# Convergence Rate Analysis in Limit Theorems for Nonlinear Functionals of the
Second Wiener Chaos

Gi-Ren Liu  Gi-Ren Liu
Department of Mathematics
National Cheng Kung University, Tainan, Taiwan<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to analyze the distribution distance between
random vectors derived from the magnitude of the analytic wavelet transform of
the squared envelopes of Gaussian processes and their large-scale limits. When
the Hurst index of the underlying Gaussian process falls below 1/2, the large-
scale limit takes the form of a combination of the second and fourth Wiener
chaos. In this case, we employ the non-Stein approach to determine the
convergence rate in the Kolmogorov metric. In scenarios where the Hurst index
of the underlying Gaussian process exceeds 1/2, the large-scale limit is a
correlated chi-distributed random process with two degrees of freedom. The
convergence rate in the Wasserstein metric is derived through the application
of Malliavin calculus and multidimensional Stein’s method. Although the large-
scale limits are the same for Hurst indices within the interval (1/2,1),
notable differences in the upper bounds of convergence rates are observed
among Hurst indices falling within the subintervals (1/2,3/4) and (3/4,1).

Keywords: rate of convergence; central limit theorems; non-central limit
theorems; analytic wavelet transform; long-range dependence; multidimensional
Stein’s method; Malliavin calculus.

##  1\. Introduction

This paper investigates the distribution of a random process derived from a
composition of local and nonlocal transformations applied to a stationary
random process. The composition, denoted as $U[s]$ and illustrated in Figure
1, involves the complex modulus and the analytic wavelet transform at the
scale $s>0$ formulated via the Hilbert transform [14, 23, 31]. Both the
transformation $U[s]$ and its square $U^{2}[s]$, often referred to as the
scalogram, have a wide range of applications in signal processing [24, 39] and
play a significant role in memory parameter estimation [13, 16, 33]. For the
purpose of establishing a mathematical foundation for the convolutional neural
network, the transformations $\\{U[s]\\}_{s>0}$ are conceived as neurons,
forming the second-order scattering network [19, 32]. In this network, the
output of a neuron $U[s^{\prime}]$, where $s^{\prime}>0$, serves as input for
another $U[s]$. To streamline numerical computation and theoretical analysis,
an alternative set of neurons $\\{U^{2}[s]\\}_{s>0}$, utilizing the squared
complex modulus as the activation function, is explored in [7, 29]. Based on
this context, we consider a chi-square random process $Y$, derived from the
squared magnitude of the analytic counterpart of a real-valued stationary
Gaussian process $X$, to serve as input for $U[s]$ and its squared form; see
Section 2.2 for details.

Analyzing the distribution of the process $U^{2}[s]Y:=\\{U^{2}[s]Y(t)\mid
t\in\mathbb{R}\\}$ provides insights that aid in understanding the
distribution of outputs from $U^{2}[s]$ applied to a clean signal in the
presence of noise. However, it is challenging to formulate a clear expression
for the distribution of $U^{2}[s]Y$. Recognizing this limitation, researchers
turn their attention to exploring large-scale limits of various functionals of
random processes [15, 17, 20]. Consequently, evaluating the convergence rate
has become a focal point for experts [6, 22, 34, 35].

$X$$X_{a}$$Y=|X_{a}|^{2}$squared
envelope$Y\star\psi_{s}$$Y\star\mathcal{H}\psi_{s}$$U[s]Y=|Y\star\psi_{s}+iY\star\mathcal{H}\psi_{s}|$analyticrepresentationWTWT
Figure 1. The relationship among notations frequently used in this paper: $X$
denotes the Gaussian input process, WT represents the wavelet transform,
$\star$ is the convolution operator, $\mathcal{H}$ stands for the Hilbert
transform, and $U[s]Y$ or its squared signifies the output.

The main contributions of this work focus on the large-scale limits of the
process $U[s]Y$ and their convergence rates. In Theorem 1, we show that when
the initial input $X$ is relatively strong long-range dependent, specifically
with a Hurst index less than 1/2, the large-scale limit of $U^{2}[s]Y$ has the
form of an additive combination of the second and fourth Wiener chaos.
Furthermore, we apply the analytical probability method introduced in the work
[4] by Anh, Vo, Nikolai Leonenko, Andriy Olenko, and Volodymyr Vaskovych to
obtain the convergence rate in the uniform (Kolmogorov) metric. In Theorem 2,
we show that when the initial input $X$ is relatively weak long-range
dependent, specifically with a Hurst index exceeding 1/2, the large-scale
limit of $U^{2}[s]Y$ is a correlated chi-square distributed random process.
For this case, we apply the work [34, 35] by Nourdin and Peccati to calculate
the convergence rate in the Wasserstein metric.

It’s noteworthy that, as indicated in the literature [3, 17, 20, 41], some
relatively weaker long-range dependent stochastic processes (referred to here
as Hurst indices between 1/2 and 1) may transition into short-range dependent
stochastic processes after undergoing local or non-local transformations, and
a distinction exists between the large-scale limits of functionals of short-
range dependent random processes and those corresponding to long-range
dependent ones. Therefore, obtaining different large-scale limits under the
conditions of Theorems 1 and 2 respectively is anticipated. From Theorem 2, we
further observe that when the scale variable $s$ tends to infinity, the upper
bound of the Wasserstein distance decreases from $s^{-1/2}L^{2}(s)$ to
$s^{0}L^{2}(s)$ following the function $s^{-1/2+2(3/4-\beta)}L^{2}(s)$ when
the strength of long-range dependence of the initial input $X$ increases
(i.e., the Hurst index $\beta$ of $X$ decreases from 3/4 to 1/2), where $L$ is
a slowly varying function. To the best of our knowledge, this relationship
between the rate of convergence and the strength of long-range dependence in
initial inputs has hardly been observed in the existing literature concerning
the central limit theorem arising from convolutional functionals of random
processes. Our findings resonates with those in the literature [9, 12] about
the total variation distance between the normalized sum of the Hermite power
of increments of a fractional Brownian motion and its limit, as well as how
the rate of convergence depends on the Hurst index of the fractional Brownian
motion.

In contrast to our previous works [25, 26, 28], we consider a transformation
with more practical significance in this work. Specifically, we replace the
conventional real-valued wavelet transform with the analytic wavelet transform
formulated via the Hilbert transform. The resulting output is a complex-valued
field indexed by time and scale variables. At each fixed scale, the magnitude
of this complex-valued field serves as the envelope of the wavelet coefficient
process, providing enhanced stability against time shifts in the underlying
time-evolution process. This transform has been extensively applied to extract
rich information from physiological signals [24] and audio recordings [2, 7]
for the purpose of intelligent classification. Because the real and imaginary
parts of the output of the analytic wavelet transform are correlated and we
need to deal with the sum of their individual square, existing results
associated with the real-valued wavelet transform cannot be applied to the
complex-valued case. Furthermore, we employ both Stein and non-Stein
approaches to calculate the upper bounds for the rates of convergence in the
Kolmogorov and Wasserstein metrics of the outputs of $U[s]$ and its square to
their large-scale limits. It is worth mentioning that we verified the positive
definiteness condition of the covariance matrix of the normal random vector
employed in the multivariate normal approximations. Different from the smooth
Wasserstein metric considered in [25, 27], this crucial verification enables
us to use the Wasserstein metric to gauge the distance between the
distributions of the outputs of $U[s]$ and their large-scale limits, as shown
in Theorem 2. Moreover, we provide an in-depth analysis of how these upper
bounds vary with the Hurst index of the initial input $X$.

The article is organized as follows. In Section 2, we recall some definitions
and formulae related to the analytic wavelet transform and the envelope of
stationary Gaussian processes. In Section 3, we state our main results,
including Theorems 1 and 2 along with key lemmas. Some conclusions are drawn
in Section 4. The proofs of our main results are given in Section 5.
Additionally, the appendix includes lemmas concerning slowly varying functions
and numerical experiments about the effect of dimension and time lag size on
the minimum eigenvalue of the covariance matrix of the normal random vector
used in the normal approximation.

##  2\. Preliminaries

###  2.1. Analytic wavelet transform

In the field of signal segment classification, considering a real-valued
process $\\{Y(t)\\}_{t\in\mathbb{R}}$ and a set of real-valued testing
functions $\\{\psi_{s}\\}_{s\in\Lambda}$, where $\Lambda$ is an index set, for
assessing the similarity between the pattern of the process $Y$ during a time
neighborhood of $t$ and $\psi_{s}$ for each $s\in\Lambda$, we consider the
convolution

(1) $\displaystyle Y\star\psi_{s}(t)=\int_{\mathbb{R}}Y(t-u)\psi_{s}(u)du,\
t\in\mathbb{R}.$

In the field of wavelet theory, $\Lambda=(0,\infty)$ and

$\displaystyle\psi_{s}(u)=\frac{1}{s}\psi(\frac{u}{s}),$

where $\psi$ is an integrable function defined on $\mathbb{R}$ satisfying
$\int_{\mathbb{R}}\psi(u)du=0$. For small values of $s$, $\psi_{s}$ quantifies
the small-scale fluctuations in $Y$ through (1). Subsequently, we refer to $s$
as the scale variable. In order to reduce the impact of small time shifting on
the value of convolution (1), one prefers to use the envelope of
$Y\star\psi_{s}$, which can be obtained from the magnitude of the following
complex wavelet transform

(2)
$\displaystyle\mathcal{W}_{s}Y(t)=Y\star\psi_{s}(t)+iY\star\mathcal{H}\psi_{s}(t),$

where $\mathcal{H}$ represents the Hilbert transform defined by

$\displaystyle\mathcal{H}\psi_{s}(t)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\psi_{s}(u)}{t-u}du.$

We denote the squared envelope of precess $Y\star\psi_{s}$ as $U^{2}[s]Y$,
i.e.,

(3) $\displaystyle
U^{2}[s]Y(t):=\left(Y\star\psi_{s}(t)\right)^{2}+\left(Y\star\mathcal{H}\psi_{s}(t)\right)^{2},$

which is known as the scalogram. We denote the Fourier transform of $\psi$ by
$\widehat{\psi}$, which is defined as

$\displaystyle\widehat{\psi}(\lambda)=\int_{\mathbb{R}}e^{-iu\lambda}\psi(u)du,\
\lambda\in\mathbb{R}.$

As stated in [14], $\widehat{\psi_{s}}(\lambda)=\widehat{\psi}(s\lambda)$ and
$\widehat{\mathcal{H}\psi_{s}}(\lambda)=-i\textup{sgn}(\lambda)\widehat{\psi}(s\lambda)$,
where $\textup{sgn}(\lambda)=\lambda/|\lambda|$ for $\lambda\neq 0$ and
$\textup{sgn}(0)=0$. Because $\psi_{s}+i\mathcal{H}\psi_{s}$ is analytic,
meaning its Fourier transform is null for negative frequencies, (2) is
referred to as the analytic wavelet transform [23, 31].

###  2.2. Envelope of stationary Gaussian processes

Consider a real-valued, mean-square continuous, and stationary Gaussian random
process $\\{X(t)\\}_{t\in\mathbb{R}}$ with a mean of zero and covariance
function denoted as $C_{X}$. It is well known that there exists a unique
nonnegative measure $F_{X}:\mathcal{B}(\mathbb{R})\rightarrow[0,\infty)$ such
that $F_{X}(\Delta)=F_{X}(-\Delta)$ for any $\Delta\in\mathcal{B}(\mathbb{R})$
and

$C_{X}(t)=\int_{\mathbb{R}}e^{i\lambda t}F_{X}(d\lambda),\ t\in\mathbb{R}.$

The measure $F_{X}$ is referred to as the spectral measure. We make the
following assumption based on the literature [4, Assumption 2].

######  Assumption 1.

The spectral measure $F_{X}$ is absolutely continuous with respect to the
Lebesgue measure with a density function $f_{X}\in L^{1}(\mathbb{R})$. It has
the form

(4) $f_{X}(\lambda)=L(|\lambda|^{-1})|\lambda|^{\beta-1},\ \lambda\neq 0,$

where $\beta\in(0,1)$ is the Hurst index of long-range dependence and
$L:(0,\infty)\rightarrow[0,\infty)$ is a locally bounded function which is
slowly varying at infinity and satisfies for sufficiently large $s$ the
condition

(5) $\displaystyle\left|1-\frac{L(\eta s)}{L(s)}\right|\leq
Cg_{\tau}(s)h_{\tau}(\eta),\ \eta\geq 1.$

In (5), $g_{\tau}(\cdot)$ is a regular varying function of index $\tau\leq 0$
with $g_{\tau}(s)\rightarrow 0$ as $s\rightarrow\infty$ and

(8) $\displaystyle h_{\tau}(\eta)=\left\\{\begin{array}[]{ll}\ln(\eta)&\
\textup{if}\ \tau=0,\\\ \tau^{-1}(\eta^{\tau}-1)&\ \textup{if}\
\tau<0.\end{array}\right.$

According to [10], $g_{\tau}(\cdot)$ is said to be a regular varying function
of index $\tau\leq 0$ if, for any $c>0$, $g_{\tau}(cs)/g_{\tau}(s)\rightarrow
c^{\tau}$ as $s\rightarrow\infty$. Under Assumption 1, according to the
Karhunen Theorem, the Gaussian process $X$ can be represented as follows

$\displaystyle X(t)=\int_{\mathbb{R}}e^{i\lambda
t}\sqrt{f_{X}(\lambda)}W(d\lambda),$

where $W$ is a complex-valued Gaussian white noise random measure on
$\mathbb{R}$ satisfying

(9) $\displaystyle W(\Delta_{1})=\overline{W(-\Delta_{1})}\ \ \textup{and}\ \
\mathbb{E}\left[W(\Delta_{1})\overline{W(\Delta_{2})}\right]=\textup{Leb}(\Delta_{1}\cap\Delta_{2})$

for any intervals $\Delta_{1},\Delta_{2}$, where Leb is the Lebesgue measure
on $\mathbb{R}$ [30].

The analytic counterpart of $X$ is a complex-valued process defined by

$\displaystyle X_{a}(t)=$ $\displaystyle X(t)+i\mathcal{H}X(t)$
$\displaystyle=$ $\displaystyle 2\int_{0}^{\infty}e^{i\lambda
t}\sqrt{f_{X}(\lambda)}W(d\lambda),\ t\in\mathbb{R},$

where the negative frequency content of $X$ is suppressed. As described in
[18, 31], from the process $X_{a}$, we can efficiently extract both the
amplitude and the phase components of $X$ through the relationship

$X(t)=\textup{Re}(|X_{a}(t)|e^{i\arg(X_{a}(t))}).$

In comparison with the phase component $\\{\arg(X_{a}(t))\mid
t\in\mathbb{R}\\}$, the process $\\{|X_{a}(t)|\mid t\in\mathbb{R}\\}$, known
as the envelope of $X$, is more stable against time shifts. Figure 2
illustrates a sample path of $X$ and its envelope $|X_{a}|$. This stability
can be advantageous in some applications [19].

From another perspective, when convolving a real-valued random process,
denoted as $\\{P(t)\mid t\in\mathbb{R}\\}$, with the analytic wavelet
$\psi_{s^{\prime}}+i\mathcal{H}\psi_{s^{\prime}}$ at scale $s^{\prime}>0$, it
automatically generates the analytic counterpart $\mathcal{W}_{s^{\prime}}P$
of the wavelet coefficient process $P\star\psi_{s^{\prime}}$. This establishes
a connection where the envelope process $|X_{a}|$ mentioned earlier can be
considered as the complex modulus of $\mathcal{W}_{s^{\prime}}P$. In other
words, the process $|X_{a}|$ (resp. $|X_{a}|^{2}$) can also be interpreted as
the output of the neuron $U[s^{\prime}]$ (resp. $U^{2}[s^{\prime}]$).

