Zilu Guo1, Jun Du1,∗, Chin-Hui Lee2, Yu Gao3, Wenbin Zhang3 ††thanks:  *
corresponding author

# Variance-Preserving-Based Interpolation Diffusion Models
for Speech Enhancement

###### Abstract

The goal of this study is to implement diffusion models for speech enhancement
(SE). The first step is to emphasize the theoretical foundation of variance-
preserving (VP)-based interpolation diffusion under continuous conditions.
Subsequently, we present a more concise framework that encapsulates both the
VP- and variance-exploding (VE)-based interpolation diffusion methods. We
demonstrate that these two methods are special cases of the proposed
framework. Additionally, we provide a practical example of VP-based
interpolation diffusion for the SE task. To improve performance and ease model
training, we analyze the common difficulties encountered in diffusion models
and suggest amenable hyper-parameters. Finally, we evaluate our model against
several methods using a public benchmark to showcase the effectiveness of our
approach.

Index Terms: speech enhancement, speech denoising, diffusion models, score-
based, interpolation diffusion

##  1 Introduction

Speech enhancement (SE) [1] has been the subject of research for several
decades with the goal of diminishing or even eliminating the noise in a noisy
speech while minimizing the distortion to speech quality. In recent years, SE
has been approached as a supervised regressive task with the assistance of
deep learning. The early attempts to deploy deep learning for SE involve
employing off-the-shelf models to predict the targets that are typically
utilized in traditional approaches. However, it is important to note that
these approaches are sub-optimal since they are not able to obtain the clean
phase of the speech. Common targets for these methods include the magnitude of
the spectrogram [2], the log magnitude (Mapping) [3], the ideal ratio mask
(IRM)[4], the spectral magnitude mask (SMM) [2], and etc. The real and
imaginary parts of the spectrogram are utilized directly as the target to
obtain the clean phase afterward [5, 6, 7]. Meanwhile, another main-stream SE
method seeks to predict clean waveforms in an end-to-end (E2E) manner [8],
rather than the spectrum. In addition to the regressive methods, some
researchers have utilized generative models for the SE, such as, VAE [9, 10],
GAN [11, 12, 13], flow [14, 15], and etc.

Recently, Diffusion models have been successful in various generative tasks,
including image generation [16, 17, 18], image editing [19], speech synthesis
[20] and etc. Diffusion models involve two processes, i.e., diffusion (or
forward) and reverse (or backward) processes. Another approach in the discrete
domain is the score-based model [18]. Both the diffusion [16] and the score-
based [18, 21] model are unified in [17, 22], and generalized to the
continuous time domain, endowing the model with more capacity. Moreover, the
authors in [17] have classified the diffusion models into two types: the
variance-preserving(VP)-based [16] and the variance-exploding(VE)-based [18]
method grounded on their intuitive properties. The VE-based approach gradually
increases the variance over time while keeping the clean component unaltered.
In contrast, VP-based diffusion attempts to preserve the amplitude with fewer
fluctuations. Besides their applications in the image processing field,
diffusion models have also been explored for SE tasks.

In their work, CDiffSE [23] proposes that a degraded signal is composed of
three components, i.e, the clean signal , the noisy and the Gaussian noise.
They suggest that the mean of the degraded signal in the diffusion process is
obtained through a linear combination of the clean and noisy speech,
specifically the linear interpolation of the two. They then apply this
interpolation method to Diffwave [20], a model for speech synthesis, for the
SE task. SRTNET [24] follows a similar approach, utilizing the interpolation
diffusion to estimate the distortion between the clean speech and an enhanced
speech predicted by a pre-trained discriminative model. However, the two-stage
models commonly have higher computational overhead. Apart from the discrete
diffusion models mentioned above, there are also several interpolation-based
methods for SE under continuous conditions. In [25, 26], the authors formulate
the theoretical foundation for the VE-based interpolation diffusion model
(VEIDM) in the continuous time field and generalize it to the continuous time
system. However, they leave the VP-based one untouched which showcases better
performance in some tasks than the VE-based diffusion. In this paper, we
attempt to apply linear interpolation to VP-based diffusion.

The rest of the paper is organized as follows. The proposed method is
introduced in Sec. 2. Specifically, we accentuate the proposed signal model,
training method, and sampling algorithms in this section. Experiments
settings, results, and analyses are presented in Sec. 3. we draw conclusions
in Sec. 4.

##  2 The proposed method

In the forward process of the vanilla VP-based diffusion model, a clean signal
(such as an image or speech) is gradually degraded by adding Gaussian noise in
steps until it reaches an approximate Gaussian noise level. In this process,
the mean of the state $t$ is an affine function of the clean signal. However,
in the VP-based interpolation diffusion, the mean is replaced with a linear
interpolation of the clean and the noisy.

Algorithm 1 Training stage

Batch size is $B$, number of training iterations $N_{\text{iter}}$

for $i=1$ to $N_{\text{iter}}$ do

Sample a batch of clean and noisy speech pairs
[$(\boldsymbol{x}_{0}^{1},\boldsymbol{y}^{1})$, …,
$(\boldsymbol{x}_{0}^{b},\boldsymbol{y}^{b})$, …,
$(\boldsymbol{x}_{0}^{B},\boldsymbol{y}^{B})]$,
$(\boldsymbol{x}_{0}^{b},\boldsymbol{y}^{b}))\sim
p(\boldsymbol{x_{0}},\boldsymbol{y})$

Sample a batch of the time indexes [$t^{1}$, $t^{2}$, …, $t^{B}]$,
$t^{b}\sim\mathcal{U}(\epsilon,1)$

Sample a batch of Gaussian noises [$\boldsymbol{z}^{1}$, $\boldsymbol{z}^{2}$,
…, $\boldsymbol{z}^{B}]$,
$\boldsymbol{z}^{b}\sim\boldsymbol{\mathcal{N}}(\boldsymbol{0},\boldsymbol{I})$

Get a batch of [$\boldsymbol{x}^{1}(t^{1})$, $\boldsymbol{x}^{2}(t^{2})$, …,
$\boldsymbol{x}^{B}(t^{2})$], $\boldsymbol{x}^{b}(t^{b})$ is sampled from
Eq.(1)

Input the [$\boldsymbol{x}^{1}(t^{1})$, $\boldsymbol{x}^{2}(t^{2})$, …,
$\boldsymbol{x}^{B}(t^{2})$] to the neural network

Get the output [$\boldsymbol{\theta}(\boldsymbol{x}^{1}(t^{1}))$,
$\boldsymbol{\theta}(\boldsymbol{x}^{2}(t^{2}))$,…,
$\boldsymbol{\theta}(\boldsymbol{x}^{B}(t^{B}))$]

Compute the loss
$\mathcal{L}=\\{\Sigma_{b=1}^{B}{||G(t^{b})\boldsymbol{\theta}(\boldsymbol{x}(t^{b})+{\boldsymbol{z}}||^{2}}\\}/B$
from Eq.(9)

Update the parameters

end for

###  2.1 The VP-based interpolation diffusion model (VPIDM)

For tasks of SE, image editing, voice conversion and etc, there is an existing
condition that holds copious information about the target. In the case of SE,
for instance, the off-the-shelf noisy speech could be implemented to guide the
diffusion process of the clean. We refer to this approach as the VP-based
interpolation diffusion model (VPIDM). The signal model for VPIDM is defined
as

$\displaystyle\boldsymbol{x}(t)={\alpha}_{t}[\lambda_{t}\boldsymbol{x}_{0}+(1-\lambda_{t})\boldsymbol{y}]+\sqrt{1-{\alpha}_{t}^{2}}\boldsymbol{z}$
(1)

where $x(t)$ is the degraded signal at $t$ time index in the diffusion
process, $\boldsymbol{x}_{0}$ is the clean signal, $\boldsymbol{y}$ is the
noisy speech, $\boldsymbol{z}$ is the Gaussian noise sampled from normal
distribution, $\alpha_{t}$ determines the diffusion process, $\lambda_{t}$ is
the slope of the interpolation process, $\alpha_{t}$ and $\lambda_{t}$ are
functions of $t$. The differential of $\boldsymbol{x}(t)$ is

$\displaystyle
d\boldsymbol{x}(t)=[\boldsymbol{x}(t)\ln^{{}^{\prime}}{(\alpha}_{t}\lambda_{t})-\boldsymbol{y}{\alpha}_{t}\ln^{{}^{\prime}}\lambda_{t}]dt+\boldsymbol{\Sigma}_{t}d\boldsymbol{w}$
(2)

where $\boldsymbol{w}$ is the stochastic process, $ln^{{}^{\prime}}[\cdot]$ is
the derivative of $ln[\cdot]$ with respect to $t$, $d[\cdot]$ is the operation
of differential. When $\boldsymbol{\Sigma}_{t}$ is a diagonal matrix, suppose
that $\boldsymbol{\Sigma}_{t}=g(t)\boldsymbol{I}$. From Eq.(5.53) in [27], we
get

$\displaystyle\frac{dG^{2}(t)}{dt}=2G^{2}(t)\ln^{{}^{\prime}}({\alpha}_{t}\lambda_{t})+g^{2}(t)$
(3)

where $g(t)$ indicates the spread speed of the stochastic process in the
derivative of $\boldsymbol{x}(t)$,
$g(t)=\sqrt{-2G^{2}(t)\ln^{{}^{\prime}}\lambda_{t}-2\ln^{{}^{\prime}}\alpha_{t}}$,
$G(t)$ is the coefficient of the Gaussian noise in $\boldsymbol{x}(t)$,
$G^{2}(t)=1-\alpha^{2}_{t}$, $t\in(0,1]$, $\ln^{{}^{\prime}}\alpha_{t}\leq 0$
and $\ln^{{}^{\prime}}\lambda_{t}\leq 0$, which means $\lambda_{t}$ and
$\alpha_{t}$ are monotonous decrease functions. In this article, all constant
superscripts represent powers, unless otherwise specified. When $t\rightarrow
0$, then $\lambda_{t}\rightarrow 1$,$\alpha_{t}\rightarrow 1$.

In principle, we hope $\lambda_{1}\rightarrow 0$, which implies that the final
state is a combination of the noisy signal and the Gaussian noise. Therefore,
the larger $-\ln^{{}^{\prime}}\lambda_{t}$ appears to be more favorable.
However, empirical evidence suggests that $-\ln^{{}^{\prime}}\lambda_{t}$
cannot be infinitely large. The reason behind this is that when we sample a
clean and $-\ln^{{}^{\prime}}\lambda_{t}$ is set to sufficiently large, the
linear interpolation tends to change quickly from $\boldsymbol{y}$ to
$\boldsymbol{x}_{0}$ over several steps, it is difficult for neural networks
to capture.

Algorithm 2 Sampling (enhancing) stage

Suppose the number of sample steps is $K$,
$t_{k}=\frac{(1-\epsilon)}{K}k+\epsilon$

Sample
$\boldsymbol{x}_{K}=\boldsymbol{x}(t_{K})=\boldsymbol{x}(1)=\alpha_{1}\boldsymbol{y}+G(1)\boldsymbol{z}$

for $k=K-1,K-2,\dots,1$ do

Input the $\boldsymbol{x}_{k+1}$, get the
$\hat{\boldsymbol{\theta}}(\boldsymbol{x}_{k+1})$

Sample a Gaussian noise $\boldsymbol{z}$,
$\boldsymbol{z}\sim\boldsymbol{\mathcal{N}}(\boldsymbol{0},\boldsymbol{I})$

Compute the $\boldsymbol{x}_{k}$ from Eq.(17)

end for

Input the $\boldsymbol{x}_{1}$, get the
$\hat{\boldsymbol{\theta}}(\boldsymbol{x}_{1})$

$\hat{\boldsymbol{x}}_{0}=\boldsymbol{x}_{1}-[f(\boldsymbol{x}_{1},\boldsymbol{y})-g_{1}^{2}\hat{\boldsymbol{\theta}}(\boldsymbol{x}_{1})]\Delta$

return $\hat{\boldsymbol{x}}_{0}$

In this paper, we adopt the similar $\alpha_{t}$ schedule of the VP-based
diffusion in [17] for the SE, i.e.,
$\alpha_{t}=e^{-0.5\int_{0}^{t}\beta(\tau)d\tau}$, where
$\beta(t)=(\beta_{\text{max}}-\beta_{\text{min}})t+\beta_{\text{min}}$,
$\beta_{\text{min}}$ controls the slope of the clean scale when $t\rightarrow
0$, $(\beta_{\text{max}}-\beta_{\text{min}})$ controls the changing speed of
$\boldsymbol{x}_{t}$ from the clean to the Gaussian.

$\displaystyle
g(t)=\sqrt{\beta(t)+2\lambda(1-e^{-\int_{0}^{t}\beta(\tau)d\tau})}$ (4)

where $\lambda_{t}=e^{-\lambda t}$. From Eq.(2), the $d\boldsymbol{x}(t)$

$\displaystyle
d\boldsymbol{x}(t)=[-(0.5\beta(t)+\lambda)\boldsymbol{x}(t)+\lambda\alpha_{t}\boldsymbol{y}]dt+g(t)d\boldsymbol{w}$
(5)

In the reverse process, the $\boldsymbol{x}(1)$ is sampled from the
distribution
$\mathcal{N}(\alpha_{1}\boldsymbol{y},\sqrt{1-\alpha_{1}^{2}}\boldsymbol{I})$.

