# A robust $\alpha$-stable central limit theorem under sublinear expectation
without integrability condition

Lianzi Jiang College of Mathematics and Systems Science, Shandong University
of Science and Technology, Qingdao, Shandong 266590, China.
<EMAIL_ADDRESS>Gechun Liang Department of Statistics, The University
of Warwick, Coventry CV4 7AL, U.K<EMAIL_ADDRESS>
Abstract. This article relaxes the integrability condition imposed in the
literature for the robust $\alpha$-stable central limit theorem under
sublinear expectation. Specifically, for $\alpha\in(0,1]$, we prove that the
normalized sums of i.i.d. non-integrable random variables
$\big{\\{}n^{-\frac{1}{\alpha}}\sum_{i=1}^{n}Z_{i}\big{\\}}_{n=1}^{\infty}$
converge in distribution to $\tilde{\zeta}_{1}$, where
$(\tilde{\zeta}_{t})_{t\in[0,1]}$ is a multidimensional nonlinear symmetric
$\alpha$-stable process with a jump uncertainty set $\mathcal{L}$. The
limiting $\alpha$-stable process is further characterized by a fully nonlinear
partial integro-differential equation (PIDE)

$\left\\{\begin{array}[c]{l}\displaystyle\partial_{t}u(t,x)-\sup\limits_{F_{\mu}\in\mathcal{L}}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}u(t,x)F_{\mu}(d\lambda)\right\\}=0,\\\
\displaystyle u(0,x)=\phi(x),\ \ \
\forall(t,x)\in[0,1]\times\mathbb{R}^{d},\end{array}\right.$

where

$\delta_{\lambda}^{\alpha}u(t,x):=\left\\{\begin{array}[c]{ll}u(t,x+\lambda)-u(t,x)-\langle
D_{x}u(t,x),\lambda\mathbbm{1}_{\\{|\lambda|\leq 1\\}}\rangle,&\alpha=1,\\\
u(t,x+\lambda)-u(t,x),&\alpha\in(0,1).\end{array}\right.$

The main tools are a weak convergence approach to obtain the limiting process,
a Lévy-Khintchine representation of the nonlinear $\alpha$-stable process and
a truncation technique to estimate the corresponding $\alpha$-stable Lévy
measures. As a byproduct, the article also provides a probabilistic approach
to prove the existence of the above fully nonlinear PIDE.

Keywords. Robust stable central limit theorem, Partial integro-differential
equation, $\alpha$-stable distribution, Sublinear expectation

MSC-classification. 60F05, 60G51, 60G52, 60G65, 45K05

##  1 Introduction

Non-Gaussian $\alpha$-stable distributions allow for large fluctuations and
are thus ideal for modeling high variability and heavy tail events. They are
the only non-Gaussian limiting distributions as attractors of normalized sums
of i.i.d. random variables
$\big{\\{}n^{-\frac{1}{\alpha}}\sum_{i=1}^{n}Z_{i}\big{\\}}_{n=1}^{\infty}$,
known as $\alpha$-stable central limit theorems. Recently, such type of
central limit theorems have been generalized to a robust case under model
uncertainty (see (2006), (2021) and (2022)) for $\alpha\in(1,2)$. The case
$\alpha\in(0,1]$ is left open because $Z_{i}$ is not integrable in such a
situation. In this paper, we study this more challenging case by establishing
a new robust $\alpha$-stable central limit theorem without integrability
condition.

The theory of robust probability and expectation has been developed by Peng
(2004, 2007, 2008(1), 2019(2)), who introduced the notion of sublinear
expectation space, called $G$-expectation space. Peng used it to evaluate
random outcomes over a family of possibly mutually singular probability
measures instead of a single probability measure. A seminal result in this
theory is Peng’s robust central limit theorem established in Peng (2008(2),
2019(1)). He showed that, under certain moment conditions, the i.i.d. sequence
$\\{(X_{i},Y_{i})\\}_{i=1}^{\infty}$ on a sublinear expectation space
$(\Omega,\mathcal{H},\mathbb{\hat{E}})$ converges in law to a $G$-distributed
random variable $(\xi,\eta)$, i.e.

$\lim_{n\rightarrow\infty}\mathbb{\hat{E}}\bigg{[}\phi\bigg{(}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}{X_{i}},\frac{1}{n}\sum_{i=1}^{n}Y_{i}\bigg{)}\bigg{]}=\mathbb{\tilde{E}}[\phi(\xi,\eta)],$

for any test function $\phi$. The $G$-distributed random variable $(\xi,\eta)$
describes volatility and mean uncertainty of a probability model. The
corresponding convergence rate was established in Song (2020) and (2019) using
Stein’s method, in Krylov (2020) using stochastic control method and in (2019)
using monotone approximation scheme method under different model assumptions.
We refer the reader to (2016); Zhang (2016); (2022) and the references therein
for more research in this field.

A general $G$-Lévy process in the setting of sublinear expectation was
introduced by (2009, 2021) to further describe Poisson jump uncertainty beyond
volatility and mean uncertainty. They built a new type of Lévy-Khintchine
representation for $G$-Lévy processes by relating to a class of fully
nonlinear partial integro-differential equations (PIDEs). Furthermore, the
case of infinite activity jumps has been studied in (2020); Kuhn (2019);
Neufeld and Nutz (2017); Nendel and Röckner (2021). An important class of
nonlinear Lévy processes with infinite activity jumps is the integrable
$\alpha$-stable process $(\zeta_{t})_{t\geq 0}$ for $\alpha\in(1,2)$, which
corresponds to a fully nonlinear PIDE driven by a family of $\alpha$-stable
Lévy measures. For the one-dimensional case, the corresponding central limit
theorem for the $\alpha$-stable process under sublinear expectation was
established in (2006), which shows that the i.i.d. sequence
$\\{Z_{i}\\}_{i=1}^{\infty}$ of _real-valued_ random variables on a sublinear
expectation space $(\Omega,\mathcal{H},\mathbb{\hat{E})}$ converges in law to
a nonlinear $\alpha$-stable distributed random variable $\zeta$ under the
integrability condition and an additional consistency condition of $Z_{i}$,
i.e.

$\lim_{n\rightarrow\infty}\mathbb{\hat{E}}\bigg{[}\phi\bigg{(}\frac{1}{\sqrt[\alpha]{n}}\sum_{i=1}^{n}Z_{i}\bigg{)}\bigg{]}=\mathbb{\tilde{E}}[\phi(\zeta)],\
\alpha\in(1,2),$

for any test function $\phi$. The corresponding convergence rate was
established in (2021) via a monotone approximation scheme.

Recently, together with Hu and Peng, the authors established a universal
robust limit theorem in (2022) by further developing the weak convergence
approach introduced in Peng (2010). This results covers Peng’s robust central
limit theorem Peng (2008(2), 2019(1)) and Bayraktar-Munk’s robust stable limit
theorem (2006) as special cases. Under certain moment conditions of
$(X_{i},Y_{i},Z_{i})$ and a consistency condition of $Z_{i}$, they proved
that, for $\alpha\in(1,2)$, the normalized sums of i.i.d. random variables
$\left\\{\left(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum_{i=1}^{n}Y_{i},\frac{1}{\sqrt[\alpha]{n}}\sum_{i=1}^{n}Z_{i}\right)\right\\}_{n=1}^{\infty}$
converge in distribution to $\tilde{L}_{1}$, where
$\tilde{L}_{t}=(\tilde{\xi}_{t},\tilde{\eta}_{t},\tilde{\zeta}_{t})$,
$t\in[0,1]$, is a multidimensional nonlinear Lévy process with an uncertainty
set $\Theta$ as a set of Lévy triplets. They further proved that the nonlinear
Lévy process can be characterized via a fully nonlinear and possibly
degenerate PIDE

$\left\\{\begin{array}[c]{l}\displaystyle\partial_{t}u(t,x,y,z)-\sup\limits_{(F_{\mu},q,Q)\in\Theta}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}u(t,x,y,z)F_{\mu}(d\lambda)\right.\\\
\displaystyle\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left.+\langle
D_{y}u(t,x,y,z),q\rangle+\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right\\}=0,\\\
\displaystyle u(0,x,y,z)=\phi(x,y,z),\ \
\forall(t,x,y,z)\in[0,1]\times\mathbb{R}^{3d},\end{array}\right.$

with $\delta_{\lambda}u(t,x,y,z):=u(t,x,y,z+\lambda)-u(t,x,y,z)-\langle
D_{z}u(t,x,y,z),\lambda\rangle$.

However, all the aforementioned works assume $\alpha\in(1,2)$, leaving the
more difficult regime $\alpha\in(0,1]$ open. There are two main challenges.
First, this case is outside the scope of the integrability conditions of Lévy
triplets in Neufeld and Nutz (2017), and the PIDE characterization of the
nonlinear $\alpha$-stable distribution is unknown to date. This leads to the
existing method in (2006) invalid, because (2006) heavily relies on the
regularity estimates of fully nonlinear PIDEs. Second, due to the lack of
integrability for the normalized sums of i.i.d. random variables, the
sufficient condition in (2022) for the existence of their weak convergence
limit fails.

We overcome the aforementioned difficulty by introducing a sublinear
expectation space with random variables not necessarily having any moments.
Hence, the random variables do not need to belong to the space but their
composition of bounded Lipschitz continuous functions do. This is sufficient
to define their distributions and study the corresponding robust central limit
theorems. We introduce a new $\delta$-moment condition on the normalized sums
of i.i.d. random variables (see the assumption (A.1) in Section 3) to
guarantee the tightness of the sequence so it has a weak convergence limit.
With an additional consistency condition (see the assumption (A.2) in Section
3), we show that the limiting process is a nonlinear symmetric $\alpha$-stable
process $(\tilde{\zeta}_{t})_{t\in[0,1]}$ such that

$\lim_{n\rightarrow\infty}\mathbb{\hat{E}}\bigg{[}\phi\bigg{(}\frac{1}{\sqrt[\alpha]{n}}\sum_{i=1}^{n}Z_{i}\bigg{)}\bigg{]}=\mathbb{\tilde{E}}[\phi(\tilde{\zeta}_{1})],\
\alpha\in(0,1],$

for any test function $\phi$. One remarkable feature of this process is that
$\tilde{\zeta}_{t}$ is non-integrable, i.e.,
$\mathbb{\tilde{E}}[|\tilde{\zeta}_{1}|]=\infty$, so one cannot simply
generalize the Lévy-Khintichine representation formula in (2022) to
$\alpha\in(0,1]$. We overcome this difficulty by using a truncation technique
to estimate the $\alpha$-stable Lévy measure, which gives a new type Lévy-
Khintichine representation for $(\tilde{\zeta}_{t})_{t\in[0,1]}$ with
$\alpha\in(0,1]$.

Together with the case $\alpha\in(1,2)$ studied in (2022), we are able to give
a complete characterization of the nonlinear and non-Gaussian $\alpha$-stable
process, for $\alpha\in(0,2)$, via a fully nonlinear PIDE

$\left\\{\begin{array}[c]{l}\displaystyle\partial_{t}u(t,x)-\sup\limits_{F_{\mu}\in\mathcal{L}}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}u(t,x)F_{\mu}(d\lambda)\right\\}=0,\\\
\displaystyle u(0,x)=\phi(x),\text{ \ \
}\forall(t,x)\in[0,1]\times\mathbb{R}^{d},\end{array}\right.$

where $\mathcal{L}$ is an $\alpha$-stable jump uncertainty set and

$\delta_{\lambda}^{\alpha}u(t,x):=\left\\{\begin{array}[c]{ll}u(t,x+\lambda)-u(t,x)-\langle
D_{x}u(t,x),\lambda\rangle,&\alpha\in(1,2),\\\ u(t,x+\lambda)-u(t,x)-\langle
D_{x}u(t,x),\lambda\mathbbm{1}_{\\{|\lambda|\leq 1\\}}\rangle,&\alpha=1,\\\
u(t,x+\lambda)-u(t,x),&\alpha\in(0,1).\end{array}\right.$

Thanks to the connection with the above fully nonlinear PIDE, we also obtain
the existence of its viscosity solution as a byproduct, i.e. the solution can
be obtained via the weak convergence limit of the normalized sums of i.i.d.
random variables. When restricting the jump uncertainty set to a singleton,
our work also complements the classical $\alpha$-stable central limit theorems
in the linear setting. See, for example, (2019, 2002, 1981, 1971, 1998); Kuske
and Keller (2001) and the references therein.

The article is organized as follows. In Section 2, we review some necessary
results about sublinear expectation for the $\alpha$-stable central limit
theorem without integrability condition. Section 3 details our main result,
discusses its connection with the classical linear case, and gives an example
highlighting the application of our main result. The proof of the main theorem
is given in Section 4 with some technical results provided in the Appendix.

##  2 Preliminaries on nonlinear $\alpha$-stable process for $\alpha\in(0,1]$

The sublinear expectation framework has been introduced in Peng (2007,
2008(1), 2010, 2019(2)). However, the framework requires that random variables
defined on a sublinear expectation space are integrable, which is too
restrictive to study robust limit theorems for $\alpha$-stable distribution
with $\alpha\in(0,1]$.