Figure 2. Illustration: a sample path of the Gaussian process $X$ (in black)
and its envelope $|X_{a}|$ (in red).

Based on this context, we consider the scenario

(10) $\displaystyle Y(t)=|X_{a}(t)|^{2},\ t\in\mathbb{R},$

for the process $Y$ considered in Section 2.1. For an analysis of level
crossings by the process $Y$, readers can refer to the literature [1]. To
facilitate the subsequent time-frequency analysis, we establish a Wiener
integral representation for $Y$ in the following lemma.

######  Lemma 1.

Under the assumption about the existence of the spectral density function
$f_{X}$ in Assumption 1, $\\{Y(t)/\|f_{X}\|_{1}\\}_{t\in\mathbb{R}}$ defined
in (10) is a chi-square distributed random process with two degrees of
freedom, and

$\displaystyle\textup{Cov}(Y(t),Y(t^{\prime}))=16\left|\int_{0}^{\infty}e^{i\lambda(t-t^{\prime})}f_{X}(\lambda)d\lambda\right|^{2},\
t,t^{\prime}\in\mathbb{R}.$

Moreover, $Y$ has the Wiener integral representation

(11) $\displaystyle
Y(t)=2\|f_{X}\|_{1}+2\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}\sqrt{f_{X}(\lambda_{1})f_{X}(\lambda_{2})}1_{\\{\lambda_{1}\lambda_{2}<0\\}}W(d\lambda_{1})W(d\lambda_{2}),$

where $W$ is a complex-valued Gaussian white noise random measure satisfying
(9) and $\int^{{}^{\prime}}$ means that the integral excludes the hyperplanes
$\lambda_{k}=\mp\lambda_{k^{{}^{\prime}}}$ for $k,k^{{}^{\prime}}\in\\{1,2\\}$
and $k\neq k^{{}^{\prime}}$.

The proof of Lemma 1 is provided in Section 5.1. Similar to the Rosenblatt
process [17] and the chi-square random field proposed in [3, Eample 2.3], the
process $Y$ is expressed as a double iterated integral with respect to the
complex-valued Gaussian white noise random measure $W$. For the wavelet
analysis of the Rosenblatt process, please refer to [8, 16]. Different from
the real-valued wavelets employed in [8, 16], the wavelets considered in this
work are complex-valued and analytic. The reason behind adopting analytic
wavelets for signal or random process analysis is their stability in magnitude
during local input translations. This stability proves advantageous in signal
classification, as signal classes typically remain invariant to local
translations [19, 32]. The interplay between the real and imaginary components
manifests in the spectral representation of the squared magnitude of the
analytic wavelet transform of $Y$, as presented in Lemma 2 below, and its
proof is provided in Section 5.2.

######  Lemma 2.

Consider the random process $Y$ defined in (10). Assuming that $\psi$ and its
Fourier transform $\widehat{\psi}$ belong to $L^{1}$, and considering
Assumption 1, for each $s>0$, the random process $\\{U^{2}[s]Y(t)\mid
t\in\mathbb{R}\\}$ has the Wiener-It$\hat{o}$ decomposition

$\displaystyle U^{2}[s]Y(t)=$
$\displaystyle\mathbb{E}\left[U^{2}[s]Y(t)\right]+\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}K_{2}(\lambda_{1},\lambda_{2})W(d\lambda_{1})W(d\lambda_{2})$
$\displaystyle+\int_{\mathbb{R}^{4}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})}K_{4}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})\overset{4}{\underset{\ell=1}{\prod}}W(d\lambda_{\ell}),$

where

$\displaystyle\mathbb{E}\left[U^{2}[s]Y(t)\right]=2^{4}\int_{\mathbb{R}^{2}}|\widehat{\psi}(s(\lambda_{1}+\lambda_{2}))|^{2}f_{X}(\lambda_{1})f_{X}(\lambda_{2})1_{\\{\lambda_{1}\lambda_{2}<0\\}}d\lambda_{1}d\lambda_{2},$
$\displaystyle K_{2}(\lambda_{1},\lambda_{2})=$ $\displaystyle
8\sqrt{f_{X}(\lambda_{1})f_{X}(\lambda_{2})}1_{\\{\lambda_{1}\lambda_{2}<0\\}}\int_{\mathbb{R}}f_{X}(\xi)\widehat{\psi}(s(\lambda_{1}+\xi))\widehat{\psi}(s(\lambda_{2}-\xi))1_{\\{(\lambda_{1}+\xi)(\lambda_{2}-\xi)<0\\}}d\xi,$

and

$\displaystyle K_{4}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=$
$\displaystyle
8\left[\overset{4}{\underset{\ell=1}{\prod}}\sqrt{f_{X}(\lambda_{\ell})}\right]\widehat{\psi}(s(\lambda_{1}+\lambda_{2}))\widehat{\psi}(s(\lambda_{3}+\lambda_{4}))$
$\displaystyle\times
1_{\\{\lambda_{1}\lambda_{2}<0\\}}1_{\\{\lambda_{3}\lambda_{4}<0\\}}1_{\\{(\lambda_{1}+\lambda_{2})(\lambda_{3}+\lambda_{4})<0\\}}.$

##  3\. Main results

In this section, we present our primary findings concerning the large-scale
limits of the process $U[s]Y$ and their convergence rates. We’ll observe that
the Hurst index of the initial input $X$ at $1/2$ serves as a threshold. When
the Hurst index falls below or exceeds this threshold, the normalized $U[s]Y$
converges to distinct limits. Moreover, the convergence rate might vary based
on the Hurst index, particularly within the interval $(1/2,3/4)$.

In the subsequent discussion, when comparing two functions, $f(s)$ and $g(s)$,
the notation $f(s)\lesssim g(s)$ implies the existence of a constant $C$ that
remains unaffected regardless of changes in the variable $s$, satisfying the
condition $f(s)\leq Cg(s)$. The value of constant $C$ may vary between
different equations.

###  3.1. Non-central limit theorems

We begin by considering a random process $Z_{s}$ obtained from the squared
magnitude of the analytic wavelet transform of $Y$:

(12) $\displaystyle Z_{s}(t)=s^{2\beta}L^{-2}(s)U^{2}[s]Y(st),$

where the time units of $Z_{s}$ are scaled by a factor of $s$ compared to the
time units of $Y$.

######  Definition 1.

Consider two random variables $V_{1}$ and $V_{2}$. The Kolmogorov distance
between the distributions of $V_{1}$ and $V_{2}$, denoted as
$d_{\textup{Kol}}(V_{1},V_{2})$, is defined as follows

$\displaystyle
d_{\textup{Kol}}(V_{1},V_{2})=\underset{z\in\mathbb{R}}{\sup}\left|P(V_{1}\leq
z)-P(V_{2}\leq z)\right|.$

Theorem 1 below concerns the convergence rate of the Kolmogorov distance
between $Z_{s}$ and its limit as $s\rightarrow\infty$.

######  Theorem 1.

Consider the random process $Y$ defined in (10). Under the assumption that the
wavelet $\psi$ and its Fourier transform $\widehat{\psi}$ belong to $L^{1}$,
and considering Assumption 1 with $0<2\beta<1$ and $-\tau<(1-2\beta)/2$, for
any $n\in\mathbb{N}$ and $c_{1},...,c_{n},t_{1},...,t_{n}\in\mathbb{R}$, the
following result holds:

(13) $\displaystyle
d_{\textup{Kol}}\left(\overset{n}{\underset{k=1}{\sum}}c_{k}Z_{s}\left(t_{k}\right),\overset{n}{\underset{k=1}{\sum}}c_{k}Z(t_{k})\right)\lesssim\left(g_{\tau}(s)\right)^{2/9},$

where $Z=Z^{(0)}+Z^{(1)}+Z^{(2)}$ with

(14) $\displaystyle
Z^{(0)}=16\int_{\mathbb{R}^{2}}|\widehat{\psi}(\lambda_{1}+\lambda_{2})|^{2}|\lambda_{1}\lambda_{2}|^{\beta-1}1_{\\{\lambda_{1}\lambda_{2}<0\\}}d\lambda_{1}d\lambda_{2},$
$\displaystyle Z^{(1)}(t)=$ $\displaystyle
8\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}\left|\lambda_{1}\lambda_{2}\right|^{\frac{\beta-1}{2}}1_{\\{\lambda_{1}\lambda_{2}<0\\}}$
(15)
$\displaystyle\times\left[\int_{\mathbb{R}}\frac{\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)}{\left|\xi\right|^{1-\beta}}1_{\\{(\lambda_{1}+\xi)(\lambda_{2}-\xi)<0\\}}d\xi\right]W(d\lambda_{1})W(d\lambda_{2}),$

and

$\displaystyle Z^{(2)}(t)=$ $\displaystyle
8\int_{\mathbb{R}^{4}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})}\frac{\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})}{|\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}|^{(1-\beta)/2}}$
(16) $\displaystyle\times
1_{\\{\lambda_{1}\lambda_{2}<0\\}}1_{\\{\lambda_{3}\lambda_{4}<0\\}}1_{\\{(\lambda_{1}+\lambda_{2})(\lambda_{3}+\lambda_{4})<0\\}}W(d\lambda_{1})W(d\lambda_{2})W(d\lambda_{3})W(d\lambda_{4}).$

######  Remark 1.

Because of Assumption 1, which states that $g_{\tau}(s)\rightarrow 0$ as
$s\rightarrow\infty$, (13) together with the Cramer-Wold device implies that
$\\{Z_{s}(t)\\}_{t\in\mathbb{R}}$ converges to $\\{Z(t)\\}_{t\in\mathbb{R}}$
in the finite-dimensional distribution sense as $s\rightarrow\infty$. On the
other hand, in addition to the condition $-\tau<(1-2\beta)/2$, if $\tau$ is
strictly negative, (13) implies that

$\displaystyle
d_{\textup{Kol}}\left(\overset{n}{\underset{k=1}{\sum}}c_{k}Z_{s}\left(t_{k}\right),\overset{n}{\underset{k=1}{\sum}}c_{k}Z\left(t_{k}\right)\right)\lesssim
s^{2\tau/9+\delta}$

for any $\delta>0$ because Theorem 1.5.3 in [10] shows that
$g_{\tau}(s)\lesssim s^{\tau+\delta}$ for any $\delta>0$.

The proof of Theorem 1 is detailed in Section 5.3, mainly relying on the
analytical probability method [4]. Additionally, the proof involves the
utilization of the following lemma concerning singular integrals and product
contractions, whose proof is provided in Section 5.4.

######  Lemma 3.

For $\nu_{1},\nu_{2},\nu_{3}\in(-\beta,1-\beta)$, $\nu_{1}+\nu_{3}<1-2\beta$,
and $\nu_{2}+\nu_{3}<1-2\beta$,

$\displaystyle\int_{\mathbb{R}^{2}}\left|\lambda_{1}\right|^{\nu_{1}+\beta-1}\left|\lambda_{2}\right|^{\nu_{2}+\beta-1}\left|\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta-\nu_{3}}}d\xi\right|^{2}d\lambda_{1}d\lambda_{2}<\infty.$

###  3.2. Central limit theorems

In the case where $2\beta>1$, we investigate the process
$\\{s^{1/2}U[s]Y(st)\mid t\in\mathbb{R}\\}$. This process represents the
normalized magnitude of the analytic wavelet transform of $Y$. It’s worth
noting that the normalization considered in this case differs from that in
(12). In the following theorem, we will show that its distribution will be
close to the chi distribution with two degrees of freedom when the scale
variable $s$ tends to infinity. For technical reason, we use the Wasserstein
metric to measure the distance between them.

######  Definition 2.

Consider two random variables $V_{1}$ and $V_{2}$. The Wasserstein distance
between the distributions of $V_{1}$ and $V_{2}$, denoted as
$d_{\textup{W}}(V_{1},V_{2})$, is defined as follows

$\displaystyle d_{\textup{W}}(V_{1},V_{2})=\underset{\|h\|_{\textup{Lip}}\leq
1}{\sup}\left|\mathbb{E}[h(V_{1})]-\mathbb{E}[h(V_{2})]\right|,$

where the supremum is taken over functions with Lipschitz constants less than
one.

######  Theorem 2.

Consider the random process $Y$ defined in (10). If the nonvanishing wavelet
$\psi$ and its Fourier transform $\widehat{\psi}$ belong to $L^{1}$ and the
assumption (4) about the spectral density function $f_{X}$ holds, then for any
$n\in\mathbb{N}$, distinct time points $t_{1},...,t_{n}\in\mathbb{R}$, and
$c_{1},...,c_{n}\in\mathbb{R}$ with $c_{1}^{2}+\cdots+c_{n}^{2}\leq 1$, when
$s\rightarrow\infty$,

$\displaystyle
d_{\textup{W}}\left(\overset{n}{\underset{k=1}{\sum}}c_{k}s^{1/2}U[s]Y(st_{k}),\overset{n}{\underset{k=1}{\sum}}c_{k}\sqrt{N_{2k-1,s}^{2}+N_{2k,s}^{2}}\right)$
(19) $\displaystyle\lesssim$
$\displaystyle\left\\{\begin{array}[]{cl}s^{-1/2}&\ \textup{for}\
\beta\in(3/4,1),\\\ s^{-1/2+2(3/4-\beta)}L^{2}(s)&\ \textup{for}\
\beta\in(1/2,3/4).\end{array}\right.$

Here, $(N_{1,s},N_{2,s},...,N_{2n-1,s},N_{2n,s})$ is a centered normal random
vector and its covariance matrix converges to $16\|f_{X}\|_{2}^{2}\Psi$ when
$s\rightarrow\infty$. The matrix
$\mathrm{\Psi}=[\Psi(\ell,\ell^{\prime})]_{1\leq\ell,\ell^{\prime}\leq 2n}$ is
positive definite for any distinct time points $t_{1},...,t_{n}\in\mathbb{R}$
and composed of

(20)
$\Psi(\ell,\ell^{\prime})=\left\\{\begin{array}[]{l}\int_{0}^{\infty}\cos((t_{k}-t_{k^{\prime}})u)|\widehat{\psi}(u)|^{2}du\
\textup{for}\ (\ell,\ell^{\prime})=(2k-1,2k^{\prime}-1)\ \textup{or}\
(2k,2k^{\prime}),\\\
\int_{0}^{\infty}\sin((t_{k}-t_{k^{\prime}})u)|\widehat{\psi}(u)|^{2}du\
\textup{for}\ (\ell,\ell^{\prime})=(2k-1,2k^{\prime}),\\\
\int_{0}^{\infty}\sin((t_{k^{\prime}}-t_{k})u)|\widehat{\psi}(u)|^{2}du\
\textup{for}\ (\ell,\ell^{\prime})=(2k,2k^{\prime}-1),\end{array}\right.$

where $k,k^{\prime}=1,2,...,n$.

######  Remark 2.

By the continuous mapping theorem, Theorem 2 shows that the process
$\\{sU^{2}[s]Y(st)\mid t\in\mathbb{R}\\}$, derived from the normalized
scalogram of the process $Y$, converges to a $\chi^{2}_{2}$-distributed random
process, denoted as $\\{F(t)\\}_{t\in\mathbb{R}}$, in the finite-dimensional
distribution sense as $s\rightarrow\infty$. It’s noteworthy that the upper
bounds of the Wasserstein distance, as shown in Theorem 2, don’t entirely
align with the standard Berry-Esseen bound in the Wasserstein metric for the
central limit theorem [11]. Specifically, Theorem 2 shows that the speed of
convergence to zero decreases from $s^{-1/2}L^{2}(s)$ to $s^{0}L^{2}(s)$
according to the function $s^{-1/2+2(3/4-\beta)}L^{2}(s)$ when the Hurst index
$\beta$ of the initial input $X$ decreases from 3/4 to 1/2. This observation
resonates with findings in the literature [9, 12] regarding the total
variation distance between the normalized sum of the Hermite power of
increments of a fractional Brownian motion and its limit, as well as how the
rate of convergence depends on the Hurst index of the fractional Brownian
motion.

