# almost sure central limit theorem for the hyperbolic Anderson model with
Lévy white noise

Raluca M. Balan, Panqiu Xia, and Guangqu Zheng  Raluca M. Balan, Department of
Mathematics and Statistics
University of Ottawa
150 Louis Pasteur Private
Ottawa, ON, K1N 6N5
Canada<EMAIL_ADDRESS>Panqiu Xia
Department of Mathematics and Statistics
Auburn University
221 Parker Hall
Auburn, Alabama 36849
United States of America<EMAIL_ADDRESS>Guangqu Zheng, Department of
Mathematical Sciences
University of Liverpool
Mathematical Sciences Building
Liverpool, L69 7ZL
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

In this paper, we present an almost sure central limit theorem (ASCLT) for the
hyperbolic Anderson model (HAM) with a Lévy white noise in a finite-variance
setting. This work complements a recent work by the first and third authors on
the (quantitative) central limit theorems for the solution to the HAM. We
present two different proofs: one uses the Clark-Ocone formula and takes
advantage of the martingale structure of the white-in-time noise, and the
other is obtained by combining the second-order Gaussian Poincaré inequality
with Ibragimov and Lifshits’ method of characteristic functions. Both
approaches are different from the one developed in the PhD thesis of C. Zheng
(2011), allowing us to establish the ASCLT without using lengthy computations
based on star contractions. Moreover, the second approach is expected to be
useful for similar studies on SPDEs with Gaussian colored-in-time noise when
the first approach, based on Itô calculus, is not applicable.

###### Key words and phrases:

Almost sure central limit theorem; hyperbolic Anderson model; Ibragimov-
Lifshits’ criterion; second-order Gaussian Poincaré inequality; Clark-Ocone
formula; Malliavin calculus; Lévy noise.

###### 2020 Mathematics Subject Classification:

60F15, 60H30

Panqiu Xia is partially supported by NSF grant DMS-2246850.

##  1\. Introduction

This short note is devoted to the study of an almost sure central limit
theorem (ASCLT) for a stochastic (linear) wave equation (1.1) on
$\mathbb{R}_{+}\times\mathbb{R}$ driven by a space-time Lévy (white)
noise.111In the literature, such an equation is called the hyperbolic Anderson
model, in analogy to the parabolic Anderson model with the wave operator
replaced by the heat operator. Many papers in the literature regard the
Anderson models as the ones with spatial-only white noise; however, many other
papers loosely included stochastic linear wave/heat equations with space-time
(colored/white) noises in the family of Anderson models. The title of this
short note indicates our stand, aligning with the previous paper [6]. The Lévy
noise can be colored in space as in [3], but in this note we focus on the case
of the space-time Lévy white noise.

In recent years, there has been an ongoing interest and active research on the
limit theorems for the random field solutions to stochastic partial
differential equations (SPDEs); see Subsection 1.1 for a brief overview. In
Subsection 1.2, we will state our main result (Theorem 1.6) after briefly
setting up the main framework. The proofs of Theorem 1.6 will be presented in
Section 2 and the proof of technical lemmas will be given in the Appendix A.

###  1.1. Central limit theorems for SPDEs

In a recent work [21], Huang, Nualart, and Viitasaari established a central
limit theorem (CLT) for the spatial integral of the solution to a stochastic
nonlinear heat equation with space-time Gaussian white noise and constant
initial condition on $\mathbb{R}_{+}\times\mathbb{R}$:
$(\partial_{t}-\tfrac{1}{2}\partial_{x}^{2})u=\sigma(u)\dot{W}$ and $u(0,x)=1$
for $(t,x)\in\mathbb{R}_{+}\times\mathbb{R}$, where $\dot{W}$ denotes the
space-time Gaussian white noise on $\mathbb{R}_{+}\times\mathbb{R}$ and the
nonlinearity $\sigma(u)$ is described by a Lipschitz function
$\sigma:\mathbb{R}\to\mathbb{R}$. More precisely, the authors of [21]
established that for fixed $t\in(0,\infty)$, the spatial integral of the
solution $u(t,x)$ over $[-R,R]$ admits Gaussian fluctuation as $R\to\infty$.
Their CLT result was partially motivated by the study of mathematical
intermittency of the random field solution. In retrospect, (a) it is not hard
to expect that the solution
$\\{u(t,x)\\}_{(t,x)\in\mathbb{R}_{+}\times\mathbb{R}}$ is spatially
stationary due to the constant initial condition and homogeneous noise; (b) it
is also natural to investigate furthermore the spatial ergodicity of $u$ that
will lead to the first-order result, a law of large number: as $R\to+\infty$,
$\frac{1}{2R}\int_{-R}^{R}u(t,x)dx\to 1$ in $L^{2}(\mathbb{P})$ and almost
surely; (c) therefore, the central limit theorem (the second-order
fluctuation) would naturally come into the picture.

Since the appearance of the work [21], there have been a growing body of
literature on similar CLT results for heat equations with various Gaussian
noises; see, for example, [22, 36, 33, 12, 13, 14, 35, 38, 39, 32]. Meanwhile,
such a program was carried out by Nualart, the third author, and their
collaborators to investigate the stochastic (nonlinear) wave equations driven
by Gaussian noises; see [16, 10, 38, 37, 5] and see also the recent works [18,
19] by Ebina on three and higher-dimensional stochastic wave equations. In a
recent paper [6] by the first and third authors, a similar program for the
SPDEs with Lévy noises has been initiated. In particular, this paper
considered the following stochastic (linear) wave equation driven by a space-
time Lévy white noise:

$\displaystyle\begin{cases}&(\partial_{t}^{2}-\partial_{x}^{2})u=u\dot{L}\\\\[5.0pt]
&\big{(}u(0,\bullet),\partial_{t}u(0,\bullet)\big{)}=(1,0)\end{cases}\quad{\rm
on}\quad(t,x)\in\mathbb{R}_{+}\times\mathbb{R},$ (1.1)

where (i) $\dot{L}$ denotes the space-time (pure-jump) Lévy noise on
$\mathbb{R}_{+}\times\mathbb{R}$ with finite variance:

  1. (i-a)

let $\mathbf{Z}\coloneqq\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{0}$,
with $\mathbb{R}_{0}\coloneqq\mathbb{R}\setminus\\{0\\}$ equipped with the
distance $d(x,y)=|x^{-1}-y^{-1}|$, and let $\mathcal{Z}$ be the Borel
$\sigma$-algebra on $\mathbf{Z}$, and let $m={\rm Leb}\times\nu$, with ${\rm
Leb}$ the Lebesgue measure on $\mathbb{R}_{+}\times\mathbb{R}$ and $\nu$ a
$\sigma$-finite measure on $\mathbb{R}_{0}$ satisfying the Lévy-measure
condition $\int_{\mathbb{R}_{0}}\min\\{1,|z|^{2}\\}\nu(dz)<\infty$ and the
finite-variance condition
$m_{2}\coloneqq\int_{\mathbb{R}_{0}}|z|^{2}\nu(dz)<\infty$;

  2. (i-b)

let $N$ be a Poisson random measure on the space $(\mathbf{Z},\mathcal{Z})$
with intensity $m$, and let $\widehat{N}=N-m$ be the compensated version of
$N$, and we put

$L(A)=\int_{A\times\mathbb{R}_{0}}z\widehat{N}(ds,dy,dz)$

for any Borel set $A\subset\mathbb{R}_{+}\times\mathbb{R}_{0}$ with ${\rm
Leb}(A)<\infty$, where the above integral $L(A)$ is defined in the Itô sense,
which is an infinitely divisible random variable with Lévy-Khintchine formula
$\mathbb{E}\big{[}e^{i\lambda L(A)}\big{]}=\exp\big{(}{\rm
Leb}(A)\int_{\mathbb{R}_{0}}(e^{i\lambda z}-1-i\lambda z)\nu(dz)\big{)}$,
$\lambda\in\mathbb{R}$;

  3. (i-c)

$\dot{L}(s,y)=L(ds,dy)$ is the formal derivative $\partial_{s}\partial_{y}L$;

(ii) the product of the unknown $u$ and the Lévy noise $\dot{L}$ is
intetpreted in the Itô sense; and in Duhamel formulation, we rewrite (1.1) in
the following integral form:

$\displaystyle
u(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}}G_{t-s}(x-y)u(s,y)L(ds,dy),$ (1.2)

with the integral understood as follows:

$\int_{\mathbb{R}\times\mathbb{R}}X(s,y)L(ds,dy)=\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}_{0}}X(s,y)z\widehat{N}(ds,dy,dz),$

whenever the right side makes sense, and
$G_{t}(x)=\frac{1}{2}\mathbf{1}_{\\{|x|<t\\}}$ denotes the fundamental
solution to the corresponding deterministic wave equation.