To resolve the issue, we introduce a triplet
$(\Omega,\mathcal{H},\mathbb{\hat{E})}$, where $\Omega$ is a given set and
$\mathcal{H}$ is a linear space of real valued functions on $\Omega$ such that
$\varphi(X_{1},\ldots,X_{n})\in\mathcal{H}$ if
$X_{1},\ldots,X_{n}\in\mathcal{H}$ for each
$\varphi\in{C_{b,Lip}}(\mathbb{R}^{n})$, the space of bounded and Lipschitz
continuous functions on $\mathbb{R}^{n}$. Then, $\mathbb{\hat{E}}$:
$\mathcal{H}\rightarrow\mathbb{R}$ is called a sublinear expectation if, for
all $X,Y\in\mathcal{H}$, it satisfies the following properties:

(i)

(Monotonicity) $\mathbb{\hat{E}}[X]\geq\mathbb{\hat{E}}[Y]$, if $X\geq Y$;

(ii)

(Constant preservation) $\mathbb{\hat{E}}[c]=c$, for $c\in\mathbb{R}$;

(iii)

(Sub-additivity)
$\mathbb{\hat{E}}[X+Y]\leq\mathbb{\hat{E}}[X]+\mathbb{\hat{E}}[Y];$

(iv)

(Positive homogeneity) $\mathbb{\hat{E}}[\lambda
X]=\lambda\mathbb{\hat{E}}[X]$, for $\lambda>0$.

By an $n$-dimensional random variable $X$ defined on
$(\Omega,\mathcal{H},\mathbb{\hat{E})}$, we mean $\varphi(X)\in\mathcal{H}$
for all $\varphi\in C_{b,Lip}(\mathbb{R}^{n})$. However, the random variable
$X$ itself is not required to be in $\mathcal{H}$, i.e. $\mathbb{\hat{E}}[X]$
may not exist. Our new framework relaxes the integrability condition imposed
on $X$ but it is sufficient to define distribution of $X$.

###### Definition 2.1

Let $X$ be a given $n$-dimensional random variable defined on a sublinear
expectation space $(\Omega,\mathcal{H},\mathbb{\hat{E})}$. Define a functional
on $C_{b,Lip}(\mathbb{R}^{n})$ by

$\mathbb{F}_{X}[\varphi]:=\mathbb{\hat{E}}[\varphi(X)],\text{ for }\varphi\in
C_{b,Lip}(\mathbb{R}^{n}).$

Then, $(\mathbb{R}^{n},C_{b,Lip}(\mathbb{R}^{n}),\mathbb{F}_{X})$ forms a
sublinear expectation space. $\mathbb{F}_{X}$ is called the distribution of
$X$ under $\mathbb{\hat{E}}$.

###### Definition 2.2

A sequence of $n$-dimensional random variables $\\{X_{i}\\}_{i=1}^{\infty}$
defined on a sublinear expectation space
$(\Omega,\mathcal{H},\mathbb{\hat{E}})$ is said to converge in distribution
(or converge in law) under $\mathbb{\hat{E}}$ if for each $\varphi\in
C_{b,Lip}(\mathbb{R}^{n})$, the sequence
$\\{\mathbb{F}_{X_{i}}[\varphi]\\}_{i=1}^{\infty}$ is a Cauchy sequence.

Two random variables $X$ and $Y$, which may be defined on different sublinear
expectation spaces, are called _identically distributed_ if
$\mathbb{\hat{E}}_{1}[\varphi(X)]=\mathbb{\hat{E}}_{2}[\varphi(Y)]$, for all
$\varphi\in C_{b,Lip}(\mathbb{R}^{n})$. It is denoted by $X\overset{d}{=}Y$.
On the other hand, $Y$ is called _independent_ from $X$ if for every
$\varphi\in C_{b,Lip}(\mathbb{R}^{m}\times\mathbb{R}^{n})$ we have

$\mathbb{\hat{E}}\left[\varphi(X,Y)\right]=\mathbb{\hat{E}}\left[\mathbb{\hat{E}}\left[\varphi(x,Y)\right]_{x=X}\right],$

which is denoted by $Y\perp\\!\\!\\!\perp X$. Notably, $Y$ is independent from
$X$ does not necessarily imply that $X$ is independent from $Y$. Then, $Y$ is
said to be an independent copy of $X$ if $Y\overset{d}{=}X$ and
$Y\perp\\!\\!\\!\perp X$.

One can readily obtain the following two results concerning the convergence in
distribution of random variables.

###### Proposition 2.3

Let $\\{X_{i}\\}_{i=1}^{\infty}$ be a sequence of $n$-dimensional random
variables defined on a sublinear expectation spaces
$(\Omega,\mathcal{H},\mathbb{\hat{E}})$. If $\\{X_{i}\\}_{i=1}^{\infty}$
converges in distribution to $X$ under $\mathbb{\hat{E}}$, i.e.,

$\lim_{i\rightarrow\infty}\mathbb{\hat{E}}[\varphi(X_{i})]=\mathbb{\tilde{E}}[\varphi(X)],\text{
for }\varphi\in C_{b,Lip}(\mathbb{R}^{n}),$

which is denoted by $X_{i}\overset{\mathcal{D}}{\rightarrow}X$. Then,

$\bar{X}_{i}\overset{\mathcal{D}}{\rightarrow}\bar{X}\text{\ and \
}X_{i}+\bar{X}_{i}\overset{\mathcal{D}}{\rightarrow}X+\bar{X},$

where $\bar{X}_{i}$ and $\bar{X}$ are independent copy of $X_{i}$ and $X$,
respectively.

###### Remark 2.4

The above results can be generalized to multiple summations. For each $i$ and
$m\in\mathbb{N}$, let $\\{X_{i}^{n}\\}_{n=1}^{m}$ be an independent copy
sequence of $X_{i}$ in the sense that $X_{i}^{1}\overset{d}{=}X_{i}$,
$X_{i}^{n+1}\overset{d}{=}X_{i}^{n}$ and
$X_{i}^{n+1}\perp\\!\\!\\!\perp(X_{i}^{1},X_{i}^{2},\ldots,X_{i}^{n})$ for
$n=1,\ldots,m-1$, and let $\\{X^{n}\\}_{n=1}^{m}$ be an independent copy
sequence of $X$ in the sense that $X^{1}\overset{d}{=}X$,
$X^{n+1}\overset{d}{=}X^{n}$ and
$X^{n+1}\perp\\!\\!\\!\perp(X^{1},X^{2},\ldots,X^{n})$ for $n=1,\ldots,m-1$.
If $X_{i}\overset{\mathcal{D}}{\rightarrow}X$, then

$\sum_{n=1}^{m}X_{i}^{n}\overset{\mathcal{D}}{\rightarrow}\sum_{n=1}^{m}X^{n}.$

We are interested in non-integrable $\alpha$-stable distributed random
variables, and they serve as the attractors of normalized sums of i.i.d.
random variables.

###### Definition 2.5

Let $\alpha\in(0,1]$. An $n$-dimensional random variable $X$ is said to be
(strictly) $\alpha$-stable under a sublinear expectation space
$(\Omega,\mathcal{H},\mathbb{\hat{E})}$ if

$aX+bY\overset{d}{=}(a^{\alpha}+b^{\alpha})^{1/\alpha}X,\text{\ for }a,b\geq
0,$

where $Y$ is an independent copy of $X$.

###### Remark 2.6

A special attention is deserved for the case $\alpha=1$. If
$|X|\in{\mathcal{H}}$, then $\mathbb{\hat{E}}[|X|]$ exists. In this situation,
$X$ is the maximum distribution as introduced in Peng (2008(2), 2019(2)).
However, if $|X|\notin\mathcal{H}$ but $\varphi(X)\in\mathcal{H}$ as
considered in this paper, then $\mathbb{\hat{E}}[|X|]$ does not exist but
$\mathbb{\hat{E}}[\varphi(X)]$ exists. In this case $X$ is fundamentally
different from the maximum distribution.

We close this section by recalling nonlinear Lévy processes under sublinear
expectations introduced in (2021) and Neufeld and Nutz (2017) and, in
particular, nonlinear symmetric $\alpha$-stable processes.

###### Definition 2.7

An $n$-dimensional càdlàg process $(X_{t})_{t\geq 0}$ defined on a sublinear
expectation space $(\Omega,\mathcal{H},\mathbb{\hat{E})}$ is called a
nonlinear Lévy process if the following properties hold.

(i)

$X_{0}=0$;

(ii)

$(X_{t})_{t\geq 0}$ has stationary increments, that is, $X_{t}-X_{s}$ and
$X_{t-s}$ are identically distributed for all $0\leq s\leq t;$

(iii)

$(X_{t})_{t\geq 0}$ has independent increments, that is, $X_{t}-X_{s}$ is
independent from $(X_{t_{1}},\ldots,X_{t_{n}})$ for each $n\in\mathbb{N}$ and
$0\leq t_{1}\leq\cdots\leq t_{n}\leq s\leq t$.

Furthermore, $X$ is called a nonlinear symmetric $\alpha$-stable process if
${X}_{t}\overset{d}{=}t^{1/\alpha}X_{1}$ and $X_{t}\overset{d}{=}-X_{t}$.

##  3 Main results

Let $\alpha\in(0,1]$, $(\underline{\Lambda},\overline{\Lambda})$ for some
$\underline{\Lambda},\overline{\Lambda}>0$, and $F_{\mu}$ be the
$\alpha$-stable Lévy measure on
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$,

$F_{\mu}(B)=\int_{S}\mu(dz)\int_{0}^{\infty}\mathbbm{1}_{B}(rz)\frac{dr}{r^{1+\alpha}},\text{
\ for }B\in\mathcal{B}(\mathbb{R}^{d}),$ (3.1)

where $\mu$ is a symmetric finite measure (cf. Sato Sato (1999)) on the unit
sphere $S=\\{z\in\mathbb{R}^{d}:|z|=1\\}$. Introduce a _jump uncertainty set_
:

$\mathcal{L}=\left\\{F_{\mu}\ \text{measure on
}\mathbb{R}^{d}:\mu(S)\in(\underline{\Lambda},\overline{\Lambda})\right\\}.$
(3.2)

Let $\\{Z_{i}\\}_{i=1}^{\infty}$ be an i.i.d. non-integrable sequence of
$\mathbb{R}^{d}$-valued random variables defined on a sublinear expectation
space $(\Omega,\mathcal{H},\mathbb{\hat{E}})$ in the sense that
$\mathbb{\hat{E}}[|Z_{i}|]=\infty$, $Z_{i+1}$ $\overset{d}{=}Z_{i}$ and
$Z_{i+1}$ is independent from $(Z_{1},\ldots,Z_{i})$ for each
$i\in\mathbb{N}$.

We impose the following assumptions throughout the paper.

(A1)

($\delta$-Moment condition) For $S_{n}:=\sum\limits_{i=1}^{n}Z_{i}$,
$M_{\delta}:=\sup\limits_{n}\mathbb{\hat{E}}[|n^{-\frac{1}{\alpha}}S_{n}|^{\delta}]<\infty$,
for some $0<\delta<\alpha$.

(A2)

(Consistency condition) For each $\varphi\in C_{b}^{3}(\mathbb{R}^{d})$, the
space of functions on $\mathbb{R}^{d}$ with uniformly bounded derivatives up
to the order $3$, satisfies

$\frac{1}{s}\bigg{|}\mathbb{\hat{E}}\big{[}\varphi(x+s^{\frac{1}{\alpha}}Z_{1})-\varphi(x)\big{]}-s\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(x)F_{\mu}(d\lambda)\bigg{|}\leq
l(s)\rightarrow 0$

uniformly on $x\in\mathbb{R}^{d}$ as $s\rightarrow 0$, where $l$ is a function
on $[0,1]$ and

$\delta_{\lambda}^{\alpha}\varphi(x):=\left\\{\begin{array}[c]{ll}\varphi(x+\lambda)-\varphi(x)-\langle
D\varphi(x),\lambda\mathbbm{1}_{\\{|\lambda|\leq 1\\}}\rangle,&\alpha=1,\\\
\varphi(x+\lambda)-\varphi(x),&\alpha\in(0,1).\end{array}\right.$

###  3.1 Discussion on the $\delta$-moment and consistency conditions
(A1)-(A2)

The $\delta$-moment condition (A1) is inspired by the existing moment
conditions imposed in the literature. Indeed, for the robust central limit
theorem, the second moment condition $\mathbb{\hat{E}}[|Z_{i}|^{2}]<\infty$ is
equivalent to $M_{2}<\infty$ with $\alpha=2$, and for the robust
$\alpha$-stable central limit theorem in (2022) with $\alpha\in(1,2)$, the
moment condition therein is $M_{1}<\infty$. For $\alpha\in(0,1]$, since in
this case the $\alpha$-stable distributions are non-integrable, the
$\delta$-moment condition $M_{\delta}<\infty$ for some $0<\delta<\alpha$ is
natural. While the assumption (A1) guarantees the weak convergence of the
i.i.d. sequence $\\{Z_{i}\\}_{i=1}^{\infty}$, the assumption (A2) is
essentially a consistency condition for the distribution of $Z_{i}$. The
consistency condition has been widely used in the numerical analysis of the
monotone approximation schemes for nonlinear PDEs Barles and Jakobsen (2002,
2005, 2007).