The proof of Theorem 2 relies on the multivariate Stein method and the
Malliavin calculus [34, 35]. There are two key points in this approach. The
first concerns the covariance matrices of the normal random vectors used for
the normal approximation. Remark 5.1.4 and Theorem 6.1.1 in [34] highlight the
challenge in ensuring the positive definiteness of the covariance matrices. By
using the observation (60) along with the characterization of strictly
positive definite functions in [42, Corollary 6.9], we have proven the
positive definiteness of the covariance matrix for
$(N_{1,s},N_{2,s},...,N_{2n-1,s},N_{2n,s})$. The effects of the number of
sampling points $n$ and the time lag for $t_{1},...,t_{n}$ on the minimum
eigenvalue of the matrix $\Psi$ are explored in Appendix B. The second key
point is about the estimate of the norms of product contractions (69) of the
integrand functions of multiple Wiener integrals. Lemma 4 below plays a
crucial role in estimating these norms. Because Lemma 4 has not appeared in
existing literature, its proof is provided in Section 5.6.

######  Lemma 4.

If the wavelet $\psi$ and its Fourier transform $\widehat{\psi}$ belong to
$L^{1}$ and $\beta\in(1/2,3/4)$, then for any
$\mu_{1},\mu_{2},\mu_{3},\mu_{4}\in\\{-1,1\\}$, the integral below

(21)
$\displaystyle\mathcal{K}:=\int_{\mathbb{R}^{4}}\frac{|\widehat{\psi}(x)||\widehat{\psi}(y)||\widehat{\psi}(z)||\widehat{\psi}(x+y-z)|}{|u|^{1-\beta-\delta\mu_{1}}|x+y-u|^{1-\beta-\delta\mu_{2}}|x-u|^{1-\beta-\delta\mu_{3}}|z-u|^{1-\beta-\delta\mu_{4}}}\,du\,dx\,dy\,dz$

is finite for sufficiently small $\delta>0$.

######  Corollary 1.

Under the hypotheses of Theorem 2, the covariance function
$\textup{Cov}\left(F(t),F(t^{\prime})\right)$, where
$t,t^{\prime}\in\mathbb{R}$, for the limiting process of
$\\{sU^{2}[s]Y(st)\\}_{t\in\mathbb{R}}$ can be expressed as follows

$\displaystyle
2^{10}\|f_{X}\|_{2}^{4}\left[\left(\int_{0}^{\infty}\cos((t-t^{\prime})u)|\widehat{\psi}(u)|^{2}du\right)^{2}+\left(\int_{0}^{\infty}\sin((t-t^{\prime})u)|\widehat{\psi}(u)|^{2}du\right)^{2}\right].$

##  4\. Concluding remarks

For the nonlinear transformation, depicted in Figure 1, from Theorem 1 and
Theorem 2, we observed that the Hurst index of the initial input $X$ at $1/2$
acts as a dividing point. When the Hurst index falls below or exceeds this
threshold, the rescaled version of $U[s]Y$ converges to distinct limits. The
former (i.e., $s^{2\beta}L^{-2}(s)U^{2}[s]Y(s\cdot)$) converges to an additive
combination of the second and fourth Wiener chaos, while the latter (i.e.,
$sU^{2}[s]Y(s\cdot)$) tends towards a chi-square random process. Besides, we
also conducted a systematic analysis about the relationship between the
convergence rate of the large scaling limits and the strength of long-range
dependence in the initial input $X$. About the upper bound of the distance
between the distributions of the rescaled $U[s]Y$ and its limit, our
observations indicate that the speed of convergence to zero diminishes as the
strength of long-range dependence in the initial input $X$ increases.

##  5\. Proofs

###  5.1. Proof of Lemma 1

First of all, $\\{Y(t)\\}_{t\in\mathbb{R}}$ can be rewritten as
$Y(t)=|X(t)+i\widetilde{X}(t)|^{2}$, where

$\displaystyle\widetilde{X}(t)=-i\int_{\mathbb{R}}e^{i\lambda
t}\textup{sgn}(\lambda)\sqrt{f_{X}(\lambda)}W(d\lambda).$

From (9), we know that $\widetilde{X}$ is also a real-valued Gaussian process
and

(26)
$\displaystyle\mathbb{E}\left[\begin{array}[]{cc}X(t)X(t^{\prime})&X(t)\widetilde{X}(t^{\prime})\\\
\widetilde{X}(t)X(t^{\prime})&\widetilde{X}(t)\widetilde{X}(t^{\prime})\end{array}\right]=2\int_{0}^{\infty}\left[\begin{array}[]{lr}\cos(\lambda(t-t^{{}^{\prime}}))&-\sin(\lambda(t-t^{{}^{\prime}}))\\\
\sin(\lambda(t-t^{\prime}))&\cos(\lambda(t-t^{{}^{\prime}}))\end{array}\right]f_{X}(\lambda)d\lambda.$

Especially,
$\mathbb{E}[X^{2}(t)]=\mathbb{E}[\widetilde{X}^{2}(t)]=\|f_{X}\|_{1}$ and
$\mathbb{E}[X(t)\widetilde{X}(t)]=0$, which implies the independence between
$X(t)$ and $\widetilde{X}(t)$. Hence,
$Y(t)/\|f_{X}\|_{1}=(X^{2}(t)+\widetilde{X}^{2}(t))/\|f_{X}\|_{1}$ follows a
chi-square distribution with two degrees of freedom.

By the product formula [30],

(27) $\displaystyle
X^{2}(t)=\|f_{X}\|_{1}+\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{i(\lambda_{1}+\lambda_{2})t}\sqrt{f_{X}(\lambda_{1})f_{X}(\lambda_{2})}W(d\lambda_{1})W(d\lambda_{2})$

and

(28)
$\displaystyle\widetilde{X}^{2}(t)=\|f_{X}\|_{1}-\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{i(\lambda_{1}+\lambda_{2})t}\textup{sgn}(\lambda_{1}\lambda_{2})\sqrt{f_{X}(\lambda_{1})f_{X}(\lambda_{2})}W(d\lambda_{1})W(d\lambda_{2}).$

The representation (11) is obtained by combining (27) and (28). The covariance
function of $Y$ follows from (26). More precisely,

$\displaystyle\textup{Cov}(Y(t),Y(t^{\prime}))=\textup{Cov}(X^{2}(t)+\widetilde{X}^{2}(t),X^{2}(t^{\prime})+\widetilde{X}^{2}(t^{\prime}))$
$\displaystyle=$ $\displaystyle
2\left\\{[\textup{Cov}(X(t),X(t^{\prime}))]^{2}+[\textup{Cov}(\widetilde{X}(t),\widetilde{X}(t^{\prime}))]^{2}+[\textup{Cov}(\widetilde{X}(t),X(t^{\prime}))]^{2}+[\textup{Cov}(\widetilde{X}(t),\widetilde{X}(t^{\prime}))]^{2}\right\\}$
$\displaystyle=$ $\displaystyle
16\left|\int_{0}^{\infty}e^{i\lambda(t-t^{\prime})}f_{X}(\lambda)d\lambda\right|^{2}.$

###  5.2. Proof of Lemma 2

By the spectral representation of $Y$ in (11), the stochastic Fubini theorem
[38, Theorem 2.1], and $\widehat{\psi_{s}}(\lambda)=\widehat{\psi}(s\lambda)$,

(29) $\displaystyle
Y\star\psi_{s}(t)=\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}k(\lambda_{1},\lambda_{2})W(d\lambda_{1})W(d\lambda_{2}),$

where

$\displaystyle
k(\lambda_{1},\lambda_{2})=2\sqrt{f_{X}(\lambda_{1})f_{X}(\lambda_{2})}1_{\\{\lambda_{1}\lambda_{2}<0\\}}\widehat{\psi}(s(\lambda_{1}+\lambda_{2})).$

By the product formula [30],

$\displaystyle\left(Y\star\psi_{s}(t)\right)^{2}=$ $\displaystyle
2\|k\|_{2}^{2}+4\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}k\otimes_{1}k(\lambda_{1},\lambda_{2})W(d\lambda_{1})W(d\lambda_{2})$
(30)
$\displaystyle+\int_{\mathbb{R}^{4}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})}k\otimes
k(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})\overset{4}{\underset{\ell=1}{\prod}}W(d\lambda_{\ell}),$

where

$\displaystyle
k\otimes_{1}k(\lambda_{1},\lambda_{2})=\int_{\mathbb{R}}k(\lambda_{1},\xi)k(\lambda_{2},-\xi)d\xi$

and $k\otimes
k(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=k(\lambda_{1},\lambda_{2})k(\lambda_{3},\lambda_{4})$.

By the convolution theorem,
$\widehat{\mathcal{H}\psi_{s}}(\lambda)=-i\textup{sgn}(\lambda)\widehat{\psi}(s\lambda)$,
where $\textup{sgn}(\lambda)=\lambda/|\lambda|$ for $\lambda\neq 0$ and
$\textup{sgn}(0)=0$. In parallel with (29) and (30),

$\displaystyle
Y\star\mathcal{H}\psi_{s}(t)=-i\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}\textup{sgn}(\lambda_{1}+\lambda_{2})k(\lambda_{1},\lambda_{2})W(d\lambda_{1})W(d\lambda_{2})$

and

$\displaystyle\left(Y\star\mathcal{H}\psi_{s}(t)\right)^{2}=$ $\displaystyle
2\|k\|_{2}^{2}+4\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}\widetilde{k}\otimes_{1}\widetilde{k}(\lambda_{1},\lambda_{2})W(d\lambda_{1})W(d\lambda_{2})$
(31)
$\displaystyle+\int_{\mathbb{R}^{4}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})}\widetilde{k}\otimes\widetilde{k}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})\overset{4}{\underset{\ell=1}{\prod}}W(d\lambda_{\ell}),$

where
$\widetilde{k}(\lambda_{1},\lambda_{2})=-i\textup{sgn}(\lambda_{1}+\lambda_{2})k(\lambda_{1},\lambda_{2}).$
The proof is finished by combining (30) and (31).

###  5.3. Proof of Theorem 1

Here, we focus on the case where $n=1$, $c_{1}=1$, and $t_{1}=t$ to simplify
notation. The following proof can be extended to the general case for $n\geq
2.$ First of all, we denote $Z_{s}$ as the sum of three components: an
$s$-dependent constant $Z_{s}^{(0)}$, and two random processes, $Z_{s}^{(1)}$
and $Z_{s}^{(2)}$, where

(32) $\displaystyle Z_{s}^{(0)}=$ $\displaystyle
s^{2\beta}L^{-2}(s)\mathbb{E}[U^{2}[s]Y(st)],$ $\displaystyle
Z_{s}^{(1)}(t)=s^{2\beta}L^{-2}(s)\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{ist(\lambda_{1}+\lambda_{2})}K_{2}(\lambda_{1},\lambda_{2})W(d\lambda_{1})W(d\lambda_{2}),$

and

$\displaystyle
Z_{s}^{(2)}(t)=s^{2\beta}L^{-2}(s)\int_{\mathbb{R}^{4}}^{{}^{\prime}}e^{ist(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4})}K_{4}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})\overset{4}{\underset{\ell=1}{\prod}}W(d\lambda_{\ell}),$

where $K_{2}$ and $K_{4}$ are defined in Lemma 2. Consider two random
processes $V_{1}$ and $V_{2}$, and denote $V_{1}\overset{d}{=}V_{2}$ to
signify that their finite-dimensional distributions are identical. Utilizing
the self-similarity property
$W(s^{-1}d\lambda)\overset{d}{=}s^{-1/2}W(d\lambda)$ (see also [41, Lemma
6.1]), we have

(33) $\displaystyle
Z_{s}(t)=Z^{(0)}_{s}+Z^{(1)}_{s}(t)+Z^{(2)}_{s}(t)\overset{d}{=}\widetilde{Z}_{s}(t),$

where
$\widetilde{Z}_{s}=Z^{(0)}_{s}+\widetilde{Z}^{(1)}_{s}+\widetilde{Z}^{(2)}_{s}$
with

$\displaystyle\widetilde{Z}^{(1)}_{s}(t)=$ $\displaystyle
s^{2\beta-1}L^{-2}(s)\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}K_{2}(\lambda_{1}/s,\lambda_{2}/s)W(d\lambda_{1})W(d\lambda_{2})$
$\displaystyle=$ $\displaystyle
8s^{2\beta-2}L^{-2}(s)\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}\sqrt{f_{X}\left(\frac{\lambda_{1}}{s}\right)f_{X}\left(\frac{\lambda_{2}}{s}\right)}1_{\\{\lambda_{1}\lambda_{2}<0\\}}$
$\displaystyle\times\left[\int_{\mathbb{R}}f_{X}\left(\frac{\xi}{s}\right)\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)1_{\\{(\lambda_{1}+\xi)(\lambda_{2}-\xi)<0\\}}d\xi\right]W(d\lambda_{1})W(d\lambda_{2})$
$\displaystyle=$ $\displaystyle
8s^{2\beta-2}L^{-2}(s)\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}L^{1/2}\left(\frac{s}{|\lambda_{1}|}\right)L^{1/2}\left(\frac{s}{|\lambda_{2}|}\right)\left|\frac{\lambda_{1}}{s}\right|^{\frac{\beta-1}{2}}\left|\frac{\lambda_{2}}{s}\right|^{\frac{\beta-1}{2}}1_{\\{\lambda_{1}\lambda_{2}<0\\}}$
$\displaystyle\times\left[\int_{\mathbb{R}}L\left(\frac{s}{|\xi|}\right)\left|\frac{\xi}{s}\right|^{\beta-1}\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)1_{\\{(\lambda_{1}+\xi)(\lambda_{2}-\xi)<0\\}}d\xi\right]W(d\lambda_{1})W(d\lambda_{2})$
$\displaystyle=$ $\displaystyle
8\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}\frac{L^{1/2}\left(s/|\lambda_{1}|\right)}{L^{1/2}(s)}\frac{L^{1/2}\left(s/|\lambda_{2}|\right)}{L^{1/2}(s)}\left|\lambda_{1}\lambda_{2}\right|^{\frac{\beta-1}{2}}1_{\\{\lambda_{1}\lambda_{2}<0\\}}$
(34)
$\displaystyle\times\left[\int_{\mathbb{R}}\frac{L\left(s/|\xi|\right)}{L(s)}\frac{\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)}{\left|\xi\right|^{1-\beta}}1_{\\{(\lambda_{1}+\xi)(\lambda_{2}-\xi)<0\\}}d\xi\right]W(d\lambda_{1})W(d\lambda_{2})$

and

$\displaystyle\widetilde{Z}_{s}^{(2)}(t)=$ $\displaystyle
s^{2(\beta-1)}L^{-2}(s)\int_{\mathbb{R}^{4}}^{{}^{\prime}}e^{it(\lambda_{1}+\cdots+\lambda_{4})}K_{4}(\lambda_{1}/s,...,\lambda_{4}/s)W(d\lambda_{1})W(d\lambda_{2})W(d\lambda_{3})W(d\lambda_{4})$
$\displaystyle=$ $\displaystyle
8\int_{\mathbb{R}^{4}}^{{}^{\prime}}e^{it(\lambda_{1}+\cdots+\lambda_{4})}\left[\overset{4}{\underset{\ell=1}{\prod}}\frac{L^{1/2}(s/\lambda_{\ell})}{L^{1/2}(s)}\right]1_{\\{\lambda_{1}\lambda_{2}<0\\}}1_{\\{\lambda_{3}\lambda_{4}<0\\}}1_{\\{(\lambda_{1}+\lambda_{2})(\lambda_{3}+\lambda_{4})<0\\}}$
(35)
$\displaystyle\times\frac{\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})}{|\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}|^{(1-\beta)/2}}W(d\lambda_{1})W(d\lambda_{2})W(d\lambda_{3})W(d\lambda_{4}).$