It is known that the equation (1.1) admits a unique solution $\\{u(t,x):t\geq
0,x\in\mathbb{R}\\}$ that is an $\mathbb{F}$-predictable process satisfying
(1.2) and
$\sup\big{\\{}\mathbb{E}[|u(t,x)|^{2}]:(t,x)\in[0,T]\times\mathbb{R}\\}<\infty$
for any finite $T>0$; see [4, Theorem 1.1]. Here,
$\mathbb{F}=\\{\mathcal{F}_{t}\\}_{t\in\mathbb{R}_{+}}$ denotes the natural
filtration generated by the Lévy noise $\dot{L}$.222More precisely, let
$\mathcal{F}^{0}_{t}$ be the $\sigma$-algebra generated by the random
variables $N([0,s]\times A\times B)$ with $s\in[0,t]$ and ${\rm
Leb}([0,s]\times A)+\nu(B)<\infty$. And let
$\mathcal{F}_{t}=\sigma\big{(}\mathcal{F}_{t}^{0}\cup\mathcal{N}\big{)}$ be
the $\sigma$-algebra generated by $\mathcal{F}_{t}^{0}$ and the set
$\mathcal{N}$ of $\mathbb{P}$-null sets. This gives us a filtration
$\mathbb{F}=\\{\mathcal{F}_{t}\\}_{t\in\mathbb{R}_{+}}$.  Motivated by recent
work on Gaussian fluctuations of spatial integrals of SPDEs, the first and
third authors established in [6] the first result on spatial ergodicity and
CLT results for SPDEs with Lévy noise. The goal of the current note is to
further complement [6] by providing a corresponding almost sure central limit
theorem (ASCLT). We will state below several results from [6] after fixing
some notations, and state our main Theorem 1.6 (i.e. ASCLT) in Subsection 1.2.

$\bullet$ Notations. In this paper, all random objects are defined on a rich
common probability space $(\Omega,\mathcal{F},\mathbb{P})$, and $\mathbb{E}$,
${\rm Var},{\rm Cov}$ stand for the associated expectation, variance,
covariance operators respectively. We use
$\|X\|_{p}\coloneqq\|X\|_{L^{p}(\Omega)}$ for any $p\in[1,\infty)$ and real-
valued random variable $X$, and we write $\|f\|_{\infty}$ for the essential-
sup norm for any measurable $f:\mathbb{R}\to\mathbb{R}$. The symbol
“$\Longrightarrow$” indicates the weak convergence of probability measures.
Throughout this paper, $\delta_{z}$ and $\gamma$ stand for the Dirac mass at
$z\in\mathbb{R}$ and the standard Gaussian measure on $\mathbb{R}$
respectively. We write $Y\sim\gamma$ to mean that $Y$ is a standard normal
random variable. For any two (probability) measures $\mu_{1}$ and $\mu_{2}$ on
$\mathbb{R}$, we define the Fortet-Mourier distance

$\displaystyle d_{\rm
FM}(\mu_{1},\mu_{2}):=\sup_{\phi}\Big{|}\int_{\mathbb{R}}\phi(x)\mu_{1}(dx)-\int_{\mathbb{R}}\phi(x)\mu_{2}(dx)\Big{|},$
(1.3)

where the supremum runs over all bounded Lipschitz functions $\phi$ with
$\|\phi\|_{\infty}+\|\phi^{\prime}\|_{\infty}\leq 1$. It is a well-known fact
that the distance $d_{\rm FM}$ characterizes the weak convergence of
probability measures on $\mathbb{R}$; see [17, Theorem 11.3.3]. Another two
stronger metrics, the Wasserstein distance $d_{\rm Wass}$ and the Kolmogorov
distance $d_{\rm Kol}$, are defined, respectively, by

$\displaystyle d_{\rm Wass}(\mu_{1},\mu_{2})$
$\displaystyle\coloneqq\sup_{\|\phi^{\prime}\|_{\infty}\leq
1}\Big{|}\int_{\mathbb{R}}\phi(x)\mu_{1}(dx)-\int_{\mathbb{R}}\phi(x)\mu_{2}(dx)\Big{|},$
(1.4) $\displaystyle d_{\rm Kol}(\mu_{1},\mu_{2})$
$\displaystyle\coloneqq\sup_{t\in\mathbb{R}}|\mu_{1}((-\infty,t])-\mu_{2}((-\infty,t])|.$
(1.5)

For two random variables $X,Y$, we also use $d_{\rm Wass}(X,Y)$ to denote the
Wasserstein distance between their distributional measures. On the $L^{2}$
probability space generated by the Poisson random measure $N$, one can develop
the Malliavin calculus; see e.g. [6, Section 2] for more details and for any
unexplained notations. In particular, we recall that for
$F\in\mathbb{D}^{1,2}$ Malliavin differentiable, $D_{\xi}F$ denotes the
Malliavin derivative $DF\in L^{2}(\Omega;L^{2}(\mathbf{Z},m))$ valued at
$\xi\in\mathbf{Z}$, where for convenience, we often write $\xi,m(d\xi)$ as
short hand notations for
$\xi=(r,y,z)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{0}$ and
$m(d\xi)=dr\times dy\times\nu(dz)$ respectively, whenever no ambiguity arises.
For two functions $g(t)$ and $h(t)$, $g(t)\asymp h(t)$ means that
$0<c_{1}\leq\liminf_{t\to+\infty}g(t)/h(t)\leq\limsup_{t\to+\infty}g(t)/h(t)\leq
c_{2}<\infty$ for some constants $c_{1}$ and $c_{2}$.

Now we are ready to state several results from [6, Theorem 1.1 and Proposition
3.2].

######  Proposition 1.1.

Fix $t_{0}\in(0,\infty)$. For any $\theta>0$, we put

$\displaystyle
F_{\theta}:=\int_{-\theta}^{\theta}\big{[}u(t_{0},x)-1\big{]}dx\quad\text{and}\quad\sigma_{\theta}:=\sqrt{{\rm
Var}(F_{\theta})}.$ (1.6)

Recall $G_{t}(x)=\frac{1}{2}\mathbf{1}_{\\{|x|<t\\}}$, and define

$m_{p}\coloneqq\int_{\mathbb{R}_{0}}|z|^{p}\nu(dz),\quad p\in[1,\infty).$

Then, the following statements hold.

(i) Suppose $m_{2}<\infty$. Then, $F_{\theta}\in\mathbb{D}^{1,2}$ and
$\sigma_{\theta}>0$ for every $\theta>0$ with
$\sigma_{\theta}\asymp\sqrt{\theta}$ as $\theta\to+\infty$.

(ii) Assume in addition that $m_{2+2\alpha}<\infty$ for some $\alpha\in[0,1]$.
Then,

$\displaystyle\|D_{r_{1},y,z}u(t_{0},x)\|_{2+2\alpha}\lesssim
G_{t_{0}-r_{1}}(x-y)|z|,$ and $\displaystyle\begin{aligned}
\|D_{r_{2},y_{2},z_{2}}D_{r_{1},y_{1},z_{1}}u(t_{0},x)\|_{2+2\alpha}\lesssim
G_{t_{0}}(x-y_{1})G_{t_{0}}(y_{1}-y_{2})\end{aligned}$ (1.7)

for any $r_{1},r_{2}\in(0,t_{0})$.

(iii) Assume both $m_{1+\alpha}$ and $m_{2+2\alpha}$ are finite for some
$\alpha\in(0,1]$.333In particular, this forces $m_{2}$ to be finite.  Then,
$d_{\rm
Wass}(\frac{1}{\sigma_{\theta}}F_{\theta},Y)\lesssim\theta^{-\frac{\alpha}{1+\alpha}}$
for any $\theta>0$, where $Y\sim\gamma$ is a standard normal random variable.