In the following, we will show that when our attention is confined to the
classical linear case, the $\delta$-moment condition and consistency condition
turn out to be mild and are more general than the usual assumptions imposed in
the literature. For simplicity, we consider the one-dimensional case. Let
$\zeta\ $be a classical symmetric $\alpha$-stable random variable with Lévy
triplet $(F_{\mu},0,0)$. In this case, the symmetric finite measure $\mu$ in
$F_{\mu}$ is supported on $\\{1,-1\\}$ with $k:=\mu\\{1\\}=\mu\\{-1\\}$.
Theorem 2.6.7 from Ibragimov and Linnik (1971) indicates that

$\frac{1}{\sqrt[\alpha]{n}}\sum_{i=1}^{n}Z_{i}\overset{\mathcal{D}}{\rightarrow}\zeta\text{,
\ as}\ n\rightarrow\infty$

if $Z_{1}$ has the cumulative distribution function

$F_{Z_{1}}(x)=\left\\{\begin{array}[c]{ll}\displaystyle\left[k/\alpha+\beta_{1}(x)\right]\frac{1}{|x|^{\alpha}},&x<0,\\\
\displaystyle
1-\left[k/\alpha+\beta_{2}(x)\right]\frac{1}{x^{\alpha}},&x>0,\end{array}\right.$
(3.3)

where $\beta_{1}:$ $(-\infty,0]$ $\rightarrow\mathbb{R}$ and
$\beta_{2}:[0,\infty)\rightarrow\mathbb{R}$ are functions satisfying

$\lim_{x\rightarrow-\infty}\beta_{1}(x)=\lim_{x\rightarrow\infty}\beta_{2}(x)=0.$

Under the condition (3.3), it follows that

$E[|Z_{1}|]\geq\int_{1}^{\infty}P(|Z_{1}|>z)dz=\frac{2k}{\alpha}\int_{1}^{\infty}z^{-\alpha}dz+\int_{1}^{\infty}\frac{\beta_{1}(-z)+\beta_{2}(z)}{z^{\alpha}}dz=\infty.$

To verify the $\delta$-moment condition (A1), for given $n>0$ and
$0<\delta<\alpha$, we define an approximation scheme
$u_{n}:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ recursively by

$\begin{array}[c]{l}u_{n}(t,z)=|z|^{\delta},\text{ \ if\ }t\in[0,1/n),\\\
u_{n}(t,z)=E[u_{n}(t-1/n,z+n^{-1/\alpha}Z_{1})]\text{, \ if\
}t\in[1/n,1].\end{array}$

We further assume that

$\beta_{i},i=1,2,\text{are continuously differentiable functions defined on
}(-\infty,0]\text{ and\ }[0,\infty)\text{, respectively.}$ (3.4)

Then, $u_{n}(1,0)=n^{-\frac{\delta}{\alpha}}E[|S_{n}|^{\delta}]$ and
$u_{n}(0,0)=0$. Using the regularity estimates in Appendix A.2 with
$\mathcal{L}$ being a singleton, it can be readily verified that (A1) holds.

Next, we will show that (A2) also holds. For this, we further assume that
there exist some constants $C>0$ and $\gamma\geq 0$ such that

$|\beta_{i}(x)|\leq\frac{C}{|x|^{\gamma}}\text{, \ }x\neq 0.$ (3.5)

The above condition also appears in (2019) under the classical linear case. In
the case of $\alpha=1$, we need to further impose that

$\beta_{1}(x)=\beta_{2}(-x)\text{, \ for }x<0.$ (3.6)

###### Proposition 3.1

Under the classical linear expectation, suppose that the conditions
(3.3)-(3.5) hold. Then, (A2) holds for $\alpha\in(0,1)$. If we further assume
that (3.6) holds, then (A2) holds for $\alpha=1$.

Proof. _(i)_ $\alpha\in(0,1)$. For $\varphi\in C_{b}^{3}(\mathbb{R})$ and
$s\in[0,1]$, from (3.3), by changing variables, we can derive that

$\displaystyle\frac{1}{s}\bigg{|}E\big{[}\varphi(x+s^{\frac{1}{\alpha}}Z_{1})-\varphi(x)\big{]}-s\int_{\mathbb{R}}\delta_{\lambda}^{\alpha}\varphi(x)F_{\mu}(d\lambda)\bigg{|}$
$\displaystyle=\bigg{|}\int_{\mathbb{-\infty}}^{0}\delta_{\lambda}^{\alpha}\varphi(x)\big{[}\alpha\beta_{1}(s^{-\frac{1}{\alpha}}\lambda)-\beta_{1}^{\prime}(s^{-\frac{1}{\alpha}}\lambda)s^{-\frac{1}{\alpha}}\lambda\big{]}|\lambda|^{-\alpha-1}d\lambda\
$ $\displaystyle\text{ \ \
}+\int_{0}^{\infty}\delta_{\lambda}^{\alpha}\varphi(x)\big{[}\alpha\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)-\beta_{2}^{\prime}(s^{-\frac{1}{\alpha}}\lambda)s^{-\frac{1}{\alpha}}\lambda\big{]}\lambda^{-\alpha-1}d\lambda\bigg{|}$

for all $x\in\mathbb{R}^{d}$. We only consider the integral above along the
positive half-line, and similarly for the integral along the negative half-
line. Set

$\displaystyle\bigg{|}\int_{0}^{\infty}\delta_{\lambda}^{\alpha}\varphi(x)[\alpha\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)-\beta_{2}^{\prime}(s^{-\frac{1}{\alpha}}\lambda)s^{-\frac{1}{\alpha}}\lambda]\lambda^{-\alpha-1}d\lambda\bigg{|}$
$\displaystyle=\bigg{|}\int_{0}^{1}\delta_{\lambda}^{\alpha}\varphi(x)\big{[}\alpha\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)-\beta_{2}^{\prime}(s^{-\frac{1}{\alpha}}\lambda)s^{-\frac{1}{\alpha}}\lambda\big{]}\lambda^{-\alpha-1}d\lambda\bigg{|}$
$\displaystyle\text{ \ \
}+\bigg{|}\int_{1}^{\infty}\delta_{\lambda}^{\alpha}\varphi(x)\big{[}\alpha\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)-\beta_{2}^{\prime}(s^{-\frac{1}{\alpha}}\lambda)s^{-\frac{1}{\alpha}}\lambda\big{]}\lambda^{-\alpha-1}d\lambda\bigg{|}:=\mathcal{I}+\mathcal{II}\text{.}$

For the part $\mathcal{I}$, using integration by parts and the dominated
convergence theorem, one gets

$\displaystyle\mathcal{I}$
$\displaystyle=\bigg{|}-\delta_{1}^{\alpha}\varphi(x)\beta_{2}(s^{-\frac{1}{\alpha}})+\int_{0}^{1}D\varphi(x+\lambda)\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)\lambda^{-\alpha}d\lambda\bigg{|}$
$\displaystyle\leq
2\left\|\varphi\right\|_{\infty}|\beta_{2}(s^{-\frac{1}{\alpha}})|+\left\|D\varphi\right\|_{\infty}\int_{0}^{1}|\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)|\lambda^{-\alpha}d\lambda\rightarrow
0\text{, \ as }s\rightarrow 0\text{.}$

For the part $\mathcal{II}$, when $\gamma\in(1-\alpha,\infty)$, it follows
from integration by parts and (3.5) that

$\displaystyle\mathcal{II}$
$\displaystyle=\bigg{|}\delta_{1}^{\alpha}\varphi(x)\beta_{2}(s^{-\frac{1}{\alpha}})+\int_{1}^{\infty}D\varphi(x+\lambda)\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)\lambda^{-\alpha}d\lambda\bigg{|}$
$\displaystyle\leq
2\left\|\varphi\right\|_{\infty}|\beta_{2}(s^{-\frac{1}{\alpha}})|+\left\|D\varphi\right\|_{\infty}\int_{1}^{\infty}|\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)|\lambda^{-\alpha}d\lambda\rightarrow
0\text{, \ as }s\rightarrow 0\text{.}$

When $\gamma\in[0,1-\alpha]$, we choose some $N_{0}>1$ such that
$|\beta_{2}(x)|\leq C$, for $|x|\geq N_{0}$. Then, it follows that

$\displaystyle\bigg{|}\int_{N_{0}}^{\infty}\delta_{\lambda}^{\alpha}\varphi(x)\big{[}\alpha\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)-\beta_{2}^{\prime}(s^{-\frac{1}{\alpha}}\lambda)s^{-\frac{1}{\alpha}}\lambda\big{]}\lambda^{-\alpha-1}d\lambda\bigg{|}$
$\displaystyle\leq
2\left\|\varphi\right\|_{\infty}\int_{N_{0}}^{\infty}\big{|}\alpha\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)-\beta_{2}^{\prime}(s^{-\frac{1}{\alpha}}\lambda)s^{-\frac{1}{\alpha}}\lambda\big{|}\lambda^{-\alpha-1}d\lambda$
$\displaystyle=2\left\|\varphi\right\|_{\infty}|\beta_{2}(s^{-\frac{1}{\alpha}}N_{0})|N_{0}^{-\alpha}\leq
2C\left\|\varphi\right\|_{\infty}N_{0}^{-\alpha},$

and

$\displaystyle\bigg{|}\int_{1}^{N_{0}}\delta_{\lambda}^{\alpha}\varphi(x)\big{[}\alpha\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)-\beta_{2}^{\prime}(s^{-\frac{1}{\alpha}}\lambda)s^{-\frac{1}{\alpha}}\lambda\big{]}\lambda^{-\alpha-1}d\lambda\bigg{|}$
$\displaystyle=\bigg{|}\delta_{1}^{\alpha}\varphi(x)\beta_{2}(s^{-\frac{1}{\alpha}})-\delta_{N_{0}}^{\alpha}\varphi(x)\beta_{2}(s^{-\frac{1}{\alpha}}N_{0})N_{0}^{-\alpha}+\int_{1}^{N_{0}}D\varphi(x+\lambda)\beta_{2}(s^{-\frac{1}{\alpha}}\lambda)\lambda^{-\alpha}d\lambda\bigg{|}$
$\displaystyle\leq
2\left\|\varphi\right\|_{\infty}|\beta_{2}(s^{-\frac{1}{\alpha}})|+2C\left\|\varphi\right\|_{\infty}N_{0}^{-\alpha}+\frac{1}{1-\alpha}\left\|D\varphi\right\|_{\infty}\sup_{|\lambda|\geq
s^{-1/\alpha}}|\beta_{2}(\lambda)|N_{0}^{1-\alpha}.$

By letting $N_{0}=(\sup\limits_{|\lambda|\geq
s^{-1/\alpha}}|\beta_{2}(\lambda)|)^{-1}$, we obtain that

$\mathcal{II}\leq
2\left\|\varphi\right\|_{\infty}|\beta_{2}(s^{-\frac{1}{\alpha}})|+\left[4C\left\|\varphi\right\|_{\infty}+\frac{1}{1-\alpha}\left\|D\varphi\right\|_{\infty}\right]\sup\limits_{|\lambda|\geq
s^{-1/\alpha}}|\beta_{2}(\lambda)|^{\alpha}\rightarrow 0\text{, \ as
}s\rightarrow 0\text{.}$

_(ii)_ $\alpha=1$. We further impose that $\beta_{1}(x)=\beta_{2}(-x)$ for
$x<0$. Then, for any $a>0$, we have $E[Z_{1}\mathbbm{1}_{\\{|Z_{1}|\leq
a\\}}]=0$, which implies that for $\varphi\in C_{b}^{3}(\mathbb{R})$ and
$s\in[0,1]$,

$\displaystyle\frac{1}{s}\bigg{|}E\big{[}\varphi(x+sZ_{1})-\varphi(x)\big{]}-s\int_{\mathbb{R}}\delta_{\lambda}^{1}\varphi(x)F_{\mu}(d\lambda)\bigg{|}$
$\displaystyle=\frac{1}{s}\bigg{|}E\big{[}\varphi(x+sZ_{1})-\varphi(x)-D\varphi(x)sZ_{1}\mathbbm{1}_{\\{|sZ_{1}|\leq
1\\}}\big{]}-s\int_{\mathbb{R}}\delta_{\lambda}^{1}\varphi(x)F_{\mu}(d\lambda)\bigg{|}$
$\displaystyle=\bigg{|}\int_{\mathbb{-\infty}}^{0}\delta_{\lambda}^{1}\varphi(x)\big{[}\alpha\beta_{2}(-s^{-1}\lambda)+\beta_{2}^{\prime}(-s^{-1}\lambda)s^{-1}\lambda\big{]}|\lambda|^{-\alpha-1}d\lambda$
$\displaystyle\text{ \ \
}+\int_{0}^{\infty}\delta_{\lambda}^{1}\varphi(x)\big{[}\alpha\beta_{2}(s^{-1}\lambda)-\beta_{2}^{\prime}(s^{-1}\lambda)s^{-1}\lambda\big{]}\lambda^{-\alpha-1}d\lambda\bigg{|},$

for all $x\in\mathbb{R}^{d}$. Similar to the above process, consider the
integral above along the positive half-line, and denote

$\displaystyle\bigg{|}\int_{0}^{\infty}\delta_{\lambda}^{1}\varphi(x)\big{[}\beta_{2}(s^{-1}\lambda)-\beta_{2}^{\prime}(s^{-1}\lambda)s^{-1}\lambda\big{]}\lambda^{-2}d\lambda\bigg{|}$
$\displaystyle=\bigg{|}\int_{0}^{1}\delta_{\lambda}^{1}\varphi(x)\big{[}\beta_{2}(s^{-1}\lambda)-\beta_{2}^{\prime}(s^{-1}\lambda)s^{-1}\lambda\big{]}\lambda^{-2}d\lambda\bigg{|}$
$\displaystyle\text{ \ \
}+\bigg{|}\int_{1}^{\infty}\delta_{\lambda}^{1}\varphi(x)\big{[}\beta_{2}(s^{-1}\lambda)-\beta_{2}^{\prime}(s^{-1}\lambda)s^{-1}\lambda\big{]}\lambda^{-2}d\lambda\bigg{|}:=\mathcal{I}+\mathcal{II}\text{.}$

Then, we can deduce that

$\displaystyle\mathcal{I}$
$\displaystyle=\bigg{|}-\delta_{1}^{1}\varphi(x)\beta_{2}(s^{-1})+\int_{0}^{1}(D\varphi(x+\lambda)-D\varphi(x))\beta_{2}(s^{-1}\lambda)\lambda^{-1}d\lambda\bigg{|}$
$\displaystyle\leq(2\left\|\varphi\right\|_{\infty}+\left\|D\varphi\right\|_{\infty})|\beta_{2}(s^{-1})|+\left\|D^{2}\varphi\right\|_{\infty}\int_{0}^{1}|\beta_{2}(s^{-1}\lambda)|d\lambda\rightarrow
0\text{, \ as }s\rightarrow 0\text{,}$

and

$\mathcal{II}\leq
2\left\|\varphi\right\|_{\infty}|\beta_{2}(s^{-1})|+\left\|D\varphi\right\|_{\infty}\int_{1}^{\infty}|\beta_{2}(s^{-1}\lambda)|\lambda^{-1}d\lambda\rightarrow
0\text{, \ as }s\rightarrow 0\text{.}$

To sum up, the assumption (A2) holds.