By (33) and [37, Lemma 1.8], for any $\varepsilon>0$,

$\displaystyle
d_{\textup{Kol}}\left(Z_{s}(t),Z(t)\right)=d_{\textup{Kol}}\left(\widetilde{Z}_{s}(t),Z(t)\right)$
$\displaystyle=$ $\displaystyle
d_{\textup{Kol}}\left(Z(t)+(Z^{(0)}_{s}-Z^{(0)})+(\widetilde{Z}_{s}^{(1)}(t)-Z^{(1)}(t))+(\widetilde{Z}_{s}^{(2)}(t)-Z^{(2)}(t)),Z(t)\right)$
$\displaystyle\leq$
$\displaystyle\overset{2}{\underset{\ell=1}{\sum}}P\left(|\widetilde{Z}^{(\ell)}_{s}(t)-Z^{(\ell)}(t)|\geq\frac{\varepsilon}{2}\right)+d_{\textup{Kol}}\left(Z(t)+Z^{(0)}_{s}-Z^{(0)},Z(t)\right)$
$\displaystyle+d_{\textup{Kol}}\left(Z(t)+\varepsilon,Z(t)\right)$
$\displaystyle\leq$
$\displaystyle\frac{4}{\varepsilon^{2}}\overset{2}{\underset{\ell=1}{\sum}}\mathbb{E}[|\widetilde{Z}^{(\ell)}_{s}(t)-Z^{(\ell)}(t)|^{2}]+d_{\textup{Kol}}\left(Z(t)+Z^{(0)}_{s}-Z^{(0)},Z(t)\right)$
(36) $\displaystyle+d_{\textup{Kol}}\left(Z(t)+\varepsilon,Z(t)\right).$

$\bullet$ Upper bound of
$\mathbb{E}[|\widetilde{Z}^{(1)}_{s}(t)-Z^{(1)}(t)|^{2}]$: By the definitions
of $Z^{(1)}_{s}$ and $\widetilde{Z}^{(1)}$ in (15) and (34),

$\displaystyle\widetilde{Z}^{(1)}_{s}(t)-Z^{(1)}(t)=$ $\displaystyle
8\int_{\mathbb{R}^{2}}^{{}^{\prime}}e^{it(\lambda_{1}+\lambda_{2})}\left|\lambda_{1}\lambda_{2}\right|^{\frac{\beta-1}{2}}$
$\displaystyle\times\int_{\mathbb{R}}D_{1}(\lambda_{1},\lambda_{2},\xi)\frac{\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)}{\left|\xi\right|^{1-\beta}}d\xi\,W(d\lambda_{1})W(d\lambda_{2}),$

where

$\displaystyle
D_{1}(\lambda_{1},\lambda_{2},\xi)=\left[\left(\frac{L\left(s/|\lambda_{1}|\right)}{L(s)}\right)^{1/2}\left(\frac{L\left(s/|\lambda_{2}|\right)}{L(s)}\right)^{1/2}\frac{L\left(s/|\xi|\right)}{L(s)}-1\right]1_{\\{\lambda_{1}\lambda_{2}<0\\}}1_{\\{(\lambda_{1}+\xi)(\lambda_{2}-\xi)<0\\}}.$

By the isometry property of multiple Wiener integers,

$\displaystyle\mathbb{E}\left[|\widetilde{Z}^{(1)}_{s}(t)-Z^{(1)}(t)|^{2}\right]$
(37) $\displaystyle=$ $\displaystyle
128\int_{\mathbb{R}^{2}}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left|\int_{\mathbb{R}}D_{1}(\lambda_{1},\lambda_{2},\xi)\frac{\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)}{\left|\xi\right|^{1-\beta}}d\xi\right|^{2}d\lambda_{1}d\lambda_{2}.$

Applying the inequality $|xy-1|\leq|x^{2}-1|+|y^{2}-1|$ for $x,y\geq 0$
repeatedly yields

$\displaystyle|D_{1}(\lambda_{1},\lambda_{2},\xi)|\leq$
$\displaystyle\left|\left(\frac{L\left(s/|\lambda_{1}|\right)}{L(s)}\right)^{1/2}\left(\frac{L\left(s/|\lambda_{2}|\right)}{L(s)}\right)^{1/2}\frac{L\left(s/|\xi|\right)}{L(s)}-1\right|$
(38) $\displaystyle\leq$
$\displaystyle\left|\frac{L^{2}\left(s/|\lambda_{1}|\right)}{L^{2}(s)}-1\right|+\left|\frac{L^{2}\left(s/|\lambda_{2}|\right)}{L^{2}(s)}-1\right|+\left|\frac{L^{2}\left(s/|\xi|\right)}{L^{2}(s)}-1\right|.$

By (37) and (38),

$\displaystyle\mathbb{E}\left[|\widetilde{Z}^{(1)}_{s}(t)-Z^{(1)}(t)|^{2}\right]\lesssim
I_{1}+I_{2},$

where

$\displaystyle
I_{1}=\int_{\mathbb{R}^{2}}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left|\frac{L^{2}\left(s/|\lambda_{1}|\right)}{L^{2}(s)}-1\right|^{2}\left[\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}$

and

$\displaystyle
I_{2}=\int_{\mathbb{R}^{2}}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left[\int_{\mathbb{R}}\left|\frac{L^{2}\left(s/|\xi|\right)}{L^{2}(s)}-1\right|\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}.$

Let $B$ be the interval $[-1,1]$, and let $B^{c}$ denote its complement. Then,
$I_{1}=I_{1,1}+I_{1,2}$, where

$\displaystyle
I_{1,1}=\int_{\mathbb{R}}\int_{B}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left|\frac{L^{2}\left(s/|\lambda_{1}|\right)}{L^{2}(s)}-1\right|^{2}\left[\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}$

and

$\displaystyle I_{1,2}=$
$\displaystyle\int_{\mathbb{R}}\int_{B^{c}}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left|\frac{L^{2}\left(s/|\lambda_{1}|\right)}{L^{2}(s)}-1\right|^{2}\left[\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}.$

By Lemma 5 in Appendix A,

$\displaystyle I_{1,1}\lesssim
g_{\tau}^{2}(s)\int_{\mathbb{R}}\int_{B}\left|\lambda_{1}\right|^{\beta-1-2\delta}\left|\lambda_{2}\right|^{\beta-1}\left[\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}$

for any $\delta>0$. The integral above is finite if $2\beta<1$ and $\delta$ is
sufficiently small, as demonstrated by Lemma 3. By Lemma 6 in Appendix A,

$\displaystyle I_{1,2}\lesssim
g_{\tau}^{2}(s)\int_{\mathbb{R}}\int_{B^{c}}\left|\lambda_{1}\right|^{\beta-1-2\tau+2\delta}\left|\lambda_{2}\right|^{\beta-1}\left[\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}$

for any $\delta>0$. The integral above is finite if $-\tau<(1-2\beta)/2$ and
$\delta$ is sufficiently small, as demonstrated by Lemma 3. Therefore, under
the condition $-\tau<(1-2\beta)/2$,

(39) $\displaystyle I_{1}\lesssim g_{\tau}^{2}(s).$

Observe that $I_{2}\leq 2(I_{2,1}+I_{2,2}),$ where

$\displaystyle
I_{2,1}=\int_{\mathbb{R}^{2}}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left[\int_{B}\left|\frac{L^{2}\left(s/|\xi|\right)}{L^{2}(s)}-1\right|\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}$

and

$\displaystyle
I_{2,2}=\int_{\mathbb{R}^{2}}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left[\int_{B^{c}}\left|\frac{L^{2}\left(s/|\xi|\right)}{L^{2}(s)}-1\right|\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}.$

By Lemma 5,

(40) $\displaystyle I_{2,1}\lesssim
g_{\tau}^{2}(s)\int_{\mathbb{R}^{2}}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left[\int_{B}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta+\delta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}$

for any $\delta>0$. Referring to Lemma 3, the integral in (40) is finite when
$2\beta<1$ and $\delta$ is sufficiently small. On the other hand, by Lemma 6,

$\displaystyle I_{2,2}\lesssim
g_{\tau}^{2}(s)\int_{\mathbb{R}^{2}}\left|\lambda_{1}\lambda_{2}\right|^{\beta-1}\left[\int_{B^{c}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta+\tau-\delta}}d\xi\right]^{2}d\lambda_{1}d\lambda_{2}$

for any $\delta>0$, where the integral is finite when $-\tau<1-2\beta$ and
$\delta$ is sufficiently small as shown in Lemma 3. Therefore,

(41) $I_{2}\lesssim g_{\tau}^{2}(s)$

under the condition $-\tau<1-2\beta$.

Combining (39) and (41) yields

(42)
$\displaystyle\mathbb{E}\left[|\widetilde{Z}^{(1)}(t)-Z^{(1)}(t)|^{2}\right]\lesssim
g_{\tau}^{2}(s)$

under the condition $-\tau<(1-2\beta)/2$.

$\bullet$ Upper bound of
$\mathbb{E}[|\widetilde{Z}^{(2)}_{s}(t)-Z^{(2)}(t)|^{2}]$: By the definitions
of $Z^{(2)}$ in (16) and $\widetilde{Z}^{(2)}_{s}$ in (35),

$\displaystyle\widetilde{Z}^{(2)}_{s}(t)-Z^{(2)}(t)=$ $\displaystyle
8\int_{\mathbb{R}^{4}}^{{}^{\prime}}e^{it(\lambda_{1}+\cdots+\lambda_{4})}D_{2}(\lambda_{1},...,\lambda_{4})$
$\displaystyle\times\frac{\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})}{|\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}|^{(1-\beta)/2}}W(d\lambda_{1})W(d\lambda_{2})W(d\lambda_{3})W(d\lambda_{4}),$

where

$\displaystyle
D_{2}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=\left[\overset{4}{\underset{\ell=1}{\prod}}\frac{L^{1/2}(s/\lambda_{\ell})}{L^{1/2}(s)}-1\right]1_{\\{\lambda_{1}\lambda_{2}<0\\}}1_{\\{\lambda_{3}\lambda_{4}<0\\}}1_{\\{(\lambda_{1}+\lambda_{2})(\lambda_{3}+\lambda_{4})<0\\}}.$

By the isometry property of multiple Wiener integrals and the Minkowski
inequality,

(43)
$\displaystyle\mathbb{E}\left[|\widetilde{Z}^{(2)}_{s}(t)-Z^{(2)}(t)|^{2}\right]\leq
64\cdot
4!\int_{\mathbb{R}^{4}}|D_{2}(\lambda_{1},...,\lambda_{4})|^{2}\frac{|\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})|^{2}}{|\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}|^{1-\beta}}d\lambda_{1}d\lambda_{2}d\lambda_{3}d\lambda_{4}.$

Applying the inequality

$\displaystyle|D_{2}(\lambda_{1},...,\lambda_{4})|\leq\overset{4}{\underset{\ell=1}{\sum}}\left|\frac{L^{2}\left(s/|\lambda_{\ell}|\right)}{L^{2}(s)}-1\right|$

to (43) yields

(44)
$\displaystyle\mathbb{E}\left[|\widetilde{Z}^{(2)}_{s}(t)-Z^{(2)}(t)|^{2}\right]\lesssim\int_{\mathbb{R}^{4}}\left|\frac{L^{2}\left(s/|\lambda_{1}|\right)}{L^{2}(s)}-1\right|^{2}\frac{|\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})|^{2}}{|\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}|^{1-\beta}}d\lambda_{1}d\lambda_{2}d\lambda_{3}d\lambda_{4}.$

Denote

$\displaystyle
J_{1}=\int_{\mathbb{R}^{3}}\int_{B}\left|\frac{L^{2}\left(s/|\lambda_{1}|\right)}{L^{2}(s)}-1\right|^{2}\frac{|\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})|^{2}}{|\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}|^{1-\beta}}d\lambda_{1}d\lambda_{2}d\lambda_{3}d\lambda_{4},$

where $B$ is the interval $[-1,1]$, and

$\displaystyle
J_{2}=\int_{\mathbb{R}^{3}}\int_{B^{c}}\left|\frac{L^{2}\left(s/|\lambda_{1}|\right)}{L^{2}(s)}-1\right|^{2}\frac{|\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})|^{2}}{|\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}|^{1-\beta}}d\lambda_{1}d\lambda_{2}d\lambda_{3}d\lambda_{4}.$

By Lemma 5, for sufficiently small $\delta>0$,

$\displaystyle J_{1}\lesssim$ $\displaystyle
g_{\tau}^{2}(s)\int_{\mathbb{R}^{3}}\int_{B}\frac{|\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})|^{2}}{|\lambda_{1}|^{1-\beta+2\delta}|\lambda_{2}\lambda_{3}\lambda_{4}|^{1-\beta}}d\lambda_{1}d\lambda_{2}d\lambda_{3}d\lambda_{4}$
(45) $\displaystyle\lesssim$ $\displaystyle
g_{\tau}^{2}(s)\int_{\mathbb{R}}\frac{|\widehat{\psi}(u)|^{2}}{|u|^{1-2\beta+2\delta}}du\int_{\mathbb{R}}\frac{|\widehat{\psi}(v)|^{2}}{|v|^{1-2\beta}}dv<\infty.$

By Lemma 6, for any $\delta>0$,

$\displaystyle J_{2}\lesssim$ $\displaystyle
g_{\tau}^{2}(s)\int_{\mathbb{R}^{3}}\int_{B^{c}}\frac{|\widehat{\psi}(\lambda_{1}+\lambda_{2})\widehat{\psi}(\lambda_{3}+\lambda_{4})|^{2}}{|\lambda_{1}|^{1-\beta+2\tau-2\delta}|\lambda_{2}\lambda_{3}\lambda_{4}|^{1-\beta}}d\lambda_{1}d\lambda_{2}d\lambda_{3}d\lambda_{4}$
(46) $\displaystyle\lesssim$ $\displaystyle
g_{\tau}^{2}(s)\int_{\mathbb{R}}\frac{|\widehat{\psi}(u)|^{2}}{|u|^{1-2\beta+2\tau-2\delta}}du\int_{\mathbb{R}}\frac{|\widehat{\psi}(v)|^{2}}{|v|^{1-2\beta}}dv,$

where the integrals are finite when $-\tau<(1-2\beta)/2$ and $\delta$ is
sufficiently small. By substituting the estimates (45) and (46) into (44), we
obtain

(47)
$\displaystyle\mathbb{E}\left[|\widetilde{Z}^{(2)}_{s}(t)-Z^{(2)}(t)|^{2}\right]\lesssim
g_{\tau}^{2}(s)$

under the condition $-\tau<(1-2\beta)/2$.