###  2.2 The loss function and the training stage

For unconditional diffusion, the neural network is trained for predicting
$\boldsymbol{\nabla}_{\boldsymbol{x}}\ln(p_{t}(\boldsymbol{x}))$. This is
equivalent to optimizing the following cost function

$\displaystyle\mathcal{L}=\mathbb{E}_{t,\boldsymbol{x}_{0},\boldsymbol{x}(t)}\\{W\cdot||\boldsymbol{\theta}(\boldsymbol{x}(t))-\boldsymbol{\nabla}_{\boldsymbol{x}}\ln(p_{t}(\boldsymbol{x}))||^{2}\\}$
(6)

where $\boldsymbol{\theta}(\boldsymbol{x}(t))$ is the output of the neural
network. For interpolation-based diffusion,

$\displaystyle
p_{t}(\boldsymbol{x})=p(\boldsymbol{x}(t)|\boldsymbol{x}_{0},\boldsymbol{y})=\mathcal{N}(\boldsymbol{m}(\boldsymbol{x}_{0},\boldsymbol{y});G(t)\boldsymbol{I})$
(7)

where $p_{t}(\boldsymbol{x})$ is the conditional probability density function
of $\boldsymbol{x}(t)$,
$\boldsymbol{m}(\boldsymbol{x}_{0},\boldsymbol{y})={\alpha}_{t}[\lambda_{t}\boldsymbol{x}_{0}+(1-\lambda_{t})\boldsymbol{y}]$
is the mean of $p_{t}(\boldsymbol{x})$.

$\displaystyle\boldsymbol{\nabla}_{\boldsymbol{x}}\ln(p_{t}(\boldsymbol{x}))=\boldsymbol{\nabla}_{\boldsymbol{x}}[-\frac{||\boldsymbol{x}(t)-\boldsymbol{m}(\boldsymbol{x}_{0},\boldsymbol{y})||^{2}}{2G^{2}(t)}]=-\frac{\boldsymbol{z}}{G(t)}$
(8)

here
$\boldsymbol{x}(t)-\boldsymbol{m}(\boldsymbol{x}_{0},\boldsymbol{y})=G(t)\boldsymbol{z}$.
Then we get the loss function

$\displaystyle\mathcal{L}=\mathbb{E}\\{W\cdot||\boldsymbol{\theta}(\boldsymbol{x}(t))+\frac{\boldsymbol{z}}{G(t)})||^{2}\\}=\mathbb{E}||G(t)\boldsymbol{\theta}(\boldsymbol{x}(t))+\boldsymbol{z})||^{2}$
(9)

We follow the settings in [17, 18, 25], utilize the weighted loss, and set
$W=G^{2}(t)$ for better performance.

The training algorithm is shown in Alg. 1, where $\epsilon$ represents the
minimum sample time, where the superscript $b$ in $[\cdot]^{b}$ denotes the
$b$-th sample of a batch, batch size is $B$,
$p(\boldsymbol{x_{0}},\boldsymbol{y})$ denotes the joint probability density
function of the clean and noisy pair, i.e.,
$\boldsymbol{x}_{0},\boldsymbol{y}$.

Table 1: Comparison of the VEIDM, VPIDM, and IDM.

| The state equations | The stochastic differential equations
---|---|---
VEIDM [25] |  $\displaystyle\boldsymbol{x}(t)=\lambda_{t}\boldsymbol{x}_{0}+(1-\lambda_{t})\boldsymbol{y}+\sqrt{\ln(\frac{\sigma_{\text{max}}}{\sigma_{\text{min}}})\frac{\sigma_{\text{min}}^{2}((\sigma_{\text{max}}/\sigma_{\text{min}})^{2t}-e^{-2\lambda t})}{\lambda+\ln(\sigma_{\text{max}}/\sigma_{\text{min}})}}\boldsymbol{z}$ (10) |  $\displaystyle d\boldsymbol{x}(t)=\lambda(\boldsymbol{y}-\boldsymbol{x}(t))dt+\sigma_{\text{min}}(\frac{\sigma_{\text{max}}}{\sigma_{\text{min}}})^{t}\sqrt{2\ln(\frac{\sigma_{\text{max}}}{\sigma_{\text{min}}})}d\boldsymbol{w}$ (11)
VPIDM |  $\displaystyle\boldsymbol{x}(t)={\alpha}_{t}[\lambda_{t}\boldsymbol{x}_{0}+(1-\lambda_{t})\boldsymbol{y}]+\sqrt{1-{\alpha}_{t}^{2}}\boldsymbol{z}$ (12) |  $\displaystyle d\boldsymbol{x}(t)=[\boldsymbol{x}(t)\ln^{{}^{\prime}}{(\alpha}_{t}\lambda_{t})-\boldsymbol{y}{\alpha}_{t}\ln^{{}^{\prime}}\lambda_{t}]dt+\boldsymbol{\Sigma}_{t}d\boldsymbol{w}$ (13)
IDM |  $\displaystyle\boldsymbol{x}(t)={\alpha}_{t}[\lambda_{t}\boldsymbol{x}_{0}+(1-\lambda_{t})\boldsymbol{y}]+G(t)\boldsymbol{z}$ (14) |  $\displaystyle d\boldsymbol{x}(t)=\boldsymbol{x}(t)d\ln{(\alpha}_{t}\lambda_{t})-\boldsymbol{y}{\alpha}_{t}d\ln\lambda_{t}+g(t)d\boldsymbol{w}$ (15)

###  2.3 The reverse process for sampling a clean

From [17, 28], the reverse process is also a diffusion process which can be
represented as

$\displaystyle
d\boldsymbol{x}(t)=[f(\boldsymbol{x}(t),\boldsymbol{y})-g(t)^{2}\boldsymbol{\theta}(\boldsymbol{x}(t))]dt+g(t)d\bar{\boldsymbol{w}}$
(16)

where
$f(\boldsymbol{x}(t),\boldsymbol{y})=\boldsymbol{x}(t)\ln^{{}^{\prime}}{(\alpha}_{t}\lambda_{t})-\boldsymbol{y}{\alpha}_{t}\ln^{{}^{\prime}}\lambda_{t}$
for the VPIDM and $\bar{\boldsymbol{w}}$ is another stochastic process but has
the same distribution with $\boldsymbol{w}$. Typically, the continuous process
is discretized in the sampling stage. $\Delta=\frac{1-\epsilon}{K}$ is set and
let $\boldsymbol{x}_{k}=\boldsymbol{x}(\frac{k(1-\epsilon)}{K}+\epsilon)$ and
$g_{k}=g(\frac{k(1-\epsilon)}{K}+\epsilon)\sqrt{\Delta}$. The sampling stage
is elaborated in Alg. 2.

$\displaystyle\boldsymbol{x}_{k-1}=\boldsymbol{x}_{k}-[f(\boldsymbol{x}_{k},\boldsymbol{y})-g_{k}^{2}\boldsymbol{\theta}(\boldsymbol{x}_{k})]\Delta+g_{k}{\boldsymbol{z}}$
(17)

where the subscript $k$ of $\boldsymbol{x}_{k}$ and $g_{k}$ denote the
discrete sampling time index, $\boldsymbol{x}_{k}$ and $g_{k}$ represent the
discrete samplings of $\boldsymbol{x}(t)$ and $g(t)$.

###  2.4 Comparison with the VEIDM

Furthermore, the proposed VPIDM and the VEIDM proposed in [25] can be
concluded as

$\displaystyle
d\boldsymbol{x}(t)=\boldsymbol{x}(t)d\ln{(\alpha}_{t}\lambda_{t})-\boldsymbol{y}{\alpha}_{t}d\ln\lambda_{t}+g(t)d\boldsymbol{w}$
(18)

$\displaystyle\boldsymbol{x}(t)={\alpha}_{t}[\lambda_{t}\boldsymbol{x}_{0}+(1-\lambda_{t})\boldsymbol{y}]+G(t)\boldsymbol{z}$
(19)

The relation between $G$ and $g$ is constrained by Eq.(3) which is referred to
as the interpolation diffusion model (IDM). When $G^{2}(t)=1-\alpha^{2}_{t}$,
the interpolation diffusion becomes a VP-based method. In the case of VE-based
interpolation diffusion, $\alpha_{t}$ in Eq.(18) and (19) is constant $1$.
Substitute $\alpha_{t}$ with constant $1$ and solve the ordinary differential
equation in Eq.(3), we get $G(t)$

$\displaystyle
G^{2}(t)=\lambda^{2}_{t}G(0)^{2}+\lambda^{2}_{t}\int_{0}^{t}{g^{2}(\tau)}/{\lambda^{2}_{\tau}}d\tau$
(20)

In [25], The interpolation coefficient $\lambda_{t}$ is defined as
$e^{-\lambda t}$. It can be verified that the VEIDM, which use $G(t)$ from
Eq.(10), and the $g(t)$ from Eq.(11), satisfies Eq.(20) as a special case.
Additionally, a comparison is made between the VEIDM, the proposed VPIDM, and
IDM in Tab.1.

In the reverse process, the initial sample of the VEIDM in [25] is taken as
$\boldsymbol{y}+G(1)\boldsymbol{z}$, rather than the ground truth
$\lambda_{1}\boldsymbol{x}_{0}+(1-\lambda_{1})\boldsymbol{y}+G(1)\boldsymbol{z}$
because obtaining the clean is not possible at this stage. However, this can
result in damage to the enhanced speech. To quantify this effect, we define
the initial error (IE)

$\displaystyle\textit{IE}_{\text{VEIDM}}$
$\displaystyle=[\boldsymbol{y}+G(1)\boldsymbol{z}]-[\lambda_{1}\boldsymbol{x}_{0}+(1-\lambda_{1})\boldsymbol{y}+G(1)\boldsymbol{z}]$
(21) $\displaystyle=\lambda_{1}(\boldsymbol{y}-\boldsymbol{x}_{0})$ (22)

Whereas, the reverse of the proposed start from
$\alpha_{1}\boldsymbol{y}+G(1)\boldsymbol{z}$, and the truth is
$\alpha_{1}(\lambda_{1}\boldsymbol{x}_{0}+(1-\lambda_{1})\boldsymbol{y})+G(1)\boldsymbol{z}$.
The IE is

$\displaystyle\textit{IE}_{\text{VPIDM}}$
$\displaystyle=\alpha_{1}\lambda_{1}(\boldsymbol{y}-\boldsymbol{x}_{0})$ (23)

When the same $\lambda_{t}$ is utilized in the Sgmse+ and VPIDM,
$\textit{IE}_{\text{VPIDM}}\ll\textit{IE}_{\text{Sgmse+}}$, where
$\alpha_{1}\rightarrow 0$. That is, the VPIDM has a smaller IE than the VEIDM.

##  3 Experiments

###  3.1 Training settings

We conduct our experiments on the publicly available benchmark, i.e.,
VoiceBank-DEMAND(VBD) [29] Metrics in [30, 31, 32], i.e., SI-SDR, SI-SIR, SI-
SAR, PESQ, CSIG, CBAK, COVL, are adopted to compare the performance to other
state-of-the-art methods. $25$ speech clips from the test dataset are selected
randomly as the validation dataset. We train the neural network for 120
epochs. The best checkpoint is saved when PESQ is in its optimal state during
the validation phase.

We use the neural network proposed in [26] as our backbone model, which is
originally introduced in [17] for the image generation task. We treat the
complex spectrum as a real-valued tensor to circumvent complex-valued
computation where the real and imaginary parts of the complex are represented
as two channels of the tensor. The tensor is scaled to ensure that its
amplitude approximately falls within the range of $-1$ to $1$ before being fed
into the neural network. We follow the scaling function described in [25]
where given a complex-valued spectrum
$\boldsymbol{x}(t)=|\boldsymbol{x}(t)|e^{\angle\boldsymbol{x}(t)}$, the scaled
$[\boldsymbol{x}(t)]^{s}=a|\boldsymbol{x}(t)|^{c}e^{\angle\boldsymbol{x}(t)}$,
here $a,c$ are two hyper-parameters, $[\cdot]^{s}$ means the operation of the
scaling function. We find that $a50^{c}\approx 1,0<c\leq 1$ is of avail for
the performance and $c$ can not be set too small, as it compresses the dynamic
range of the signal drastically and makes it more difficult to learn the
structure of clean spectrum. In this paper, we empirically set $a=0.15$ and
$c=0.5$, $\beta_{\text{min}}=0.1$, $\beta_{\text{max}}=2$ and
$\lambda_{t}=e^{-\lambda t}$, $\lambda=1.5$, $G(t)=\sqrt{1-\alpha_{t}^{2}}$.

Figure 1: The log spectrogram of the STFT magnitudes. a) the clean
$\boldsymbol{x}_{0}$ spectrogram; b) the noisy $\boldsymbol{y}$ spectrogram;
c) the spectrogram of the estimated clean from the Sgmse+ (2); d) the
spectrogram of the estimated clean from the VPIDM.

###  3.2 Results and analyses

For the continuum diffusion model, we typically sample $t$ from the uniform
distribution $\mathcal{U}(\epsilon,1)$, $\epsilon$ denotes the time index of
the first state after the clean. Ideally, we want to set $\epsilon$ as small
as possible. However, when $\epsilon$ is too small, it can lead to
difficulties in achieving convergence during training and may result in
unstable fluctuations in the optimization process.

Table 2: Illustration of several metrics over different settings for VPIDM on
the VBD dataset.