###  3.2 Robust $\alpha$-stable central limit theorem

We are ready to state the main result of this paper, which is dubbed as _a
robust_ $\alpha$_-stable central limit theorem_ under sublinear expectation.

###### Theorem 3.2

Suppose that the assumptions (A1)-(A2) hold. Then, there exists a nonlinear
Lévy process $(\tilde{\zeta}_{t})_{t\in[0,1]}$, associated with the jump
uncertainty set $\mathcal{L}$ such that for any $\phi\in
C_{b,Lip}(\mathbb{R}^{d})$,

$\lim_{n\rightarrow\infty}\mathbb{\hat{E}}\left[\phi\left(\frac{S_{n}}{\sqrt[\alpha]{n}}\right)\right]=\mathbb{\tilde{E}}[\phi(\tilde{\zeta}_{1})]=u^{\phi}(1,0),$

where $u^{\phi}$ is the unique viscosity solution of the following fully
nonlinear PIDE

$\left\\{\begin{array}[c]{l}\displaystyle\partial_{t}u(t,x)-\sup\limits_{F_{\mu}\in\mathcal{L}}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}u(t,x)F_{\mu}(d\lambda)\right\\}=0,\\\
\displaystyle u(0,x)=\phi(x),\text{\ \ \
}\forall(t,x)\in[0,1]\times\mathbb{R}^{d},\end{array}\right.$ (3.7)

where

$\delta_{\lambda}^{\alpha}u(t,x):=\left\\{\begin{array}[c]{ll}u(t,x+\lambda)-u(t,x)-\langle
D_{x}u(t,x),\lambda\mathbbm{1}_{\\{|\lambda|\leq 1\\}}\rangle,&\alpha=1,\\\
u(t,x+\lambda)-u(t,x),&\alpha\in(0,1).\end{array}\right.$

Furthermore, the limiting process $(\tilde{\zeta}_{t})_{t\in[0,1]}$ is a
nonlinear symmetric $\alpha$-stable process, i.e.
$\tilde{\zeta}_{t}\overset{d}{=}t^{1/\alpha}\tilde{\zeta}_{1}$ and
$\tilde{\zeta}_{t}\overset{d}{=}-\tilde{\zeta}_{t}$, for any $0\leq t\leq 1$.

###  3.3 An example

We discuss a concrete example to illustrates how to construct the i.i.d.
sequence $\\{Z_{i}\\}_{i=1}^{\infty}$ and their corresponding sublinear
expectation space, as well as the rationality of the assumptions (A1)-(A2).

For simplicity, we consider the case $d=1$. Given
$\underline{\Lambda},\overline{\Lambda}>0$. Let $F_{\mu}$ be the Lévy measure
with the symmetric measure $\mu$ concentrated on the points $S_{0}=\\{1,-1\\}$
and

$\mathcal{L}=\left\\{F_{\mu}\ \text{measure on
}\mathbb{R}^{2}:\mu(\chi)\in(\underline{\Lambda},\overline{\Lambda}),\
\text{for }\chi\in S_{0}\right\\}.$

Denote $K=(\underline{\Lambda},\overline{\Lambda})$ and
$k=\mu\\{1\\}=\mu\\{-1\\}$. Let $0<\delta<\alpha$ and $f(n)$ be a non-negative
function on $\mathbb{N}$ tending to 0 as $n\rightarrow\infty$. For each $k\in
K$, let $W_{k}$ be a classical random variable such that

(i)

$W_{k}$ has a cumulative distribution function

$F_{W_{k}}(x)=\left\\{\begin{array}[c]{ll}\displaystyle\left[k/\alpha+\beta_{k}(-x)\right]\frac{1}{|x|^{\alpha}},&x<0,\\\
\displaystyle
1-\left[k/\alpha+\beta_{k}(x)\right]\frac{1}{x^{\alpha}},&x>0,\end{array}\right.$

where $\beta_{k}:[0,\infty)\rightarrow\mathbb{R}$ are continuously
differentiable functions such that there exist $\gamma\geq 0$ and $C>0$
satisfying

$|\beta_{k}(x)|\leq\frac{C}{|x|^{\gamma}}\text{, \ }x\neq 0.$

(ii)

the following quantities are less than $f(n)$ for all $n$:

$\begin{array}[c]{lll}\displaystyle|\beta_{k}(n^{1/\alpha})|,\text{ \
}&\displaystyle\int_{1}^{\infty}\frac{|\beta_{k}(n^{1/\alpha}x)|}{x^{1+\alpha-\delta}}dx,\text{
\
}&\displaystyle\int_{0}^{1}\frac{|\beta_{k}(n^{1/\alpha}x)|}{x^{\alpha}}dx.\end{array}$

Let $\Omega=\mathbb{R}$ and $\mathcal{H}_{0}$ be the space of bounded and
continuous functions on $\mathbb{R}$. For each $X=f(x)\in\mathcal{H}_{0}$,
define the sublinear expectation

$\mathbb{\hat{E}}[X]=\sup_{k\in K}\int_{\mathbb{R}}f(x)dF_{W_{k}}(x).$

Denote by $\mathcal{H}$ the completion of $\mathcal{H}_{0}$ under the norm$\
\left\|X\right\|:=\mathbb{\hat{E}}[|X|]$. $\mathbb{\hat{E}[\cdot]}$ can be
extended continuously to $\mathcal{H}$, on which it is a sublinear expectation
and is still denoted as $\big{(}\Omega,\mathcal{H},\mathbb{\hat{E}}\big{)}$,
see (2011) and Peng (2019(2)) for a detailed discussion.

Consider an $\mathbb{R}$-valued random variable $Z(z)=z$, $z\in\mathbb{R}$.
Clearly, $Z^{\delta,m}:=|Z|^{\delta}\wedge m\in\mathcal{H}_{0}$ for any $m>0$.
We claim that $\left\\{Z^{\delta,m}\right\\}_{m=1}^{\infty}$ is a Cauchy
sequence, which yields that
$\mathbb{\hat{E}}[|Z|^{\delta}]:=\lim\limits_{m\rightarrow\infty}\mathbb{\hat{E}}[Z^{\delta,m}]$
and $|Z|^{\delta}\in\mathcal{H}_{0}$. Indeed, for any $Z^{\delta,m}$,
$Z^{\delta,n}\in\mathcal{H}$, we have

$\displaystyle\mathbb{\hat{E}}[|Z^{\delta,m}-Z^{\delta,n}|]$
$\displaystyle\leq\mathbb{\hat{E}}\left[\left||Z|^{\delta}\wedge
m-|Z|^{\delta}\right|\right]+\mathbb{\hat{E}}\left[\left||Z|^{\delta}\wedge
n-|Z|^{\delta}\right|\right]$ $\displaystyle\leq\sup_{k\in
K}\int_{\mathbb{R}}|z|^{\delta}\mathbbm{1}_{\\{|z|^{\delta}>m\\}}dF_{W_{k}}(z)+\sup_{k\in
K}\int_{\mathbb{R}}|z|^{\delta}\mathbbm{1}_{\\{|z|^{\delta}>n\\}}dF_{W_{k}}(z)$
$\displaystyle=2m^{\frac{\delta-\alpha}{\delta}}\sup_{k\in
K}\left\\{\frac{k}{\alpha-\delta}+\delta\int_{1}^{\infty}\frac{\beta_{k}(m^{1/\delta}z)}{z^{1+\alpha-\delta}}dz+\beta_{k}(m^{1/\delta})\right\\}$
$\displaystyle\text{ \ \ }+2n^{\frac{\delta-\alpha}{\delta}}\sup_{k\in
K}\left\\{\frac{k}{\alpha-\delta}+\delta\int_{1}^{\infty}\frac{\beta_{k}(n^{1/\delta}z)}{z^{1+\alpha-\delta}}dz+\beta_{k}(n^{1/\delta})\right\\}$

tending to 0, as $m,n\rightarrow\infty$. A similar calculation shows that
$|Z|\notin\mathcal{H}$.

Next, we will verify the assumption (A1). Let $\\{Z_{i}\\}_{i=1}^{\infty}$ be
a sequence of i.i.d. $\mathbb{R}$-valued random variables in the sense that
$Z_{1}\overset{d}{=}Z$, $Z_{i+1}\overset{d}{=}Z_{i}$, and
$Z_{i+1}\perp\\!\\!\\!\perp(Z_{1},Z_{2},\ldots,Z_{i})$ for each
$i\in\mathbb{N}$. For given $n>0$, define an approximation scheme
$u_{n}:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ recursively by

$\begin{array}[c]{l}u_{n}(t,z)=|z|^{\delta},\text{ \ if\ }t\in[0,1/n),\\\
u_{n}(t,z)=\mathbb{\hat{E}}[u_{n}(t-1/n,z+n^{-1/\alpha}Z_{1})]\text{, \ if\
}t\in[1/n,1].\end{array}$ (3.8)

Then,
$u_{n}(1,0)=n^{-\frac{\delta}{\alpha}}\mathbb{\hat{E}}[|S_{n}|^{\delta}]$ and
$u_{n}(0,0)=0$. For any $t,s\in[0,1]$ and $z\in\mathbb{R}$, we deduce the
following estimate, whose proof is postponed to the Appendix A.2 due to its
technicality

$\left|u_{n}(t,z)-u_{n}(s,z)\right|\leq
I_{n}(|t-s|^{\delta/2}+n^{-\delta/2}),$

where $I_{n}<\infty$ as $n\rightarrow\infty$. It follows that

$u_{n}(1,0)=u_{n}(1,0)-u_{n}(0,0)<\infty\text{, \ as }n\rightarrow\infty,$

which implies (A1) holds.

For the assumption (A2), following along similar arguments as in Proposition
3.1, we can derive that for $\varphi\in C_{b}^{3}(\mathbb{R})$,

$\frac{1}{s}\bigg{|}\mathbb{\hat{E}}\big{[}\varphi(z+s^{\frac{1}{\alpha}}Z_{1})-\varphi(z)\big{]}-s\sup_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{2}}\delta_{\lambda}^{\alpha}\varphi(z)F_{\mu}(d\lambda)\bigg{|}\rightarrow
0,$

uniformly on $z\in\mathbb{R}$ as $s\rightarrow 0$.

###### Remark 3.3

Assume that $\mathcal{L}=\\{F_{\mu}\\}$ is a singleton and
$\\{Z_{i}\\}_{i=1}^{\infty}$ is a sequence of i.i.d. random variables with a
Pareto distribution in (2002, 1998); Kuske and Keller (2001), i.e., the
probability density is

$p(x)=\frac{\alpha}{2|x|^{\alpha+1}}\mathbbm{1}_{[1,\infty)}(|x|).$

It is easy to check that the above distribution satisfies the conditions
(i)-(ii). Theorem 3.2 implies that $\frac{1}{\sqrt[\alpha]{n}}S_{n}$ converges
in distribution to $\tilde{\zeta}_{1}$, as $n\rightarrow\infty$, where
$\tilde{\zeta}_{1}$ is a classical symmetric $\alpha$-stable distribution with
Lévy triplet $(F_{\mu},0,0)$.

##  4 Proof of Theorem 3.2

###  4.1 Outline of the proof

For the reader’s convenience, we outline the main steps below with the
detailed proof provided in the rest of this section.

(i) By using the notions of tightness and weak compactness, we first construct
a nonlinear Lévy process $(\tilde{\zeta}_{t})_{t\in[0,1]}$ on some sublinear
expectation space $(\tilde{\Omega},Lip(\tilde{\Omega}),\mathbb{\tilde{E}})$ in
Section 4.2, which is generated by the weak convergence limit of the sequences
$\big{\\{}(t/n)^{1/\alpha}S_{n}\big{\\}}_{n=1}^{\infty}$ for $t\in[0,1]$.

(ii) To link the nonlinear Lévy process $(\tilde{\zeta}_{t})_{t\in[0,1]}$ with
the fully nonlinear PIDE (3.7), a key step is to give the characterization of

$\lim\limits_{s\rightarrow
0}\mathbb{\tilde{E}}[\varphi(\tilde{\zeta}_{s})]s^{-1}\text{ for }\varphi\in
C_{b}^{3}(\mathbb{R}^{d})\text{ with }\varphi(0)=0.$

It follows from a new estimate for the $\alpha$-stable Lévy measure and the
squeeze theorem in Section 4.3.