Note that the positivity of $\sigma_{\theta}$ for $\theta>0$ can be seen from
(4.13), (4.14), and exact computation of the expression
$\Phi_{1}(t,t;\textup{w})$ in [6]. The bound (1.7) follows immediately from
[6, (3.24)] and triangle inequality (e.g.
$\mathbf{1}_{\\{|x-y_{1}|<t_{0}-r_{1}\\}}\mathbf{1}_{\\{|y_{1}-y_{2}|<r_{1}-r_{2}\\}}\leq\mathbf{1}_{\\{|x-y_{2}|<t_{0}-r_{2}\\}}$).
One of the key ingredients for part (iii) is the following second-order
Poincaré inequality that goes back to Last, Peccati, and Schulte’s paper [26],
and has been recently improved by Trauthwein [42].

######  Proposition 1.2.

([42, Theorem 3.4]) Suppose $F\in\mathbb{D}^{1,2}$ is centered with variance
$b^{2}$. Then, with $Y\sim\gamma$ and for any $p,q\in(1,2]$, we have

$\displaystyle d_{\rm
Wass}(F,Y)\leq\frac{2}{\sqrt{\pi}}|1-b^{2}|+\gamma_{1}+\gamma_{2}+\gamma_{3},$
(1.8)

where

$\displaystyle\begin{aligned}
\gamma_{1}&\coloneqq\frac{2^{\frac{2}{p}+\frac{1}{2}}}{\sqrt{\pi}}\bigg{(}\int_{\mathbf{Z}}\bigg{[}\int_{\mathbf{Z}}\|D_{\xi_{2}}F\|_{2p}\|D_{\xi_{1}}D_{\xi_{2}}F\|_{2p}\,m(d\xi_{2})\bigg{]}^{p}m(d\xi_{1})\bigg{)}^{\frac{1}{p}},\\\
\gamma_{2}&\coloneqq\frac{2^{\frac{2}{p}-\frac{1}{2}}}{\sqrt{\pi}}\bigg{(}\int_{\mathbf{Z}}\bigg{[}\int_{\mathbf{Z}}\|D_{\xi_{1}}D_{\xi_{2}}F\|^{2}_{2p}\,m(d\xi_{2})\bigg{]}^{p}m(d\xi_{1})\bigg{)}^{\frac{1}{p}},\\\
\gamma_{3}&\coloneqq
2\int_{\mathbf{Z}}\|D_{\xi}F\|_{q+1}^{q+1}\,m(d\xi);\end{aligned}$ (1.9)

see also [6, Proposition 2.8].

######  Remark 1.3.

The above Proposition 1.2 will be combined with Ibragimov-Lifshits’ method of
characteristic functions (Proposition 2.3) to provide the second proof of
Theorem 1.6 in Section 2. It is important to point out that the random
variable $F$ in (1.8) may not have unit variance, whereas the original
statement of [42, Theorem 3.4] gives an estimate for $d_{\rm
Wass}(\widehat{F},Y)$ with $\widehat{F}=(F-\mathbb{E}(F))/\sqrt{{\rm
Var}(F)}$. This seemingly minor modification is crucial for obtaining (2.14):
applying directly [42, Theorem 3.4] would require a uniform (positive) lower
bound for the variance $V_{\theta,\textup{w}}$ in (2.15), which is never the
case because of $V_{\theta,\theta}=0$. Instead, one can redo the proof of
Theorems 3.2–3.4 in [42] keeping (D.6) on page 34 unchanged but replacing
(D.19) on page 35 by

$G=\underbrace{\int_{\mathbb{Y}}D_{y}F\,\mathbb{E}\big{[}D_{y}F|\eta_{y}\big{]}\,\bar{\lambda}(dy)-b^{2}}_{\text{mean
zero; see (D.20).}}+b^{2}-1.$

Then the same arguments would lead to the bound in (1.8). We leave the details
for interested readers. Such a minor modification of the bound in (1.8) is
quite standard within the Stein’s method ([40, 15, 34]).

###  1.2. Almost sure central limit theorems

The almost sure central limit theorem (ASCLT) was first formulated by Paul
Lévy in his book [28, page 270] without a proof. This result had not gained
much attention until various authors re-discovered it in the eighties ([20,
11, 41, 25]). The simplest form of ASCLT can be stated as follows. Let
$\\{X_{n}\\}_{n\geq 1}$ be an independent and identically distributed sequence
of centered random variable with the unit variance, then for
$\mathbb{P}$-almost every $\omega\in\Omega$,

$\displaystyle\frac{1}{\log
n}\sum_{k=1}^{n}\frac{1}{k}\delta_{\frac{S_{k}}{\sqrt{k}}(\omega)}\Longrightarrow\gamma\quad\text{as
$n\to+\infty$. }$ (1.10)

In other words, the Gaussian asymptotic behavior can be observed along a
generic trajectory via this logarithmic average. It was observed in several
papers [25, 9, 1, 27] that the above hypothesis of i.i.d. sequence of random
variables with mean zero and unit variance is optimal for ASCLT. See also [8,
24] for more details. Let us first state a few definitions, one of which
generalizes (1.10).

######  Definition 1.4.

(i) [Discrete version] A sequence $\\{F_{k}\\}_{k\in\mathbb{N}}$ of real
random variables is said to satisfy the ASCLT if for $\mathbb{P}$-almost every
$\omega\in\Omega$,

$\displaystyle\mu_{N}^{\omega}\coloneqq\frac{1}{\log
N}\sum_{k=1}^{N}\frac{1}{k}\delta_{F_{k}(\omega)}\Longrightarrow\gamma\quad\text{as
$N\to\infty$. }$ (1.11)

(ii) [Continuum version] A family $\\{F_{y}\\}_{y\geq 1}$ of real random
variables is said to satisfy the ASCLT if for $\mathbb{P}$-almost every
$\omega\in\Omega$, the map $y\mapsto F_{y}(\omega)$ is almost surely
measurable and 444The (random) measure $\nu_{T}^{\omega}$ can be understood by
first testing it with $f\in C_{c}(\mathbb{R})$ (continuous function with
compact support), i.e.,  $\langle\nu_{T}^{\omega},f\rangle=\frac{1}{\log
T}\int_{1}^{T}f\big{(}F_{y}(\omega)\big{)}\frac{dy}{y},\quad(\star)$ which
defines a positive linear functional on $C_{c}(\mathbb{R})$, and therefore, as
a consequence of Riesz’s representation theorem, $\nu_{T}^{\omega}$ defined in
(1.12) makes sense as the unique Radon measure satisfying $(\star)$.

$\displaystyle\nu_{T}^{\omega}\coloneqq\frac{1}{\log
T}\int_{1}^{T}\delta_{F_{y}(\omega)}\frac{dy}{y}\Longrightarrow\gamma\quad\text{as
$T\to+\infty$.}$ (1.12)

In the current note, we aim to establish an ASCLT for the spatial integrals of
the solution to (1.1). Then, it is more natural for us to proceed with the
continuum version of Definition 1.4, while the discrete analogue can be dealt
with in the same way.

######  Remark 1.5.

(i) For simplicity, we take the logarithmic average in (1.11) and (1.12). In
fact, one can consider more general weights, for example, using
$(d_{k},D_{N})$ in place of $(\frac{1}{k},\log N)$, where $d_{k}>0$,
$D_{N}=d_{1}+\dots+d_{N}$ with $D_{N+1}/D_{N}\to 1$ as $N\to+\infty$; see also
the discussion in [25, page 204].