Settings | PESQ $\uparrow$ | SISDR $\uparrow$ | SISIR $\uparrow$ | SISAR $\uparrow$
---|---|---|---|---
$\epsilon=1\cdot 10^{-2}$ | $3.01$ | ${18.3}$ | $\mathbf{31.9}$ | $18.6$
$\epsilon=3\cdot 10^{-2}$ | $3.02$ | $\mathbf{18.9}$ | $30.1$ | ${19.4}$
$\epsilon=4\cdot 10^{-2}$ | $\mathbf{3.13}$ | $18.7$ | $28.6$ | $19.3$
$\epsilon=5\cdot 10^{-2}$ | $2.86$ | $18.6$ | $27.6$ | $19.4$
$\epsilon=6\cdot 10^{-2}$ | $2.95$ | $18.7$ | $26.6$ | $\mathbf{19.7}$
$\epsilon=7\cdot 10^{-2}$ | $1.77$ | $16.1$ | $24.5$ | $16.9$

In Fig.2, we vary the value of $\epsilon$ for the model and observe the
training loss with training steps. The results show that when the $\epsilon$
is too small, the training loss exhibits lots of fluctuations and becomes
difficult to converge. In our view, the reason is that, when $\epsilon$ is too
small, the model is required to estimate the target in a wider range of SNR
conditions. We conduct experiments to check the model's ability to predict the
target in low SNR conditions using only one state ($t=\epsilon$) as the input
for the model. However, the model can not predict the target well when
$\epsilon\leq 10^{-2}$. That is, when $t\rightarrow 0$, $\alpha_{t}\rightarrow
1$, $\lambda_{t}\rightarrow 1$, then
$\boldsymbol{x}(t)\approx\boldsymbol{x}_{0}+\sqrt{1-\alpha_{t}^{2}}\boldsymbol{z}$.
From the perspective of Gaussian noise, the $\text{SNR}\approx
10\log_{10}(\frac{1-\alpha_{t}^{2}}{1})\approx
10\log_{10}(\beta_{\text{min}}t)$. Therefore, when
$t=10^{-5},10^{-4},10^{-3},10^{-2}$, the model predicts the target at about
$-60$dB, $-50$dB, $-40$dB, $-30$dB SNR, respectively. Based on these results,
we choose $\epsilon\geq 10^{-2}$. We treat the $\epsilon$ as the minimum
resolution of the model, so we set the number of sample steps in the reverse
process to $K\approx[\frac{1}{\epsilon}]$. As a result, for
$\epsilon=[10^{-2},3\cdot 10^{-2},4\cdot 10^{-2},5\cdot 10^{-2},6\cdot
10^{-2},10^{-1}]$, the number of sampling steps are $100,30,25,20,15,10$,
respectively. According to Tab.2, the best PESQ is achieved when
$\epsilon=4\cdot 10^{-2}$, which corresponds to $25$ sample steps. The model
demonstrates strong performance across various evaluation metrics when the
number of sample steps exceeds $15$, but when the sample steps are less than
$15$, the performance drops significantly.

Figure 2: The training loss when $t\sim\mathcal{U}(\epsilon,1)$. Table 3: The
proposed method versus some SOTA methods with respect to different metrics.

Model | PESQ $\uparrow$ | CSIG $\uparrow$ | CBAK $\uparrow$ | COVL $\uparrow$
---|---|---|---|---
noisy | $1.97$ | $3.35$ | $2.44$ | $2.64$
PFP [33] | $\mathbf{3.15}$ | $4.18$ | $\mathbf{3.60}$ | $3.67$
MetricGAN [12] | $2.86$ | $3.99$ | $3.18$ | $3.42$
MetricGAN$+$ [13] | $3.15$ | $4.14$ | $3.16$ | $3.64$
CDiffuSE [23] | $2.52$ | $3.72$ | $2.91$ | $3.10$
SRTNet [24] | $2.69$ | $4.12$ | $3.19$ | $3.39$
CDSE [34] | $2.77$ | $3.91$ | $3.32$ | $3.33$
Sgmse+ (1) [26] | $2.80$ | $4.10$ | $3.24$ | $3.44$
Sgmse+ (2) [26] | $2.93$ | $4.12$ | $3.37$ | $3.51$
VPIDM | $3.13$ | $\mathbf{4.63}$ | $3.41$ | $\mathbf{3.94}$

In Tab.3, we compare our model to several SOTA methods. Sgmse+ (1) denotes the
vanilla model without the corrector, while Sgmse+ (2) includes the corrector.
The proposed model achieves the best CSIG, indicating the least speech
distortion. This demonstrates that our generative model has learned the
closest clean distribution to the ground truth that can preserve clean speech
best. Sgmse+ (1) in [26] is the VE-based interpolation diffusion with $30$
sampling steps. The proposed method with fewer steps, i.e., $25$ steps
($\epsilon=4\cdot 10^{-2}$), achieves $0.3$ PESQ increment over Sgmse+ (1). To
further improve the performance, Sgmse+ (2) in [26] implements a corrector
which requires $60$ total steps. The proposed requires less than half steps of
the Sgmse+ (2) and obtains a $0.2$ PESQ improvement. In fact, the scaling of
the signal, i.e., $\alpha_{t}$, on the right side of Eq.(1) is a type of data
augmentation that is beneficial for the model to learn the intrinsic structure
of clean speech. However, as shown in Fig.1, Sgmse+ (2) causes the neural
network to consider some noises in $\boldsymbol{y}$ as the clean when starting
from $\boldsymbol{y}+g(1)\boldsymbol{z}$ in the sampling stage. Therefore,
some noises in the noisy speech are not effaced thoroughly. From the two black
ellipses in Fig.1, we can see that the Sgmse+ (2) has residual noise in
$0-0.6$(s) and $2-2.1$(s) time intervals.

##  4 Conclusions

In this paper, we present the VP-based interpolation diffusion model in a
continuous time system and summarize the VE- and VP-based interpolation
diffusion models into a more concise framework called the IDM. The VE- and VP-
based interpolation diffusion models serve as examples of the IDM. While we
only apply the VPIDM to the SE task to showcase our proposed method, it is
worth noting that the approach is a general method that can be used for other
tasks as well.

##  5 Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China under Grant 62171427. We also thank Midea Group Co., Ltd for funding
this work

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# Testing for the extent of instability in 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 unit root issues in time series analysis. It has been
known for a long time that unit root tests may be flawed when a series
although stationary has a root close to unity. That motivated recent papers
dedicated to autoregressive processes where the bridge between stability and
instability is expressed by means of time-varying coefficients. In this vein
the process we consider has a companion matrix $A_{n}$ with spectral radius
$\rho(A_{n})<1$ satisfying $\rho(A_{n})\rightarrow 1$, a situation that we
describe as ‘nearly unstable’. The question we investigate is the following:
given an observed path supposed to come from a nearly-unstable process, is it
possible to test for the ‘extent of instability’, i.e. to test how close we
are to the unit root? In this regard, we develop a strategy to evaluate
$\alpha$ and to test for $\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ against
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ when $\rho(A_{n})$ lies in an inner
$O(n^{-\alpha})$-neighborhood of the unity, for some $0<\alpha<1$. Empirical
evidence is given (on simulations and real time series) about the advantages
of the flexibility induced by such a procedure compared to the usual unit root
tests and their binary responses. As a by-product, we also build a symmetric
procedure for the usually left out situation where the dominant root lies
around $-1$.

###### Key words and phrases:

Nearly unstable autoregressive process, Unit root test, Asymptotic behavior,
Martingales.

##  1\. Introduction and Motivation

This paper is devoted to the well-known unit root issue, still at the heart of
many concerns in finance, econometrics, and more generally in time series
analysis, at the origine, when undetected, of spurious regressions through
unreliable estimates. In order to investigate the stochastic non-stationarity
of an observed autoregressive path, two strategies are frequently encountered.
The first one consists in estimating the coefficient behind the (supposed)
unit root, and the second one directly looks for a hidden random walk in the
disturbance. In a very particular case, Dickey and Fuller (1979) first tackled
the unit root issue by considering an AR$(1)$ model with a linear trend and
concluded that, when there is a unit root in the generating process, the
least-squares estimator of the autoregressive parameter has an asymmetric
distribution that can be written as a functional of a family of detrended
Wiener processes, actually depending on the trend. Such a result had been
anticipated in the Gaussian case by White (1958) in the absence of trend and,
among numerous other studies, Chan and Wei (1988) improved the assumptions on
the noise by considering a sequence of martingale differences (with sufficient
moments). The behavior of the associated $t$-statistic has been tabulated by
Dickey and Fuller (1979) for testing purposes, see also MacKinnon (1991). The
ADF test (Augmented Dickey-Fuller) is probably the most commonly used nowadays
to evaluate the presence of a unit root in a causal ARMA$(p,q)$ process. It
was formalized by Said and Dickey (1981) when $p$ and $q$ are known, and by
Said and Dickey (1984) under a truncated approximation of the AR$(\infty)$
expression, therefore working for any $p$ and $q$. At the basis of their study
is the following differentiated form of an AR$(k)$ process with parameter
$\theta$,

(1.1) $\forall\,n\geqslant 1,\hskip 14.22636pt\Delta
X_{n}=(\theta_{0}-1)X_{n-1}+\sum_{i=1}^{k-1}\delta_{i}\,\Delta
X_{n-i}+\varepsilon_{n}$

where $\delta_{i}=-(\theta_{i+1}+\ldots+\theta_{k})$ and
$\theta_{0}=\theta_{1}+\ldots+\theta_{k}$, so that $(\Delta X_{n})$ is
stationary under the null hypothesis of unit root
$\mathcal{H}_{0}:``\theta_{0}=1"$ (actually trend-stationary if a trend is
added in the model). That was an inspiration for us, and we will find a
similar reasoning in this paper. The asymptotic behavior of the estimate is
still the one of the seminal work of Dickey and Fuller (1979) as long as
$k=O(n^{1/3})$ but the power of the test is impacted and, as it is explained
by Schwert (1989) or by Ng and Perron (1995), some distorsions may occur for
badly troncated processes. Let us also mention the non-parametric approach
suggested by Phillips (1987b) and deepened by Phillips and Perron (1988) in
which all correlation phenomenon are postponed to the noise of the model,
which is now supposed to be strongly mixing. Overall there is an abundant
literature on unit roots tests, we have just summarized here some important
topics. The interested reader will find in-depth studies for example in
Bhargava (1986), Perron (1988), Schmidt and Phillips (1992), etc. From a
practical point of view, Nelson and Plosser (1982) had already highlighted the
presence of unit roots in a number of macroeconomic series via the Dickey-
Fuller strategy. Although not directly related to this paper, we found it
useful to mention the complementary approach consisting in testing for trend-
stationarity instead of integration. The noise is now assumed to include a
random walk whose variance is zero under the null hypothesis and among such
procedures, the most famous seem to be the KPSS test of Kwiatkowski et al.
(1992) and the quite similar LMC test of Leybourne and McCabe (1994). Here
again there is an ample literature but too far from our scope to dwell on it,
let us instead return to unit roots.

In the context of autoregressive processes, the substantial works of Lai and
Wei (1983) and Chan and Wei (1988) helped to establish that the least-squares
estimator is strongly consistent wherever the characteristic roots lie, but
with very different convergence rates and limit distributions whether the
process is stationary (stable), integrated (unstable), or even explosive.
However, De Jong et al. (1992) observed that the unit root tests may be
empirically less powerful than their stable counterparts when, although very
close to it, the spectral radius remains less than 1, and some econometricians
are not fully convinced by the compatibility between the theoretical impacts
of this discontinuity – from $n^{1/2}$-normality to $n-$asymmetry while the
spectral radius continuously moves towards the unit root – and real-word data.
These are strong arguments in favor of attempts to bridge stability and
instability that motivated numerous studies devoted to intermediate models,
including ours. Most of them involve a first-order process with time-varying
coefficients $(X_{n,\,k})$, generated by the triangular form

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

This is an easy-to-use relevant case to understand the underlying dynamics
since to focus on the inner neighborhood of the unit root, we may only
consider a sequence of coefficients satisfying $|\theta_{n}|<1$ for all
$n\geqslant 1$ but $|\theta_{n}|\rightarrow 1$, so that the model is always
stable but increasingly close to instability. Chan and Wei (1987), Phillips
(1987a), Giraitis and Phillips (2006) and later Phillips and Magdalinos (2007)
were interested in such a model and, among other conclusions, it was
established that the least-squares estimator is asymptotically normal around
$\theta_{n}$ with rate $(nv_{n})^{1/2}$ when $\theta_{n}=1-c/v_{n}$, for any
$1\ll v_{n}\ll n$ and $c>0$. The junction is not perfect – there is still a
discontinuity in the asymptotic variance when $v_{n}=1$ and in the asymptotic
distribution when $v_{n}=n$ – but these are promising results. Indeed, by
setting $v_{n}=n^{\alpha}$ for some $0<\alpha<1$, the gap between stability
and instability is filled with rates of the form $n^{1/2}\ll
n^{(1+\alpha)/2}\ll n$. The recent studies of Park (2003), Buchmann and Chan
(2013), Proïa (2020) and Badreau and Proïa (2023) aimed at finding a kind of
generalization to nearly-unstable AR$(p)$ processes. In the latter, the
process $(\Phi_{n,\,k})$ is assumed to follow the VAR${}_{p}(1)$ triangular
form

(1.3) $\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636pt\Phi_{n,\,k}=A_{n}\,\Phi_{n,\,k-1}+E_{k}$

where $(E_{k})$ is a $p$-vectorial noise and

(1.4)
$A_{n}=\begin{pmatrix}\theta_{n,\,1}&\theta_{n,\,2}&\ldots&\theta_{n,\,p}\\\
\ddots&&&0\\\ &I_{p-1}&&\vdots\\\ &&\ddots&0\end{pmatrix}$

is the associated $p\times p$ companion matrix with spectral radius satisfying
$\rho(A_{n})<1$ for all $n\geqslant 1$ and $\rho(A_{n})\rightarrow 1$.
Provided suitable assumptions of moments, the least-squares estimator is still
shown to be consistent and asymptotically normal around $\theta_{n}$ with
mixed rates (see Thm. 3 and 4), and it is deduced as a corollary that, when
$\rho(A_{n})=1-c/v_{n}$,

(1.5)
$\sqrt{n\,v_{n}}\,\big{\langle}L_{n},\,\widehat{\theta}_{n}-\theta_{n}\big{\rangle}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,h_{0}^{2}\big{)}\hskip
14.22636pt\text{with}\hskip
14.22636ptL_{n}^{\,T}=\left(\lambda_{1}^{i-1}\rho(A_{n})^{1-i}\right)_{1\,\leqslant\,i\,\leqslant\,p}$

where $\lambda_{1}=\pm 1$ (depending on the sign of the unit root in the
unstable limit process) and the asymptotic variance $h_{0}^{2}>0$ is given as
a function of $c$ and the eigenvalues of $A$, the limit matrix of $A_{n}$.
However, although this result and the ones that may be found in the references
above are very useful for understanding the behavior of the process and the
estimator in the inner neighborhood of the unit circle, they definitely lack
practical application.