(iii) Once the representation of the nonlinear Lévy process is established,
with the help of nonlinear stochastic analysis techniques and viscosity
solution methods, Theorem 3.2 is a consequence of the dynamic programming
principle in Section 4.4 by defining
$u(t,x)=\mathbb{\tilde{E}}[\phi(x+\tilde{\zeta}_{t})]$, for
$(t,x)\in[0,1]\times\mathbb{R}^{d}$.

(iv) Finally, we show the scaling and symmetric properties of
$(\tilde{\zeta}_{t})_{t\in[0,1]}$ in Section 4.5, so it is indeed a nonlinear
symmetric $\alpha$-stable process.

###  4.2 The construction of the nonlinear $\alpha$-stable process

Let $\tilde{\Omega}=D_{0}^{d}[0,1]$ be the space of all
$\mathbb{R}^{d}$-valued paths $(\omega_{t})_{t\in[0,1]}$ with $\omega_{0}=0$,
equipped with the Skorohod topology, where $D_{0}^{d}[0,1]$ is the space of
$\mathbb{R}^{d}$-valued càdlàg paths. Consider the canonical process
$\tilde{\zeta}_{t}(\omega)=\omega_{t}$, $t\in[0,1]$, for
$\omega\in\tilde{\Omega}$. Set

$Lip(\tilde{\Omega})=\left\\{\varphi(\tilde{\zeta}_{t_{1}},\ldots,\tilde{\zeta}_{t_{n}}-\tilde{\zeta}_{t_{n-1}}):\forall
0\leq t_{1}<t_{2}<\cdots<t_{n}\leq 1,\varphi\in C_{b,Lip}(\mathbb{R}^{d\times
n})\right\\}.$

We have the following theorem, which can be regarded as a Donsker theorem for
the nonlinear Lévy process $(\tilde{\zeta}_{t})_{t\in[0,1]}$.

###### Theorem 4.1

Assume that (A1) holds. Then, there exists a sublinear expectation
$\mathbb{\tilde{E}}$ on $(\tilde{\Omega},Lip(\tilde{\Omega}))$ such that the
sequence $\\{n^{-1/\alpha}S_{n}\\}_{n=1}^{\infty}$ converges in distribution
to $\tilde{\zeta}_{1}$, where $(\tilde{\zeta}_{t})_{t\in[0,1]}$ is a nonlinear
Lévy process on $(\tilde{\Omega},Lip(\tilde{\Omega}),\mathbb{\tilde{E}})$.

The proof of the above theorem depends on the following lemma.

###### Lemma 4.2

Assume that (A1) holds. For $\phi\in C_{b,Lip}(\mathbb{R}^{d})$, let

$\mathbb{\hat{F}}[\phi]:=\sup_{n}\mathbb{\hat{E}}\left[\phi\left(\frac{S_{n}}{\sqrt[\alpha]{n}}\right)\right].$

Then, the sublinear expectation $\mathbb{\hat{F}}$ on
$(\mathbb{R}^{d},C_{b,Lip}(\mathbb{R}^{d}))$ is tight in the sense of
Definition A.1 in the Appendix.

Proof. It is clear that $\mathbb{\hat{F}}$ is a sublinear expectation on
$(\mathbb{R}^{d},C_{b,Lip}(\mathbb{R}^{d}))$. Now we show that
$\mathbb{\hat{F}}$ is tight. For any $N>0$, define

$\varphi_{N}(x)=\left\\{\begin{array}[c]{ll}1,&|x|>N,\\\
|x|-N+1,&N-1\leq|x|\leq N,\\\ 0,&|x|<N-1.\end{array}\right.$

One can easily check that $\varphi_{N}\in C_{b,Lip}(\mathbb{R})$ and

$\mathbbm{1}_{\\{|x|>N\\}}\leq\varphi_{N}(x)\leq\mathbbm{1}_{\\{|x|>N-1\\}}\leq\frac{|x|^{\delta}}{(N-1)^{\delta}},$

for $0<\delta<\alpha$. Thus, for each $\varepsilon>0$, we can choose
$N_{0}>\sqrt[\delta]{M_{\delta}/\varepsilon}+1$ such that

$\mathbb{\hat{F}}[\varphi_{N_{0}}(z)]=\sup_{n}\mathbb{\hat{E}}\left[\varphi_{N_{0}}\left(\frac{S_{n}}{\sqrt[\alpha]{n}}\right)\right]\leq\frac{M_{\delta}}{(N_{0}-1)^{\delta}}<\varepsilon,$

where we have used the assumption (A1). This proves the desired result.

Proof of Theorem 4.1. We denote $\bar{S}_{n}=n^{-1/\alpha}S_{n}$. Seeing that
$\mathbb{\hat{F}}$ is tight and

$\mathbb{\hat{E}}[\phi(\bar{S}_{n})]-\mathbb{\hat{E}}[\phi^{\prime}(\bar{S}_{n})]\leq\mathbb{\hat{F}}[\phi-\phi^{\prime}],\text{
for }\phi,\phi^{\prime}\in C_{b,Lip}(\mathbb{R}^{d}),$

by Corollary A.5, there exists a subsequence
$\\{\bar{S}_{n_{i}}\\}_{i=1}^{\infty}\subset\\{\bar{S}_{n}\\}_{n=1}^{\infty}$
which converges in distribution to some $\zeta_{1}$ in
$(\Omega,\mathcal{H},\mathbb{\hat{E}}_{1})$. For the above convergent
subsequence $\\{\bar{S}_{n_{i}}\\}_{i=1}^{\infty}$, it is clear that for an
arbitrarily increasing integers of $\\{\tilde{n}_{i}\\}_{i=1}^{\infty}$ such
that $|\tilde{n}_{i}-n_{i}|\leq 1$, both
$\\{\bar{S}_{n_{i}}\\}_{i=1}^{\infty}$ and
$\\{\bar{S}_{\tilde{n}_{i}}\\}_{i=1}^{\infty}$ converges in distribution to
the same limit. Thus, without loss of generality, we assume that $n_{i}$,
$i=1,2,\ldots$, are all even numbers and decompose into two parts:

$\bar{S}_{n_{i}}=\frac{1}{\sqrt[\alpha]{2}}(n_{i}/2)^{-\frac{1}{\alpha}}S_{n_{i}/2}+\frac{1}{\sqrt[\alpha]{2}}(n_{i}/2)^{-\frac{1}{\alpha}}(S_{n_{i}}-S_{n_{i}/2}):=\
\bar{S}_{n_{i}/2}^{1/2}+(\bar{S}_{n_{i}}-\bar{S}_{n_{i}/2}^{1/2}),$

where $\bar{S}_{n}^{t}:=(t/n)^{1/\alpha}S_{n}$ for $t\in[0,1)$. For the first
part, applying the same argument again, we prove that there exists a
subsequence $\big{\\{}\bar{S}_{n_{i}^{1}/2}^{1/2}\big{\\}}_{i=1}^{\infty}$
$\subset\big{\\{}\bar{S}_{n_{i}/2}^{1/2}\big{\\}}_{i=1}^{\infty}\ $such that
$\big{\\{}\bar{S}_{n_{i}^{1}/2}^{1/2}\big{\\}}_{i=1}^{\infty}$ converging in
distribution to $\zeta_{1/2}$. Since
$\bar{S}_{n_{i}^{1}}-\bar{S}_{n_{i}^{1}/2}^{1/2}$ is an independent copy of
$\bar{S}_{n_{i}^{1}/2}^{1/2}$, by Proposition 2.3, we know that

$\bar{S}_{n_{i}^{1}}-\bar{S}_{n_{i}^{1}/2}^{1/2}\overset{\mathcal{D}}{\rightarrow}\bar{\zeta}_{1/2}\text{
\ and \
}\bar{S}_{n_{i}^{1}}\overset{\mathcal{D}}{\rightarrow}\zeta_{1/2}+\bar{\zeta}_{1/2},$

where $\bar{\zeta}_{1/2}$ is an independent copy of $\zeta_{1/2}$. In
addition, $\bar{S}_{n_{i}^{1}}\overset{\mathcal{D}}{\rightarrow}\zeta_{1}$.
Thus $\zeta_{1}\overset{d}{=}\zeta_{1/2}+\bar{\zeta}_{1/2}.$

Repeating the previous procedure for $\bar{S}_{n_{i}^{1}/2}^{1/2}$, we can
define random variable $\zeta_{1/4}$. Proceeding in this way, one can obtain
$\zeta_{1/2^{m}}$ in $(\Omega,\mathcal{H},\mathbb{\hat{E}}_{1})$,
$m\in\mathbb{N}$, such that for each $\zeta_{1/2^{m}}$ there exists a
convergent sequence $\\{\bar{S}_{n_{i}^{m}/2^{m}}^{1/2^{m}}\\}_{i=1}^{\infty}$
converging in distribution to it. Finally, using the random variables
$\zeta_{1/2^{m}}(m\in\mathbb{N)}$, we may construct a sublinear expectation
$\mathbb{\tilde{E}}$ on $(\tilde{\Omega},Lip(\tilde{\Omega}))$ such that the
canonical process $(\tilde{\zeta}_{t})_{t\in[0,1]}$ is a nonlinear Lévy
process.

Indeed, we will construct a sublinear expectation
$\mathbb{\tilde{E}}:Lip(\tilde{\Omega})\rightarrow\mathbb{R}$ such that the
canonical process $(\tilde{\zeta}_{t})_{t\in[0,1]}$ is a nonlinear Lévy
process. For each $m\geq 0$, let $\tau_{m}=2^{-m}$,

$\begin{array}[c]{l}\displaystyle\mathcal{H}^{m}=\big{\\{}\varphi\big{(}\tilde{\zeta}_{\tau_{m}},\tilde{\zeta}_{2\tau_{m}}-\tilde{\zeta}_{\tau_{m}},\ldots,\tilde{\zeta}_{2^{m}\tau_{m}}-\tilde{\zeta}_{(2^{m}-1)\tau_{m}}\big{)}:\forall\varphi\in
C_{b,Lip}\big{(}\mathbb{R}^{d\times 2^{m}}\big{)}\big{\\}}\text{, \ for }m\geq
1,\\\
\displaystyle\mathcal{H}^{0}=\big{\\{}\phi\big{(}\tilde{\zeta}_{1}\big{)}:\forall\phi\in
C_{b,Lip}\big{(}\mathbb{R}^{d}\big{)}\big{\\}}.\end{array}$

Let $\\{\zeta_{\tau_{m}}^{n}\\}_{n=1}^{\infty}$ be a sequence of i.i.d.
$\mathbb{R}^{d}$-valued random variables on
$(\Omega,\mathcal{H},\mathbb{\hat{E}}_{1})$ in the sense that
$\zeta_{\tau_{m}}^{1}\overset{d}{=}\zeta_{\tau_{m}}$,
$\zeta_{\tau_{m}}^{n+1}\overset{d}{=}\zeta_{\tau_{m}}^{n}$ and
$\zeta_{\tau_{m}}^{n+1}\perp\\!\\!\\!\perp(\zeta_{\tau_{m}}^{1},\zeta_{\tau_{m}}^{2},\ldots,\zeta_{\tau_{m}}^{n})$
for each $n\in\mathbb{N}$. For given $m\geq 1$,
$\phi\big{(}\tilde{\zeta}_{n\tau_{m}}-\tilde{\zeta}_{(n-1)\tau_{m}}\big{)}$
with $1\leq n\leq 2^{m}$, and $\phi\in C_{b,Lip}(\mathbb{R}^{d})$, define

$\mathbb{\tilde{E}}^{m}\big{[}\phi\big{(}\tilde{\zeta}_{n\tau_{m}}-\tilde{\zeta}_{(n-1)\tau_{m}}\big{)}\big{]}=\mathbb{\hat{E}}_{1}\big{[}\phi(\zeta_{\tau_{m}}^{n})\big{]}.$

For
$\varphi\big{(}\tilde{\zeta}_{\tau_{m}},\ldots,\tilde{\zeta}_{2^{m}\tau_{m}}-\tilde{\zeta}_{(2^{m}-1)\tau_{m}}\big{)}\in\mathcal{H}^{m}$,
for some $\varphi\in C_{b,Lip}\big{(}\mathbb{R}^{d\times 2^{m}}\big{)}$,
define

$\mathbb{\tilde{E}}^{m}\big{[}\varphi\big{(}\tilde{\zeta}_{\tau_{m}},\ldots,\tilde{\zeta}_{2^{m}\tau_{m}}-\tilde{\zeta}_{(2^{m}-1)\tau_{m}}\big{)}\big{]}=\varphi_{0},$

where $\varphi_{0}$ is defined iteratively through

$\begin{array}[c]{c}\displaystyle\varphi_{2^{m}-1}(x_{1},x_{2},\ldots,x_{2^{m}-1})=\mathbb{\tilde{E}}^{m}\big{[}\varphi\big{(}x_{1},x_{2},\ldots,x_{2^{m}-1},\tilde{\zeta}_{2^{m}\tau_{m}}-\tilde{\zeta}_{(2^{m}-1)\tau_{m}}\big{)}\big{]}\\\
\displaystyle\varphi_{2^{m}-2}(x_{1},x_{2},\ldots,x_{2^{m}-2})=\mathbb{\tilde{E}}^{m}\big{[}\varphi_{2^{m}-1}\big{(}x_{1},x_{2},\ldots,x_{2^{m}-2},\tilde{\zeta}_{(2^{m}-1)\tau_{m}}-\tilde{\zeta}_{(2^{m}-2)\tau_{m}}\big{)}\big{]}\\\
\displaystyle\vdots\\\
\displaystyle\varphi_{1}(x_{1})=\mathbb{\tilde{E}}^{m}\big{[}\varphi_{2}\big{(}x_{1},\tilde{\zeta}_{2\tau_{m}}-\tilde{\zeta}_{\tau_{m}}\big{)}\big{]}\\\
\displaystyle\varphi_{0}=\mathbb{\tilde{E}}^{m}\big{[}\varphi_{1}\big{(}\tilde{\zeta}_{\tau_{m}}\big{)}\big{]}.\end{array}$

For $\phi\in C_{b,Lip}(\mathbb{R}^{d})$, we also define

$\mathbb{\tilde{E}}^{0}\big{[}\phi\big{(}\tilde{\zeta}_{1}\big{)}\big{]}=\mathbb{\hat{E}}_{1}\big{[}\phi(\zeta_{1})\big{]}.$

From the above definition we know that
$(\tilde{\Omega},\mathcal{H}^{m},\mathbb{\tilde{E}}^{m})$ is a sublinear
expectation space under which
$\tilde{\zeta}_{t}-\tilde{\zeta}_{s}\overset{d}{=}$ $\tilde{\zeta}_{t-s}$ and
$\tilde{\zeta}_{t}-\tilde{\zeta}_{s}\perp\\!\\!\\!\perp(\tilde{\zeta}_{t_{1}},\ldots,\tilde{\zeta}_{t_{i}})$
for each $t_{i},s,t\in\mathcal{D}_{m}[0,1]:=\\{l2^{-m}:0\leq l\leq
2^{m},l\in\mathbb{N}\\}$ with $t_{i}\leq s\leq t$. Also, it can be checked
that $\mathbb{\tilde{E}}^{m}[\cdot]$ is consistent, i.e., for each $m\geq 0$,
$\mathbb{\tilde{E}}^{m+1}[\cdot]=\mathbb{\tilde{E}}^{m}[\cdot]$ on
$\mathcal{H}^{m}$.

