive definite as shown
in the proof of Lemma 3.6 when $A$ has only real eigenvalues. To apply the
central limit theorem for arrays of martingales, see e.g. Thm. 1 of Sec. 8 in
Pollard (1984), it remains to show that the Lindeberg’s condition is
satisfied, in other words that

(3.28) $\forall\,\epsilon>0,\hskip
14.22636pt\frac{1}{n}\sum_{k=1}^{n}\mathbb{E}\big{[}(m_{n,\,k}^{(a)})^{2}\,\mathbbm{1}_{\\{|m_{n,\,k}^{(a)}|\,>\,\epsilon\sqrt{n}\\}}\,|\,\mathcal{F}_{n,\,k-1}\big{]}~{}\overset{p}{\longrightarrow}~{}0.$

To prove (3.28), we can first see that, for any $1\leqslant k\leqslant n$,

$\displaystyle\mathbb{E}\big{[}(m_{n,\,k}^{(a)})^{2}\,\mathbbm{1}_{\\{|m_{n,\,k}^{(a)}|\,>\,\epsilon\sqrt{n}\\}}\,|\,\mathcal{F}_{n,\,k-1}\big{]}$
$\displaystyle=$ $\displaystyle T^{(a)}_{n,\,k-1}\,\xi_{n,\,k}$
$\displaystyle\leqslant$ $\displaystyle
T^{(a)}_{n,\,k-1}\,\sup_{1\,\leqslant\,k\,\leqslant\,n}\xi_{n,\,k}$

where
$T^{(a)}_{n,\,k-1}\,\overset{\Delta}{=}\,a^{\,T}\,V_{n}^{\,1/2}\,P_{n}^{-1}\,\Phi_{n,\,k-1}\,\Phi_{n,\,k-1}^{\,T}(P_{n}^{-1})^{\,T}\,V_{n}^{\,1/2}\,a\,>\,0$
and with, by Hölder’s and Markov’s inequalities,

$\displaystyle\xi_{n,\,k}$ $\displaystyle\,\overset{\Delta}{=}\,$
$\displaystyle\mathbb{E}\big{[}\varepsilon_{k}^{\,2}\,\mathbbm{1}_{\\{|m_{n,\,k}^{(a)}|\,>\,\epsilon\sqrt{n}\\}}\,|\,\mathcal{F}_{n,\,k-1}\big{]}$
$\displaystyle\leqslant$
$\displaystyle\eta_{\nu}^{\,2/(2+\nu)}\,\mathbb{P}\big{(}(m_{n,\,k}^{(a)})^{2}\,>\,\epsilon^{\,2}n\,|\,\mathcal{F}_{n,\,k-1}\big{)}^{\nu/(2+\nu)}\leqslant~{}C_{st}\left(\frac{T^{(a)}_{n,\,k-1}}{n}\right)^{\\!\nu/(2+\nu)}$

in which $\eta_{\nu}$ is the moment of order $2+\nu$ of $(\varepsilon_{k})$,
as defined in the statement of the theorem. But choosing $k=n$, (3.27)
directly entails

(3.29) $\xi_{n,\,n}~{}\overset{p}{\longrightarrow}~{}0\hskip
14.22636pt\text{so that}\hskip
14.22636pt\xi^{\,\sharp}_{n,\,n}\,\overset{\Delta}{=}\,\sup_{1\,\leqslant\,k\,\leqslant\,n}\xi_{n,\,k}~{}\overset{p}{\longrightarrow}~{}0$

according to Lem. 1.3.20 of Duflo (1997) adapted to convergence in
probability. Finally,

$\frac{1}{n}\sum_{k=1}^{n}\mathbb{E}\big{[}(m_{n,\,k}^{(a)})^{2}\,\mathbbm{1}_{\\{|m_{n,\,k}^{(a)}|\,>\,\epsilon\sqrt{n}\\}}\,|\,\mathcal{F}_{n,\,k-1}\big{]}\,\leqslant\,\left(\frac{1}{n}\sum_{k=1}^{n}T^{(a)}_{n,\,k-1}\right)\xi^{\,\sharp}_{n,\,n}$

and, via (3.24) and (3.29), Lindeberg’s condition (3.28) is satisfied.
Together with (3.27), this is sufficient to establish that