The question we wish to investigate in this paper is the following: given an
observed path supposed to come from a nearly-unstable process (e.g. doubtful
ADF/KPSS diagnoses), is it possible to test for the ‘extent of instability’,
i.e. to admit that the process is stable but to test how close we are to the
unit root? Setting $\rho(A_{n})=1-c/v_{n}$ with $v_{n}=n^{\alpha}$, the
question is clearly related to the choice of $\alpha$ and may reduce to a test
of $\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ against
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ for some test value $\alpha_{0}$. In
addition, evaluating $\alpha$ is likely to provide adjustments in the
confidence intervals associated with the autoregressive parameters. This is an
innovative approach of the unit root issue (to the best of our knowledge),
especially from a practical point of view. In Section 2, we give some specific
features of the model and the hypotheses before developing the test procedure
just mentioned. We will see that, as a corollary of our reasonings, we obtain
a version of the asymptotic normality for a corrected statistic closely
related to the least-squares estimation that does not need a mixing of the
rates matrix, actually deepening Badreau and Proïa (2023, Thm. 4). An
empirical study is provided in Section 3, first on simulated data and then on
real time series – the famous dataset coming from Nelson and Plosser (1982) –
in order to assess the performances of our procedure and justify its utility.
Finally, we prove our results in Section 4.

##  2\. A (Quasi-)Unit Root Test

Let us consider the triangular model (1.3), where

$\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636pt\Phi^{\,T}_{n,\,k}=\begin{pmatrix}X_{n,\,k}&\cdots&X_{n,\,k-p+1}\end{pmatrix}\hskip
14.22636pt\text{and}\hskip
14.22636ptE^{\,T}_{k}=\begin{pmatrix}\varepsilon_{k}&0&\cdots&0\end{pmatrix}.$

The process $(\varepsilon_{k})$ is assumed to be a white noise with finite
variance $\sigma^{2}>0$, and the arbitrary initial value $\Phi_{n,\,0}$ is
square-integrable and independent of the noise. The companion matrix $A_{n}$
of the process is given in (1.4) and has spectrum
$\\{\lambda_{n,\,i}\\}_{1\,\leqslant\,i\,\leqslant\,p}$ where
$\lambda_{n,\,1}$ is the one with largest modulus. The following technical
hypotheses are intended to reflect our understanding of ‘near instability’.

  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 eigenvalues111The complex eigenvalues are sorted according to
their modulus, in descending order, with ties broken by lexicographic order,
also in descending order.
$0\,<\,|\lambda_{p}|\,\leqslant\,\ldots\,\leqslant\,|\lambda_{2}|\,\leqslant\,|\lambda_{1}|\,=\,\rho(A)=1$,
and the top-right element of $A$ is non-zero ($\theta_{p}\neq 0$).

  2. (H2)

Number of unit roots. There is exactly one unit root in $A$ ($\lambda_{1}=\pm
1$ but $|\lambda_{2}|<1$).

  3. (H3)

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

$\forall\,n\geqslant 1,\hskip 14.22636pt\rho(A_{n})=1-\frac{c}{n^{\alpha}}$

for some $c>0$ and $0<\alpha<1$.

For any $d\geqslant 1$, the usual $d\times d$ matrices

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

will be frequently encountered and the first vector of the canonical basis of
$\mathbb{R}^{d}$ will be denoted by $e_{d}^{\,T}=(1,\,0,\,\ldots,\,0)$.

######  Remark 2.1.

For readability purposes, we prove the results as if the limit process had a
positive unit root (that is, $\lambda_{1}=1$), so that $\lambda_{n,\,1}$ is
equal222To be rigorous, this is true only for $n$ sufficiently large. to the
spectral radius $\rho_{n}=\rho(A_{n})$ of the companion matrix. Nevertheless
they remain valid for $\lambda_{1}=-1$, at the cost of some adjustments that
will be clarified alongside (see Section 4.6). The parameters $c$ and $\alpha$
have a very different role to play: $c$ is a variance parameter and is clearly
less important than $\alpha$, which is a rate parameter. That is the reason
why we assume that $c$ is known (e.g. $c=1$ in the empirical section) and why
we focus on $\alpha$.

Like Park (2003), it is convenient to express the process under the
hierarchical form

(2.2) $\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636pt\left\\{\begin{array}[]{lcl}X_{n,\,k}&=&\lambda_{n,\,1}\,X_{n,\,k-1}+V_{n,\,k}\\\
V_{n,\,k}&=&\beta_{n,\,1}\,V_{n,\,k-1}+\ldots+\beta_{n,\,p-1}\,V_{n,\,k-p+1}+\varepsilon_{k}\end{array}\right.$

where we merely set $V_{n,\,k}=\varepsilon_{k}$ for $p=1$. As it can be easily
seen by factoring the autoregressive polynomial of $(X_{n,\,k})$ by
$(1-\lambda_{n,\,1}z)$, the companion matrix $\bar{A}_{n}$ associated with
$(V_{n,\,k})$ has spectrum
$\\{\lambda_{n,\,i}\\}_{2\,\leqslant\,i\,\leqslant\,p}$. Under (H1) and (H2),
it is therefore a stable AR$(p-1)$ process with time-varying coefficients
$\beta_{n,\,i}\rightarrow\beta_{i}$, such that $\bar{A}_{n}\rightarrow\bar{A}$
and $\rho(\bar{A}_{n})\rightarrow\rho(\bar{A})=|\lambda_{2}|<1$. So the limit
process is itself stable and its (normalized) covariance matrix is given by

(2.3)
$\Sigma_{p-1}=\sum_{k\,\geqslant\,0}\bar{A}^{\,k}\,K_{p-1}\,(\bar{A}^{\,T})^{k}.$

But that also leads to another representation of the process, closely related
to (1.1) mentioned in the introduction,

(2.4) $\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636ptX_{n,\,k}=\lambda_{n,\,1}\,X_{n,\,k-1}+\sum_{i=1}^{p-1}\beta_{n,\,i}\left(X_{n,\,k-i}-\lambda_{n,\,1}\,X_{n,\,k-i-1}\right)+\varepsilon_{k}$

from which we will benefit in our following results. In consequence, it is
straightforward to see that
$\theta_{n}^{\,T}=(\theta_{n,\,1},\,\ldots,\,\theta_{n,\,p})$ and
$\beta_{n}^{\,T}=(\beta_{n,\,1},\,\ldots,\,\beta_{n,\,p-1})$ are such that

(2.5) $\forall n\geqslant 1,\hskip
14.22636pt\theta_{n}=J_{n}\,\beta_{n}+\lambda_{n,\,1}\,e_{p}\hskip
14.22636pt\text{where}\hskip 14.22636ptJ_{n}=\begin{pmatrix}1&&\\\
-\lambda_{n,\,1}&\ddots&\\\ &\ddots&1\\\ &&-\lambda_{n,\,1}\end{pmatrix}\hskip
14.22636pt(=0~{}\text{if $p=1$})$

is a rectangular matrix of size $p\times(p-1)$. To emphasize that the
‘artificial’ $V_{n,\,k}$ depends on $\alpha$, from now on we will use
$V_{n,\,k}(\alpha)$ instead and we will do the same for all confusing
notation. Accordingly, let

$\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636pt\Psi_{n,\,k}^{\,T}(\alpha)=\begin{pmatrix}X_{n,\,k}&V_{n,\,k}(\alpha)&\cdots&V_{n,\,k-p+2}(\alpha)\end{pmatrix}$

where the arbitrary initial value $\Psi_{n,\,0}(\alpha)$ is square-integrable,
so that from (2.4),

(2.6) $\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636ptX_{n,\,k}=\vartheta_{n}^{\,T}(\alpha)\,\Psi_{n,\,k-1}(\alpha)+\varepsilon_{k}$

where
$\vartheta_{n}^{\,T}(\alpha)=(\lambda_{n,\,1},\,\beta_{n,\,1},\,\ldots,\,\beta_{n,\,p-1})$.
Let also

(2.7) $S_{n,\,n}=\sum_{k=0}^{n}\Phi_{n,\,k}\,\Phi_{n,\,k}^{\,T}\hskip
14.22636pt\text{and}\hskip
14.22636ptT_{n,\,n}(\alpha)=\sum_{k=0}^{n}\Psi_{n,\,k}(\alpha)\,\Psi_{n,\,k}^{\,T}(\alpha)$

together with the rate matrix of size $p\times p$, that shall help to control
the convergence of $S_{n,\,n}$ and $T_{n,\,n}(\alpha)$, given by

(2.8)
$C_{n}(\alpha)=\textnormal{diag}\left(\frac{1}{1-\rho_{n}(\alpha)},\,1,\,\ldots,\,1\right).$

The sum of squares $\sum_{k}\|E_{k}\|^{2}$ in (1.3) is obviously minimized by

(2.9)
$\widehat{\theta}_{n}=S_{n,\,n-1}^{\,-1}\sum_{k=1}^{n}\Phi_{n,\,k-1}\,X_{n,\,k}$

whereas in (2.6), it is minimized (as a function of $\alpha$), by

(2.10)
$\widehat{\vartheta}_{n}(\alpha)=T_{n,\,n-1}^{\,-1}(\alpha)\sum_{k=1}^{n}\Psi_{n,\,k-1}(\alpha)\,X_{n,\,k}.$

Note that if $p=1$, then
$\widehat{\vartheta}_{n}(\alpha)=\widehat{\vartheta}_{n}$ coincides with
$\widehat{\theta}_{n}$. After this contextualization, we are now ready to
define a statistic that will be useful to achieve our objectives (recall that
$\lambda_{1}=\pm 1$ means that the limit process has a unit root located at
$\pm 1$), which is given, provided that
$\lambda_{1}\widehat{v}_{n}(\alpha)<1$, by

(2.11) $\forall\,n>1,\,\hskip 14.22636pt\widehat{\alpha}_{n}(\alpha)=\frac{\ln
c-\ln\big{(}1-\lambda_{1}\widehat{v}_{n}(\alpha)\big{)}}{\ln n}$

where $\widehat{v}_{n}(\alpha)=e_{p}^{\,T}\,\widehat{\vartheta}_{n}(\alpha)$
is the first component of $\widehat{\vartheta}_{n}(\alpha)$.

######  Theorem 2.1.

Assume that (H1), (H2) and (H3) hold, and that
$\mathbb{E}[|\varepsilon_{1}|^{\,2+\nu}]=\eta_{\nu}<+\infty$ for some $\nu>0$.
Then,

(2.12) $\forall\,0<\alpha<1,\hskip 14.22636pt(\ln
n)\,\sqrt{n^{1-\alpha}}\,\big{(}\widehat{\alpha}_{n}(\alpha)-\alpha)~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,2\,c^{-1}\pi_{11}^{-2}\big{)}$

where

(2.13)
$\pi_{11}=\frac{\lambda_{1}^{\,p-1}}{\prod_{j=2}^{p}(1-\lambda_{j})}\hskip
14.22636pt(=1~{}\text{if $p=1$}).$

###### Proof.

See Section 4.3. ∎

Now choose some $\alpha/2<\alpha_{0}<1$ (in particular, any
$1/2\leqslant\alpha_{0}<1$ is suitable), and consider the test statistic

(2.14) $Z_{n}^{\,2}(\alpha_{0})=\frac{c\,\widehat{\pi}_{n}^{\,2}}{2}\,(\ln
n)^{2}\,n^{1-\alpha_{0}}\,\big{(}\widehat{\alpha}_{n}(\alpha_{0})-\alpha_{0}\big{)}^{2}$

where

$\widehat{\pi}_{n}=\frac{\lambda_{1}^{\,p-1}}{\prod_{i=2}^{p}(1-\widehat{\lambda}_{n,\,i})}\hskip
14.22636pt(=1~{}\text{if $p=1$})$

and $\widehat{\lambda}_{n,\,i}$ is the $i$-th eigenvalue of the estimated
companion matrix $\widehat{A}_{n}$ built by plugging $\widehat{\theta}_{n}$
into $A_{n}$, ordered as indicated in the footnote 1. Whenever
$\widehat{\alpha}_{n}(\alpha_{0})$ is not defined for the reason mentioned
above, we set $Z_{n}^{\,2}(\alpha_{0})=+\infty$ by convention so that
$\mathcal{H}_{0}$ is rejected. The following result is devoted to the
asymptotic behavior of $Z_{n}^{\,2}$ under the null
$\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ as well as the alternative
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$.