$\bullet$ Upper bound of $d_{\textup{Kol}}\left(Z(t)+\varepsilon,Z(t)\right)$:
According to Definition 1 for the Kolmogorov distance,

$\displaystyle d_{\textup{Kol}}\left(Z(t)+\varepsilon,Z(t)\right)=$
$\displaystyle\underset{z\in\mathbb{R}}{\sup}\left|P(Z(t)+\varepsilon\leq
z)-P(Z(t)\leq z)\right|$ $\displaystyle=$
$\displaystyle\underset{z\in\mathbb{R}}{\sup}\left|P(Z(t)\leq
z-\varepsilon)-P(Z(t)\leq z)\right|$ $\displaystyle=$
$\displaystyle\underset{z\in\mathbb{R}}{\sup}P(z-\varepsilon<Z(t)\leq z)$ (48)
$\displaystyle=$
$\displaystyle\underset{z\in\mathbb{R}}{\sup}P(z-\varepsilon/2<Z(t)\leq
z+\varepsilon/2).$

Because $Z(t)$ is a random variable defined as the sum of the second Wiener
chaos, $Z^{(1)}(t)$, as defined in (15), and the fourth Wiener chaos,
$Z^{(2)}(t)$, as defined in (16), there exists a constant $C>0$ such that

(49) $\displaystyle P\left(\left|Z(t)-z\right|\leq\varepsilon/2\right)\leq
C\left\\{\mathbb{E}\left[|Z(t)-z|^{2}\right]\right\\}^{-1/8}\varepsilon^{1/4}$

for any $\varepsilon>0$ and $z\in\mathbb{R}$. We skip the proof of (49)
because it can be derived using the method employed in the proofs of Theorem
3.1 and Lemma 4.3 in [36], as well as the proof of Lemma 4 in [4].
Substituting (49) into (48) yields

(50) $\displaystyle d_{\textup{Kol}}\left(Z(t)+\varepsilon,Z(t)\right)\leq
C\underset{z\in\mathbb{R}}{\sup}\left\\{\mathbb{E}\left[|Z(t)-z|^{2}\right]\right\\}^{-1/8}\varepsilon^{1/4}=C\left\\{\mathbb{E}\left[|Z(t)|^{2}\right]\right\\}^{-1/8}\varepsilon^{1/4}.$

$\bullet$ Upper bound of
$d_{\textup{Kol}}\left(Z(t)+Z^{(0)}_{s}-Z^{(0)},Z(t)\right)$: By (50),

(51) $\displaystyle
d_{\textup{Kol}}\left(Z(t)+Z^{(0)}_{s}-Z^{(0)},Z(t)\right)\lesssim|Z^{(0)}_{s}-Z^{(0)}|^{1/4}.$

From (32) and Lemma 2,

$\displaystyle Z_{s}^{(0)}=$ $\displaystyle
2^{4}s^{2\beta}L^{-2}(s)\int_{\mathbb{R}^{2}}|\widehat{\psi}(s(\lambda_{1}+\lambda_{2}))|^{2}f_{X}(\lambda_{1})f_{X}(\lambda_{2})1_{\\{\lambda_{1}\lambda_{2}<0\\}}d\lambda_{1}d\lambda_{2}$
$\displaystyle=$ $\displaystyle
2^{4}\int_{\mathbb{R}^{2}}|\widehat{\psi}(\lambda_{1}+\lambda_{2})|^{2}\frac{L(s/|\lambda_{1}|)}{L(s)}\frac{L(s/|\lambda_{2}|)}{L(s)}|\lambda_{1}\lambda_{2}|^{\beta-1}1_{\\{\lambda_{1}\lambda_{2}<0\\}}d\lambda_{1}d\lambda_{2}.$

By the definition of $Z^{(0)}$ in (14),

(52) $\displaystyle|Z_{s}^{(0)}-Z^{(0)}|\leq
2^{4}\int_{\mathbb{R}^{2}}|\widehat{\psi}(\lambda_{1}+\lambda_{2})|^{2}\left|\frac{L(s/|\lambda_{1}|)}{L(s)}\frac{L(s/|\lambda_{2}|)}{L(s)}-1\right||\lambda_{1}\lambda_{2}|^{\beta-1}d\lambda_{1}d\lambda_{2}.$

By the inequality

$\left|\frac{L(s/|\lambda_{1}|)}{L(s)}\frac{L(s/|\lambda_{2}|)}{L(s)}-1\right|\leq\left|\frac{L^{2}(s/|\lambda_{1}|)}{L^{2}(s)}-1\right|+\left|\frac{L^{2}(s/|\lambda_{2}|)}{L^{2}(s)}-1\right|,$

for any $\delta>0$, we can further estimate (52) as follows

$\displaystyle|Z_{s}^{(0)}-Z^{(0)}|\leq$ $\displaystyle
2^{5}\int_{\mathbb{R}^{2}}|\widehat{\psi}(\lambda_{1}+\lambda_{2})|^{2}\left|\frac{L^{2}(s/|\lambda_{1}|)}{L^{2}(s)}-1\right||\lambda_{1}\lambda_{2}|^{\beta-1}d\lambda_{1}d\lambda_{2}$
$\displaystyle\lesssim$ $\displaystyle
g_{\tau}(s)\int_{\mathbb{R}}\int_{B}|\widehat{\psi}(\lambda_{1}+\lambda_{2})|^{2}|\lambda_{1}|^{\beta-1-\delta}|\lambda_{2}|^{\beta-1}d\lambda_{1}d\lambda_{2}$
$\displaystyle+g_{\tau}(s)\int_{\mathbb{R}}\int_{B^{c}}|\widehat{\psi}(\lambda_{1}+\lambda_{2})|^{2}|\lambda_{1}|^{\beta-1-\tau+\delta}|\lambda_{2}|^{\beta-1}d\lambda_{1}d\lambda_{2},$

where the last inequality follows from Lemma 5 and Lemma 6. According to Lemma
3, when $2\beta<1$ and $-\tau<1-2\beta$, the integrals above are finite for
sufficiently small $\delta>0$. Hence, $|Z_{s}^{(0)}-Z^{(0)}|\lesssim
g_{\tau}(s).$ As a result, (51) implies that

(53) $\displaystyle
d_{\textup{Kol}}\left(Z(t)+Z^{(0)}_{s}-Z^{(0)},Z(t)\right)\lesssim
g_{\tau}^{1/4}(s).$

Finally, by substituting the estimates from (42), (47), (50), and (53) into
(36), we obtain

(54) $\displaystyle d_{\textup{Kol}}\left(Z_{s}(t),Z(t)\right)\lesssim$
$\displaystyle\varepsilon^{-2}g_{\tau}^{2}(s)+\varepsilon^{1/4}+g_{\tau}^{1/4}(s)$

under the condition $-\tau<(1-2\beta)/2$. By choosing $\varepsilon$ such that
$\varepsilon^{-2}g_{\tau}^{2}(s)=\varepsilon^{1/4}$, achieved by setting
$\varepsilon=g^{8/9}_{\tau}(s)$, and noting that $g_{\tau}(s)\rightarrow 0$ as
$s\rightarrow\infty$, we can rewrite (54) as

$\displaystyle d_{\textup{Kol}}\left(Z_{s}(t),Z(t)\right)\lesssim
g^{2/9}_{\tau}(s)+g_{\tau}^{1/4}(s)\lesssim g^{2/9}_{\tau}(s).$

###  5.4. Proof of Lemma 3

By H$\ddot{\textup{o}}$lder’s inequality,

$\displaystyle\left|\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta-\nu_{3}}}d\xi\right|^{2}\leq\left(\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)|^{2}}{\left|\xi\right|^{1-\beta-\nu_{3}}}d\xi\right)\left(\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{2}-\xi)|^{2}}{\left|\xi\right|^{1-\beta-\nu_{3}}}d\xi\right).$

Hence,

$\displaystyle\int_{\mathbb{R}^{2}}\left|\lambda_{1}\right|^{\nu_{1}+\beta-1}\left|\lambda_{2}\right|^{\nu_{2}+\beta-1}\left|\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)\widehat{\psi}(\lambda_{2}-\xi)|}{\left|\xi\right|^{1-\beta-\nu_{3}}}d\xi\right|^{2}d\lambda_{1}d\lambda_{2}$
$\displaystyle\leq$
$\displaystyle\left[\int_{\mathbb{R}}\left|\lambda_{1}\right|^{\nu_{1}+\beta-1}\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{1}+\xi)|^{2}}{\left|\xi\right|^{1-\beta-\nu_{3}}}d\xi
d\lambda_{1}\right]\left[\int_{\mathbb{R}}\left|\lambda_{2}\right|^{\nu_{2}+\beta-1}\int_{\mathbb{R}}\frac{|\widehat{\psi}(\lambda_{2}-\xi)|^{2}}{\left|\xi\right|^{1-\beta-\nu_{3}}}d\xi
d\lambda_{2}\right]$ $\displaystyle=$
$\displaystyle\left[\int_{\mathbb{R}}\frac{|\widehat{\psi}(z)|^{2}}{\left|z\right|^{1-2\beta-(\nu_{1}+\nu_{3})}}dz\right]\left[\int_{\mathbb{R}}\frac{|\widehat{\psi}(z)|^{2}}{\left|z\right|^{1-2\beta-(\nu_{2}+\nu_{3})}}dz\right]<\infty,$

where the last equal sign follows from the Riesz composition formula [5,
Appendix A], which holds under the conditions $0<\nu_{k}+\beta<1$ for
$k\in\\{1,2,3\\}$ and $0<\nu_{\ell}+\nu_{3}+2\beta<1$ for $\ell\in\\{1,2\\}$.

###  5.5. Proof of Theorem 2

By the definition of $U^{2}[s]Y$ in (3),
$sU^{2}[s]Y(st_{k})=(F_{2k-1,s})^{2}+(F_{2k,s})^{2},\ k=1,2,...,n,$ where
$F_{2k-1,s}=s^{1/2}Y\star\psi_{s}(st_{k})$ and
$F_{2k,s}=s^{1/2}Y\star\mathcal{H}\psi_{s}(st_{k})$. By Lemma 1 and the
identity
$\widehat{\mathcal{H}\psi}(\lambda)=-i\textup{sgn}(\lambda)\widehat{\psi}(\lambda)$,
we can express $F_{2k-1,s}$ and $F_{2k,s}$ as follows:

$\displaystyle
F_{\bullet,s}=\int_{\mathbb{R}^{2}}^{{}^{\prime}}f_{\bullet,s}(\lambda_{1},\lambda_{2})W(d\lambda_{1})W(d\lambda_{2}),$

where

(55) $\displaystyle
f_{2k-1,s}(\lambda_{1},\lambda_{2})=2s^{1/2}e^{ist_{k}(\lambda_{1}+\lambda_{2})}\widehat{\psi}(s(\lambda_{1}+\lambda_{2}))\sqrt{f_{X}(\lambda_{1})f_{X}(\lambda_{2})}1_{\\{\lambda_{1}\lambda_{2}<0\\}}$

and

$\displaystyle
f_{2k,s}(\lambda_{1},\lambda_{2})=-i2s^{1/2}e^{ist_{k}(\lambda_{1}+\lambda_{2})}\textup{sgn}(\lambda_{1}+\lambda_{2})\widehat{\psi}(s(\lambda_{1}+\lambda_{2}))\sqrt{f_{X}(\lambda_{1})f_{X}(\lambda_{2})}1_{\\{\lambda_{1}\lambda_{2}<0\\}}.$

Let
$\mathbf{F}_{s}=(F_{1,s},F_{2,s},...,F_{2n-1,s},F_{2n,s})\in\mathbb{R}^{2n}$,
and define $\mathbf{C}_{s}\in\mathbb{R}^{2n\times 2n}$ as its covariance
matrix. Additionally, let $\mathbf{N}_{s}$ be a $2n$-dimensional normal random
vector with mean zero and covariance matrix
$\mathbf{C}_{s}=[C_{s}(\ell,\ell^{\prime})]_{1\leq\ell,\ell^{\prime}\leq 2n}$,
represented as $(N_{1,s},N_{2,s},...,N_{2n-1,s},N_{2n,s})$. Recalling the
definition of the Wasserstein distance between $\mathbf{F}_{s}$ and
$\mathbf{N}_{s}$:

$\displaystyle
d_{\textup{W}}(\mathbf{F}_{s},\mathbf{N}_{s})=\underset{\|h\|_{\textup{Lip}}\leq
1}{\textup{sup}}\left|\mathbb{E}[h(\mathbf{F}_{s})]-\mathbb{E}[h(\mathbf{N}_{s})]\right|,$

where the supremum is taken over continuous functions
$h:\mathbb{R}^{2n}\rightarrow\mathbb{R}$ with
$|h(\mathbf{x})-h(\mathbf{y})|\leq|\mathbf{x}-\mathbf{y}|$ for
$\mathbf{x},\mathbf{y}\in\mathbb{R}^{2n}$. For any function
$g:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $\|g\|_{\textup{Lip}}\leq 1$
and any $(c_{1},...,c_{n})\in\mathbb{R}^{n}$ with
$c_{1}^{2}+\cdots+c_{n}^{2}\leq 1$, the mapping

$(x_{1},x_{2},...,x_{2n-1},x_{2n})\in\mathbb{R}^{2n}\mapsto
g\left(\overset{n}{\underset{k=1}{\sum}}c_{k}\sqrt{x_{2k-1}^{2}+x_{2k}^{2}}\right)$

is continuous with a Lipschitz constant less than one. Hence,

$\displaystyle
d_{\textup{W}}\left(\overset{n}{\underset{k=1}{\sum}}c_{k}s^{1/2}U[s]Y(st_{k}),\overset{n}{\underset{k=1}{\sum}}c_{k}\sqrt{N_{2k-1,s}^{2}+N_{2k,s}^{2}}\right)$
$\displaystyle=$ $\displaystyle
d_{\textup{W}}\left(\overset{n}{\underset{k=1}{\sum}}c_{k}\sqrt{F_{2k-1,s}^{2}+F_{2k,s}^{2}},\overset{n}{\underset{k=1}{\sum}}c_{k}\sqrt{N_{2k-1,s}^{2}+N_{2k,s}^{2}}\right)$
$\displaystyle=$ $\displaystyle\underset{\|g\|_{\textup{Lip}}\leq
1}{\textup{sup}}\left|\mathbb{E}\left[g\left(\overset{n}{\underset{k=1}{\sum}}c_{k}\sqrt{F_{2k-1,s}^{2}+F_{2k,s}^{2}}\right)\right]-\mathbb{E}\left[g\left(\overset{n}{\underset{k=1}{\sum}}c_{k}\sqrt{N_{2k-1,s}^{2}+N_{2k,s}^{2}}\right)\right]\right|$
(56) $\displaystyle\leq$ $\displaystyle
d_{\textup{W}}(\mathbf{F}_{s},\mathbf{N}_{s}).$

According to [35, Theorem 6.1.1],

(57) $\displaystyle
d_{\textup{W}}(\mathbf{F}_{s},\mathbf{N}_{s})\leq\sqrt{2}\|\mathbf{C}_{s}^{-1}\|_{\textup{op}}\|\mathbf{C}_{s}\|_{\textup{op}}^{1/2}\sqrt{\overset{2n}{\underset{\ell,\ell^{\prime}=1}{\sum}}\mathbb{E}\left[\left(C_{s}(\ell,\ell^{\prime})-\langle
DF_{\ell^{\prime},s},-DL^{-1}F_{\ell,s}\rangle_{\overline{H}}\right)^{2}\right]},$

where $\|\mathbf{C}_{s}^{-1}\|_{\textup{op}}$ represents the operator norm of
the inverse matrix of $\mathbf{C}_{s}$, $D$ denotes the Malliavin derivative,
$L^{-1}$ signifies the pseudo-inverse of the infinitesimal generator $L$ for
the Ornstein-Uhlenbeck semigroup, and $\overline{H}=\\{f\in
L^{2}(\mathbb{R})\mid f(-\lambda)=\overline{f(\lambda)}\ \textup{for all}\
\lambda\in\mathbb{R}\\}$ is a complex Hilbert space equipped with the inner
product $\langle
f,g\rangle_{\overline{H}}=\int_{\mathbb{R}}f(\lambda)\overline{g(\lambda)}d\lambda.$