Next, we denote

$\mathcal{H}^{\infty}=\bigcup_{m\geq 0}\mathcal{H}^{m}.$

Obviously, $\mathcal{H}^{\infty}\subset Lip(\tilde{\Omega})$. For any
$\chi\in\mathcal{H}^{\infty}$, there exists an $m_{0}\in\mathbb{N}$ such that
$\chi\in\mathcal{H}^{m_{0}}$, define

$\mathbb{\tilde{E}}[\chi]:=\mathbb{\tilde{E}}^{m_{0}}[\chi].$

Since $\mathbb{\tilde{E}}^{m}[\cdot]$ is consistent,
$\mathbb{\tilde{E}}:\mathcal{H}^{\infty}\rightarrow\mathbb{R}$ is a well-
defined sublinear expectation.

We shall extend the sublinear expectation
$\mathbb{\tilde{E}}:\mathcal{H}^{\infty}\rightarrow\mathbb{R}$ to
$\mathbb{\tilde{E}}:Lip(\tilde{\Omega})\rightarrow\mathbb{R}$ and still use
$\mathbb{\tilde{E}}$ for simplicity. Denote
$\mathcal{D}_{\infty}[0,1]:=\cup_{m\geq 0}\mathcal{D}_{m}[0,1]$. For each
$\varphi(\tilde{\zeta}_{t_{1}},\ldots,\tilde{\zeta}_{t_{n}}-\tilde{\zeta}_{t_{n-1}})\in
Lip(\tilde{\Omega})$ with $\varphi\in C_{b,Lip}(\mathbb{R}^{d\times n})$, for
each $t_{k}\in[0,1]$, $1\leq k\leq n$, we choose a sequence
$\\{t_{k}^{i}\\}_{i=1}^{\infty}\in\mathcal{D}_{\infty}[0,1]$ such that
$t_{k}^{i}<t_{k+1}^{i}$ and $t_{k}^{i}\downarrow t_{k}$ as
$i\rightarrow\infty$. Define

$\mathbb{\tilde{E}}\big{[}\varphi\big{(}\tilde{\zeta}_{t_{1}},\ldots,\tilde{\zeta}_{t_{n}}-\tilde{\zeta}_{t_{n-1}}\big{)}\big{]}=\lim_{i\rightarrow\infty}\mathbb{\tilde{E}}\big{[}\varphi\big{(}\tilde{\zeta}_{t_{1}^{i}},\ldots,\tilde{\zeta}_{t_{n}^{i}}-\tilde{\zeta}_{t_{n-1}^{i}}\big{)}\big{]}.$

It can be verified that the limit does not depend on the choice of
$\\{t_{k}^{i}\\}_{i=1}^{\infty}$. Indeed, for two descending sequences
$\\{t_{k}^{i}\\}_{i=1}^{\infty}$ and
$\\{t_{k}^{i^{\prime}}\\}_{i^{\prime}=1}^{\infty}$ with the same limit $t_{k}$
as $i,i^{\prime}\rightarrow\infty$, we assume that
$t_{k}^{i}-t_{k}^{i^{\prime}}:=l_{k}\tau_{m_{k}}$ for some
$m_{k}\in\mathbb{N}$ and $1\leq l_{k}\leq 2^{m_{k}}$. From the construction of
$\tilde{\zeta}_{\cdot}$ and Remark 2.4, there exists a convergent sequence
$\\{\bar{S}_{n_{j}^{\ast}}^{\tau_{m_{k}}}\\}_{j=1}^{\infty}$ such that for
$\varphi\in C_{b,Lip}(\mathbb{R}^{d\times n})$

$\begin{array}[c]{l}\displaystyle\left|\mathbb{\tilde{E}}\big{[}\varphi\big{(}\tilde{\zeta}_{t_{1}^{i}},\ldots,\tilde{\zeta}_{t_{n}^{i}}-\tilde{\zeta}_{t_{n-1}^{i}}\big{)}\big{]}-\mathbb{\tilde{E}}\big{[}\varphi\big{(}\tilde{\zeta}_{t_{1}^{i^{\prime}}},\ldots,\tilde{\zeta}_{t_{n}^{i^{\prime}}}-\tilde{\zeta}_{t_{n-1}^{i^{\prime}}}\big{)}\big{]}\right|\\\
\displaystyle\leq
L_{\varphi}\mathbb{\tilde{E}}\big{[}\big{(}\sum_{k=1}^{n}|\tilde{\zeta}_{t_{k}^{i}}-\tilde{\zeta}_{t_{k-1}^{i}}-\tilde{\zeta}_{t_{k}^{i^{\prime}}}+\tilde{\zeta}_{t_{k-1}^{i^{\prime}}}|\big{)}\wedge
N_{\varphi}\big{]}\\\ \displaystyle\leq
2L_{\varphi}\sum_{k=1}^{n}\mathbb{\tilde{E}}\big{[}|\tilde{\zeta}_{t_{k}^{i}-t_{k}^{i^{\prime}}}|\wedge
N_{\varphi}\big{]}\\\
\displaystyle=2L_{\varphi}\sum\limits_{k=1}^{n}\mathbb{\hat{E}}_{1}\big{[}|\zeta_{\tau_{m_{k}}}^{1}+\cdots+\zeta_{\tau_{m_{k}}}^{l_{k}}|\wedge
N_{\varphi}\big{]}\\\
\displaystyle=2L_{\varphi}\sum_{k=1}^{n}\lim\limits_{j\rightarrow\infty}\mathbb{\hat{E}}\big{[}|\bar{S}_{l_{k}n_{j}^{\ast}}^{t_{k}^{i}-t_{k}^{i^{\prime}}}|\wedge
N_{\varphi}\big{]}\\\
\displaystyle=2L_{\varphi}\sum_{k=1}^{n}\lim\limits_{j\rightarrow\infty}\mathbb{\hat{E}}\big{[}\big{|}(t_{k}^{i}-t_{k}^{i^{\prime}})^{1/\alpha}(l_{k}n_{j}^{\ast})^{-1/\alpha}S_{l_{k}n_{j}^{\ast}}\big{|}\wedge
N_{\varphi}\big{]}\\\ \displaystyle\leq
2L_{\varphi}\sum_{k=1}^{n}|t_{k}^{i}-t_{k}^{i^{\prime}}|^{\delta/\alpha}\lim\limits_{j\rightarrow\infty}\mathbb{\hat{E}}\big{[}|(l_{k}n_{j}^{\ast})^{-1/\alpha}S_{l_{k}n_{j}^{\ast}}|^{\delta}\big{]}N_{\varphi}^{1-\delta}\\\
\displaystyle\leq
2L_{\varphi}M_{\delta}N_{\varphi}^{1-\delta}\sum_{k=1}^{n}|t_{k}^{i}-t_{k}^{i^{\prime}}|^{\delta/\alpha}\\\
\displaystyle\rightarrow 0\text{, \ as
}i,i^{\prime}\rightarrow\infty,\end{array}$

where $L_{\varphi}>0$ is the Lipschitz constant of $\varphi$,
$K_{\varphi}:=\left\|\varphi\right\|_{\infty}$ and
$N_{\varphi}:=\frac{2K_{\varphi}}{L_{\varphi}}$. Also,
$\mathbb{\tilde{E}}:Lip(\tilde{\Omega})\rightarrow\mathbb{R}$ is a well-
defined sublinear expectation, that is, if
$\varphi(\tilde{\zeta}_{t_{1}},\ldots,$
$\tilde{\zeta}_{t_{n}}-\tilde{\zeta}_{t_{n-1}})=\varphi^{\prime}(\tilde{\zeta}_{t_{1}},\ldots,\tilde{\zeta}_{t_{n}}-\tilde{\zeta}_{t_{n-1}})$
with $\varphi,\varphi^{\prime}\in C_{b,Lip}(\mathbb{R}^{d\times n})$, then

$\mathbb{\tilde{E}}\big{[}\varphi\big{(}\tilde{\zeta}_{t_{1}},\ldots,\tilde{\zeta}_{t_{n}}-\tilde{\zeta}_{t_{n-1}}\big{)}\big{]}=\mathbb{\tilde{E}}\big{[}\varphi^{\prime}\big{(}\tilde{\zeta}_{t_{1}},\ldots,\tilde{\zeta}_{t_{n}}-\tilde{\zeta}_{t_{n-1}}\big{)}\big{]}.$

Moreover, for each $t_{i},s,t\in[0,1]\ $with $t_{i}\leq s\leq t$,
$\tilde{\zeta}_{t}-\tilde{\zeta}_{s}\overset{d}{=}$ $\tilde{\zeta}_{t-s}$ and
$\tilde{\zeta}_{t}-\tilde{\zeta}_{s}\perp\\!\\!\\!\perp(\tilde{\zeta}_{t_{1}},\ldots,\tilde{\zeta}_{t_{i}})$
under $\mathbb{\tilde{E}}$. Thus,
$(\tilde{\Omega},Lip(\tilde{\Omega}),\mathbb{\tilde{E}})$ is a sublinear
expectation space on which the canonical process
$(\tilde{\zeta}_{t})_{t\in[0,1]}$ is a nonlinear Lévy process.

Finally, note that the distribution of $\tilde{\zeta}_{1}$ is uniquely
determined by $u(1,0)$, where $u$ is the unique viscosity solution of the
fully nonlinear PIDE (4.4) (see Theorem 4.7). We complete the proof by further
claiming that

$\lim_{n\rightarrow\infty}\mathbb{\hat{E}}[\phi(\bar{S}_{n})]=\mathbb{\tilde{E}}[\phi(\tilde{\zeta}_{1})]\text{,
\ for\ }\phi\in C_{b,Lip}(\mathbb{R}^{d}).$

Suppose not. We can find a sequence
$\\{\bar{S}_{\tilde{n}}\\}_{\tilde{n}=1}^{\infty}\subset\\{\bar{S}_{n}\\}_{n=1}^{\infty}$
such that for any subsequence of
$\\{\bar{S}_{\tilde{n}}\\}_{\tilde{n}=1}^{\infty}$ does not converge in
distribution to $\tilde{\zeta}_{1}$. However, using Lemma 4.2 for
$\\{\bar{S}_{\tilde{n}}\\}_{\tilde{n}=1}^{\infty}$, we derive from the above
process that there exists a subsequence
$\\{\bar{S}_{\tilde{n}_{i}}\\}_{i=1}^{\infty}\subset\\{\bar{S}_{\tilde{n}}\\}_{\tilde{n}=1}^{\infty}$
which converges in distribution to $\tilde{\zeta}_{1}$, which induces a
contradiction. The proof is completed.

###  4.3 Lévy-Khintchine representation of nonlinear $\alpha$-stable process

For $\alpha\in(1,2)$, the Lévy-Khintchine representation for a nonlinear
$\alpha$-stable process has been established in (2022). However, the Lévy-
Khintchine representation for a nonlinear $\alpha$-stable process with
$\alpha\in(0,1]$ is still lacking. It turns out such a representation is
crucial for the corresponding robust limit theorem.