######  Theorem 2.2.

Assume that (H1), (H2) and (H3) hold, and that
$\mathbb{E}[|\varepsilon_{1}|^{\,2+\nu}]=\eta_{\nu}<+\infty$ for some $\nu>0$.
Then, under $\mathcal{H}_{0}:``\alpha=\alpha_{0}"$, the test statistic (2.14)
satisfies

(2.15) $Z_{n}^{\,2}(\alpha_{0})~{}\overset{d}{\longrightarrow}~{}\chi_{1}^{2}$

where $\chi_{1}^{2}$ has a chi-square distribution with one degree of freedom.
If we additionally assume that $\alpha<2\alpha_{0}$, then under
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$,

(2.16) $Z_{n}^{\,2}(\alpha_{0})~{}\overset{p}{\longrightarrow}~{}+\infty.$

###### Proof.

See Section 4.4. ∎

From now on, we replace the assumption $\alpha<2\alpha_{0}$ by the stronger
one $\alpha_{0}\geqslant 1/2$ (this will be justified in the empirical
section). As a corollary, for some $\alpha_{0}\geqslant 1/2$, the rejection of
$\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ in favor of
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ will happen as soon as
$Z_{n}^{\,2}(\alpha_{0})>z_{1-\epsilon}$ where $z_{1-\epsilon}$ is the
quantile of order $1-\epsilon$ of the $\chi_{1}^{2}$ distribution. All the
empirical study of Section 3 is based on this rule. To evaluate the extent of
instability, we may also look for

(2.17)
$\widehat{\alpha}_{max}=\inf\,\\{\alpha_{0}\in\mathcal{A}\,:\,\text{$\mathcal{H}_{0}$
is not rejected}\\}$

where $\mathcal{A}\subset\,[1/2,1[$ is a grid of test values (by convention
$\inf\emptyset=+\infty$ so that our conclusion is that the series is
integrated if $\mathcal{H}_{0}$ is always rejected and $\alpha\geqslant 1/2$).
It remains to generalize the procedure to the less common situation where the
limit process has a negative unit root, that is $\lambda_{1}=-1$. Actually,
Theorems 2.1 and 2.2 are still valid: $\lambda_{1}$ already appears in the
first component of $\vartheta_{n}(\alpha)$ in (2.6), i.e.
$\lambda_{n,\,1}=\lambda_{1}\rho_{n}$ (see footnote 2), and in
$\widehat{\alpha}_{n}(\alpha)$ given in (2.11). The reader is referred to
Section 4.6 for more technical details. As an interesting consequence of our
study, it is now possible to build a consistent and asymptotically normal
estimator of $\theta_{n}$ with no mixing rates, deepening in that sense
Badreau and Proïa (2023, Thm. 4). On the basis of (2.5), let

(2.18)
$\widetilde{\theta}_{n}=\widehat{J}_{n}\,\widehat{\beta}_{n}+\widehat{v}_{n}\,e_{p}$

where $\widehat{J}_{n}$ is $J_{n}$ with $\lambda_{n,\,1}$ estimated by
$\widehat{v}_{n}=\widehat{v}_{n}(\alpha)$ and $\widehat{\beta}_{n}$ is
$\widehat{\vartheta}_{n}(\alpha)$ deprived of its first component (we drop the
dependency on $\alpha$ in the notation of that last part). Then, we have the
following result.

######  Corollary 2.3.

Assume that (H1), (H2) and (H3) hold, and that
$\mathbb{E}[|\varepsilon_{1}|^{\,2+\nu}]=\eta_{\nu}<+\infty$ for some $\nu>0$.
Then,

(2.19)
$\big{\|}\widetilde{\theta}_{n}-\theta_{n}\big{\|}~{}\overset{p}{\longrightarrow}~{}0\hskip
14.22636pt\text{and}\hskip
14.22636pt\sqrt{n}\,\big{(}\widetilde{\theta}_{n}-\theta_{n}\big{)}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,J\,\Sigma_{p-1}^{-1}J^{\,T}\big{)}$

where $\Sigma_{p-1}$ is given in (2.3) and

$J=\begin{pmatrix}1&&\\\ -\lambda_{1}&\ddots&\\\ &\ddots&1\\\
&&-\lambda_{1}\end{pmatrix}\hskip 14.22636pt(=0~{}\text{if $p=1$}).$

###### Proof.

See Section 4.5. ∎

The asymptotic normality is degenerate in the sense that
$J\,\Sigma_{p-1}^{-1}J^{\,T}$ is not invertible (as a $p\times p$ matrix of
rank at most $p-1$). This may be illustrated by the relation
$v_{p}^{\,T}\,J=0$ where
$v_{p}^{\,T}=(\lambda_{1}^{i-1})_{1\,\leqslant\,i\,\leqslant\,p}$ so that
(2.19) becomes

$\sqrt{n}\,v_{p}^{\,T}\big{(}\widetilde{\theta}_{n}-\theta_{n}\big{)}~{}\overset{p}{\longrightarrow}~{}0$

and a parallel can be made with (1.5). It is not surprising for $p=1$ since it
is well-known that in that case,
$|\widetilde{\theta}_{n}-\theta_{n}|=O_{p}(n^{-(1+\alpha)/2})$, see e.g.
Phillips and Magdalinos (2007).

######  Remark 2.2.

Strictly speaking, $\widetilde{\theta}_{n}$ – as well as
$\widehat{\theta}_{n}$ – is a consistent estimation of $\theta$, the limit of
$\theta_{n}$. But it can only be asymptotically normal under the stronger
hypothesis $\|\theta_{n}-\theta\|=o_{p}(n^{-1/2})$, which requires in
particular that $\alpha>1/2$.

##  3\. Empirical Study

This section is dedicated to the empirical performances of the test procedure
described in Theorem 2.2 and the selection rule (2.17).

###  3.1. On simulated data

Let $\rho_{n}=1-n^{-\alpha}=\pm\lambda_{n,\,1}$ with $1/2\leqslant\alpha<1$.
For each simulation, $\lambda_{n,\,2},\ldots,\lambda_{n,\,p}$ are randomly
chosen in the interval $[-\rho_{n}+0.1,\,\rho_{n}-0.1]$, and a process of size
$n$ is generated with a standard Gaussian noise. The test of
$\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ against
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ is done via Theorem 2.2 for
$\alpha_{0}\in\mathcal{A}=\\{1/2+k/50,\,k=0,\ldots,24\\}$. The rejection
frequencies are evaluated on the basis of 5000 replications for each
$\alpha_{0}$. The distribution of $Z_{n}^{\,2}(\alpha_{0})$ and the empirical
power of the test with 5% type I risk are represented on Figure 1
($\alpha=2/3$), Figure 2 ($\alpha=3/4$) and Figure 3 ($\alpha=4/5$), for
$\lambda_{1}=1$, $n=1000$ and $p\in\\{1,2,3,4\\}$. To get an idea of the
impact of smaller samples and a negative unit root, similar experiments are
conducted on Figure 4 ($\lambda_{1}=1$, $\alpha=4/5$, $p=1$, $n=250$) and
Figure 5 ($\lambda_{1}=-1$, $\alpha=4/5$, $p=1$, $n=1000$). The light-gray
items (at the right of the vertical line) correspond to $\alpha_{0}>\alpha$,
situations beyond the scope of our study that we decided anyway to incorporate
in the simulations for informational purposes. Finally, the distribution of
$\widehat{\alpha}_{max}$ over 5000 replications is outlined for
$\alpha\in\\{2/3,3/4,4/5\\}$ with $n=250$ and $n=1000$, on Figure 6. For the
latter simulations, we have adopted the rule (2.17) with 5% type I risk.

Figure 1. Empirical distribution of $Z_{n}^{\,2}(\alpha_{0})$ for $n=1000$,
$\lambda_{1}=1$, $\alpha=2/3$, $\alpha_{0}\in\mathcal{A}$ and
$p\in\\{1,2,3,4\\}$ on the left. The corresponding rejection frequencies of
$\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ against
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ (5% level) are represented on the
right. Figure 2. Empirical distribution of $Z_{n}^{\,2}(\alpha_{0})$ for
$n=1000$, $\lambda_{1}=1$, $\alpha=3/4$, $\alpha_{0}\in\mathcal{A}$ and
$p\in\\{1,2,3,4\\}$ on the left. The corresponding rejection frequencies of
$\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ against
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ (5% level) are represented on the
right. Figure 3. Empirical distribution of $Z_{n}^{\,2}(\alpha_{0})$ for
$n=1000$, $\lambda_{1}=1$, $\alpha=4/5$, $\alpha_{0}\in\mathcal{A}$ and
$p\in\\{1,2,3,4\\}$ on the left. The corresponding rejection frequencies of
$\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ against
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ (5% level) are represented on the
right.

From the simulations of Figures 1–3, we observe that $p$ has a negligible
influence on the asymptotic behaviors when $\lambda_{1}=1$. Under
$\mathcal{H}_{0}$, the rejection rate is $\approx 5\%$ as expected and the
power of the test is quite satisfying under $\mathcal{H}_{1}$ ($\geqslant
75\%$ for $\alpha_{0}\leqslant\alpha-0.2$, $\approx 100\%$ for
$\alpha_{0}\leqslant\alpha-0.3$). The decrease in the rejection rates is
rather similar for $\alpha_{0}$ growing from $1/2$ to $\alpha$, whether
$\alpha=2/3$, $\alpha=3/4$ or $\alpha=4/5$. However, we can see that the
trickier alternatives where $\alpha_{0}\leqslant\alpha-0.1$ are underestimated
($\approx 25\%$): this is due to the fine mesh of $\mathcal{A}$, it is
difficult to discern substantial differences in the statistic between values
of $\alpha_{0}$ that increases 0.02 by 0.02 and overall, the numerical results
are encouraging.

Figure 4. Empirical distribution of $Z_{n}^{\,2}(\alpha_{0})$ for $n=250$,
$\lambda_{1}=1$, $\alpha=4/5$, $\alpha_{0}\in\mathcal{A}$ and $p=1$ on the
left. The corresponding rejection frequencies of
$\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ against
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ (5% level) are represented on the
right.

In terms of sample sizes, the results are obviously impacted when we set
$n=250$ as can be seen on Figure 4 with $p=1$ and $\alpha=4/5$ ($\geqslant
50\%$ for $\alpha_{0}\leqslant\alpha-0.2$), but the global behavior follows
the same lines. The real time series of next subsection are even smaller
($n=80$ to $n=129$), that will mainly result in larger confidence intervals.

Figure 5. Empirical distribution of $Z_{n}^{\,2}(\alpha_{0})$ for $n=1000$,
$\lambda_{1}=-1$, $\alpha=4/5$, $\alpha_{0}\in\mathcal{A}$ and $p=1$ on the
left. The corresponding rejection frequencies of
$\mathcal{H}_{0}:``\alpha=\alpha_{0}"$ against
$\mathcal{H}_{1}:``\alpha>\alpha_{0}"$ (5% level) are represented on the
right.

Under the hypothesis that the limit process has a negative unit root, Figure 5
highlights the same efficiency but we must precise here that, unlike
$\lambda_{1}=1$, more experiments than those presented here showed a notable
degradation with $p$: the decrease is faster and the statistic falls below
$z_{1-\epsilon}$ too quickly. However, despite the lack of power observed for
$p\geqslant 2$, this remains an innovative approach since the ADF, KPSS, and
all usual tests for integration or trend-stationarity are unable to detect
instability coming from negative unit roots.

Figure 6. Empirical distribution of $\widehat{\alpha}_{max}$ (5% level) for
$n=250$ (left) and $n=1000$ (right), $\lambda_{1}=1$,
$\alpha\in\\{2/3,3/4,4/5\\}$, $\alpha_{0}\in\mathcal{A}$ and $p=1$.

As expected, the selected values of $\alpha$ on Figure 6 are underestimated,
on average: for $n=1000$, we can roughly say that $\widehat{\alpha}_{max}$ is
close to $(\alpha-0.10)$, yet greater than $(\alpha-0.2)$ with an overwhelming
probability. It is a direct consequence of the construction rule (2.17), and
of course an unbiased strategy may be an interesting trail for future studies
(even if the procedure is asymptotically unbiased). In particular, this bias
together with $\inf\mathcal{A}=1/2$ explains the initial peak on smaller
samples.

###  3.2. On a real dataset

The extended dataset NelPlo is available as open-
source333https://www.rdocumentation.org/packages/tseries/versions/0.9-9/topics/NelPlo
and contains 14 macroeconomic series ending in 1988. According to Nelson and
Plosser (1982), those data are known to be either integrated (in the sense of
the ADF test) or trend-stationary with spectral radius close to 1, so let us
approach them from the nearly-unstable point of view. It is important to note
that the data of NelPlo are longer than the original ones so that our
conclusions may be slightly different. The statistical protocol itself is
different: $p$ is chosen by looking at the partial autocorrelations of the
differentiated series, then $\widehat{\alpha}_{max}$ whose 95%-confidence
interval derives from Theorem 2.1 is selected according to (2.17), and no
trend is estimated. See Table 1 below for the numerical outcomes.

Series | $n$ | $p$ | Selected $\alpha$ | (Quasi-)unit root
---|---|---|---|---
Consumer prices | 129 | 4 | $+\infty$ | 1
Industrial production | 129 | 6 | $[0.50,\,0.87]$ | $[0.91,\,0.99]$
Nominal GNP | 80 | 2 | $[0.50,\,0.60]$ | $[0.89,\,0.93]$
Velocity | 120 | 1 | $[0.50,\,0.67]$ | $[0.91,\,0.96]$
Employment | 99 | 5 | $[0.50,\,0.65]$ | $[0.90,\,0.95]$
Interest rate | 89 | 4 | $[0.50,\,0.60]$ | $[0.89,\,0.93]$
Wages | 89 | 2 | $[0.50,\,0.59]$ | $[0.89,\,0.93]$
GNP deflator | 100 | 6 | $[0.50,\,0.54]$ | $[0.90,\,0.92]$
Money stock | 100 | 3 | $[0.50,\,0.53]$ | $[0.90,\,0.91]$
Real GNP | 80 | 4 | $[0.50,\,0.86]$ | $[0.89,\,0.98]$
Common stock prices | 118 | 6 | $+\infty$ | 1
Real per capita GNP | 80 | 2 | $[0.50,\,0.76]$ | $[0.89,\,0.96]$
Real wages | 89 | 2 | $[0.50,\,0.80]$ | $[0.89,\,0.97]$
Unemployment rate | 99 | 3 | $+\infty$ | 1
Table 1. NelPlo dataset from the nearly-unstable point of view. For each
series, $\widehat{\alpha}_{max}$ coming from (2.17) is given with a confidence
interval computed with a type I risk of 5%. The estimated spectral radius is
also precised.