(ii) Due to the separability of the real line $\mathbb{R}$, one can construct
a _countable_ family $\\{\phi_{n}\\}_{n\in\mathbb{N}}$ of real bounded
Lipschitz functions on $\mathbb{R}$ such that
$\\{\phi_{n}\\}_{n\in\mathbb{N}}$ forms a separating class for the weak
convergence of probability measures on $\mathbb{R}$; see, for example, [24,
Proposition 2.2]. As a consequence, the validity of (1.12) (for
$\mathbb{P}$-almost every $\omega\in\Omega$) is equivalent to each of the
following statements:

$\displaystyle\textup{(a)}\quad d_{\rm
FM}(\nu^{\omega}_{T},\gamma)\xrightarrow[T\to+\infty]{\rm
almost\,\,surely}0,\quad\text{with $d_{\rm FM}$ defined as in \eqref{FM}};$
$\displaystyle\textup{(b)}\quad\textup{$\forall f\in C_{b}(\mathbb{R})$,
almost surely,}\quad\frac{1}{\log
T}\int_{1}^{T}\frac{1}{\theta}f(F_{\theta})\,d\theta\xrightarrow{T\to+\infty}\int_{\mathbb{R}}f(x)\gamma(dx);$
here $C_{b}(\mathbb{R})$ is the set of real bounded continuous functions on
$\mathbb{R}$, and (b) is equivalent to the seemingly stronger statement
$\displaystyle\textup{(b')}\quad\textup{almost surely, $\forall f\in
C_{b}(\mathbb{R})$,}\quad\frac{1}{\log
T}\int_{1}^{T}\frac{1}{\theta}f(F_{\theta})\,d\theta\xrightarrow{T\to+\infty}\int_{\mathbb{R}}f(x)\gamma(dx).$
In view of [34, Proposition C.3.2], the above statements are also equivalent
to the following statements: $\displaystyle\textup{(c)}\quad\textup{almost
surely, $\forall t\in\mathbb{R}$,}\quad\frac{1}{\log
T}\int_{1}^{T}\frac{1}{\theta}\mathbf{1}_{\\{F_{\theta}\leq
t\\}}\,d\theta\xrightarrow{T\to+\infty}\gamma((-\infty,t]);$
$\displaystyle\textup{(d)}\quad d_{\rm
Kol}(\nu^{\omega}_{T},\gamma)\xrightarrow[T\to+\infty]{\rm
almost\,\,surely}0,\quad\text{with $d_{\rm Kol}$ defined as in \eqref{KOL}}.$

The usual CLT does not require the random variables to be defined on a common
probability space, while the ASCLT does; and furthermore, the convergence to
normality at a fast rate does not necessarily imply the ASCLT. This can be
readily confirmed by, for example, setting $F_{k}\equiv Y\sim\gamma$ for all
$k$, leading to a clear violation of the ASCLT (1.11). In other words, a finer
understanding of the whole family of random variables is necessary to
investigate the ASCLT.

Now let us state our main result.

######  Theorem 1.6.

Let the notations in Proposition 1.1 prevail and let
$\widetilde{F}_{\theta}=F_{\theta}/\sigma_{\theta}$ for $\theta>0$. Assume
that $m_{1+\alpha}+m_{2+2\alpha}<\infty$ for some $\alpha\in(0,1]$. Then,
$\\{\widetilde{F}_{\theta}\\}_{\theta\geq 1}$ satisfy the ASCLT.

We will provide two different proofs in the following.

##  2\. Two proofs of Theorem 1.6

###  2.1. Proof of Theorem 1.6 via the Clark-Ocone formula

In view of Remark 1.5-(ii), it suffices to show

$\displaystyle\frac{1}{\log
T}\int_{1}^{T}\frac{1}{\theta}f(\widetilde{F}_{\theta})d\theta\xrightarrow[T\to+\infty]{a.s.}\int_{\mathbb{R}}f(x)\gamma(dx)$
(2.1)

for any bounded Lipschitz function $f:\mathbb{R}\to\mathbb{R}$ with Lipschitz
constant ${\rm Lip}(f)$. By the CLT result in Proposition 1.1,
$\mathbb{E}[f(\widetilde{F}_{\theta})]\to\int_{\mathbb{R}}f(x)\gamma(dx)$ as
$\theta\to+\infty$. Then, (2.1) is equivalent to

$\displaystyle\frac{1}{\log
T}\int_{1}^{T}\frac{1}{\theta}\big{(}f(\widetilde{F}_{\theta})-\mathbb{E}[f(\widetilde{F}_{\theta})]\big{)}d\theta\xrightarrow[T\to+\infty]{a.s.}0.$
(2.2)

Put

$\displaystyle H_{\theta}\coloneqq
f(\widetilde{F}_{\theta})-\mathbb{E}[f(\widetilde{F}_{\theta})].$ (2.3)

Since $H_{\theta}$ is $\mathcal{F}_{t_{0}}$-measurable, one can deduce from
the Clark-Ocone formula (see, for example, [6, Lemma 2.5]) that

$\displaystyle
H_{\theta}=\int_{0}^{t_{0}}\int_{\mathbb{R}}\int_{\mathbb{R}_{0}}\mathbb{E}\big{[}D_{r,y,z}H_{\theta}|\mathcal{F}_{r}\big{]}\widehat{N}(dr,dy,dz),$
(2.4)

which is an Itô integral with respect to the compensated Poisson random
measure $\widehat{N}$, and $t_{0}\in(0,\infty)$ is fixed as in (1.6). Then, we
deduce from the Itô isometry, the chain rule (e.g. [6, (2.42)]), Cauchy-
Schwarz inequality, and Jensen inequality that for $\theta<\textup{w}$,

$\displaystyle\big{|}\mathbb{E}[H_{\theta}H_{\textup{w}}]\big{|}$
$\displaystyle\leq\frac{{\rm
Lip}^{2}(f)}{\sigma_{\theta}\sigma_{\textup{w}}}\int_{0}^{t_{0}}\int_{\mathbb{R}}\int_{\mathbb{R}_{0}}\|D_{r,y,z}F_{\theta}\|_{2}\cdot\|D_{r,y,z}F_{\textup{w}}\|_{2}\,drdy\nu(dz);$
(2.5)

while Proposition 1.1-(ii), together with Minkowski’s inequality, implies that

$\displaystyle\begin{aligned}
\|D_{r,y,z}F_{\theta}\|_{2}&\leq\int_{-\theta}^{\theta}\|D_{r,y,z}u(t_{0},x)\|_{2}\,dx\\\
&\lesssim|z|\int_{-\theta}^{\theta}G_{t_{0}-r}(x-y)dx.\end{aligned}$ (2.6)

Therefore, it follows from (2.5), (2.6), Proposition 1.1-(i), and the
following simple inequalities

$G_{t_{0}-r}(\bullet)\leq G_{t_{0}}(\bullet),\quad
G_{t_{0}}(x_{1}-y)G_{t_{0}}(x_{2}-y)\leq
G_{t_{0}}(x_{1}-y)G_{2t_{0}}(x_{1}-x_{2})$

that

$\displaystyle\begin{aligned}
\big{|}\mathbb{E}[H_{\theta}H_{\textup{w}}]\big{|}&\lesssim\frac{1}{\sqrt{\theta\textup{w}}}\int_{0}^{t_{0}}\int_{\mathbb{R}}\int_{\mathbb{R}_{0}}|z|^{2}\left(\int_{-\theta}^{\theta}G_{t_{0}-r}(x_{1}-y)dx_{1}\right)\\\
&\qquad\qquad\quad\cdot\left(\int_{-\textup{w}}^{\textup{w}}G_{t_{0}-r}(x_{2}-y)dx_{2}\right)drdy\nu(dz)\\\
&\leq\frac{m_{2}t_{0}}{\sqrt{\theta\textup{w}}}\int_{\mathbb{R}}\left(\int_{-\theta}^{\theta}G_{t_{0}}(x_{1}-y)dx_{1}\right)\cdot\left(\int_{-\textup{w}}^{\textup{w}}G_{2t_{0}}(x_{2}-x_{1})dx_{2}\right)dy\\\
&\leq
4m_{2}t^{3}_{0}\big{(}\theta/\textup{w}\big{)}^{\frac{1}{2}}\quad\text{for
$\theta<\textup{w}$,}\end{aligned}$ (2.7)

where the last step is obtained by performing integration in the order of
$dx_{2},dy$, then $dx_{1}$.