$\forall\,a\in\mathbb{R}^{p}\backslash\\{0\\},\hskip
14.22636pt\frac{M_{n}^{(a)}}{\sqrt{n}}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}(0,\,\sigma^{2}\,a^{\,T}Ha).$

Because $a$ is arbitrary, we can now make use of the Cramér-Wold device to get
the $p$-vectorial convergence

(3.30)
$\frac{M_{n}}{\sqrt{n}}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}_{p}(0,\,\sigma^{2}\,H)$

with

$M_{n}=V_{n}^{\,1/2}\,P_{n}^{-1}\,\sum_{k=1}^{n}\Phi_{n,\,k-1}\,\varepsilon_{k}.$

Coming back to (3.24) and (3.25), and applying Slutsky’s lemma, the proof is
complete for the real case. If the spectrum of $A$ contains complex
eigenvalues, we shall exploit the fact that the first row of $P_{n}^{-1}$
(say, $\ell_{n}^{\,T}$) still contains real entries, as it is established in
Lemma 3.2. The scalar array of $\mathcal{F}_{n,\,k}$-martingale to use is now

$\forall\,n\geqslant 1,\hskip 14.22636ptM_{n}=\sum_{k=1}^{n}m_{n,\,k}$

with

$\forall\,1\leqslant k\leqslant n,\hskip
14.22636ptm_{n,\,k}=\sqrt{1-\rho_{n}}\,\ell_{n}^{\,T}\,\Phi_{n,\,k-1}\,\varepsilon_{k}.$

Similarly, the predictable quadratic variation of $M_{n}$ is

$\langle
M\rangle_{n}=\sigma^{2}\,(1-\rho_{n})\,\ell_{n}^{\,T}\,\left[\sum_{k=1}^{n}\Phi_{n,\,k-1}\,\Phi_{n,\,k-1}^{\,T}\right]\ell_{n}.$

From (3.24),

(3.31) $\frac{\langle
M\rangle_{n}}{n}~{}\overset{p}{\longrightarrow}~{}\sigma^{4}\,\frac{\pi_{11}^{2}}{2}\,>\,0\hskip
14.22636pt\text{so that}\hskip
14.22636pt\frac{M_{n}}{\sqrt{n}}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\left(0,\,\sigma^{4}\,\frac{\pi_{11}^{2}}{2}\right)$

by the same reasoning, the real and positive limit being the top-left element
of $H$ (times $\sigma^{2}$). Since $L_{n}$ as defined in the statement of the
theorem is the first column of $P_{n}$, it follows that
$L_{n}^{\,T}(P_{n}^{\,T})^{-1}=e_{p}^{\,T}$. Thus, by (3.25) and noting that
$v_{n}\,(1-\rho_{n})=c$ under (H3),

$\displaystyle\sqrt{n\,v_{n}}\,\big{\langle}L_{n},\,\widehat{\theta}_{n}-\theta_{n}\big{\rangle}$
$\displaystyle=$
$\displaystyle\sqrt{n\,v_{n}\,(1-\rho_{n})}\,e_{p}^{\,T}\left(V_{n}^{\,1/2}\,P_{n}^{-1}\,\frac{S_{n,\,n-1}}{n}\,(P_{n}^{-1})^{\,T}\,V_{n}^{\,1/2}\right)^{-1}$
$\displaystyle\hskip 14.22636pt\hskip
14.22636pt\times\left(V_{n}^{\,1/2}\,P_{n}^{-1}\frac{Z_{n}}{n}\,\right)e_{p}$
$\displaystyle=$
$\displaystyle\sqrt{c}\,e_{p}^{\,T}\left[\left(V_{n}^{\,1/2}\,P_{n}^{-1}\,\frac{S_{n,\,n-1}}{n}\,(P_{n}^{-1})^{\,T}\,V_{n}^{\,1/2}\right)^{-1}-H^{-1}\right]$
$\displaystyle\hskip 14.22636pt\hskip
14.22636pt\times\left(V_{n}^{\,1/2}\,P_{n}^{-1}\frac{Z_{n}}{\sqrt{n}}\,\right)e_{p}$
$\displaystyle\hskip 14.22636pt\hskip
14.22636pt+~{}\frac{2\sqrt{c}\,M_{n}}{\sigma^{2}\,\pi_{11}^{2}\,\sqrt{n}}~{}\,\overset{\Delta}{=}\,~{}T_{1,\,n}+T_{2,\,n}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\left(0,\,\frac{2c}{\pi_{11}^{2}}\right)$

exploiting (3.31), the block diagonal structure of $H$ which directly gives
$e_{p}^{\,T}H^{-1}=2\,e_{p}^{\,T}/(\sigma^{2}\,\pi_{11}^{2})$, Slutsky’s lemma
and the fact that $\|T_{1,\,n}\|=o_{p}(\|T_{2,\,n}\|)$. The proof is now
complete. ∎