Although the statistical context is different, our results are rather
consistent with Table 5 of Nelson and Plosser (1982). According to these
authors, all series except ‘Unemployment rate’ are integrated. In fact it
seems that among them, only ‘Consumer prices’ and ‘Common stock prices’ have a
unit root whereas the nearly-unstable approach may be suitable for the others.
To be complete, it should be noted that ‘Unemployment rate’ is a point of
divergence since our procedure concludes that it is integrated. Two possible
reasons for this: either it is nearly-unstable with $\alpha<1/2$ (in
accordance with the original study where the spectral radius is $0.706$), or
the change in the size (from $n=81$ to $n=99$) generates more instability.
This has not been investigated as part of this paper but, actually, this is
the main trail to improve the research on a broader scale. In other words it
would be interesting to unveil the limit behavior of $Z_{n}^{\,2}$ with no
constraints on $\alpha$. The light-gray boxplots of Figures 1–3 suggest that
$\mathcal{H}_{1}:``\alpha<\alpha_{0}"$ does not lead to a divergence but to
some kind of stabilization away from zero (in distribution ?). But a spectral
radius close to unity together with $\alpha<1/2$ must correspond to high
values of $n$, questioning the practical interest (a series with moderate size
is more likely stable in that context). On the other hand, when
$\alpha\geqslant 1/2$, a bilateral test would undoubtedly refine the selection
of $\widehat{\alpha}_{max}$ and contribute to eliminate the observed bias, so
this is an important issue. Another degree of freedom not exploited here that
would deserve an in-depth study is the estimation of $c$, especially when $n$
is small: in that case $c$ prevails over $\alpha$ and choosing it by cross-
validation or direct estimation seems wiser than imposing $c=1$. Also,
authorizing $p=p_{n}$ to grow to infinity could lead to the formalization of
nearly-unstable ARMA$(p,q)$ processes under an hypothesis of invertibility at
fixed $n$ and some restrictions on $p_{n}$ – like the strategy of ADF. The
results of this paper shall then probably be generalized without much
difficulty to that wider class of models.

##  4\. Technical Developments

To deal with martingale terms, we need to consider the triangular array of
filtrations

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

obviously satisfying
$\mathcal{F}_{n,\,0}\subseteq\mathcal{F}_{n,\,1}\subseteq\ldots\subseteq\mathcal{F}_{n,\,n}$
for each $n\geqslant 1$.

###  4.1. Known results

The two lemmas given in this section may be found with far more details in
Badreau and Proïa (2023).

######  Lemma 4.1.

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, $\|P_{n}\|$ and $\|P_{n}^{-1}\|$ are bounded and there exists an
invertible matrix $P$ such that $P_{n}\rightarrow P$ and
$P_{n}^{-1}\rightarrow P^{-1}$.

Note that because its spectrum is a subset of the one of $A_{n}$, the
companion matrix $\bar{A}_{n}$ associated with $(V_{n,\,k})$ is also
diagonalizable for all $n>n_{0}$ and the lemma still holds when replacing
$A_{n}$ by $\bar{A}_{n}$ and obviously $p$ by $p-1$. This will be a key
argument to justify (4.3) in the next section.

######  Lemma 4.2.

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

$(1-\rho_{n}(\alpha))\,\frac{S_{n,\,n}}{n}~{}\overset{p}{\longrightarrow}~{}\Gamma_{p}\,\overset{\Delta}{=}\,\sigma^{2}\ell\left(\lambda_{1}^{i+j}\right)_{1\,\leqslant\,i,j\,\leqslant\,p}\hskip
14.22636pt\text{with}\hskip 14.22636pt\ell=\frac{\pi_{11}^{2}}{2}>0,$

where $S_{n,\,n}$ is given in (2.7) and $\pi_{i1}$ ($i=1,\ldots,p$) are the
(non-zero) entries of the first column of $P^{-1}$. Moreover,

$C_{n}^{-\frac{1}{2}}(\alpha)\,P_{n}^{-1}\,\frac{S_{n,\,n}}{n}\,(P_{n}^{\,T})^{-1}C_{n}^{-\frac{1}{2}}(\alpha)~{}\overset{p}{\longrightarrow}~{}\sigma^{2}\,\begin{pmatrix}\frac{\pi_{11}^{2}}{2}&0&\ldots&0\\\
0&\frac{\pi_{21}^{2}}{1-\lambda_{2}^{2}}&\ldots&\frac{\pi_{21}\pi_{p1}}{1-\lambda_{2}\lambda_{p}}\\\
\vdots&\vdots&\ddots&\vdots\\\
0&\frac{\pi_{p1}\pi_{21}}{1-\lambda_{p}\lambda_{2}}&\ldots&\frac{\pi_{p1}^{2}}{1-\lambda_{p}^{2}}\end{pmatrix}\,\overset{\Delta}{=}\,H_{p}$

where $C_{n}(\alpha)$ is defined in (2.8). In addition, $H_{p}$ is invertible
(and positive definite if $A$ has only real eigenvalues).

###  4.2. Technical lemmas

######  Lemma 4.3.

Assume that (H1), (H2) and (H3) hold and that
$\mathbb{E}[\varepsilon_{1}^{\,2}]=\sigma^{2}<+\infty$. Then, as $n$ tends to
infinity, for any $\alpha/2<\alpha_{0}\leqslant\alpha$,

$C_{n}^{-\frac{1}{2}}(\alpha)\,\frac{T_{n,\,n}(\alpha_{0})}{n}\,C_{n}^{-\frac{1}{2}}(\alpha)~{}\overset{p}{\longrightarrow}~{}\Upsilon_{p}\,\overset{\Delta}{=}\,\sigma^{2}\,\textnormal{diag}\left(\ell,\,\Sigma_{p-1}\right)$

where $T_{n,\,n}(\alpha)$ is defined in (2.7), $C_{n}(\alpha)$ in (2.8), and
where $\ell$ and $\Sigma_{p-1}$ are defined in Lemma 4.2 and (2.3),
respectively. In addition, $\Upsilon_{p}$ is positive definite.

###### Proof.

Note that $S_{n,\,n}$ and $T_{n,\,n}(\alpha)$ share the same top-left element
(which actually does not depend on $\alpha$), so we directly get

(4.2)
$\frac{1-\rho_{n}(\alpha)}{n}\sum_{k=0}^{n}X^{2}_{n,\,k}=\frac{c}{n^{1+\alpha}}\sum_{k=0}^{n}X^{2}_{n,\,k}~{}\overset{p}{\longrightarrow}~{}\sigma^{2}\ell.$

Moreover, coming back to the reasoning of Badreau and Proïa (2023), especially
Lem. 13–15 and Prop. 2, and adapting it to the present case (see the remark
after Lemma 4.1),

(4.3)
$\frac{1}{n}\sum_{k=0}^{n}\Lambda_{n,\,k}(\alpha)\,\Lambda_{n,\,k}^{\,T}(\alpha)~{}\overset{p}{\longrightarrow}~{}\sigma^{2}\,\Sigma_{p-1}\hskip
14.22636pt\text{where}\hskip
14.22636pt\Lambda_{n,\,k}^{\,T}(\alpha)=\begin{pmatrix}V_{n,\,k}(\alpha)&\cdots&V_{n,\,k-p+2}(\alpha)\end{pmatrix}.$

Indeed, the equivalence

$\frac{1}{n}\sum_{k=0}^{n}\Lambda_{n,\,k}(\alpha)\,\Lambda_{n,\,k}^{\,T}(\alpha)~{}\overset{p}{\sim}~{}\left(\sum_{k\,\geqslant\,0}\bar{A}_{n}^{\,k}\,K_{p-1}\,(\bar{A}_{n}^{\,T})^{\,k}\right)\frac{1}{n}\sum_{k=1}^{n}\varepsilon_{k}^{\,2},$

the law of large numbers together with the fact that $\rho(\bar{A}_{n})<1$ and
$\rho(\bar{A})<1$ lead to (4.3). It remains to treat the (symmetrical) non-
diagonal blocks. First,

$\displaystyle\sum_{k=0}^{n}\Lambda^{\\!(p)}_{n,\,k}(\alpha)\,X_{n,\,k}$
$\displaystyle=$
$\displaystyle\Lambda^{\\!(p)}_{n,\,0}(\alpha)X_{n,\,0}+\left[\sum_{k=1}^{n}\left(\Phi_{n,\,k}-\rho_{n}(\alpha)\,\Phi_{n,\,k-1}\right)\Phi^{\,T}_{n,\,k}\right]e_{p}$
$\displaystyle=$
$\displaystyle\left[S_{n,\,n}\left(I_{p}-\rho_{n}(\alpha)A_{n}^{\,T}\right)-\rho_{n}(\alpha)M_{n}+r_{n}\right]e_{p}$

where $\Lambda^{\\!(p)}_{n,\,k}$ is the $p$-dimensional version of
$\Lambda_{n,\,k}$ (with last element $V_{n,\,k-p+1}(\alpha)$),

$M_{n}=\sum_{k=1}^{n}\Phi_{n,\,k-1}\,E^{\,T}_{k}$

and $r_{n}$ is made of isolated terms. It is already proven (Lem. 14–15) that
$\|M_{n}\|$ and $\|r_{n}\|$ are $o_{p}(n)$. As an intermediate result, we can
see that for $n>n_{0}$,

$\displaystyle
C_{n}^{\frac{1}{2}}(\alpha)\,P_{n}^{\,T}\left(I_{p}-\rho_{n}(\alpha)A_{n}^{\,T}\right)$
$\displaystyle=$ $\displaystyle
C_{n}^{\frac{1}{2}}(\alpha)\left(I_{p}-\rho_{n}(\alpha)D_{n}\right)P_{n}^{\,T}$
$\displaystyle~{}\overset{p}{\longrightarrow}~{}$
$\displaystyle\textnormal{diag}\left(0,(1-\lambda_{i})_{2\,\leqslant\,i\,\leqslant\,p}\right)P^{\,T}\,\overset{\Delta}{=}\,\Delta_{p}\,P^{\,T}.$

Hence,

(4.6)
$\displaystyle\frac{\sqrt{1-\rho_{n}(\alpha)}}{n}\,S_{n,\,n}\left(I_{p}-\rho_{n}(\alpha)A_{n}^{\,T}\right)$
$\displaystyle=$ $\displaystyle
P_{n}\,\frac{C_{n}^{\frac{1}{2}}(\alpha)\sqrt{c}}{n^{\frac{\alpha}{2}}}\left[C_{n}^{-\frac{1}{2}}(\alpha)\,P_{n}^{-1}\,\frac{S_{n,\,n}}{n}\,(P_{n}^{\,T})^{-1}C_{n}^{-\frac{1}{2}}(\alpha)\right]$
$\displaystyle\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt\times\left[C_{n}^{\frac{1}{2}}(\alpha)\,P_{n}^{\,T}\left(I_{p}-\rho_{n}(\alpha)A_{n}^{\,T}\right)\right]$
$\displaystyle~{}\overset{p}{\longrightarrow}~{}$ $\displaystyle
P\,K_{p}\,H_{p}\,\Delta_{p}\,P^{\,T}$

where the block diagonal matrix $H_{p}$ is given in Lemma 4.2 and $\Delta$ is
defined above. But $K_{p}$, $H_{p}$ and $\Delta_{p}$ have particular
structures consisting of zeros in key locations and implying the following
pattern,

$P\,K_{p}\,H_{p}\,\Delta_{p}=\begin{pmatrix}*&0&\ldots&0\\\
\vdots&\vdots&&\vdots\\\ \vdots&\vdots&&\vdots\\\
*&0&\ldots&0\end{pmatrix}\begin{pmatrix}0&\ldots&\ldots&0\\\
\vdots&*&\ldots&*\\\ \vdots&\vdots&&\vdots\\\ 0&*&\ldots&*\end{pmatrix}=0.$

The combination of (4.2), (4.3), (4.6) and the remarks that follow lead to the
convergence

(4.7)
$C_{n}^{-\frac{1}{2}}(\alpha)\,\frac{T_{n,\,n}(\alpha)}{n}\,C_{n}^{-\frac{1}{2}}(\alpha)~{}\overset{p}{\longrightarrow}~{}\Upsilon_{p}$

that is, when $\alpha_{0}$ takes the ‘true’ value $\alpha$. We now wish to
generalize the result to $T_{n,\,n}(\alpha_{0})$ for any
$\alpha/2<\alpha_{0}\leqslant\alpha$. Regarding this, note that

(4.8)
$\Psi_{n,\,k}(\alpha_{0})=\Psi_{n,\,k}(\alpha)+c\left(n^{-\alpha_{0}}-n^{-\alpha}\right)\dot{\Phi}_{n,\,k}$

where

$\forall\,n\geqslant 1,\,\forall\,1\leqslant k\leqslant n,\hskip
14.22636pt\dot{\Phi}^{\,T}_{n,\,k}=\begin{pmatrix}0&X_{n,\,k-1}&\cdots&X_{n,\,k-p+1}\end{pmatrix}.$

Thus, we have the decomposition

(4.9) $T_{n,\,n}(\alpha_{0})=T_{n,\,n}(\alpha)+R_{n}$

with

$\displaystyle R_{n}$ $\displaystyle=$ $\displaystyle
c\left(n^{-\alpha_{0}}-n^{-\alpha}\right)\left[\sum_{k=0}^{n}\Psi_{n,\,k}(\alpha)\,\dot{\Phi}_{n,\,k}^{\,T}+\sum_{k=0}^{n}\dot{\Phi}_{n,\,k}\Psi_{n,\,k}^{\,T}(\alpha)\right]$
$\displaystyle\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt+~{}c^{2}\left(n^{-\alpha_{0}}-n^{-\alpha}\right)^{2}\sum_{k=0}^{n}\dot{\Phi}_{n,\,k}\,\dot{\Phi}_{n,\,k}^{\,T}~{}\,\overset{\Delta}{=}\,~{}R_{n,\,1}+R_{n,\,1}^{\,T}+R_{n,\,2}.$