Hence, the proof of (2.2) is done by invoking the following Lemma 2.1. ∎

######  Lemma 2.1.

Suppose $\\{H_{\theta}\\}_{\theta>0}$ is a family of real random variables
that are uniformly bounded with

$\big{|}\mathbb{E}[H_{\theta}H_{\textup{w}}]\big{|}\leq
C_{\beta}\big{(}\theta/\textup{w}\big{)}^{\beta}$

for any $0<\theta<\textup{w}$, where the exponent $\beta>0$ and the constant
$C_{\beta}$ do not depend on $(\theta,\textup{w})$. Assume also that
$\theta\mapsto H_{\theta}$ is a measurable function almost surely. Then,

$\displaystyle L_{T}\coloneqq\frac{1}{\log
T}\int_{1}^{T}\frac{1}{\theta}H_{\theta}d\theta\xrightarrow[T\to+\infty]{a.s.}0.$
(2.8)

A discrete analogue of Lemma 2.1 can be formulated in a straightforward
manner. Lemma 2.1 follows essentially from a Borel-Cantelli argument, and for
the sake of completeness, we present a short proof in the Appendix A.

######  Remark 2.2.

As one has already seen in the first proof of Theorem 1.6, an application of
Lemma 2.1 for
$H_{\theta}=f(\widetilde{F}_{\theta})-\mathbb{E}\big{[}f(\widetilde{F}_{\theta})\big{]}$
with $f$ bounded Lipschitz is sufficient. A crucial point is the usage of the
Clark-Ocone formula for representing $H_{\theta}$ as an Itô integral (2.4) to
take advantage of the moment estimate of Malliavin derivatives (Proposition
1.1). This approach has been used in the papers [29, 30, 31] for similar
studies on SPDEs driven by Gaussian noises. It is clear that if the noise is
not white in time (like the situations in [5, 35]), the strategy, based on
Itô’s calculus, is not applicable any more. This motivates us to provide
another proof based on a combination of the Ibragimov-Lifshits’ method of
characteristic functions and the second-order Gaussian Poincaré inequalities;
see also Remark 2.4 for a further discussion.

###  2.2. Proof of Theorem 1.6 based on the Ibragimov-Lifshits’ criterion

Let us first state a variant of the Ibragimov-Lifshits’ result.

######  Proposition 2.3.

([23, Ibragimov-Lifshits’ criterion]) A family of real-valued random variables
$\\{F_{\theta}\\}_{\theta\geq 1}$ satisfies the ASCLT if $\theta\mapsto
F_{\theta}$ is measurable almost surely, and the following inequality holds

$\displaystyle\sup_{|s|\leq
T}\int_{2}^{\infty}\dfrac{\mathbb{E}\big{[}|\mathbf{K}_{t}(s)|^{2}\big{]}}{t\log
t}dt<\infty,$ (2.9)

for any finite $T>0$, where

$\displaystyle\mathbf{K}_{t}(s)\coloneqq\frac{1}{\log
t}\int_{1}^{t}\frac{1}{\theta}\big{(}e^{isF_{\theta}}-e^{-s^{2}/2}\big{)}d\theta,\quad
t\in(1,\infty).$ (2.10)

In Ibragimov-Lifshits’ original paper [23], the criterion is proved for the
discrete version (see Definition 1.4-(i)). Although the proof for the
continuum version is almost identical, we provide a sketchy proof of
Proposition 2.3 for readers’ convenience in the Appendix A.

Let $F_{\theta}$ be defined as in (1.6) and $\mathbf{K}_{t}(s)$ be given by
(2.10) with $F_{\theta}$ replaced by
$\widetilde{F}_{\theta}=\frac{1}{\sigma_{\theta}}F_{\theta}$. By expanding
$|\mathbf{K}_{t}(s)|^{2}$, we write

$\displaystyle|\mathbf{K}_{t}(s)|^{2}$ $\displaystyle=\frac{1}{(\log
t)^{2}}\int_{[1,t]^{2}}\frac{1}{\theta\textup{w}}\big{(}e^{is\widetilde{F}_{\theta}}-e^{-\frac{s^{2}}{2}}\big{)}\big{(}e^{-is\widetilde{F}_{\textup{w}}}-e^{-\frac{s^{2}}{2}}\big{)}d\theta
d\textup{w}$ $\displaystyle=\frac{1}{(\log
t)^{2}}\int_{[1,t]^{2}}\frac{1}{\theta\textup{w}}\big{(}e^{is(\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}})}+e^{-s^{2}}-e^{is\widetilde{F}_{\theta}}e^{-\frac{s^{2}}{2}}-e^{-is\widetilde{F}_{\textup{w}}}e^{-\frac{s^{2}}{2}}\big{)}d\theta
d\textup{w}$ $\displaystyle=\mathbb{I}_{t}(s)-e^{-\frac{s^{2}}{2}}\text{I
I}_{t}(s),$

where

$\displaystyle\begin{aligned} \mathbb{I}_{t}(s)&\coloneqq\frac{1}{(\log
t)^{2}}\int_{[1,t]^{2}}\frac{1}{\theta\textup{w}}\Big{(}e^{is(\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}})}-e^{-s^{2}}\Big{)}\,d\theta
d\textup{w},\\\ \text{I I}_{t}(s)&\coloneqq\frac{1}{\log
t}\int_{1}^{t}\frac{1}{\theta}\big{(}e^{is\widetilde{F}_{\theta}}+e^{-is\widetilde{F}_{\theta}}-2e^{-\frac{s^{2}}{2}}\big{)}\,d\theta.\end{aligned}$
(2.11)

Therefore, thanks to Proposition 2.3, it suffices to show that

$\displaystyle
A_{1}(s)\coloneqq\int_{2}^{\infty}\frac{\mathbb{E}\big{[}\mathbb{I}_{t}(s)\big{]}}{t\log
t}dt\quad\text{and}\quad
A_{2}(s)\coloneqq\int_{2}^{\infty}\frac{\mathbb{E}\big{[}\text{I
I}_{t}(s)\big{]}}{t\log t}dt,\quad s\in[-T,T]$

are both uniformly bounded for any given $T>0$.

$\star$ Estimation of the $A_{2}$ term. Applying Euler’s formula
$e^{iz}+e^{-iz}=2\cos(z)$, $z\in\mathbb{R}$, and
$\mathbb{E}[e^{isY}]=e^{-\frac{s^{2}}{2}}$ for $Y\sim\gamma$, one can write,
with $\phi_{s}(z)=2\cos(sz)$, that

$\displaystyle\begin{aligned}
\big{|}\mathbb{E}\big{(}e^{is\widetilde{F}_{\theta}}+e^{-is\widetilde{F}_{\theta}}-2e^{-\frac{s^{2}}{2}}\big{)}\big{|}&=\big{|}\mathbb{E}\big{(}\phi_{s}(\widetilde{F}_{\theta})-\phi_{s}(Y)\big{)}\big{|}\leq
2|s|\cdot d_{\rm Wass}(\widetilde{F}_{\theta},Y),\end{aligned}$ (2.12)

where we used the definition of Wasserstein distance (1.4) and the fact that
$\phi_{s}$ is a Lipschitz function with Lipschitz constant bounded by
$2|s|\leq 2T$. Then, it follows from (2.11), (2.12), and Proposition 1.1-(iii)
that

$\displaystyle\begin{aligned} \sup_{|s|\leq T}\big{|}\mathbb{E}[\text{I
I}_{t}(s)]\big{|}&\lesssim_{T}\frac{1}{\log
t}\int_{1}^{\infty}\frac{d\theta}{\theta^{1+\frac{\alpha}{1+\alpha}}}\lesssim_{T}\frac{1}{\log
t}.\end{aligned}$

Therefore, it holds that $\sup\\{|A_{2}(s)|:s\in[-T,T]\\}<\infty$ for any
finite $T>0$.