######  Remark 3.4.

If we replace (H2) (H21) by (H2) (H22) in Theorems 2.3 and 2.4, the results
still hold with slight adjustments. In particular, we have now to consider

$V_{n}=\textnormal{diag}(1-\lambda_{n,\,1},\,1+\lambda_{n,\,2},\,1,\,\ldots,\,1)$

as rates, and this implies different limits in (3.23) and (3.24). The main
consequence is that the top-left $1\times 1$ block of $H_{0}$ is changed into
a $2\times 2$ block given by

$\begin{pmatrix}\frac{\pi_{11}^{2}}{2}&0\\\
0&\frac{\pi_{21}^{2}}{2}\end{pmatrix}.$

This new submatrix is obviously real and positive definite (see Lemma 3.3 with
$k=2$), and all the subsequent reasoning is perfectly similar.

##  4\. Applications and simulations

To illustrate the asymptotic normality of Theorem 2.4, we will use the
marginal convergence stated in Corollary 2.6, i.e.

(4.1)
$Z_{n}^{\,2}\,\overset{\Delta}{=}\,\frac{\pi_{11}^{\,2}\,n\,v_{n}}{2\,c}\,\left[\sum_{i=1}^{p}\frac{1}{(\lambda_{1}\rho_{n})^{\,i-1}}\,\big{(}\widehat{\theta}_{n,\,i}-\theta_{n,\,i}\big{)}\right]^{\,2}~{}\overset{d}{\longrightarrow}~{}\chi^{2}_{1}$

including positive and negative unit root situations ($\lambda_{1}=\pm 1$).
According to (H3), we set $\rho_{n}=1-c/v_{n}$ with $c=1$ and
$v_{n}=n^{\alpha}$ for some $0<\alpha<1$. For each simulation,
$\lambda_{n,\,2},\ldots,\lambda_{n,\,p}$ are randomly chosen out of
$[-|\lambda_{n,\,1}|+\epsilon,\,|\lambda_{n,\,1}|-\epsilon]$ for
$\epsilon=0.1$, then the process is generated with $n=5000$ and $Z_{n}^{\,2}$
is computed. The experiment is repeated 3000 times. On Figure 1, the
simulations are conducted with $p\in\\{2,3,4\\}$ for $\alpha=1/2$,
$\lambda_{1}\in\\{-1,1\\}$ and $\sigma^{2}=1$. The empirical distributions of
$Z_{n}^{\,2}$ are superimposed with the ‘true’ $\chi_{1}^{2}$ density (in
red), and the frequencies of observed values greater than $6.5$ are indicated.
On Figure 2, the simulations are conducted with $p=3$,
$\lambda_{1}\in\\{-1,1\\}$, $\alpha\in\\{1/5,1/4,1/3,1/2\\}$ and
$\sigma^{2}=1$. The empirical cumulative distributions of $Z_{n}^{\,2}$ are
superimposed with the ‘true’ $\chi_{1}^{2}$ cumulative distribution (in red).
Figure 3 displays the same experiments with $\alpha\in\\{1/2,2/3,3/4,4/5\\}$.
Now let us comment our observations. One can see that there are a few more
outliers than expected: $\approx 2.5\%$ of observed values greater than 6.5
whereas $\mathbb{P}(\chi_{1}^{2}>6.5)\approx 1.1\%$. This is due to the fact
that, as we have noticed, the convergence may be slow (see the perspectives in
the conclusion) and the Gaussian distribution tails remain somewhat
overloaded. It is particularly clear on Figure 2 when $\alpha$ is close to 0
and, unsurprisingly, this phenomenon is widely attenuated when we choose
larger values of $n$. The best results are obtained for $\alpha=1/2$ and we
observe a drift which occurs faster when $\alpha$ tends to 0 (Figure 2) than
when $\alpha$ tends to 1 (Figure 3). This is probably a consequence of the
well-known fact that the rates of convergence are faster in an unstable model
that in a stable one, as we recalled in the introduction. On the whole, we may
roughly say that for the moderate values of $\alpha$ (between 1/3 and 2/3,
say), the simulations are convincing and clearly fit with the theoretical
behavior.