The direct computation shows that
$\Psi_{n,\,k}(\alpha)\,\dot{\Phi}_{n,\,k}^{\,T}$ contains a column of zeros, a
line made of $X_{n,\,k}\,X_{n,\,k-i}$ ($1\leqslant i\leqslant p-1$), and a
$(p-1)\times(p-1)$ block of terms of the form
$V_{n,\,k-j}(\alpha)\,X_{n,\,k-i}$ ($0\leqslant j\leqslant p-2$, $1\leqslant
i\leqslant p-1$). For $2\leqslant i\leqslant p-1$, according to the first
expression in (2.2),

$\displaystyle\rho_{n}^{2i}(\alpha)\,X_{n,\,k-i}$
$\displaystyle=\rho_{n}^{2i-1}(\alpha)\left(X_{n,\,k-i+1}-V_{n,\,k-i+1}(\alpha)\right)$
$\displaystyle=\rho_{n}^{i+1}(\alpha)\,X_{n,\,k-1}-\sum_{j=1}^{i-1}\rho_{n}^{2i-j}(\alpha)\,V_{n,\,k-i+j}(\alpha)$

which entails that, for any $0\leqslant u\leqslant p-2$,

(4.10)
$\displaystyle\rho_{n}^{2i}(\alpha)\sum_{k=1}^{n}V_{n,\,k-u}(\alpha)\,X_{n,\,k-i}$
$\displaystyle=$
$\displaystyle\rho_{n}^{i+1}(\alpha)\sum_{k=1}^{n}V_{n,\,k-u}(\alpha)\,X_{n,\,k-1}$
$\displaystyle\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt-~{}\sum_{j=1}^{i-1}\rho_{n}^{2i-j}(\alpha)\sum_{k=1}^{n}V_{n,\,k-u}(\alpha)\,V_{n,\,k-i+j}(\alpha).$

The first part of this equation is $o_{p}(n^{1+\alpha/2})$ and the second part
is a finite sum of $O_{p}(n)$, see for instance (4.7). Thus, via Lemma 4.2,

(4.11)
$\left\|C_{n}^{-\frac{1}{2}}(\alpha)\,\frac{R_{n,\,1}}{n}\,C_{n}^{-\frac{1}{2}}(\alpha)\right\|=O_{p}\left(\frac{n^{-\alpha_{0}}-n^{-\alpha}}{n^{1+\frac{\alpha}{2}}}\,n^{1+\alpha}\right)=o_{p}(1)$

as soon as $\alpha_{0}>\alpha/2$. In the same way, $R_{n,\,2}$ is closely
related to the covariance matrix $S_{n,\,n}$ but with zeros on the first line
and column. So,

(4.12)
$\left\|C_{n}^{-\frac{1}{2}}(\alpha)\,\frac{R_{n,\,2}}{n}\,C_{n}^{-\frac{1}{2}}(\alpha)\right\|=O_{p}\left(\frac{(n^{-\alpha_{0}}-n^{-\alpha})^{2}}{n}\,n^{1+\alpha}\right)=o_{p}(1)$

provided that $\alpha_{0}>\alpha/2$. Using (4.7), (4.9), (4.11) and (4.12),
the convergence is now established. Since $\ell>0$ (Lemma 4.2), the positive-
definiteness of $\Upsilon_{p}$ relies on the invertibility of $\Sigma_{p-1}$,
which can be found e.g. in Brockwell and Davis (1991) in the context of a
stable AR$(p-1)$ process with $\sigma^{2}>0$. ∎

######  Lemma 4.4.

Assume that (H1), (H2) and (H3) hold and that
$\mathbb{E}[|\varepsilon_{1}|^{\,2+\nu}]=\eta_{\nu}<+\infty$. Then, as $n$
tends to infinity,

(4.13)
$\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha)\left(\widehat{\vartheta}_{n}(\alpha)-\vartheta_{n}(\alpha)\right)~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,\sigma^{2}\,\Upsilon_{p}^{-1}\big{)}$

where the limit covariance $\Upsilon_{p}$ is given in Lemma 4.3. Moreover, for
any $\alpha/2<\alpha_{0}<\alpha$,

(4.14)
$\left\|\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha_{0})\left(\widehat{\vartheta}_{n}(\alpha_{0})-\vartheta_{n}(\alpha_{0})\right)\right\|~{}\overset{p}{\longrightarrow}~{}+\infty.$

###### Proof.

Developing
$X_{n,\,k}=\vartheta_{n}^{\,T}(\alpha)\Psi_{n,\,k-1}(\alpha)+\varepsilon_{k}$
in (2.10), the estimation error satisfies

(4.15)
$\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha)\left(\widehat{\vartheta}_{n}(\alpha)-\vartheta_{n}(\alpha)\right)=\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha)\,T_{n,\,n-1}^{\,-1}(\alpha)\sum_{k=1}^{n}\Psi_{n,\,k-1}(\alpha)\,\varepsilon_{k}.$

The sequence $(\Psi_{n,\,k-1}(\alpha)\,\varepsilon_{k})_{n,\,k}$ is a
martingale-difference array w.r.t. $\mathcal{F}_{n,\,k}$ defined in (4.1) so,
for all $u\in\mathbb{R}^{p}\backslash\\{0\\}$,

(4.16)
$M_{n,\,n}^{(u)}=u^{\,T}\,\frac{C_{n}^{-\frac{1}{2}}(\alpha)}{\sqrt{n}}\sum_{k=1}^{n}\Psi_{n,\,k-1}(\alpha)\,\varepsilon_{k}$

is a martingale array having predictable quadratic variation

(4.17) $\displaystyle\langle M^{(u)}\rangle_{n,\,n}$ $\displaystyle=$
$\displaystyle\sigma^{2}\,u^{\,T}C_{n}^{-\frac{1}{2}}(\alpha)\,\frac{T_{n,\,n-1}(\alpha)}{n}\,C_{n}^{-\frac{1}{2}}(\alpha)\,u$
$\displaystyle~{}\overset{p}{\longrightarrow}~{}$
$\displaystyle\sigma^{2}\,u^{\,T}\Upsilon_{p}\,u>0$

by the convergence in Lemma 4.3 (with $\alpha_{0}=\alpha$) and the positive
definiteness of $\Upsilon_{p}$. It is now necessary to establish the
Lindeberg’s condition in order to apply the central limit theorem for arrays
of martingales (see e.g. Thm. 1 of Sec. 8 of Pollard (1984)), namely that

(4.18) $\forall\,\epsilon>0,\hskip
14.22636pt\sum_{k=1}^{n}\mathbb{E}\left[(\Delta
M_{n,\,k}^{(u)})^{2}\,\mathbbm{1}_{\\{|\Delta
M_{n,\,k}^{(u)}|\,>\,\epsilon\\}}\,|\,\mathcal{F}_{n,\,k-1}\right]~{}\overset{p}{\longrightarrow}~{}0$

where $\Delta M_{n,\,k}^{(u)}=M_{n,\,k}^{(u)}-M_{n,\,k-1}^{(u)}$. But for any
$1\leqslant k\leqslant n$,

(4.19) $\mathbb{E}\big{[}(\Delta M_{n,\,k}^{(u)})^{2}\,\mathbbm{1}_{\\{|\Delta
M_{n,\,k}^{(u)}|\,>\,\epsilon\\}}\,|\,\mathcal{F}_{n,\,k-1}\big{]}~{}=~{}\varphi^{(u)}_{n,\,k-1}(\alpha)\,\frac{\xi_{n,\,k}}{n}~{}\leqslant~{}\varphi^{(u)}_{n,\,k-1}(\alpha)\,\frac{\xi^{*}_{n}}{n}$

where
$\varphi^{(u)}_{n,\,k-1}(\alpha)=u^{\,T}C_{n}^{-1/2}(\alpha)\,\Psi_{n,\,k-1}(\alpha)\,\Psi^{\,T}_{n,\,k-1}(\alpha)\,C_{n}^{-1/2}(\alpha)\,u$,

$\xi_{n,\,k}=\mathbb{E}\big{[}\varepsilon_{k}^{\,2}\,\mathbbm{1}_{\\{|\Delta
M_{n,\,k}^{(u)}|\,>\,\epsilon\\}}\,|\,\mathcal{F}_{n,\,k-1}\big{]}\hskip
14.22636pt\text{and}\hskip
14.22636pt\xi^{*}_{n}=\sup_{1\,\leqslant\,k\,\leqslant\,n}\xi_{n,\,k}.$

Applying Hölder’s and Markov’s inequalities,

$\displaystyle\forall\,1\leqslant k\leqslant n,\hskip 14.22636pt\xi_{n,\,k}$
$\displaystyle\leqslant$
$\displaystyle\eta_{\nu}^{\frac{2}{2+\nu}}\,\mathbb{P}\big{(}(\Delta
M_{n,\,k}^{(u)})^{2}>\epsilon^{2}\,|\,\mathcal{F}_{n,\,k-1}\big{)}^{\frac{\nu}{2+\nu}}$
$\displaystyle\leqslant$
$\displaystyle\eta_{\nu}^{\frac{2}{2+\nu}}\left(\frac{\sigma^{2}\,\varphi^{(u)}_{n,\,k-1}(\alpha)}{\epsilon^{2}\,n}\right)^{\\!\frac{\nu}{2+\nu}}~{}\overset{p}{\longrightarrow}~{}0$

from Lemma 4.3 (with $\alpha_{0}=\alpha$), where $\eta_{\nu}$ is the moment of
order $2+\nu$ of the white noise $(\varepsilon_{k})$. Thus
$\xi^{*}_{n}=o_{p}(1)$, see e.g. Lem. 1.3.20 of Duflo (1997). Together with
(4.19) and again Lemma 4.3, (4.18) follows. Consequently, the martingale
defined in (4.16) satisfies

$M_{n,\,n}^{(u)}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}(0,\,\sigma^{2}\,u^{\,T}\Upsilon_{p}\,u)$

and because that holds for any $u\neq 0$, by the Cramér-Wold device,

(4.20)
$\frac{C_{n}^{-\frac{1}{2}}(\alpha)}{\sqrt{n}}\sum_{k=1}^{n}\Psi_{n,\,k-1}(\alpha)\,\varepsilon_{k}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,\sigma^{2}\,\Upsilon_{p}).$

Combined with (4.17), Slutsky’s lemma and (4.15),

$\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha)\left(\widehat{\vartheta}_{n}(\alpha)-\vartheta_{n}(\alpha)\right)~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,\sigma^{2}\,\Upsilon_{p}^{-1}\big{)}.$

That concludes the proof of the first part of the lemma. For the other part,
note that using (2.6), (4.8) and (2.10), we have the decomposition

(4.21)
$\displaystyle\widehat{\vartheta}_{n}(\alpha_{0})-\vartheta_{n}(\alpha_{0})$
$\displaystyle=\left[\vartheta_{n}(\alpha)-\vartheta_{n}(\alpha_{0})\right]+T_{n,\,n-1}^{\,-1}(\alpha_{0})\sum_{k=1}^{n}\Psi_{n,\,k-1}(\alpha_{0})\,\varepsilon_{k}$
$\displaystyle\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt+~{}c\left(n^{-\alpha}-n^{-\alpha_{0}}\right)T_{n,\,n-1}^{\,-1}(\alpha_{0})\sum_{k=1}^{n}\Psi_{n,\,k-1}(\alpha_{0})\,\dot{\Phi}_{n,\,k-1}^{\,T}\,\vartheta_{n}(\alpha)$
$\displaystyle\,\overset{\Delta}{=}\,\tilde{R}_{n,\,1}+\tilde{R}_{n,\,2}+\tilde{R}_{n,\,3}.$

The reasoning we have just developed also applies to $\tilde{R}_{n,\,2}$ via
the same lines and tools, and as a result we obtain the convergence

(4.22)
$\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha)\,\tilde{R}_{n,\,2}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,\sigma^{2}\,\Upsilon_{p}^{-1}\big{)}.$

That leads to the first intermediate step,

(4.23)
$\left|e_{p}^{\,T}\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha_{0})\,\tilde{R}_{n,\,2}\right|~{}=~{}\left|\sqrt{c\,n^{\alpha_{0}-\alpha}}\,e_{p}^{\,T}\,\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha)\,\tilde{R}_{n,\,2}\right|~{}=~{}o_{p}(1)$

as soon as $\alpha_{0}<\alpha$. We can consider the rest of the decomposition
similarly, that is

(4.24)
$\left|e_{p}^{\,T}\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha_{0})\left(\tilde{R}_{n,\,1}+\tilde{R}_{n,\,3}\right)\right|=\left|\,\sqrt{c\,n^{1-\alpha_{0}}}\left(1-n^{\alpha_{0}-\alpha}\right)\right||1-u_{n}|$

where the first part is clearly divergent when $\alpha_{0}<\alpha$ and

$u_{n}\,\overset{\Delta}{=}\,e_{p}^{\,T}\,T_{n,\,n-1}^{\,-1}(\alpha_{0})\left[\sum_{k=1}^{n}\Psi_{n,\,k-1}(\alpha_{0})\,\dot{\Phi}_{n,\,k-1}^{\,T}\right]\vartheta_{n}(\alpha).$

As in the proof of Lemma 4.3, it is easy to see that
$\Psi_{n,\,k-1}(\alpha_{0})\,\dot{\Phi}_{n,\,k-1}^{\,T}$ is a $p\times p$
matrix made of a first column of zeros, terms of the form
$X_{n,\,k-1}\,X_{n,\,k-j}$ on the first line and
$V_{n,\,k-i}(\alpha_{0})\,X_{n,\,k-j}$ in the bottom-right block $(1\leqslant
i\leqslant p-1,\,2\leqslant j\leqslant p)$, broken down as