$\star$ Estimation of the $A_{1}$ term. We first write

$\displaystyle\sup_{|s|\leq T}A_{1}(s)$ $\displaystyle\leq
2\sqrt{2}T\int_{2}^{\infty}\frac{1}{t(\log
t)^{3}}\bigg{(}\int_{[1,t]^{2}}\frac{1}{\theta\textup{w}}\,d_{\rm
Wass}\big{(}\tfrac{\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}}}{\sqrt{2}},Y\big{)}\,d\theta
d\textup{w}\bigg{)}dt$
$\displaystyle=4\sqrt{2}T\int_{2}^{\infty}\frac{1}{t(\log
t)^{3}}\bigg{(}\int_{1<\theta<\textup{w}<t}\frac{1}{\theta\textup{w}}\,d_{\rm
Wass}\big{(}\tfrac{\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}}}{\sqrt{2}},Y\big{)}\,d\theta
d\textup{w}\bigg{)}dt,$ (2.13)

where we used $d_{\rm Wass}(-X,Y)=d_{\rm Wass}(X,Y)$ for $Y\sim\gamma$ and the
following bound

$\big{|}\mathbb{E}\big{[}e^{is(\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}})}-e^{-s^{2}}\big{]}\big{|}=\big{|}\mathbb{E}\big{[}e^{i\sqrt{2}s(\frac{\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}}}{\sqrt{2}})}-e^{i\sqrt{2}sY}\big{]}\big{|}\leq
2\sqrt{2}|s|d_{\rm
Wass}\big{(}\tfrac{\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}}}{\sqrt{2}},Y\big{)}.$

We claim that there exist positive real numbers $\beta_{1},\beta_{2}$, and
$\beta_{3}$ such that

$\displaystyle d_{\rm
Wass}\big{(}\tfrac{\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}}}{\sqrt{2}},Y\big{)}\lesssim\theta^{-\beta_{1}}+\textup{w}^{-\beta_{2}}+(\theta/\textup{w})^{\beta_{3}}$
(2.14)

for $1<\theta<\textup{w}<\infty$. It is then easy to deduce the finiteness of
(2.13) from the claim (2.14). The rest of the proof is then devoted to
verifying the claim (2.14).

Observe that $\widetilde{F}_{\theta}$ and $\widetilde{F}_{\textup{w}}$ are
centered with unit variance, and thus,

$\displaystyle
V_{\theta,\textup{w}}\coloneqq\mathbb{E}\Big{[}\big{(}\tfrac{\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}}}{\sqrt{2}}\big{)}^{2}\Big{]}=1-{\rm
Cov}(\widetilde{F}_{\theta},\widetilde{F}_{\textup{w}}).$ (2.15)

It follows from Proposition 1.2 with (2.15) that

$\displaystyle d_{\rm
Wass}\big{(}\tfrac{\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}}}{\sqrt{2}},Y\big{)}\leq|{\rm
Cov}(\widetilde{F}_{\theta},\widetilde{F}_{\textup{w}})|+\sum_{j=1}^{3}\gamma_{j}(\theta,\textup{w}),$
(2.16)

where $\gamma_{j}(\theta,\textup{w})$, $j=1,2,3$, are defined as in (1.9) with
$F=\tfrac{\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}}}{\sqrt{2}}$ and
$p=1+\alpha$, $q=\min\\{2,1+2\alpha\\}$ for some $\alpha\in(0,1]$; namely,
ignoring the constants

$\displaystyle\gamma_{1}(\theta,\textup{w}):=\bigg{[}\int_{\mathbf{Z}}\bigg{(}\int_{\mathbf{Z}}\big{\|}D_{\xi_{2}}(\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}})\big{\|}_{2p}\big{\|}D_{\xi_{1}}D_{\xi_{2}}(\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}})\big{\|}_{2p}\,m(d\xi_{2})\bigg{)}^{1+\alpha}m(d\xi_{1})\bigg{]}^{\frac{1}{1+\alpha}},$
$\displaystyle\gamma_{2}(\theta,\textup{w}):=\bigg{[}\int_{\mathbf{Z}}\bigg{(}\int_{\mathbf{Z}}\big{\|}D_{\xi_{1}}D_{\xi_{2}}(\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}})\big{\|}^{2}_{2p}\,m(d\xi_{2})\bigg{)}^{1+\alpha}m(d\xi_{1})\bigg{]}^{\frac{1}{1+\alpha}},$
and
$\displaystyle\gamma_{3}(\theta,\textup{w}):=\int_{\mathbf{Z}}\big{\|}D_{\xi}(\widetilde{F}_{\theta}-\widetilde{F}_{\textup{w}})\big{\|}_{q+1}^{q+1}\,m(d\xi).$

Step (i). The estimate for ${\rm
Cov}(\widetilde{F}_{\theta},\widetilde{F}_{\textup{w}})$ is already there, if
one notices that ${\rm
Cov}(\widetilde{F}_{\theta},\widetilde{F}_{\textup{w}})=\mathbb{E}[H_{\theta}H_{\textup{w}}]$,
where $H$ is defined as in (2.3) with $f(x)=x$ for all $x\in\mathbb{R}$. As a
consequence of (2.7), for all $0<\theta<\textup{w}$,555To limit the length of
this note, here we take a shortcut by using directly (2.7) although this does
not really fit the spirit of the second proof. Instead of an application of
(2.7), we have the explicit chaos expansion of $\widetilde{F}_{\theta}$ that
can be used to verify the bound (2.17) via a direct estimation for each chaos
with the orthogonality relation of Wiener chaoses; we leave this for
interested readers. It is worth pointing out that this approach based on the
chaos expansion requires $f(x)=x$ for all $x\in\mathbb{R}$ in (2.3), and the
obtention of (2.7) for general $f$ crucially relies on the Clark-Ocone formula
for $f(\widetilde{F}_{\theta})$, which is not available in the colored-in-time
setting.

$\displaystyle|{\rm
Cov}(\widetilde{F}_{\theta},\widetilde{F}_{\textup{w}})|\lesssim(\theta/\textup{w})^{\frac{1}{2}}.$
(2.17)

Step (ii). In this step we provide the estimate for $\gamma_{1}$. Using the
triangle inequality, we have

$\displaystyle\begin{aligned}
\gamma_{1}^{1+\alpha}(\theta,\textup{w})&\lesssim\mathbf{T}_{1}(\theta,\theta)+\mathbf{T}_{1}(\theta,\textup{w})+\mathbf{T}_{1}(\textup{w},\theta)+\mathbf{T}_{1}(\textup{w},\textup{w}),\end{aligned}$
(2.18)

where

$\mathbf{T}_{1}(\theta,\textup{w})\coloneqq\int_{\mathbf{Z}}\bigg{[}\int_{\mathbf{Z}}\|D_{\xi_{2}}\widetilde{F}_{\theta}\|_{2p}\|D_{\xi_{1}}D_{\xi_{2}}\widetilde{F}_{\textup{w}}\|_{2p}\,m(d\xi_{2})\bigg{]}^{1+\alpha}m(d\xi_{1}).$

The two terms $\mathbf{T}_{1}(\theta,\theta)$ and
$\mathbf{T}_{1}(\textup{w},\textup{w})$ have been already dealt with in [6],
and we have

$\displaystyle\mathbf{T}_{1}(\theta,\theta)\lesssim\theta^{-\alpha}\quad\text{and}\quad\mathbf{T}_{1}(\textup{w},\textup{w})\lesssim\textup{w}^{-\alpha};$
(2.19)

see equations (4.21)–(4.24) therein. For the other two terms in (2.18), we can
address them similarly. Using Proposition 1.1-(ii), we can write the following
inequities analogous to (2.6),

$\displaystyle\begin{aligned}
\|D_{r,y,z}F_{\theta}\|_{2p}&\lesssim|z|\cdot\int_{-\theta}^{\theta}G_{t_{0}}(x-y)dx,\\\
\|D_{r_{1},y_{1},z_{1}}D_{r_{2},y_{2},z_{2}}F_{\textup{w}}\|_{2p}&\lesssim|z_{1}z_{2}|\cdot\int_{-\textup{w}}^{\textup{w}}G_{t_{0}}(x-y_{1})G_{t_{0}}(y_{2}-y_{1})dx,\end{aligned}$
(2.20)

where $t_{0}$ is fixed as in Proposition 1.1; see also equations (4.18)–(4.20)
in [6]666The bounds in (2.20) follow easily from those in [6, (4.18)–(4.20)]
and triangle inequality with the fact $G_{t_{0}-r}\leq G_{t_{0}}$. .