Figure 1. Empirical distributions of $Z_{n}^{\,2}$ for $p=2$ (left), $p=3$
(middle) and $p=4$ (right), with $\lambda_{1}=-1$ (top) and $\lambda_{1}=1$
(bottom), on the basis of $3000$ experiments of size $n=5000$ with
$\alpha=1/2$. The red curve is the $\chi_{1}^{2}$ distribution. The
frequencies of observed values greater than $6.5$ are indicated, to be
compared with $\mathbb{P}(\chi_{1}^{2}>6.5)\approx 1.1\%$. Figure 2. Empirical
cumulative distributions of $Z_{n}^{\,2}$ for $p=3$ with $\lambda_{1}=-1$
(left), $\lambda_{1}=1$ (right) and $\alpha\in\\{1/5,1/4,1/3,1/2\\}$, on the
basis of $3000$ experiments of size $n=5000$. The red curve is the
$\chi_{1}^{2}$ cumulative distribution. Figure 3. Empirical cumulative
distributions of $Z_{n}^{\,2}$ for $p=3$ with $\lambda_{1}=-1$ (left),
$\lambda_{1}=1$ (right), $\alpha\in\\{1/2,2/3,3/4,4/5\\}$, on the basis of
$3000$ experiments of size $n=5000$. The red curve is the $\chi_{1}^{2}$
cumulative distribution.

To sum up, aligned with the work of Phillips and Magdalinos (2007), this study
completes the moderate deviations of Proïa (2020) by providing a sharp
analysis of the asymptotic behavior of the OLS between the rates $\sqrt{n}$
and $n$ corresponding to the stable and unstable AR$(p)$ processes,
respectively, when the unit root in $A$ is either positive or negative (and
even when both unit roots are present). The same conclusion prevails
concerning the lack of continuity at the boundaries: $\alpha\rightarrow 0^{+}$
and $\alpha=0$ do not match and the same is true for $\alpha\rightarrow 1^{-}$
and $\alpha=1$. However, by focusing on the inner neighborhood of the unit
root, nearly-unstable time-varying AR processes are instructive, especially
considering the extension of the asymptotic normality when $\rho_{n}$ does not
converge ‘too fast’ to 1. There are still many improvements to be made, the
main one being whether a unit root test can be derived. For the processes
which admit a decomposition of the form

$\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636pt\left\\{\begin{array}[]{lcl}X_{n,\,k}&=&\beta_{n}\,X_{n,\,k-1}+W_{k}\\\
(W_{k})&\sim&\textnormal{stable AR}(p-1),\end{array}\right.$

an identification with (1.2) is possible and shows that $(X_{n,\,k})$ is in
fact a time-varying AR$(p)$ process such that if $\beta_{n}$ converges to
$\beta$, then all its coefficients converge at the same rate. In this subclass
of processes, Park (2003) suggests a theoretical procedure to test for
$\mathcal{H}_{0}$ : “$\beta=1$” vs $\mathcal{H}_{1}$ : “$\beta<1$” provided
that $|\beta_{n}-\beta|\asymp 1/v_{n}$. The authors are pretty convinced that
this could be extended to all the nearly unstable processes covered by (1.2)
and that such a trail is likely to outperform the usual Dickey-Fuller tests
for unit root. This is a work in progress. In addition, the simulations
highlight an asymmetry in the convergence rates: we observed, on many more
experiments than those presented here, that the asymptotic distribution
deteriorates less quickly when $\alpha$ moves away from 1/2 towards 1 than
towards 0 (see again Figures 2 and 3). It should therefore be interesting to
investigate more deeply the convergence rate of the OLS. Finally, it would be
instructive but very challenging to look at the behavior of the OLS on the
outer neighborhood of the unit root, that is, for explosive but nearly
unstable AR$(p)$ processes, as it is already done for $p=1$.