$\displaystyle\rho_{n}^{2j}(\alpha)\sum_{k=1}^{n}V_{n,\,k-i}(\alpha_{0})\,X_{n,\,k-j}$
$\displaystyle=$
$\displaystyle\rho_{n}^{j+1}(\alpha)\sum_{k=1}^{n}V_{n,\,k-i}(\alpha_{0})\,X_{n,\,k-1}$
$\displaystyle\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt-~{}\sum_{s=1}^{j-1}\rho_{n}^{2j-s}(\alpha)\sum_{k=1}^{n}V_{n,\,k-i}(\alpha_{0})\,V_{n,\,k-j+s}(\alpha)$

as it is done in (4.10). As soon as $\alpha/2<\alpha_{0}<\alpha$, Lemma 4.3
entails that the first part of this equation is $o_{p}(n^{1+\alpha/2})$ and,
using Cauchy–Schwarz inequality,

$\left|\sum_{k=1}^{n}V_{n,\,k-i}(\alpha_{0})\,V_{n,\,k-j+l}(\alpha)\right|~{}\leqslant~{}\sqrt{\sum_{k=1}^{n}V^{\,2}_{n,\,k-i}(\alpha_{0})}\,\sqrt{\sum_{k=1}^{n}V^{\,2}_{n,\,k-j+l}(\alpha)}$

where both sums are $O_{p}(n)$. Finally, it is straightforward to obtain the
simplification

(4.25)
$u_{n}~{}\overset{p}{\longrightarrow}~{}e_{p}^{\,T}\,\Upsilon_{p}^{-1}\,\sigma^{2}\,\ell\,\begin{pmatrix}0&1&\ldots&1\\\
\vdots&0&\ldots&0\\\ \vdots&\vdots&&\vdots\\\ 0&0&\ldots&0\\\
\end{pmatrix}\,\begin{pmatrix}1\\\ \vdots\\\ \beta_{i}\\\ \vdots\\\
\end{pmatrix}~{}=~{}\sum_{i=1}^{p-1}\beta_{i}\,\overset{\Delta}{=}\,s_{\beta}$

where $\ell$ and $\Upsilon$ are defined in Lemmas 4.2 and 4.3, respectively.
Note that $1-s_{\beta}$ is obviously non-zero since by stability, the
polynomial $\Theta_{\beta}(z)=1-\beta_{1}z-\ldots-\beta_{p-1}z^{p-1}$ has no
zero on the unit circle. Going back to (4.23) and (4.24), we then have

(4.26)
$\left|e_{p}^{\,T}\sqrt{n}\,C_{n}^{\frac{1}{2}}(\alpha_{0})\left(\widehat{\vartheta}_{n}(\alpha_{0})-\vartheta_{n}(\alpha_{0})\right)\right|~{}\overset{p}{\longrightarrow}~{}+\infty$

which concludes the proof. ∎

###  4.3. Proof of Theorem 2.1.

Taking the first line of the convergence in Lemma 4.4,

(4.27)
$\sqrt{n^{1+\alpha}}\,\big{(}\widehat{v}_{n}(\alpha)-\rho_{n}(\alpha)\big{)}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,c\,\ell^{-1}\big{)}$

where we recall that $\widehat{v}_{n}(\alpha)$ is the first component of
$\widehat{\vartheta}_{n}(\alpha)$, $\rho_{n}(\alpha)=1-c\,n^{-\alpha}$ and
$\ell>0$ is given in Lemma 4.2. Now define the sequence of
$\mathcal{C}^{1}$-mappings from $]0,\,1[$ to $\mathbb{R}$ given by

$\forall\,n>1,\hskip 14.22636ptg_{n}(x)=\frac{\ln c-\ln\left(1-x\right)}{\ln
n}$

so that $\alpha=g_{n}(\rho_{n}(\alpha))$ and
$\widehat{\alpha}_{n}(\alpha)=g_{n}(\widehat{v}_{n}(\alpha))$. Using Taylor’s
expansion, there is a sequence $(\vartheta^{*}_{n})$ such that

$\widehat{\alpha}_{n}(\alpha)=\alpha+\big{(}\widehat{v}_{n}(\alpha)-\rho_{n}(\alpha)\big{)}\,g_{n}^{\prime}(\vartheta^{*}_{n})$

with, for all $n>1$,
$(\rho_{n}(\alpha)\wedge\widehat{v}_{n}(\alpha))<\vartheta^{*}_{n}<(\rho_{n}(\alpha)\vee\widehat{v}_{n}(\alpha))$.
So,

(4.28) $(\ln
n)\,\sqrt{n^{1-\alpha}}\,\left(\widehat{\alpha}_{n}(\alpha)-\alpha\right)=\sqrt{n^{1+\alpha}}\,\big{(}\widehat{v}_{n}(\alpha)-\rho_{n}(\alpha)\big{)}\,\frac{n^{-\alpha}}{1-\vartheta^{*}_{n}}.$

We also note that

$n^{\alpha}\,\big{(}1-\rho_{n}(\alpha)\vee\widehat{v}_{n}(\alpha)\big{)}~{}<~{}n^{\alpha}\,\big{(}1-\vartheta^{*}_{n}\big{)}~{}<~{}n^{\alpha}\,\big{(}1-\rho_{n}(\alpha)\wedge\widehat{v}_{n}(\alpha)\big{)}$

and

$\displaystyle n^{\alpha}\,\big{(}1-\widehat{v}_{n}(\alpha)\big{)}$
$\displaystyle=$ $\displaystyle
n^{\alpha}\,\big{(}1-\rho_{n}(\alpha)\big{)}+n^{\alpha}\,\big{(}\rho_{n}(\alpha)-\widehat{v}_{n}(\alpha)\big{)}$
$\displaystyle=$ $\displaystyle
c+O_{p}\big{(}\sqrt{n^{\alpha-1}}\big{)}~{}\overset{p}{\longrightarrow}~{}c$

from Lemma 4.4 and since $\alpha<1$. That clearly implies that

(4.29)
$n^{\alpha}\,\big{(}1-\vartheta^{*}_{n}\big{)}~{}\overset{p}{\longrightarrow}~{}c.$

The result follows from (4.27), (4.28) and (4.29) together with Slutsky’s
lemma. ∎

###  4.4. Proof of Theorem 2.2.

First, a direct computation of the entries of $P^{-1}$ shows that

$\pi_{11}=\frac{1}{\prod_{j=2}^{p}(1-\lambda_{j})}\hskip
14.22636pt(=1~{}\text{if $p=1$}),$

see (2.13) with $\lambda_{1}=1$. We know by Rem. 2.2 in Badreau and Proïa
(2023) that the OLS of $\theta_{n}$ defined in (2.9) is consistent for
$\theta$, so obviously

$\widehat{A}_{n}~{}\overset{p}{\longrightarrow}~{}A$

also holds. Since the eigenvalues of a square matrix are continuous functions
of its entries, see e.g. Thm. 2.4.9.2 of Horn and Johnson (2012), it follows
by the continuous mapping theorem that

$\widehat{\pi}_{n}~{}\overset{p}{\longrightarrow}~{}\pi_{11}.$

Then, taking for $\alpha_{0}$ the ‘true’ value $\alpha$ and using Slutsky’s
lemma, (2.15) is a straightforward consequence of Theorem 2.1. Now suppose
that $\alpha/2<\alpha_{0}<\alpha$. Coming back to (4.28) with $\alpha_{0}$,
through the reasoning of the previous proof, there is a sequence
$(\vartheta^{**}_{n})$ such that

(4.30) $(\ln
n)\,\sqrt{n^{1-\alpha_{0}}}\,\left(\widehat{\alpha}_{n}(\alpha_{0})-\alpha_{0}\right)=\sqrt{n^{1+\alpha_{0}}}\,\big{(}\widehat{v}_{n}(\alpha_{0})-\rho_{n}(\alpha_{0})\big{)}\,\frac{n^{-\alpha_{0}}}{1-\vartheta^{**}_{n}}$

where
$(\rho_{n}(\alpha_{0})\wedge\widehat{v}_{n}(\alpha_{0}))<\vartheta^{**}_{n}<(\rho_{n}(\alpha_{0})\vee\widehat{v}_{n}(\alpha_{0}))$.
Using again the decomposition (4.21), we first obtain with (4.22),

$\big{|}n^{\alpha_{0}}\,e_{p}^{\,T}\tilde{R}_{n,\,2}\big{|}=O_{p}\big{(}\sqrt{n^{2\alpha_{0}-1-\alpha}}\big{)}~{}\overset{p}{\longrightarrow}~{}0$

since $2\alpha_{0}<\alpha+1$. Then,

$n^{\alpha_{0}}\,e_{p}^{\,T}\left(\tilde{R}_{n,\,1}+\tilde{R}_{n,\,3}\right)=c\left(1-n^{\alpha_{0}-\alpha}\right)(1-u_{n})~{}\overset{p}{\longrightarrow}~{}c\,(1-s_{\beta})$

where $u_{n}$ has been introduced in (4.24) and its limit behavior established
in (4.25). Hence,

$n^{\alpha_{0}}\,\big{(}1-\widehat{v}_{n}(\alpha_{0})\big{)}=c+n^{\alpha_{0}}\,\big{(}\rho_{n}(\alpha_{0})-\widehat{v}_{n}(\alpha_{0})\big{)}~{}\overset{p}{\longrightarrow}~{}c\,s_{\beta}.$

This is sufficient to get

(4.31)
$\liminf_{n\,\rightarrow\,+\infty}\,\frac{n^{-\alpha_{0}}}{|1-\vartheta^{**}_{n}|}~{}\geqslant~{}\frac{1}{c\,(1\vee|s_{\beta}|)}~{}>~{}0.$

Finally, (4.26) entails

(4.32)
$\left|\sqrt{n^{1+\alpha_{0}}}\,\big{(}\widehat{v}_{n}(\alpha_{0})-\rho_{n}(\alpha_{0})\big{)}\right|~{}\overset{p}{\longrightarrow}~{}+\infty.$

The combination of (4.30), (4.31) and (4.32) leads to (2.16), which concludes
the proof. ∎

###  4.5. Proof of Corollary 2.3.

First, note that

$\displaystyle\sqrt{n}\,\big{(}\widetilde{\theta}_{n}-\theta_{n}\big{)}$
$\displaystyle=$
$\displaystyle\sqrt{n}\,\big{(}\widehat{J}_{n}-J_{n}\big{)}\,\big{(}\widehat{\beta}_{n}-\beta_{n}\big{)}+\sqrt{n}\,\big{(}\widehat{J}_{n}-J_{n}\big{)}\,\beta_{n}$
$\displaystyle\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt+~{}\sqrt{n}\,J_{n}\,\big{(}\widehat{\beta}_{n}-\beta_{n}\big{)}+\sqrt{n}\,\big{(}\widehat{v}_{n}-\lambda_{n,\,1}\big{)}\,e_{p}.$

In fact, the proof rests entirely on Lemma 4.4. The first line of the
asymptotic normality gives
$|\widehat{v}_{n}-\lambda_{n,\,1}|=O_{p}(n^{-(1+\alpha)/2})=o_{p}(n^{-1/2})$
and obviously $\|\widehat{J}_{n}-J_{n}\|=o_{p}(n^{-1/2})$. Now from the
remaining part of (4.13),

$\sqrt{n}\,\big{(}\widehat{\beta}_{n}-\beta_{n}\big{)}~{}\overset{d}{\longrightarrow}~{}\mathcal{N}\big{(}0,\,\Sigma_{p-1}^{-1}\big{)}$

where the (invertible) covariance $\Sigma_{p-1}$ is given in (2.3). So
$\|\widehat{J}_{n}-J_{n}\|\,\|\widehat{\beta}_{n}-\beta_{n}\|=o_{p}(n^{-1/2})$
and the result follows from (H1), that is from $J_{n}\rightarrow J$. ∎

###  4.6. The case of a negative unit root

Our results have all been established for $\lambda_{1}=1$ but they remain true
in a context of a negative unit root, at the cost of some slight adjustments.
In what follows, it is thus implied that $\lambda_{1}=-1$ so that we may now
consider, without loss of generality,
$\lambda_{n,\,1}=\lambda_{1}\,\rho_{n}(\alpha)=-\rho_{n}(\alpha)$. So, in the
proof of Lemma 4.3, $\rho_{n}(\alpha)$ must be replaced by $-\rho_{n}(\alpha)$
in (4.2) and (4.10), and that leads to
$\Delta_{p}=\textnormal{diag}\left(0,(1+\lambda_{i})_{2\,\leqslant\,i\,\leqslant\,p}\right)$
in (4.2). Later, we may also note that $c$ becomes $-c$ in (4.9) and (4.21).
The sequence $(u_{n})$ in (4.24) now converges to

$s_{\beta}=\sum_{i=1}^{p-1}(-1)^{i}\beta_{i}=1-\Theta_{\beta}(-1)\neq 0.$

It is interesting to note that, for $\lambda_{1}=\pm 1$, we have in general
terms

$1-u_{n}~{}\overset{p}{\longrightarrow}~{}\Theta_{\beta}(\lambda_{1})$

where $\Theta_{\beta}(z)=1-\beta_{1}z-\ldots-\beta_{p-1}z^{p-1}$ has no zero
on the unit circle, by stability: the sign of the asymptotic behavior of
$Z_{n}$ under $\mathcal{H}_{1}$ thus depends on the sign of
$\Theta_{\beta}(\lambda_{1})\neq 0$. Finally, by considering the
$\mathcal{C}^{1}$-mappings from $]\\!-1,\,0[$ to $\mathbb{R}$ given by

$\forall\,n>1,\hskip 14.22636ptg_{n}(x)=\frac{\ln c-\ln\left(1+x\right)}{\ln
n},$

one can see that $\alpha=g_{n}(-\rho_{n}(\alpha))$ and
$\widehat{\alpha}_{n}(\alpha)=g_{n}(\widehat{v}_{n}(\alpha))$ as defined in
(2.11). Theorems 2.1 and 2.2 easily follow.

Acknowledgements. This research benefited from the support of the ANR project
‘Efficient inference for large and high-frequency data’ (ANR-21-CE40-0021).

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