Recall that $F_{\mu}$ is the $\alpha$-stable Lévy measure given in (3.1), and
$\mathcal{L}$ is the set of $\alpha$-stable Lévy measures on $\mathbb{R}^{d}$
satisfying (3.2). Denote

$\mathcal{K}_{\alpha}:=\left\\{\begin{array}[c]{ll}\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}|\lambda|^{2}\wedge
1F_{\mu}(d\lambda),&\alpha=1,\\\
\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}|\lambda|\wedge
1F_{\mu}(d\lambda),&\alpha\in(0,1).\end{array}\right.$ (4.1)

It follows from the Sato-type result (see (Sato, 1999, Remark 14.4)) that
$\mathcal{K}_{\alpha}<\infty.$

###### Lemma 4.3

For each $\varphi\in C_{b}^{3}(\mathbb{R}^{d})$, we have for
$x,x^{\prime}\in\mathbb{R}^{d}$,

$\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\left|\delta_{\lambda}^{\alpha}\varphi(x^{\prime})-\delta_{\lambda}^{\alpha}\varphi(x)\right|F_{\mu}(d\lambda)\leq
C_{\alpha}|x^{\prime}-x|^{\delta},$

where

$C_{\alpha}=\left\\{\begin{array}[c]{ll}\mathcal{K}_{\alpha}\big{(}4\left\|D\varphi\right\|_{\infty}^{\delta}\left\|\varphi\right\|_{\infty}^{1-\delta}+2\left\|D^{3}\varphi\right\|_{\infty}^{\delta}\left\|D^{2}\varphi\right\|_{\infty}^{1-\delta}\big{)},&\alpha=1,\\\
\mathcal{K}_{\alpha}\big{(}4\left\|D\varphi\right\|_{\infty}^{\delta}\left\|\varphi\right\|_{\infty}^{1-\delta}+2\left\|D^{2}\varphi\right\|_{\infty}^{\delta}\left\|D\varphi\right\|_{\infty}^{1-\delta}\big{)},&\alpha\in(0,1).\end{array}\right.$

Proof. When $\alpha\in(0,1)$, we have for $x\in\mathbb{R}^{d}$,

$\delta_{\lambda}^{\alpha}\varphi(x)=\varphi\left(x+\lambda\right)-\varphi(x)=\int_{0}^{1}\langle
D\varphi(x+\theta\lambda),\lambda\rangle d\theta.$

Then, it follows that for $x,x^{\prime}\in\mathbb{R}^{d}$,

$\displaystyle\int_{|\lambda|\leq
1}\left|\delta_{\lambda}^{\alpha}\varphi(x^{\prime})-\delta_{\lambda}^{\alpha}\varphi(x)\right|F_{\mu}(d\lambda)$
$\displaystyle=\int_{|\lambda|\leq 1}\left|\int_{0}^{1}\langle
D\varphi(x^{\prime}+\theta\lambda)-D\varphi(x+\theta\lambda),\lambda\rangle
d\theta\right|F_{\mu}(d\lambda)$ $\displaystyle\leq\int_{|\lambda|\leq
1}|\lambda|F_{\mu}(d\lambda)\left\|D^{2}\varphi\right\|_{\infty}\left(|x^{\prime}-x|\wedge\frac{2\left\|D\varphi\right\|_{\infty}}{\left\|D^{2}\varphi\right\|_{\infty}}\right)$
$\displaystyle\leq\int_{|\lambda|\leq
1}|\lambda|F_{\mu}(d\lambda)\left\|D^{2}\varphi\right\|_{\infty}^{\delta}(2\left\|D\varphi\right\|_{\infty})^{1-\delta}|x^{\prime}-x|^{\delta},$

and

$\displaystyle\int_{|\lambda|>1}\left|\delta_{\lambda}^{\alpha}\varphi(x^{\prime})-\delta_{\lambda}^{\alpha}\varphi(x)\right|F_{\mu}(d\lambda)$
$\displaystyle=\int_{|\lambda|>1}\left|\varphi(x^{\prime}+\lambda)-\varphi(x+\lambda)-(\varphi(x^{\prime})-\varphi(x))\right|F_{\mu}(d\lambda)$
$\displaystyle\leq
2\int_{|\lambda|>1}F_{\mu}(d\lambda)\left\|D\varphi\right\|_{\infty}\left(|x^{\prime}-x|\wedge\frac{2\left\|\varphi\right\|_{\infty}}{\left\|D\varphi\right\|_{\infty}}\right)$
$\displaystyle\leq
2\int_{|\lambda|>1}F_{\mu}(d\lambda)\left\|D\varphi\right\|_{\infty}^{\delta}(2\left\|\varphi\right\|_{\infty})^{1-\delta}|x^{\prime}-x|^{\delta}.$

When $\alpha=1$, we have for $x\in\mathbb{R}^{d}$,

$\delta_{\lambda}^{1}\varphi(x)=\int_{0}^{1}\int_{0}^{1}\langle
D^{2}\varphi(x+\tau\theta\lambda)\lambda,\lambda\rangle\theta d\tau
d\theta\text{, \ for }|\lambda|\leq 1,$

and $\delta_{\lambda}^{1}\varphi(x)=\varphi\left(x+\lambda\right)-\varphi(x)$,
for $|\lambda|>1$. Similarly, we can deduce that

$\displaystyle\int_{|\lambda|\leq
1}\left|\delta_{\lambda}^{1}\varphi(x^{\prime})-\delta_{\lambda}^{1}\varphi(x)\right|F_{\mu}(d\lambda)$
$\displaystyle=\int_{|\lambda|\leq
1}\left|\int_{0}^{1}\int_{0}^{1}\langle(D^{2}\varphi(x^{\prime}+\tau\theta\lambda)-D^{2}\varphi(x+\tau\theta\lambda))\lambda,\lambda\rangle\theta
d\tau d\theta\right|F_{\mu}(d\lambda)$ $\displaystyle\leq\int_{|\lambda|\leq
1}|\lambda|^{2}F_{\mu}(d\lambda)\left\|D^{3}\varphi\right\|_{\infty}^{\delta}(2\left\|D^{2}\varphi\right\|_{\infty})^{1-\delta}|x^{\prime}-x|^{\delta},$

and

$\int_{|\lambda|>1}\left|\delta_{\lambda}^{1}\varphi(x^{\prime})-\delta_{\lambda}^{1}\varphi(x)\right|F_{\mu}(d\lambda)\leq
2\int_{|\lambda|>1}F_{\mu}(d\lambda)\left\|D\varphi\right\|_{\infty}^{\delta}(2\left\|\varphi\right\|_{\infty})^{1-\delta}|x^{\prime}-x|^{\delta}.$

The proof is completed.

The following estimate is crucial to our main result.

###### Theorem 4.4

Assume that (A1)-(A2) hold. Then, for $\varphi\in C_{b}^{3}(\mathbb{R}^{d})$
and $s\in[0,1]$,

$\lim_{n\rightarrow\infty}\bigg{|}\mathbb{\hat{E}}\bigg{[}\varphi\left(x+(s/n)^{\frac{1}{\alpha}}S_{n}\right)\bigg{]}-\varphi(x)-s\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(x)F_{\mu}(d\lambda)\bigg{|}=o(s),$

uniformly on $x\in\mathbb{R}^{d}$, where $o(s)/s\rightarrow 0$ as
$s\rightarrow 0$.

Proof. Because $Z_{n}$ is independent from $Z_{1},\dots,Z_{n-1}$, we have

$\displaystyle\mathbb{\hat{E}}\bigg{[}\varphi\left(x+(s/n)^{\frac{1}{\alpha}}S_{n}\right)\bigg{]}-\varphi(x)-s\epsilon(x)$
$\displaystyle=\mathbb{\hat{E}}\bigg{[}\left.\mathbb{\hat{E}}\bigg{[}\varphi\left(x+(s/n)^{\frac{1}{\alpha}}(\omega_{n-1}+Z_{n})\right)\bigg{]}\right|_{\begin{subarray}{c}z_{1}=Z_{1}\\\
\cdots\\\ z_{n-1}=Z_{n-1}\end{subarray}}\bigg{]}-s\epsilon(x)-\varphi(x),$

where

$\omega_{n-1}:=\sum\limits_{k=1}^{n-1}z_{k}\ \ \ \text{and}\ \ \
\epsilon(x):=\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(x)F_{\mu}(d\lambda).$

Thanks to the assumptions (A1)-(A2) and Lemma 4.3, we deduce that

$\displaystyle\mathbb{\hat{E}}\bigg{[}\varphi\left(x+(s/n)^{\frac{1}{\alpha}}(\omega_{n-1}+Z_{n})\right)\bigg{]}$
$\displaystyle=\mathbb{\hat{E}}\bigg{[}\varphi\left(x+(s/n)^{\frac{1}{\alpha}}(\omega_{n-1}+Z_{n})\right)-\varphi\left(x+(s/n)^{\frac{1}{\alpha}}\omega_{n-1}\right)$
$\displaystyle\text{ \ \
}-\frac{s}{n}\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi\left(x+(s/n)^{\frac{1}{\alpha}}\omega_{n-1}\right)F_{\mu}(d\lambda)\bigg{]}$
$\displaystyle\text{ \ \
}+\frac{s}{n}\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi\left(x+(s/n)^{\frac{1}{\alpha}}\omega_{n-1}\right)F_{\mu}(d\lambda)$
$\displaystyle\text{ \ \
}-\frac{s}{n}\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi\left(x\right)F_{\mu}(d\lambda)+\varphi\left(x+(s/n)^{\frac{1}{\alpha}}\omega_{n-1}\right)+\frac{s}{n}\epsilon(x)$
$\displaystyle\leq\frac{s}{n}l\left(\frac{s}{n}\right)+C\left(\frac{s}{n}\right)^{1+\frac{\delta}{\alpha}}\left|\omega_{n-1}\right|^{\delta}+\varphi\left(x+(s/n)^{\frac{1}{\alpha}}\omega_{n-1}\right)+\frac{s}{n}\epsilon(x),$

which implies the following one-step estimate

$\displaystyle\mathbb{\hat{E}}\left[\varphi\left(x+(s/n)^{\frac{1}{\alpha}}S_{n}\right)\right]$
$\displaystyle\leq\mathbb{\hat{E}}\left[\varphi\left(x+(s/n)^{\frac{1}{\alpha}}S_{n-1}\right)\right]+\frac{s}{n}\epsilon(x)+\frac{s}{n}l\left(\frac{s}{n}\right)+CM_{\delta}s^{1+\frac{\delta}{\alpha}}\frac{1}{n}\left(\frac{n-1}{n}\right)^{\frac{\delta}{\alpha}}.$

Repeating the above process recursively, we obtain that

$\mathbb{\hat{E}}\left[\varphi\left(x+(s/n)^{\frac{1}{\alpha}}S_{n}\right)\right]\leq\varphi\left(x\right)+s\epsilon(x)+sl\left(\frac{s}{n}\right)+CM_{\delta}s^{1+\frac{\delta}{\alpha}}\frac{1}{n}\sum\limits_{k=1}^{n-1}\left(\frac{k}{n}\right)^{\frac{\delta}{\alpha}}.$

Analogously, we have

$\mathbb{\hat{E}}\left[\varphi\left(x+(s/n)^{\frac{1}{\alpha}}S_{n}^{3}\right)\right]\geq\varphi\left(x\right)+s\epsilon(x)-sl\left(\frac{s}{n}\right)-CM_{\delta}s^{1+\frac{\delta}{\alpha}}\frac{1}{n}\sum\limits_{k=1}^{n-1}\left(\frac{k}{n}\right)^{\frac{\delta}{\alpha}}.$

Thus,

$\lim_{n\rightarrow\infty}\left|\mathbb{\hat{E}}\left[\varphi\left(x+(s/n)^{\frac{1}{\alpha}}S_{n}\right)\right]-\varphi(x)-s\epsilon(x)\right|\leq
CM_{\delta}s^{1+\frac{\delta}{\alpha}}\frac{\alpha}{\delta+\alpha},$

where we have used the fact that

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{k=1}^{n-1}\left(\frac{k}{n}\right)^{\frac{\delta}{\alpha}}=\frac{\alpha}{\delta+\alpha}.$

This implies the desired result.

Denote

$\mathfrak{F}=\left\\{\varphi\in
C_{b}^{3}(\mathbb{R}^{d}):\varphi(0)=0\right\\}$

In the following, we shall present the characterization of
$\lim\limits_{s\rightarrow
0}\mathbb{\tilde{E}}[\varphi(\tilde{\zeta}_{s})]s^{-1}$ for
$\varphi\in\mathfrak{F}$, which can be regarded as a new type of Lévy-
Khintchine representation for the nonlinear pure jump Lévy process
$(\tilde{\zeta}_{t})_{t\in[0,1]}$. It will play an important role in
establishing the related PIDE in Section 4.4.

###### Theorem 4.5

Assume that (A1)-(A2) hold. Then, for each $\varphi\in\mathfrak{F}$,

$\lim\limits_{s\rightarrow
0}\mathbb{\tilde{E}}[\varphi(\tilde{\zeta}_{s})]s^{-1}=\sup_{F_{\mu}\in\mathcal{L}}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(0)F_{\mu}(d\lambda)\right\\}.$

Proof. For each $\varphi\in\mathfrak{F}$ and $s\in[0,1]$, Theorem 4.1 shows
that there exists a sequence
$\\{s_{k}\\}_{k=1}^{\infty}\subset\mathcal{D}_{\infty}[0,1]$ satisfying
$s_{k}\downarrow s$ as $k\rightarrow\infty$ and a convergent sequence
$\big{\\{}(s_{k}/n_{i}^{\ast})^{\frac{1}{\alpha}}S_{n_{i}^{\ast}}\big{\\}}_{i=1}^{\infty}$
for each $s_{k}$, such that

$\mathbb{\tilde{E}}[\varphi(\tilde{\zeta}_{s})]=\lim_{k\rightarrow\infty}\lim_{i\rightarrow\infty}\mathbb{\hat{E}}\big{[}\varphi((s_{k}/n_{i}^{\ast})^{\frac{1}{\alpha}}S_{n_{i}^{\ast}})\big{]}.$

Under the assumption (A2), it follows from Theorem 4.4 that

$\displaystyle\bigg{|}\mathbb{\tilde{E}}[\varphi(\tilde{\zeta}_{s})]-\varphi(0)-s\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(0)F_{\mu}(d\lambda)\bigg{|}$
(4.2)
$\displaystyle\leq\lim_{k\rightarrow\infty}\bigg{|}\lim_{i\rightarrow\infty}\mathbb{\hat{E}}\big{[}\varphi((s_{k}/n_{i}^{\ast})^{\frac{1}{\alpha}}S_{n_{i}^{\ast}})\big{]}-\varphi(0)-s_{k}\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(0)F_{\mu}(d\lambda)\bigg{|}=o(s),$

which implies that

$\displaystyle\lim\limits_{s\rightarrow
0}\mathbb{\tilde{E}}[\varphi(\tilde{\zeta}_{s})]s^{-1}$
$\displaystyle=\lim_{s\rightarrow
0}\bigg{(}\mathbb{\tilde{E}}[\varphi(\tilde{\zeta}_{s})]-\varphi(0)-s\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(0)F_{\mu}(d\lambda)\bigg{)}s^{-1}+\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(0)F_{\mu}(d\lambda)$
$\displaystyle\leq\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(0)F_{\mu}(d\lambda),$

and similarly,

$\lim\limits_{s\rightarrow
0}\mathbb{\tilde{E}}[\varphi(\tilde{\zeta}_{s})]s^{-1}\geq\sup\limits_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\varphi(0)F_{\mu}(d\lambda).$

Therefore, by the squeeze theorem, we complete the proof.