Acknowledgements. This research benefited from the support of the ANR project
‘Efficient inference for large and high-frequency data’ (ANR-21-CE40-0021).
The authors sincerely thank the anonymous reviewer and the associate editor
for their comments and references which have clearly contributed to the
improvement of the paper.

## References

  * Beran et al. (2013) J. Beran, Y. Feng, S. Ghosh, and R. Kulik.  _Long-Memory Processes_.  Probabilistic Properties and Statistical Methods. Springer, Heidelberg, 2013.
  * Brockwell and Davis (1991) P. J. Brockwell and R. A. Davis.  _Time series: Theory and Methods (Second Edition)_.  Springer Series in Statistics. Springer, New York, 1991.
  * Buchmann and Chan (2013) B. Buchmann and N. H. Chan.  Unified asymptotic theory for nearly unstable AR$(p)$ processes.  _Stoch. Proc. Appl._ , 123:952–985, 2013.
  * Chan and Wei (1987) N. H. Chan and C. Z. Wei.  Asymptotic inference for nearly nonstationary AR$(1)$ processes.  _Ann. Stat._ , 15:1050–1063, 1987.
  * Chan and Wei (1988) N. H. Chan and C. Z. Wei.  Limiting distributions of least squares estimates of unstable autoregressive processes.  _Ann. Stat._ , 16:367–401, 1988.
  * Duflo (1997) M. Duflo.  _Random iterative models_.  Applications of Mathematics (vol. 34), New York. Springer-Verlag, Berlin, 1997.
  * Giraitis and Phillips (2006) L. Giraitis and P. C. B. Phillips.  Uniform limit theory for stationary autoregression.  _J. Time Ser. Anal._ , 27:51–60, 2006.
  * Horn and Johnson (2012) R. A. Horn and C. R. Johnson.  _Matrix Analysis (Second Edition)_.  Cambridge University Press, Cambridge, New-York, 2012.
  * Horváth and Trapani (2016) L. Horváth and L. Trapani.  Statistical inference in a random coefficient panel model.  _J. Econometrics._ , 193:54–75, 2016.
  * Jiang and Wei (2003) T. Jiang and M. Wei.  On solutions of the matrix equations ${X-AXB=C}$ and ${X-A\bar{X}B=C}$.  _Linear Algebra Appl._ , 367:225–233, 2003.
  * Lai and Wei (1983) T. L. Lai and C. Z. Wei.  Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters.  _J. Multivariate Anal._ , 13:1–23, 1983.
  * Ling (2004) S. Ling.  Estimation and testing stationarity for double-autoregressive models.  _J. R. Stat. Soc. B._ , 66:63–78, 2004.
  * Miao et al. (2015) Y. Miao, Y. Wang, and G. Yang.  Moderate deviation principles for empirical covariance in the neighbourhood of the unit root.  _Scand. J. Stat._ , 42:234–255, 2015.
  * Nielsen and Rahbek (2014) H. B. Nielsen and A. Rahbek.  Unit root vector autoregression with volatility induced stationarity.  _J. Empir. Financ._ , 29:144–167, 2014.
  * Park (2003) J. W. Park.  _Weak unit roots_.  Department of Economics, Rice University, 2003.
  * Phillips (1987) P. C. B. Phillips.  Towards a unified asymptotic theory for autoregression.  _Biometrika._ , 74:535–547, 1987.
  * Phillips and Lee (2015) P. C. B. Phillips and J. H. Lee.  Limit theory for VARs with mixed roots near unity.  _Economet. Rev._ , 34:1034–1055, 2015.
  * Phillips and Magdalinos (2007) P. C. B. Phillips and T. Magdalinos.  Limit theory for moderate deviations from a unit root.  _J. Econometrics._ , 136:115–130, 2007.
  * Pollard (1984) D. Pollard.  _Convergence of Stochastic Processes_.  Springer, Berlin, 1984.
  * Proïa (2020) F. Proïa.  Moderate deviations in a class of stable but nearly unstable processes.  _J. Stat. Plan. Inference._ , 208:66–81, 2020.
  * Trapani (2021) L. Trapani.  Testing for strict stationarity in a Random Coefficient AutoRegression.  _Economet. Rev._ , 40:220–256, 2021.