Denote

$\displaystyle
f^{*2}(u)=\int_{\mathbb{R}}f_{X}(u-v)f_{X}(v)1_{\\{(u-v)v<0\\}}dv,\
u\in\mathbb{R},$

which is an even and continuous function. By the orthogonal property (9), for
any $\ell,\ell^{\prime}\in\\{1,2,...,n-1,2n\\}$,

(58)
$\frac{C_{s}(\ell,\ell^{\prime})}{16}=\left\\{\begin{array}[]{l}\int_{0}^{\infty}\cos((t_{k}-t_{k^{\prime}})u)|\widehat{\psi}(u)|^{2}f^{*2}(u/s)du\
\textup{if}\ (\ell,\ell^{\prime})=(2k-1,2k^{\prime}-1)\ \textup{or}\
(2k,2k^{\prime}),\\\
\int_{0}^{\infty}\sin((t_{k}-t_{k^{\prime}})u)|\widehat{\psi}(u)|^{2}f^{*2}(u/s)du\
\textup{if}\ (\ell,\ell^{\prime})=(2k-1,2k^{\prime}),\\\
\int_{0}^{\infty}\sin((t_{k^{\prime}}-t_{k})u)|\widehat{\psi}(u)|^{2}f^{*2}(u/s)du\
\textup{if}\ (\ell,\ell^{\prime})=(2k,2k^{\prime}-1)\end{array}\right.$

for $k\in\\{1,2,...,n-1,,n\\}$. Define a matrix
$\mathbf{C}^{(n)}_{s}\in\mathbb{C}^{n\times n}$, whose components are given by

(59)
$C^{(n)}_{s}(k,k^{\prime})=16\int_{0}^{\infty}e^{i(t_{k}-t_{k^{\prime}})u}|\widehat{\psi}(u)|^{2}f^{*2}(u/s)du,\
k,k^{\prime}=1,2,...,n.$

For any $\mathbf{v}=[v_{1}\ v_{2}\ \cdots\ v_{2n-1}\
v_{2n}]\in\mathbb{R}^{2n},$ we denote $\mathbf{v}^{(n)}=[v_{1}+iv_{2}\ \cdots\
v_{2n-1}+iv_{2n}]\in\mathbb{C}^{n}.$ We notice that the structure (58) for the
matrix $\mathbf{C}_{s}$ enables us to derive the following equality

(60)
$\displaystyle\mathbf{v}\mathbf{C}_{s}\mathbf{v}^{\top}=\mathbf{v}^{(n)}\mathbf{C}^{(n)}_{s}(\mathbf{v}^{(n)})^{*},$

where $\top$ denotes the transpose, and $*$ represents the complex conjugate.
Because there exists a threshold $\widetilde{s}$ such that
$|\widehat{\psi}(\cdot)|^{2}f^{*2}(\cdot/s)1_{[0,\infty)}(\cdot)$ is a
nonvanishing, nonnegative and integrable function for any $s>\widetilde{s}$,
the Fourier transform of
$|\widehat{\psi}(\cdot)|^{2}f^{*2}(\cdot/s)1_{[0,\infty)}(\cdot)$ is strictly
positive definite [42, Corollary 6.9] at least for $s>\widetilde{s}$. Hence,
$\mathbf{C}^{(n)}_{s}$ is positive definite for $s>\widetilde{s}$. This
implies that for any nonzero vector $\mathbf{v}^{(n)}\in\mathbb{C}^{n}$,
$\mathbf{v}^{(n)}\mathbf{C}^{(n)}_{s}(\mathbf{v}^{(n)})^{*}>0$ for
$s>\widetilde{s}$. This property, combined with the observation (60), implies
the positive definiteness of $\mathbf{C}_{s}$ for $s>\widetilde{s}$, thus
ensuring that $\|\mathbf{C}_{s}^{-1}\|_{\textup{op}}$ in (57) is finite for
$s>\widetilde{s}$.

Under the condition $2\beta>1$, for any $u\in\mathbb{R}$,
$f^{*2}(u/s)\rightarrow\|f_{X}\|_{2}^{2}$ as $s\rightarrow\infty$. Hence,

$\underset{s\rightarrow\infty}{\lim}C_{s}(\ell,\ell^{\prime})=C_{\infty}(\ell,\ell^{\prime}):=16\|f_{X}\|_{2}^{2}\Psi(\ell,\ell^{\prime}),\
\ell,\ell^{\prime}\in\\{1,2,...,2n-1,2n\\},$

where $\Psi$ is defined in (20). The positive definiteness of $\Psi$ can also
be proved using the observation (60), along with the fact
$0<\int_{0}^{\infty}|\widehat{\psi}(u)|^{2}du<\infty$ and the strictly
positive definiteness of the Fourier transform of
$|\widehat{\psi}(\cdot)|^{2}$. Hence,
$\|\mathbf{C}_{s}^{-1}\|_{\textup{op}}\|\mathbf{C}_{s}\|_{\textup{op}}^{1/2}$
converges when $s\rightarrow\infty$ and (57) implies that

(61) $\displaystyle
d_{\textup{W}}(\mathbf{F}_{s},\mathbf{N}_{s})\lesssim\sqrt{\overset{2n}{\underset{\ell,\ell^{\prime}=1}{\sum}}\mathbb{E}\left[\left(C_{s}(\ell,\ell^{\prime})-\langle
DF_{\ell,s},-DL^{-1}F_{\ell^{\prime},s}\rangle_{\overline{H}}\right)^{2}\right]}.$

Next, we aim to estimate the right hand side of (61). For any $p\in\mathbb{N}$
and any function $f$ in the $p$-th tensor product of $\overline{H}$, denoted
as $\overline{H}^{\otimes p}$, let $I_{p}(f)$ represent the $p$-fold Wiener
integral of $f$ with respect to the complex-valued Gaussian white noise random
measure $W$. According to [35, Chapter 2], for
$\ell,\ell^{\prime}\in\\{1,...,2n\\}$,

(62) $\displaystyle DF_{\ell,s}=DI_{2}(f_{\ell,s})=2I_{1}(f_{\ell,s}),$

where

$\displaystyle
I_{1}(f_{\ell,s})(\cdot)=\int_{\mathbb{R}}f_{\ell,s}(\lambda,\cdot)W(d\lambda),$

and

(63) $\displaystyle
L^{-1}I_{2}(f_{\ell^{\prime},s})=-\frac{1}{2}I_{2}(f_{\ell^{\prime},s}).$

By (62), (63), and the product formula [30],

(64) $\displaystyle\langle
DF_{\ell,s},-DL^{-1}F_{\ell^{\prime},s}\rangle_{\overline{H}}=\langle
2I_{1}(f_{\ell,s}),I_{1}(f_{\ell^{\prime},s})\rangle_{\overline{H}}=2\langle
f_{\ell,s},f_{\ell^{\prime},s}\rangle_{\overline{H}^{\otimes
2}}+2I_{2}(f_{\ell,s}\widetilde{\otimes}_{1}f_{\ell^{\prime},s}),$

where

$\langle f_{\ell,s},f_{\ell^{\prime},s}\rangle_{\overline{H}^{\otimes
2}}=\int_{\mathbb{R}^{2}}f_{\ell,s}(\lambda_{1},\lambda_{2})\overline{f_{\ell^{\prime},s}(\lambda_{1},\lambda_{2})}\,d\lambda_{1}d\lambda_{2},$
(65) $\displaystyle
f_{\ell,s}\otimes_{1}f_{\ell^{\prime},s}(\lambda_{1},\lambda_{2})=\int_{\mathbb{R}}f_{\ell,s}(\lambda_{1},\eta)f_{\ell^{\prime},s}(\lambda_{2},-\eta)\,d\eta,$

and $f_{\ell,s}\widetilde{\otimes}_{1}f_{\ell^{\prime},s}$ is the canonical
symmetrization of the product contraction
$f_{\ell,s}\otimes_{1}f_{\ell^{\prime},s}$, defined as

$\displaystyle
f_{\ell,s}\widetilde{\otimes}_{1}f_{\ell^{\prime},s}(\lambda_{1},\lambda_{2})=2^{-1}\left[f_{\ell,s}\otimes_{1}f_{\ell^{\prime},s}(\lambda_{1},\lambda_{2})+f_{\ell,s}\otimes_{1}f_{\ell^{\prime},s}(\lambda_{2},\lambda_{1})\right].$

On the other hand, for $\ell,\ell^{\prime}\in\\{1,...,2n\\}$, by the isometry
property of the multiple Wiener-It$\hat{\textup{o}}$ integrals,

(66) $\displaystyle
C_{s}(\ell,\ell^{\prime})=\mathbb{E}\left[I_{2}(f_{\ell,s})I_{2}(f_{\ell^{\prime},s})\right]=2\langle
f_{\ell,s},f_{\ell^{\prime},s}\rangle_{\overline{H}^{\otimes 2}}.$

Combining (64) and (66) yields

$\displaystyle\mathbb{E}\left[\left(C_{s}(\ell,\ell^{\prime})-\langle
DF_{\ell,s},-DL^{-1}F_{\ell^{\prime},s}\rangle_{\overline{H}}\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(2I_{2}(f_{\ell,s}\widetilde{\otimes}_{1}f_{\ell^{\prime},s})\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 8\langle
f_{\ell,s}\widetilde{\otimes}_{1}f_{\ell^{\prime},s},f_{\ell,s}\widetilde{\otimes}_{1}f_{\ell^{\prime},s}\rangle_{\overline{H}^{\otimes
2}}$ (67) $\displaystyle\leq$ $\displaystyle 8\langle
f_{\ell,s}\otimes_{1}f_{\ell^{\prime},s},f_{\ell,s}\otimes_{1}f_{\ell^{\prime},s}\rangle_{\overline{H}^{\otimes
2}},$

where the inequality follows from the Minkowski inequality. Denote

$\displaystyle
f_{s}(\lambda_{1},\lambda_{2})=s^{1/2}|\widehat{\psi}(s(\lambda_{1}+\lambda_{2}))|\sqrt{f_{X}(\lambda_{1})f_{X}(\lambda_{2})}.$

According to the definition of $f_{\ell,s}$ in (55), $f_{s}\geq|f_{\ell,s}|/2$
for $\ell\in\\{1,...,2n\\}$. Hence, (67) implies that

(68)
$\displaystyle\overset{2n}{\underset{\ell,\ell^{\prime}=1}{\sum}}\mathbb{E}\left[\left(C_{s}(\ell,\ell^{\prime})-\langle
DF_{\ell,s},-DL^{-1}F_{\ell^{\prime},s}\rangle_{\overline{H}}\right)^{2}\right]\leq
2^{9}n^{2}\|f_{s}\otimes_{1}f_{s}\|_{\overline{H}^{\otimes 2}}^{2}.$

Using (56) and substituting (68) into (61), we obtain

$\displaystyle
d_{\textup{W}}\left(\overset{n}{\underset{k=1}{\sum}}c_{k}s^{1/2}U[s]Y(st_{k}),\overset{n}{\underset{k=1}{\sum}}c_{k}\sqrt{N_{2k-1,s}^{2}+N_{2k,s}^{2}}\right)\lesssim\|f_{s}\otimes_{1}f_{s}\|_{\overline{H}^{\otimes
2}}.$

To complete the proof, we need to show that
$\|f_{s}\otimes_{1}f_{s}\|_{\overline{H}^{\otimes 2}}\lesssim s^{-1/2}$ for
the case $\beta\in(3/4,1)$ and
$\|f_{s}\otimes_{1}f_{s}\|_{\overline{H}^{\otimes 2}}\lesssim
s^{-1/2+2(3/4-\beta)}L^{2}(s)$ for the case $\beta\in(1/2,3/4)$. From the
definition of the product contraction $\otimes_{1}$ in (65),

$\displaystyle\|f_{s}\otimes_{1}f_{s}\|_{\overline{H}^{\otimes 2}}^{2}=$
$\displaystyle
s^{2}\int_{\mathbb{R}^{2}}f_{X}(\lambda_{1})f_{X}(\lambda_{2})\left[\int_{\mathbb{R}}|\widehat{\psi}(s(\lambda_{1}+\eta_{1}))||\widehat{\psi}(s(\lambda_{2}-\eta_{1}))|f_{X}(\eta_{1})d\eta_{1}\right]$
$\displaystyle\times\left[\int_{\mathbb{R}}|\widehat{\psi}(s(\lambda_{1}+\eta_{2}))||\widehat{\psi}(s(\lambda_{2}-\eta_{2}))|f_{X}(\eta_{2})d\eta_{2}\right]d\lambda_{1}d\lambda_{2}$
$\displaystyle=$ $\displaystyle
s^{-1}\int_{\mathbb{R}^{3}}\left[\int_{\mathbb{R}}f_{X}(\lambda)f_{X}(\frac{x+y}{s}-\lambda)f_{X}(\frac{x}{s}-\lambda)f_{X}(\frac{z}{s}-\lambda)d\lambda\right]$
(69)
$\displaystyle\times|\widehat{\psi}(x)||\widehat{\psi}(y)||\widehat{\psi}(z)||\widehat{\psi}(x+y-z)|\,dx\,dy\,dz.$

$\bullet$ For the case $\beta\in(3/4,1)$ (i.e., $4(\beta-1)>-1$), because
$f_{X}(\lambda)=L(|\lambda|^{-1})|\lambda|^{\beta-1}$, and for any $\delta>0$,
there exists a constant $C>0$ such that $L(|\lambda|^{-1})\leq
C|\lambda|^{-\delta}$ for all $\lambda\in(-1,1)\setminus\\{0\\}$, it follows
that $\|f_{X}\|_{4}<\infty$. By the generalized H$\ddot{\textup{o}}$lder
inequality,

(70)
$\displaystyle\int_{\mathbb{R}}f_{X}(\lambda)f_{X}(\frac{x+y}{s}-\lambda)f_{X}(\frac{x}{s}-\lambda)f_{X}(\frac{z}{s}-\lambda)d\lambda\leq\|f_{X}\|_{4}^{4}.$

By applying (70) to (69), we obtain

$\displaystyle\|f_{s}\otimes_{1}f_{s}\|_{\overline{H}^{\otimes 2}}^{2}\lesssim
s^{-1}.$

$\bullet$ For the case $\beta\in(1/2,3/4)$, by considering a change of
variable $\lambda=u/s$, the expression inside the brackets in (69) can be
rewritten as

$\displaystyle\int_{\mathbb{R}}f_{X}(\lambda)f_{X}(\frac{x+y}{s}-\lambda)f_{X}(\frac{x}{s}-\lambda)f_{X}(\frac{z}{s}-\lambda)d\lambda$
$\displaystyle=$ $\displaystyle
s^{-1}\int_{\mathbb{R}}f_{X}(\frac{u}{s})f_{X}(\frac{x+y-u}{s})f_{X}(\frac{x-u}{s})f_{X}(\frac{z-u}{s})du$
$\displaystyle=$ $\displaystyle
s^{-1+4(1-\beta)}\int_{\mathbb{R}}\frac{L(s|u|^{-1})}{|u|^{1-\beta}}\frac{L(s|x+y-u|^{-1})}{|x+y-u|^{1-\beta}}\frac{L(s|x-u|^{-1})}{|x-u|^{1-\beta}}\frac{L(s|z-u|^{-1})}{|z-u|^{1-\beta}}du$
$\displaystyle=$ $\displaystyle
s^{-1+4(1-\beta)}L^{4}(s)\int_{\mathbb{R}}\left[\frac{L(s|u|^{-1})}{L(s)}\frac{L(s|x+y-u|^{-1})}{L(s)}\frac{L(s|x-u|^{-1})}{L(s)}\frac{L(s|z-u|^{-1})}{L(s)}\right]$
(71) $\displaystyle\times\frac{1}{|u(x+y-u)(x-u)(z-u)|^{1-\beta}}du.$