Now, we are ready to estimate $\mathbf{T}_{1}(\theta,\textup{w})$ by utilizing
(2.20) and Proposition 1.1-(i):

$\displaystyle\mathbf{T}_{1}(\theta,\textup{w})$
$\displaystyle\leq(\sigma_{\theta}\sigma_{\textup{w}})^{-1-\alpha}\int_{0}^{t_{0}}\int_{\mathbb{R}\times\mathbb{R}_{0}}dr_{1}dy_{1}\nu(dz_{1})\,|z_{1}|^{1+\alpha}\bigg{[}\int_{0}^{t_{0}}\int_{\mathbb{R}\times\mathbb{R}_{0}}dr_{2}dy_{2}\nu(dz_{2})\,|z_{2}|^{2}$
$\displaystyle\qquad\cdot\bigg{(}\int_{-\theta}^{\theta}G_{t_{0}}(x_{2}-y_{2})dx_{2}\bigg{)}\bigg{(}\int_{-\textup{w}}^{\textup{w}}G_{t_{0}}(x_{1}-y_{1})G_{t_{0}}(y_{2}-y_{1})dx_{1}\bigg{)}\bigg{]}^{1+\alpha}$
$\displaystyle\lesssim\frac{1}{(\theta\textup{w})^{\frac{1+\alpha}{2}}}$
$\displaystyle\int_{\mathbb{R}}dy_{1}\Big{(}\int_{\mathbb{R}}dy_{2}\int_{-\theta}^{\theta}dx_{2}\int_{-\textup{w}}^{\textup{w}}dx_{1}G_{t_{0}}(x_{2}-y_{2})G_{t_{0}}(x_{1}-y_{1})G_{t_{0}}(y_{2}-y_{1})\Big{)}^{1+\alpha},$

where in the last inequality we used the finiteness of $m_{2}$ and
$m_{1+\alpha}$ (see Footnote 3). Therefore, applying Jensen’s inequality with
the fact that the triple integral with respect to $dy_{2}dx_{2}dx_{1}$ is
uniformly bounded by $t^{3}_{0}$, one gets

$\displaystyle\mathbf{T}_{1}(\theta,\textup{w})$
$\displaystyle\lesssim\frac{t^{3\alpha}_{0}}{(\theta\textup{w})^{\frac{1+\alpha}{2}}}\int_{\mathbb{R}}dy_{1}\int_{\mathbb{R}}dy_{2}\int_{-\theta}^{\theta}dx_{2}\int_{-\textup{w}}^{\textup{w}}dx_{1}G_{t_{0}}(x_{2}-y_{2})G_{t_{0}}(x_{1}-y_{1})G_{t_{0}}(y_{2}-y_{1})$
$\displaystyle\lesssim\frac{\theta}{(\theta\textup{w})^{\frac{1+\alpha}{2}}}\leq(\theta/\textup{w})^{\frac{1+\alpha}{2}}$
(2.21)

for $1\leq\theta\leq\textup{w}$, where in the second-to-last inequality, we
have simply performed integration in the exact order of
$dx_{1},dy_{1},dy_{2},dx_{2}$. In the same way, we can obtain

$\displaystyle\mathbf{T}_{1}(\textup{w},\theta)\leq(\theta/\textup{w})^{\frac{1+\alpha}{2}}\quad\text{for
$1\leq\theta\leq\textup{w}$}.$ (2.22)

Therefore, in view of (2.18), (2.19), (2.21), and (2.22), we get

$\displaystyle\gamma_{1}(\theta,\textup{w})\lesssim\theta^{-\alpha}+\textup{w}^{-\alpha}+(\theta/\textup{w})^{\frac{1+\alpha}{2}}.$
(2.23)

Step (iii). For the term $\gamma_{2}$, we can write

$\displaystyle\begin{aligned}
\gamma_{2}^{1+\alpha}(\theta,\textup{w})&\lesssim\int_{\mathbf{Z}}\bigg{[}\int_{\mathbf{Z}}\|D_{\xi_{1}}D_{\xi_{2}}\widetilde{F}_{\theta}\|^{2}_{2p}\,m(d\xi_{2})\bigg{]}^{1+\alpha}m(d\xi_{1})\\\
&\qquad+\int_{\mathbf{Z}}\bigg{[}\int_{\mathbf{Z}}\|D_{\xi_{1}}D_{\xi_{2}}\widetilde{F}_{\textup{w}}\|^{2}_{2p}\,m(d\xi_{2})\bigg{]}^{1+\alpha}m(d\xi_{1})\lesssim\theta^{-\alpha}+\textup{w}^{-\alpha},\end{aligned}$
(2.24)

which is essentially done in [6, (4.25)–(4.27)].

Step (iv). Now we consider the last term $\gamma_{3}(\theta,\textup{w})$ with
$q=\min\\{2,1+2\alpha\\}\in(1,2].$

$\displaystyle\begin{aligned}
\gamma_{3}(\theta,\textup{w})\lesssim\int_{\mathbf{Z}}\|D_{\xi}\widetilde{F}_{\theta}\|_{q+1}^{q+1}\,m(d\xi)+\int_{\mathbf{Z}}\|D_{\xi}\widetilde{F}_{\textup{w}}\|_{q+1}^{q+1}\,m(d\xi)\lesssim\theta^{-\frac{q-1}{2}}+\textup{w}^{-\frac{q-1}{2}},\end{aligned}$
(2.25)

where the last step is essentially done in [6, (4.28)–(4.30)]. Therefore,
claim (2.14) follows from (2.16), (2.17), (2.23), (2.24), and (2.25). Hence,
the proof of Theorem 1.6 is completed.∎

######  Remark 2.4.

For this proof, we combined the Ibragimov-Lifshits’ method of characteristic
functions and the second-order Gaussian Poincaré inequalities. The strategy
was largely motivated by earlier works [7, 43, 44, 2] based on the combination
of Ibragimov-Lifshits’ method and the Malliavin-Stein approach ([34]). With
explicit chaos expansion, the papers [7, 43, 44] have established sufficient
conditions (in terms of contractions) for the ASCLT on Gaussian, Poisson, and
Rademacher settings. Our strategy has the same root as the aforementioned
references but differs by using the the second-order Gaussian Poincaré
inequalities. Such a novel combination allows one to avoid lengthy computation
of asymptotic negligibility of the (star-)contractions. We will further
illustrate this strategy in a future work.

##  Appendix A Proofs of Lemma 2.1 and Proposition 2.3

###### Proof of Lemma 2.1.

Let us first compute the second moment of $L_{T}$ in (2.8):

$\displaystyle\mathbb{E}\big{[}L_{T}^{2}\big{]}$ $\displaystyle=\frac{1}{(\log
T)^{2}}\int_{1}^{T}\int_{1}^{T}\frac{1}{\theta\textup{w}}\mathbb{E}[H_{\theta}H_{\textup{w}}]\,d\theta
d\textup{w}$ $\displaystyle\leq\frac{2}{(\log
T)^{2}}\int_{1<\theta<\textup{w}\leq
T}\frac{1}{\theta\textup{w}}C_{\beta}\big{(}\theta/\textup{w}\big{)}^{\beta}\,d\theta
d\textup{w}\leq\frac{2C_{\beta}}{\beta\log T}.$

In particular, we get (by Fubini)

$\mathbb{E}\sum_{k=1}^{\infty}L^{2}_{2^{k^{2}}}<+\infty,\quad\text{and
therefore $L_{2^{k^{2}}}\xrightarrow{a.s.}0$ as $k\to+\infty$.}$

Next, we will show the almost sure convergence along the continuum parameter
(as $T\to+\infty$).

By assumption, one can find a constant $M>0$ such that $|H_{\theta}|\leq M$,
$\forall\theta>0$, with probability one. Then, for any $T>1$, one can find
some (unique) $k=k_{T}\in\mathbb{N}_{\geq 0}$ such that

$\displaystyle 2^{k^{2}_{T}}\leq T<2^{(k_{T}+1)^{2}}.$ (A.1)

It follows that

$\displaystyle|L_{T}|$ $\displaystyle=\Big{|}\frac{\log(2^{k^{2}})}{\log
T}L_{2^{k^{2}}}+\frac{1}{\log
T}\int_{2^{k^{2}}}^{T}\frac{1}{\theta}H_{\theta}\,d\theta\Big{|}$
$\displaystyle\leq\big{|}L_{2^{k^{2}}}\big{|}+\frac{M}{\log T}\big{[}\log
T-\log(2^{k^{2}}\big{)}\big{]}\quad\text{with $k=k_{T}$ as in \eqref{KT}}$
$\displaystyle\leq\big{|}L_{2^{k^{2}}}\big{|}+\frac{M}{k^{2}}\big{[}(k+1)^{2}-k^{2}\big{]},$

which tends to zero as $T\to+\infty$ ($k=k_{T}\to+\infty$ as well). Hence the
proof is completed. ∎

###### Proof of Proposition 2.3.

The proof consists of two parts: (i) for any $s\in\mathbb{R}$,
$\mathbf{K}_{t}(s)\to 0$ almost surely as $t\to+\infty$; and (ii) for
$\mathbb{P}$-almost every $\omega\in\Omega$ and for any increasing sequence
$t_{n}\uparrow+\infty$ (as $n\to\infty$), the family of probability measures
$\\{\nu_{t_{n}}^{\omega}:n\geq 1\\}$ is tight, where we have used the notation
$\nu_{T}^{\omega}$ from (1.12). Then, we can deduce from (i) and (ii) together
that
$\mathbb{P}(\\{\omega\in\Omega:\nu^{\omega}_{T}\Longrightarrow\gamma\,\,\text{as
$T\to+\infty$}\\})=1$. That is, ASCLT holds for $\\{F_{\theta}\\}_{\theta\geq
1}$.