By substituting (71) into (69), we obtain

(72) $\displaystyle\|f_{s}\otimes_{1}f_{s}\|_{\overline{H}^{\otimes
2}}^{2}=s^{-2+4(1-\beta)}L^{4}(s)\mathcal{K}_{s},$

where

$\displaystyle\mathcal{K}_{s}=$
$\displaystyle\int_{\mathbb{R}^{4}}\left[\frac{L(s|u|^{-1})}{L(s)}\frac{L(s|x+y-u|^{-1})}{L(s)}\frac{L(s|x-u|^{-1})}{L(s)}\frac{L(s|z-u|^{-1})}{L(s)}\right]$
$\displaystyle\times\frac{|\widehat{\psi}(x)||\widehat{\psi}(y)||\widehat{\psi}(z)||\widehat{\psi}(x+y-z)|}{|u(x+y-u)(x-u)(z-u)|^{1-\beta}}\,du\,dx\,dy\,dz.$

Let $\mu=(\mu_{1},\mu_{2},\mu_{3},\mu_{4})\in\\{-1,1\\}^{4}$ be a function of
$(|u|,|x+y-u|,|x-u|,|z-u|)$ defined as follows

$\displaystyle\begin{aligned}
&\left\\{\begin{array}[]{ll}\mu_{1}=1&\textup{if}\ |u|>1,\\\
\mu_{1}=-1&\textup{if}\ |u|\leq 1,\end{array}\right.\\\
&\left\\{\begin{array}[]{ll}\mu_{2}=1&\textup{if}\ |x+y-z|>1,\\\
\mu_{2}=-1&\textup{if}\ |x+y-u|\leq 1,\end{array}\right.\end{aligned}$
$\displaystyle\begin{aligned}
&\left\\{\begin{array}[]{ll}\mu_{3}=1&\textup{if}\ |x-u|>1,\\\
\mu_{3}=-1&\textup{if}\ |x-u|\leq 1,\end{array}\right.\\\
&\left\\{\begin{array}[]{ll}\mu_{4}=1&\textup{if}\ |z-u|>1,\\\
\mu_{4}=-1&\textup{if}\ |z-u|\leq 1.\end{array}\right.\end{aligned}$

By denoting

$R_{\mu}=\\{(u,x,y,z)\in\mathbb{R}^{4}\mid|u|^{\mu_{1}}\geq 1,\
|x+y-u|^{\mu_{2}}\geq 1,\ |x-u|^{\mu_{3}}\geq 1,\ |z-u|^{\mu_{4}}\geq 1\\},$

we can express $\mathcal{K}_{s}$ as

$\displaystyle\mathcal{K}_{s}=\underset{\mu\in\\{-1,1\\}^{4}}{\sum}\mathcal{K}_{s,\mu},$

where

$\displaystyle\mathcal{K}_{s,\mu}=$
$\displaystyle\int_{\mathbb{R}_{\mu}}\left[\frac{L(s|u|^{-1})}{L(s)}\frac{L(s|x+y-u|^{-1})}{L(s)}\frac{L(s|x-u|^{-1})}{L(s)}\frac{L(s|z-u|^{-1})}{L(s)}\right]$
(73)
$\displaystyle\times\frac{|\widehat{\psi}(x)||\widehat{\psi}(y)||\widehat{\psi}(z)||\widehat{\psi}(x+y-z)|}{|u(x+y-u)(x-u)(z-u)|^{1-\beta}}\,du\,dx\,dy\,dz.$

According to [10, Theorem 1.5.3] (see also [21, p. 1474]), for any $\delta>0$,
there exists a constant $C>0$ and a threshold $s^{*}>0$ such that for any $s$
greater than $s^{*}$, the following inequalities hold

(74)
$\displaystyle\underset{0<r<1}{\sup}\frac{(rs)^{\delta}L\left(rs\right)}{s^{\delta}L(s)}\leq
C$

and

(75) $\displaystyle\underset{r\geq
1}{\sup}\frac{(rs)^{-\delta}L\left(rs\right)}{s^{-\delta}L(s)}\leq C.$

By applying (74) and (75) to (73), we obtain the estimate

$\displaystyle\mathcal{K}_{s,\mu}\leq
C^{4}\int_{\mathbb{R}_{\mu}}\frac{|\widehat{\psi}(x)||\widehat{\psi}(y)||\widehat{\psi}(z)||\widehat{\psi}(x+y-z)|}{|u|^{1-\beta-\delta\mu_{1}}|x+y-u|^{1-\beta-\delta\mu_{2}}|x-u|^{1-\beta-\delta\mu_{3}}|z-u|^{1-\beta-\delta\mu_{4}}}\,du\,dx\,dy\,dz$

for any $s$ greater than $s^{*}$. By Lemma 4, the integral above is finite for
sufficiently small $\delta>0$. Therefore, from (72), we obtain

$\displaystyle\|f_{s}\otimes_{1}f_{s}\|_{\overline{H}^{\otimes 2}}^{2}\lesssim
s^{-2+4(1-\beta)}L^{4}(s)=s^{-1+4(3/4-\beta)}L^{4}(s)$

for the case $\beta\in(1/2,3/4)$.

###  5.6. Proof of Lemma 4

For each $k\in\\{1,2,3,4\\}$, we denote $\beta_{k}=\beta+\delta\mu_{k}$.
Because $\beta\in(1/2,3/4)$, there exists a sufficiently small $\delta>0$ such
that $\beta_{k}\in(1/2,3/4)$ for all $k\in\\{1,2,3,4\\}$. Let $\beta_{k}^{+}$
be a constant greater than $\beta_{k}$ and sufficiently close to $\beta_{k}$.
By H$\ddot{\textup{o}}$lder’s inequality,

$\displaystyle\int_{\mathbb{R}}\frac{|\widehat{\psi}(z)||\widehat{\psi}(x+y-z)|}{|z-u|^{1-\beta_{4}}}dz=I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(u)\times
I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(x+y-u),$

where

$\displaystyle
I_{\beta_{4}}|\widehat{\psi}|^{2}(w):=\int_{\mathbb{R}}\frac{|\widehat{\psi}(z)|^{2}}{|w-z|^{1-\beta_{4}}}dz,$

which is an even function because
$\widehat{\psi}(-z)=\overline{\widehat{\psi}(z)}.$ Hence, (21) can be bounded
as follows

(76)
$\displaystyle\mathcal{K}\leq\int_{\mathbb{R}^{3}}\frac{|\widehat{\psi}(x)||\widehat{\psi}(y)|}{|u|^{1-\beta_{1}}|x+y-u|^{1-\beta_{2}}|x-u|^{1-\beta_{3}}}\,I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(u)\,I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(x+y-u)\,du\,dx\,dy.$

By denoting

(77) $\displaystyle
Q(w)=\int_{\mathbb{R}}|\widehat{\psi}(y)|\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w+y)}{|w+y|^{1-\beta_{2}}}\,dy,\
w\in\mathbb{R},$

we can rewrite (76) as

(78) $\displaystyle\mathcal{K}\leq$
$\displaystyle\int_{\mathbb{R}}\left[\int_{\mathbb{R}}\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(u)}{|u|^{1-\beta_{1}}}\frac{Q(x-u)}{|x-u|^{1-\beta_{3}}}\,du\right]|\widehat{\psi}(x)|\,dx.$

We will show that

(Claim 1)
$\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(\bullet)}{|\bullet|^{1-\beta_{1}}}\in
L^{2/(3-2\beta_{1}^{+}-\beta_{4}^{+})},$ (Claim 2) $Q\in
L^{2/(3-2\beta_{2}^{+}-\beta_{4}^{+})}\cap C,$

where $C$ is the set of continuous functions, and

(Claim 3) $\frac{Q(\bullet)}{|\bullet|^{1-\beta_{3}}}\in
L^{2/(5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+})}.$

Denote $p=2/(3-2\beta_{1}^{+}-\beta_{4}^{+})$ and
$r=2/(5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+})$. Because
$\beta\in(1/2,3/4)$, we have $p>4/3$ and $r>4/5$. We consider the following
two cases according to the range of $r$.

Case 1: For $r\in(4/5,1]$, Claim 2 and Claim 3 imply that
$Q(\bullet)|\bullet|^{\beta_{3}-1}\in L^{1}$. By Minkowski’s inequality for
integrals, we have the following estimate for the convolution inside the
bracket in (78)

(79)
$\displaystyle\left\|\int_{\mathbb{R}}\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(u)}{|u|^{1-\beta_{1}}}\frac{Q(\bullet-u)}{|\bullet-u|^{1-\beta_{3}}}\,du\right\|_{p}\leq\left\|\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(\bullet)}{|\bullet|^{1-\beta_{1}}}\right\|_{p}\left\|\frac{Q(\bullet)}{|\bullet|^{1-\beta_{3}}}\right\|_{1}.$

Case 2: For $r>1$, let $q$ be a constant defined by

$\frac{1}{q}=\frac{1}{p}+\frac{1}{r}-1=3-(\beta_{1}^{+}+\beta_{2}^{+}+\beta_{3}^{+}+\beta_{4}^{+}).$

Because $\beta\in(1/2,3/4)$, when $\delta$ is sufficiently small and
$\beta_{k}^{+}$ is sufficiently close to $\beta_{k}$ for each
$k\in\\{1,2,3,4\\}$, we have
$\beta_{1}^{+}+\beta_{2}^{+}+\beta_{3}^{+}+\beta_{4}^{+}\in(2,3)$. It implies
that $q\in(1,\infty)$. By Young’s inequality,

(80)
$\displaystyle\left\|\int_{\mathbb{R}}\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(u)}{|u|^{1-\beta_{1}}}\frac{Q(\bullet-u)}{|\bullet-u|^{1-\beta_{3}}}\,du\right\|_{q}\leq\left\|\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(\bullet)}{|\bullet|^{1-\beta_{1}}}\right\|_{p}\left\|\frac{Q(\bullet)}{|\bullet|^{1-\beta_{3}}}\right\|_{r}.$

The assertion $\mathcal{K}<\infty$ follows from (79), (80), the assumption
$\psi,\widehat{\psi}\in L^{1}$, and H$\ddot{\textup{o}}$lder’s inequality. In
the following, we prove Claim 1, Claim 2, and Claim 3.

Proof of Claim 1: First of all, according to the Hardy-Littlewood-Sobolev
theorem for fractional integration [40, p. 119], for any
$1<p^{\prime}<q^{\prime}<\infty$ and $\beta\in(0,1)$ with
$1/q^{\prime}=1/p^{\prime}-\beta$, there exists a constant
$A_{p^{\prime},q^{\prime}}$ such that

$\displaystyle\|I_{\beta}|\widehat{\psi}|^{2}\|_{q^{\prime}}\leq
A_{p^{\prime},q^{\prime}}\||\widehat{\psi}|^{2}\|_{p^{\prime}}.$

Because $\||\widehat{\psi}|^{2}\|_{p^{\prime}}<\infty$ for any $p^{\prime}>1$,
we have $I_{\beta}|\widehat{\psi}|^{2}\in L^{1/(1-\beta^{+})}$, where
$\beta^{+}$ is a constant greater than $\beta$, sufficiently proximate to it.
Additionally, for any $w,w^{\prime}\in\mathbb{R}$, $\beta\in(1/2,1)$, and
$\widehat{\psi}\in L^{1}\cap L^{\infty}$,

$\displaystyle\left|I_{\beta}|\widehat{\psi}|^{2}(w)-I_{\beta}|\widehat{\psi}|^{2}(w^{\prime})\right|=$
$\displaystyle\left|\int_{\mathbb{R}}\frac{1}{|z|^{1-\beta}}\left[|\widehat{\psi}(w-z)|^{2}-|\widehat{\psi}(w^{\prime}-z)|^{2}\right]dz\right|$
$\displaystyle\leq$
$\displaystyle\int_{-1}^{1}\frac{1}{|z|^{1-\beta}}\left||\widehat{\psi}(w-z)|^{2}-|\widehat{\psi}(w^{\prime}-z)|^{2}\right|dz$
$\displaystyle+\int_{\mathbb{R}}1_{\\{|z|>1\\}}\frac{1}{|z|^{1-\beta}}\left||\widehat{\psi}(w-z)|^{2}-|\widehat{\psi}(w^{\prime}-z)|^{2}\right|dz$
$\displaystyle\leq$
$\displaystyle\left[\int_{-1}^{1}\frac{dz}{|z|^{2(1-\beta)}}\right]^{1/2}\left[\int_{-1}^{1}\left||\widehat{\psi}(w-z)|^{2}-|\widehat{\psi}(w^{\prime}-z)|^{2}\right|^{2}dz\right]^{1/2}$
$\displaystyle+\int_{\mathbb{R}}\left||\widehat{\psi}(w-z)|^{2}-|\widehat{\psi}(w^{\prime}-z)|^{2}\right|dz,$

which tends to zero as $|w-w^{\prime}|\rightarrow 0$ due to the continuity of
translation in the $L^{r^{\prime}}$ norm, where $r^{\prime}\in[1,\infty)$.
Hence, for any $\beta\in(1/2,1)$,

(81) $I^{1/2}_{\beta}|\widehat{\psi}|^{2}\in L^{2/(1-\beta^{+})}\cap C,$

where $C$ represents the set of continuous functions. Because
$I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(\cdot)$ is an even function,

$\displaystyle\int_{\mathbb{R}}\left(\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w)}{|w|^{1-\beta_{1}}}\right)^{2/(3-2\beta_{1}^{+}-\beta_{4}^{+})}dw=$
$\displaystyle
2\int_{0}^{1}\left(\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w)}{|w|^{1-\beta_{1}}}\right)^{2/(3-2\beta_{1}^{+}-\beta_{4}^{+})}dw$
$\displaystyle+$ $\displaystyle
2\int_{1}^{\infty}\left(\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w)}{|w|^{1-\beta_{1}}}\right)^{2/(3-2\beta_{1}^{+}-\beta_{4}^{+})}dw:=\mathcal{I}_{1}+\mathcal{I}_{2}.$

Because $2(1-\beta_{1})/(3-2\beta_{1}^{+}-\beta_{4}^{+})\in(0,1)$ and (81)
implies the boundedness of $I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}$, we have

$\displaystyle\mathcal{I}_{1}\lesssim\int_{0}^{1}|w|^{-\frac{2(1-\beta_{1})}{3-2\beta_{1}^{+}-\beta_{4}^{+}}}dw<\infty.$

Regarding the estimation of $\mathcal{I}_{2}$, because

$\displaystyle\left(\frac{3-2\beta_{1}^{+}-\beta_{4}^{+}}{1-\beta_{4}^{+}}\right)^{-1}+\left(\frac{3-2\beta_{1}^{+}-\beta_{4}^{+}}{2(1-\beta_{1}^{+})}\right)^{-1}=1$

and

$\frac{3-2\beta_{1}^{+}-\beta_{4}^{+}}{1-\beta_{4}^{+}}>\frac{3-2\beta_{1}^{+}-\beta_{4}^{+}}{2(1-\beta_{1}^{+})}>1$

for $\beta\in(1/2,3/4)$, we apply H$\ddot{\textup{o}}$lder’s inequality to
derive

$\displaystyle\mathcal{I}_{2}=$ $\displaystyle
2\int_{1}^{\infty}\left(\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w)}{|w|^{1-\beta_{1}}}\right)^{2/(3-2\beta_{1}^{+}-\beta_{4}^{+})}dw$
$\displaystyle\lesssim$
$\displaystyle\left[\int_{1}^{\infty}\left|I_{\beta_{4}}|\widehat{\psi}|^{2}(w)\right|^{1/(1-\beta_{4}^{+})}dw\right]^{(1-\beta_{4}^{+})/(3-2\beta_{1}^{+}-\beta_{4}^{+})}\left[\int_{1}^{\infty}|w|^{-\frac{1-\beta_{1}}{1-\beta_{1}^{+}}}dw\right]^{2(1-\beta_{1}^{+})/(3-2\beta_{1}^{+}-\beta_{4}^{+})}.$

The convergence of the right-hand side of this inequality stems from (81).
Consequently, the proof of Claim 1 is concluded.