To show (i), we fix any $s\in\mathbb{R}$. Let $h>1$ and define

$I_{j}=[e^{h^{j}},e^{h^{j+1}}],\quad j\in\mathbb{N}_{\geq 0}.$

It is clear that $t\mapsto\mathbb{E}\big{[}|\mathbf{K}_{t}(s)|^{2}\big{]}$ is
continuous, and therefore there exists some (deterministic) $s_{j}\in I_{j}$
such that

$\displaystyle\begin{aligned}
\mathbb{E}\big{[}|\mathbf{K}_{s_{j}}(s)|^{2}\big{]}=\min_{t\in
I_{j}}\mathbb{E}\big{[}|\mathbf{K}_{t}(s)|^{2}\big{]}&\leq\bigg{(}\int_{I_{j}}\dfrac{1}{t\log
t}dt\bigg{)}^{-1}\int_{I_{j}}\dfrac{\mathbb{E}\big{[}|\mathbf{K}_{t}(s)|^{2}\big{]}}{t\log
t}dt\\\ &=\frac{1}{\log
h}\int_{I_{j}}\dfrac{\mathbb{E}\big{[}|\mathbf{K}_{t}(s)|^{2}\big{]}}{t\log
t}dt,\end{aligned}$ (A.2)

which is summable in $j$ by assumption (2.9). It follows from Fubini that

almost surely, $\mathbf{K}_{s_{j}}(s)\to 0$ as $j\to\infty$. (A.3)

Next, for $e^{h^{j}}\leq a<b\leq e^{h^{j+1}}$, we get

$\displaystyle\begin{aligned}
\big{|}\mathbf{K}_{a}(s)-\mathbf{K}_{b}(s)\big{|}&=\Big{|}\Big{(}\frac{1}{\log
a}-\frac{1}{\log
b}\Big{)}\int_{1}^{a}\frac{1}{\theta}e^{isF_{\theta}}\,d\theta-\frac{1}{\log
b}\int_{a}^{b}\frac{1}{\theta}e^{isF_{\theta}}\,d\theta\Big{|}\\\ &\leq
2(1-\tfrac{1}{h}).\end{aligned}$ (A.4)

It follows from (A.3) and (A.4) that almost surely,

$\displaystyle\limsup_{t\to\infty}|\mathbf{K}_{t}(s)|\leq
2(1-\tfrac{1}{h})\quad\forall h>1.$ (A.5)

Thus, part (i) is verified.

Now, let us continue with part (ii). First, we deduce from the same continuity
argument as in (A.2) that for $r>0$, there exists some (deterministic)
$s_{j}^{\ast}\in I_{j}$ (depending on $r$ as well) such that

$\displaystyle\mathbb{E}\int_{|s|\leq r}|\mathbf{K}_{s_{j}^{\ast}}(s)|^{2}ds$
$\displaystyle=\min_{t\in I_{j}}\mathbb{E}\int_{|s|\leq
r}|\mathbf{K}_{t}(s)|^{2}ds\leq\frac{1}{\log h}\int_{I_{j}}\frac{dt}{t\log
t}\bigg{(}\mathbb{E}\int_{|s|\leq r}|\mathbf{K}_{t}(s)|^{2}ds\bigg{)},$

which, together with Fubini and the assumption (2.9), implies that

$\displaystyle\mathbb{E}\sum_{j}\int_{|s|\leq
r}|\mathbf{K}_{s_{j}^{\ast}}(s)|^{2}ds\leq\frac{2r}{\log h}\sup_{|s|\leq
r}\int_{2}^{\infty}\frac{1}{t\log
t}\mathbb{E}\big{[}|\mathbf{K}_{t}(s)|^{2}\big{]}dt<+\infty.$

Thus, it holds almost surely that $\int_{|s|\leq
r}|\mathbf{K}_{s_{j}^{\ast}}(s)|^{2}ds\to 0$ as $j\to+\infty$. Therefore, we
can obtain by the same arguments as in (A.3), (A.4), and (A.5) that with
probability one, $\int_{|s|\leq r}|\mathbf{K}_{t}(s)|^{2}ds\to 0$ as
$t\to+\infty$. Then, we get from Hölder inequality that

$\displaystyle\text{almost surely},\quad\int_{|s|\leq
r}\phi_{\nu_{t}^{\omega}}(s)ds\to\int_{|s|\leq
r}\phi_{\gamma}(s)ds\quad\text{as $t\to+\infty$,}$ (A.6)

where $\phi_{\mu}(s)=\int_{\mathbb{R}}e^{isx}\mu(dx)$ denotes the
characteristic function of a probability measure $\mu$ on $\mathbb{R}$, and
$\gamma$ denotes the standard Gaussian measure on $\mathbb{R}$ in this paper.
Note that

$\displaystyle\begin{aligned}
\mu\big{(}\\{x\in\mathbb{R}:|x|\leq\tfrac{1}{r^{2}}\\}\big{)}&\geq\int_{|x|\leq
r^{-2}}\frac{\sin(rx)}{rx}\mu(dx)\quad\text{(using $|\sin(z)|\leq|z|$,
$\forall z\in\mathbb{R}$)}\\\
&\geq\int_{\mathbb{R}}\frac{\sin(rx)}{rx}\mu(dx)-r=\frac{1}{2r}\int_{-r}^{r}\phi_{\mu}(s)ds-r.\end{aligned}$
(A.7)

It follows from (A.7) and (A.6) that for $\mathbb{P}$-almost every
$\omega\in\Omega$ and for every $\varepsilon>0$,

$\displaystyle\liminf_{t\to+\infty}\nu_{t}^{\omega}(|x|\leq r^{-2})$
$\displaystyle\geq\frac{1}{2r}\int_{-r}^{r}\phi_{\gamma}(s)ds-r\geq
1-\frac{\varepsilon}{2}$

for small enough $r=r_{\varepsilon}$ that does not depend on $\omega$. For any
increasing divergent sequence $\\{t_{n}\\}$, the above bound indicates that
there is some $N=N_{\varepsilon}$ such that

$\displaystyle\nu_{t_{n}}^{\omega}(|x|\leq r^{-2}_{\varepsilon})\geq
1-\varepsilon,\quad\forall n\geq N_{\varepsilon};$ (A.8)

while by choosing another small enough $r^{\prime}_{\varepsilon}>0$, we get

$\displaystyle\nu_{t_{n}}^{\omega}(|x|\leq\tfrac{1}{(r^{\prime}_{\varepsilon})^{2}})\geq
1-\varepsilon,\quad\forall n<N_{\varepsilon}.$ (A.9)

Combining (A.8) and (A.9) yields the tightness of
$\\{\nu_{t_{n}}^{\omega}:n\geq 1\\}$. Therefore, the result in part (i)
implies that $\nu_{t_{n}}^{\omega}\Longrightarrow\gamma$ as $n\to\infty$, and
this weak convergence holds true for any increasing divergent sequence
$\\{t_{n}\\}$. Hence, we proved
$\mathbb{P}\\{\omega\in\Omega:\nu^{\omega}_{T}\Longrightarrow\gamma\,\,\text{as
$T\to+\infty$}\\}=1$. ∎

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