Proof of Claim 2: Referring to the definition of $Q$ in (77), and utilizing
Minkowski’s inequality for integrals, we derive

$\displaystyle\|Q\|_{2/(3-2\beta_{2}^{+}-\beta_{4}^{+})}\leq\|\widehat{\psi}\|_{1}\left\|\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(\bullet)}{|\bullet|^{1-\beta_{2}}}\right\|_{2/(3-2\beta_{2}^{+}-\beta_{4}^{+})}<\infty,$

where the final inequality stems from Claim 1 by substituting $\beta_{1}$ with
$\beta_{2}$. Because $\beta\in(1/2,3/4)$, both
$2/(3-2\beta_{2}^{+}-\beta_{4}^{+})$ and $2/(2\beta_{2}^{+}+\beta_{4}^{+}-1)$
are strictly greater than one. Moreover, the following equality

$\left(\frac{2}{3-2\beta_{2}^{+}-\beta_{4}^{+}}\right)^{-1}+\left(\frac{2}{2\beta_{2}^{+}+\beta_{4}^{+}-1}\right)^{-1}=1$

holds. It allows us to apply H$\ddot{\textup{o}}$lder inequality. For any
$w,w^{\prime}\in\mathbb{R}$,

$\displaystyle\left|Q(w)-Q(w^{\prime})\right|\leq$
$\displaystyle\int_{\mathbb{R}}\left|\widehat{\psi}(y)\right|\left|\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w+y)}{|w+y|^{1-\beta_{2}}}-\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w^{\prime}+y)}{|w^{\prime}+y|^{1-\beta_{2}}}\right|\,dy$
$\displaystyle\leq$
$\displaystyle\|\widehat{\psi}\|_{2/(2\beta_{2}^{+}+\beta_{4}^{+}-1)}\left\|\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w+\bullet)}{|w+\bullet|^{1-\beta_{2}}}-\frac{I^{1/2}_{\beta_{4}}|\widehat{\psi}|^{2}(w^{\prime}+\bullet)}{|w^{\prime}+\bullet|^{1-\beta_{2}}}\right\|_{2/(3-2\beta_{2}^{+}-\beta_{4}^{+})},$

which tends to zero as $|w-w^{\prime}|\rightarrow 0$ due to the continuity of
translation in the $L^{r^{\prime}}$ norm, where $r^{\prime}\in[1,\infty)$.
Hence, $Q$ is continuous everywhere, thereby concluding the proof of Claim 2.

Proof of Claim 3: Because $Q$ is continuous everywhere,

$\int_{\mathbb{R}}\left|\frac{Q(w)}{|w|^{1-\beta_{3}}}\right|^{2/(5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+})}dw\lesssim\mathcal{I}_{3}+\mathcal{I}_{4},$

where

$\mathcal{I}_{3}=\int_{0}^{1}|w|^{-\frac{2(1-\beta_{3})}{5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+}}}dw$

and

$\mathcal{I}_{4}=\int_{1}^{\infty}\left|\frac{Q(w)}{|w|^{1-\beta_{3}}}\right|^{2/(5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+})}dw.$

It is clear that $\mathcal{I}_{3}<\infty$. Because

$\left(\frac{5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+}}{3-2\beta_{2}^{+}-\beta_{4}^{+}}\right)^{-1}+\left(\frac{5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+}}{2(1-\beta_{3}^{+})}\right)^{-1}=1$

and both terms within the parentheses are greater than one, we have

(82)
$\displaystyle\mathcal{I}_{4}\leq\left[\int_{1}^{\infty}|Q(w)|^{2/(3-2\beta_{2}^{+}-\beta_{4}^{+})}dw\right]^{\frac{3-2\beta_{2}^{+}-\beta_{4}^{+}}{5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+}}}\left[\int_{1}^{\infty}|w|^{\frac{1-\beta_{3}}{1-\beta_{3}^{+}}}dw\right]^{\frac{2(1-\beta_{3}^{+})}{5-2\beta_{2}^{+}-2\beta_{3}^{+}-\beta_{4}^{+}}}.$

The convergence of the first integral in (82) follows from Claim 2, thereby
establishing the proof of Claim 3.

###  5.7. Proof of Corollary 1

Consider two mean-zero Gaussian processes, denoted by
$\\{N_{R}(t)\\}_{t\in\mathbb{R}}$ and $\\{N_{I}(t)\\}_{t\in\mathbb{R}}$, with
the following covariance matrix:

(83)
$\mathbb{E}\left[\begin{array}[]{cc}N_{R}(t)N_{R}(t^{\prime})&N_{R}(t)N_{I}(t^{\prime})\\\
N_{I}(t)N_{R}(t^{\prime})&N_{I}(t)N_{I}(t^{\prime})\end{array}\right]=2^{4}\|f_{X}\|_{2}^{2}\int_{0}^{\infty}\left[\begin{array}[]{cc}\cos((t-t^{\prime})u)&\sin((t-t^{\prime})u)\\\
\sin((t^{\prime}-t)u)&\cos((t-t^{\prime})u)\end{array}\right]|\widehat{\psi}(u)|^{2}du,$

where $t,t^{\prime}\in\mathbb{R}.$ From Theorem 2,
$F(t)\overset{d}{=}N^{2}_{R}(t)+N^{2}_{I}(t),$ where $\overset{d}{=}$ means
that the finite-dimensional distributions of $F$ and $N^{2}_{R}+N^{2}_{I}$ are
identical. From (83), we obtain that

$\displaystyle\textup{Cov}\left(F(t),F(t^{\prime})\right)=\textup{Cov}\left(N^{2}_{R}(t)+N^{2}_{I}(t),N^{2}_{R}(t^{\prime})+N^{2}_{I}(t^{\prime})\right)$
$\displaystyle=$
$\displaystyle\textup{Cov}\left(N^{2}_{R}(t),N^{2}_{R}(t^{\prime})\right)+\textup{Cov}\left(N^{2}_{R}(t),N^{2}_{I}(t^{\prime})\right)+\textup{Cov}\left(N^{2}_{I}(t),N^{2}_{R}(t^{\prime})\right)+\textup{Cov}\left(N^{2}_{I}(t),N^{2}_{I}(t^{\prime})\right)$
$\displaystyle=$ $\displaystyle
2\left[\textup{Cov}\left(N_{R}(t),N_{R}(t^{\prime})\right)^{2}+\textup{Cov}\left(N_{R}(t),N_{I}(t^{\prime})\right)^{2}+\textup{Cov}\left(N_{I}(t),N_{R}(t^{\prime})\right)^{2}+\textup{Cov}\left(N_{I}(t),N_{I}(t^{\prime})\right)^{2}\right]$
$\displaystyle=$ $\displaystyle
2^{10}\|f_{X}\|_{2}^{4}\left(\int_{0}^{\infty}\cos((t-t^{\prime})u)|\widehat{\psi}(u)|^{2}du\right)^{2}+2^{10}\|f_{X}\|_{2}^{4}\left(\int_{0}^{\infty}\sin((t-t^{\prime})u)|\widehat{\psi}(u)|^{2}du\right)^{2}.$

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##  Appendix A Lemmas about the slowly varying function $L$

In the proof of Theorem 1, several lemmas from [4] are required. Below, we
integrate them for ease of reference and application.

######  Lemma 5.

Under Assumption 1, for any $\delta>0$ and sufficiently large $s$, there
exists a constant $C>0$ such that the following inequality

$\displaystyle\left|\frac{L^{2}\left(s/|x|\right)}{L^{2}(s)}-1\right|\leq
Cg_{\tau}(s)|x|^{-\delta}$

holds for any $x\in\mathbb{R}$ with $|x|\leq 1$.

Proof: Under Assumption 1, [4, Lemma 1] proved that for any $\delta>0$ and
sufficiently large $s$, these exists a constant $C_{1}$ such that

(84) $\displaystyle\left|\frac{L^{2}\left(s/|x|\right)}{L^{2}(s)}-1\right|\leq
C_{1}g_{\tau}(s)h_{\tau}\left(\frac{1}{|x|}\right)\left(\frac{1}{|x|}\right)^{\delta}$

holds for $|x|\leq 1$. From (8), i.e.,

(87) $\displaystyle
h_{\tau}\left(|x|^{-1}\right)=\left\\{\begin{array}[]{ll}\ln(|x|^{-1})&\
\textup{if}\ \tau=0,\\\ \tau^{-1}(|x|^{-\tau}-1)&\ \textup{if}\
\tau<0,\end{array}\right.$

we have the following observation.

  * •

For $\tau=0$, there exists a constant $C_{2}>0$ such that
$h_{\tau}\left(|x|^{-1}\right)\leq C_{2}|x|^{-\delta}$ for $|x|\leq 1$.

  * •

For $\tau<0$, $h_{\tau}\left(|x|^{-1}\right)\leq C_{2}$ for $|x|\leq 1$, where
$C_{2}=|\tau|^{-1}$.

Consequently, based on (84), we derive

$\displaystyle\left|\frac{L^{2}\left(s/|x|\right)}{L^{2}(s)}-1\right|\leq
C_{1}C_{2}g_{\tau}(s)|x|^{-2\delta}$

for any $x\in\mathbb{R}$ with $|x|\leq 1$. ∎

######  Lemma 6.

Under Assumption 1, for any $\delta>0$ and sufficiently large $s$, there
exists a constant $C>0$ such that the following inequality

(88) $\displaystyle\left|\frac{L^{2}\left(s/|x|\right)}{L^{2}(s)}-1\right|\leq
Cg_{\tau}(s)|x|^{-\tau+\delta}$

holds for any $x\in\mathbb{R}$ with $|x|>1$.

Proof: First of all, we rewrite the left hand side of (88) as follows

(89)
$\displaystyle\left|\frac{L^{2}\left(s/|x|\right)}{L^{2}(s)}-1\right|=\frac{L^{2}\left(s/|x|\right)}{L^{2}(s)}\left|1-\frac{L^{2}(s)}{L^{2}\left(s/|x|\right)}\right|.$

From [10, Theorem 1.5.3], we know that if a mapping $s\mapsto R_{\rho}(s)$
varies regularly with index $\rho>0$ and it is locally bounded on
$[0,\infty)$, we have

(90) $\displaystyle\underset{s\rightarrow\infty}{\lim}\underset{0\leq r\leq
s}{\sup}\ \frac{R_{\rho}\left(r\right)}{R_{\rho}(s)}=1.$

By applying (90) to the function $s\rightarrow s^{\delta}L^{2}(s)$, which
varies regularly with index $\delta$, we obtain that there exists a constant
$C_{1}$ such that

(91) $\displaystyle\frac{L^{2}\left(s/|x|\right)}{L^{2}(s)}\leq$
$\displaystyle\left[\underset{1<|z|<\infty}{\sup}\frac{(s/|z|)^{\delta}L^{2}\left(s/|z|\right)}{s^{\delta}L^{2}(s)}\right]|x|^{\delta}\leq
C_{1}|x|^{\delta}$

for any $x\in\mathbb{R}$ with $|x|>1$. According to [4, Lemma 1], there exists
another constant $C_{2}>0$ such that

$\displaystyle\left|1-\frac{L^{2}(s)}{L^{2}\left(s/|x|\right)}\right|\leq$
$\displaystyle C_{2}g_{\tau}(s/|x|)h_{\tau}(|x|)|x|^{\delta}$ $\displaystyle=$
$\displaystyle
C_{2}g_{\tau}(s)\frac{g_{\tau}(s/|x|)}{g_{\tau}(s)}h_{\tau}(|x|)|x|^{\delta}$
$\displaystyle\leq$ $\displaystyle
C_{2}g_{\tau}(s)\left[\underset{1<|z|<\infty}{\sup}\frac{(s/|z|)^{-\tau+\delta}g_{\tau}(s/|z|)}{s^{-\tau+\delta}g_{\tau}(s)}\right]h_{\tau}(|x|)|x|^{-\tau+2\delta}$

for any $x\in\mathbb{R}$ with $|x|>1$. By applying (90) to the function
$s\rightarrow s^{-\tau+\delta}g_{\tau}(s)$, which also varies regularly with
index $\delta$, we obtain that there exists a constant $C_{3}$ such that

(92) $\displaystyle\left|1-\frac{L^{2}(s)}{L^{2}\left(s/|x|\right)}\right|\leq
C_{2}C_{3}g_{\tau}(s)h_{\tau}(|x|)|x|^{-\tau+2\delta}$

for any $x\in\mathbb{R}$ with $|x|>1$. From (8), i.e.,

(95) $\displaystyle
h_{\tau}\left(|x|\right)=\left\\{\begin{array}[]{ll}\ln(|x|)&\ \textup{if}\
\tau=0,\\\ \tau^{-1}(|x|^{\tau}-1)&\ \textup{if}\ \tau<0,\end{array}\right.$

we have the following observation.

  * •

For $\tau=0$, there exists a constant $C_{4}>0$ such that
$h_{\tau}\left(|x|\right)\leq C_{4}|x|^{\delta}$ for $|x|>1$.

  * •

For $\tau<0$, $h_{\tau}\left(|x|\right)\leq C_{4}$ for $|x|>1$, where
$C_{4}=|\tau|^{-1}$.

By substituting (91) and (92) into (89) and using the observation about
$h_{\tau}(|x|)$, we obtain

$\displaystyle\left|\frac{L^{2}\left(s/|x|\right)}{L^{2}(s)}-1\right|\leq
C_{1}C_{2}C_{3}C_{4}g_{\tau}(s)|x|^{-\tau+3\delta}$

for any $x\in\mathbb{R}$ with $|x|>1$. ∎

##  Appendix B Behavior of the minimum eigenvalue of $\Psi$

In this section, considering $n$ time points $t_{1},t_{2},...,t_{n}$, where
$t_{k}=k\Delta t$ for $k=1,2,...,n$ and $\Delta t>0$, we explore the
relationship between the minimum eigenvalue of $\Psi$ and both the time lag
size $\Delta t$ and the number of time points $n$. We consider a family of
wavelets with Fourier transforms $\widehat{\psi}$ taking the form:

$\displaystyle|\widehat{\psi}(u)|^{2}=c_{\alpha,\gamma}|u|^{\alpha}e^{-|u|^{\gamma}},\quad
u\in\mathbb{R},$

where $\alpha$ and $\gamma$ are both greater than 0 to satisfy
$\widehat{\psi}(0)=0$ and $\widehat{\psi}\in L^{1}$. The constant
$c_{\alpha,\gamma}$ serves as a normalizer to ensure that
$\int_{0}^{\infty}|\widehat{\psi}(u)|^{2}du=1$.

(a) effects of $\alpha$ and $\gamma$ ($n=5$) (b) effects of $n$
($\alpha=\gamma=1$)

Figure 3. Relationship between the minimum eigenvalue of $\Psi$ and the time
lag size $\Delta t$.

Figure 3 shows that, as the time lag size $\Delta t$ approaches zero, the
minimum eigenvalue of $\Psi$ decreases towards zero. With an increase in
$\Delta t$, the correlation among the components of the random vector
$(N_{1,s},N_{2,s},...,N_{2n-1,s},N_{2n,s})$ reduces. Consequently, the matrix
$\Psi$ tends towards the identity matrix leading the minimum eigenvalue of
$\Psi$ to approach one.