###  4.4 Connection to PIDE

In this section, we relate the nonlinear pure jump Lévy process
$(\tilde{\zeta}_{t})_{t\in[0,1]}$ to the fully nonlinear PIDE (3.7). Let
$C_{b}^{2,3}([0,1]\times\mathbb{R}^{d})$ denote the set of functions on
$[0,1]\times\mathbb{R}^{d}$ having bounded continuous partial derivatives up
to the second order in $t$ and third order in $x$, respectively. Now we give
the definition of viscosity solution for PIDE (3.7).

###### Definition 4.6

A bounded upper semicontinuous (resp. lower semicontinuous) function $u$ on
$[0,1]\times\mathbb{R}^{d}$ is called a viscosity subsolution (resp. viscosity
supersolution) of (3.7) if $u(0,\cdot)\leq\phi(\cdot)$ $($resp.
$\geq\phi(\cdot))$ and for each $(t,x)\in(0,1]\times\mathbb{R}^{d}$,

$\partial_{t}\psi(t,x)-\sup\limits_{F_{\mu}\in\mathcal{L}}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\psi(t,x)F_{\mu}(d\lambda)\right\\}\leq
0\text{ }(\text{resp. }\geq 0)$

whenever $\psi\in C_{b}^{2,3}((0,1]\times\mathbb{R}^{d})$ is such that
$\psi\geq u$ (resp. $\psi\leq u$) and $\psi(t,x)=u(t,x)$. A bounded continuous
function $u$ is a viscosity solution of (3.7) if it is both a viscosity
subsolution and supersolution.

For each $\phi\in C_{b,Lip}(\mathbb{R}^{d})$, define

$u(t,x)=\mathbb{\tilde{E}}[\phi(x+\tilde{\zeta}_{t})],\text{
}(t,x)\in[0,1]\times\mathbb{R}^{d}.$ (4.3)

###### Theorem 4.7

Suppose that the assumptions (A1)-(A2) hold. Then, the value function $u$ of
(4.3) is the unique viscosity solution of the fully nonlinear PIDE (3.7),
i.e.,

$\left\\{\begin{array}[c]{l}\displaystyle\partial_{t}u(t,x)-\sup\limits_{F_{\mu}\in\mathcal{L}}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}u(t,x)F_{\mu}(d\lambda)\right\\}=0,\\\
\displaystyle u(0,x)=\phi(x),\text{\ \ \
}\forall(t,x)\in[0,1]\times\mathbb{R}^{d},\end{array}\right.$ (4.4)

where

$\delta_{\lambda}^{\alpha}u(t,x)=\left\\{\begin{array}[c]{ll}u(t,x+\lambda)-u(t,x)-\langle
D_{x}u(t,x),\lambda\mathbbm{1}_{\\{|\lambda|\leq 1\\}}\rangle,&\alpha=1,\\\
u(t,x+\lambda)-u(t,x),&\alpha\in(0,1).\end{array}\right.$

Proof. We first show that $u$ is continuous. It is clear that $u(t,\cdot)$ is
uniformly Lipschitz continuous with the same Lipschitz constant as for $\phi$.
For each $t,s\in[0,1]$ such that $t+s\leq 1$, we obtain

$u(t+s,x)=\mathbb{\tilde{E}}[u(t,x+\tilde{\zeta}_{s})],\text{ \
}x\in\mathbb{R}^{d}.$ (4.5)

Theorem 4.1 shows that there exists a sequence
$\\{s_{k}\\}_{k=1}^{\infty}\subset\mathcal{D}_{\infty}[0,1]$ satisfying
$s_{k}\downarrow s$ as $k\rightarrow\infty$ and a convergent sequence
$\big{\\{}(s_{k}/n_{i}^{\ast})^{\frac{1}{\alpha}}S_{n_{i}^{\ast}}\big{\\}}_{i=1}^{\infty}$
for each $s_{k}$, such that

$\displaystyle\left|\mathbb{\tilde{E}}\big{[}u(t,x+\tilde{\zeta}_{s})\big{]}-u(t,x)\right|$
$\displaystyle\leq L_{\phi}\mathbb{\tilde{E}}\big{[}|\tilde{\zeta}_{s}|\wedge
N_{\phi}\big{]}$
$\displaystyle=L_{\phi}\lim\limits_{k\rightarrow\infty}\lim\limits_{i\rightarrow\infty}\mathbb{\hat{E}}\big{[}|(s_{k}/n_{i}^{\ast})^{\frac{1}{\alpha}}S_{n_{i}^{\ast}}|\wedge
N_{\phi}\big{]}$ (4.6) $\displaystyle\leq
L_{\phi}s^{\frac{\delta}{\alpha}}\sup_{i}\mathbb{\hat{E}}\big{[}|(1/n_{i}^{\ast})^{\frac{1}{\alpha}}S_{n_{i}^{\ast}}|^{\delta}\big{]}N_{\phi}^{1-\delta},$

where $L_{\phi}>0$ is the Lipschitz constant of $\phi$ and
$N_{\phi}:=\frac{2\left\|\phi\right\|_{\infty}}{L_{\phi}}$. This implies the
continuity of $u(\cdot,x)$

$|u(t+s,x)-u(t,x)|\leq
L_{\phi}M_{\delta}N_{\phi}^{1-\delta}s^{\frac{\delta}{\alpha}}.$

Next, we will prove that $u$ is the unique viscosity solution of (4.4). The
uniqueness of viscosity solution can be found in Corollary 55 in (2021). It
suffices to prove that $u$ is a viscosity subsolution, and the other case can
be proved in a similar way. Assume that $\psi$ is a smooth test function on
$(0,1]\times\mathbb{R}^{d}$ satisfying $\psi\geq u$ and
$\psi(\bar{t},\bar{x})=u(\bar{t},\bar{x})$ for some point
$(\bar{t},\bar{x})\in(0,1]\times\mathbb{R}^{d}$. For each $s\in(0,\bar{t})$,
the dynamic programming principle (4.5) shows that

$0=\mathbb{\tilde{E}}[u(\bar{t}-s,\bar{x}+\tilde{\zeta}_{s})-u(\bar{t},\bar{x})]\leq\mathbb{\tilde{E}}[\psi(\bar{t}-s,\bar{x}+\tilde{\zeta}_{s})-\psi(\bar{t},\bar{x})].$
(4.7)

Using Taylor’s expansion, we obtain that

$\displaystyle\psi(\bar{t}-s,\bar{x}+\tilde{\zeta}_{s})-\psi(\bar{t},\bar{x})$
$\displaystyle=\psi(\bar{t}-s,\bar{x}+\tilde{\zeta}_{s})-\psi(\bar{t},\bar{x}+\tilde{\zeta}_{s})+\psi(\bar{t},\bar{x}+\tilde{\zeta}_{s})-\psi(\bar{t},\bar{x})$
(4.8)
$\displaystyle=-\partial_{t}\psi(\bar{t},\bar{x})s+\psi(\bar{t},\bar{x}+\tilde{\zeta}_{s})-\psi(\bar{t},\bar{x})+\epsilon.$

where

$\epsilon=s\int_{0}^{1}[-\partial_{t}\psi(\bar{t}-\theta
s,\bar{x}+\tilde{\zeta}_{s})+\partial_{t}\psi(\bar{t},\bar{x})]d\theta.$

Since $\psi$ is the smooth function, similar to (4.6), one easily gets
$\mathbb{\tilde{E}}[|\epsilon|]\leq C_{\psi}s^{1+\frac{\delta}{\alpha}}$.
Then, it follows from (4.7)-(4.8) that

$0\leq\lim_{s\rightarrow
0}\mathbb{\tilde{E}}[\psi(\bar{t}-s,\bar{x}+\tilde{\zeta}_{s})-\psi(\bar{t},\bar{x})]s^{-1}=-\partial_{t}\psi(\bar{t},\bar{x})+\lim_{s\rightarrow
0}\mathbb{\tilde{E}}[\psi(\bar{t},\bar{x}+\tilde{\zeta}_{s})-\psi(\bar{t},\bar{x})]s^{-1}.$
(4.9)

Moreover, Theorem 4.5 implies that

$\lim_{s\rightarrow
0}\mathbb{\tilde{E}}\big{[}\psi(\bar{t},\bar{x}+\tilde{\zeta}_{s})-\psi(\bar{t},\bar{x})\big{]}s^{-1}=\sup\limits_{F_{\mu}\in\mathcal{L}}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\psi(\bar{t},\bar{x})F_{\mu}(d\lambda)\right\\}.$
(4.10)

Combining (4.9) with (4.10), we conclude that

$\partial_{t}\psi(\bar{t},\bar{x})-\sup\limits_{F_{\mu}\in\mathcal{L}}\left\\{\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}\psi(\bar{t},\bar{x})F_{\mu}(d\lambda)\right\\}\leq
0.$

The proof is completed.

###  4.5 The scaling and symmetric properties

We start with the scaling property. In view of Theorem 4.1, for each $0\leq
t\leq 1$, there exists a sequence
$\\{t_{k}\\}_{k=1}^{\infty}\subset\mathcal{D}_{\infty}[0,1]$ satisfying
$t_{k}\downarrow t$ as $k\rightarrow\infty$, such that for $\phi\in
C_{b,Lip}(\mathbb{R}^{d})$,
$\mathbb{\tilde{E}}\big{[}\phi(\tilde{\zeta}_{t_{k}})\big{]}\rightarrow\mathbb{\tilde{E}}\big{[}\phi(\tilde{\zeta}_{t})\big{]}$
as $k\rightarrow\infty$. For each fixed $t_{k}\in\mathcal{D}_{\infty}[0,1]$,
we assume that $t_{k}=l_{k}\tau_{m_{k}}=l_{k}2^{-m_{k}}$ for some
$m_{k}\in\mathbb{N}$ and $0\leq l_{k}\leq 2^{m_{k}}$. From the construction of
$\tilde{\zeta}_{t_{k}}$ and Remark 2.4, there exists a convergent sequence
$\\{\bar{S}_{n_{i}^{\ast}}^{\tau_{m_{k}}}\\}_{i=1}^{\infty}$ such that

$\mathbb{\tilde{E}}\big{[}\phi\big{(}\tilde{\zeta}_{t_{k}}\big{)}\big{]}=\mathbb{\tilde{E}}^{m_{k}}\big{[}\phi\big{(}\tilde{\zeta}_{l_{k}\tau_{m_{k}}}\big{)}\big{]}=\mathbb{\hat{E}}_{1}\big{[}\phi\big{(}\zeta_{\tau_{m_{k}}}^{1}+\cdots+\zeta_{\tau_{m_{k}}}^{l_{k}}\big{)}\big{]}=\lim_{i\rightarrow\infty}\mathbb{\hat{E}}\big{[}\phi\big{(}\bar{S}_{l_{k}n_{i}^{\ast}}^{t_{k}}\big{)}\big{]},$

for $\phi\in C_{b,Lip}(\mathbb{R}^{d})$, where

$\bar{S}_{l_{k}n_{i}^{\ast}}^{t_{k}}=(\tau_{m_{k}}/n_{i}^{\ast})^{1/\alpha}(S_{l_{k}n_{i}^{\ast}}-S_{(l_{k}-1)n_{i}^{\ast}})+\cdots+(\tau_{m_{k}}/n_{i}^{\ast})^{1/\alpha}(S_{2n_{i}^{\ast}}-S_{n_{i}^{\ast}})+(\tau_{m_{k}}/n_{i}^{\ast})^{1/\alpha}S_{n_{i}^{\ast}}.$

In addition, from Theorem 4.1, we obtain that for
$\phi(\sqrt[\alpha]{t_{k}}\cdot)\in C_{b,Lip}(\mathbb{R}^{d})$,

$\lim_{i\rightarrow\infty}\mathbb{\hat{E}}\big{[}\phi\big{(}\bar{S}_{l_{k}n_{i}^{\ast}}^{t_{k}}\big{)}\big{]}=\lim_{i\rightarrow\infty}\mathbb{\hat{E}}\big{[}\phi\big{(}\sqrt[\alpha]{t_{k}}\bar{S}_{l_{k}n_{i}^{\ast}}\big{)}\big{]}=\mathbb{\tilde{E}}\big{[}\phi\big{(}\sqrt[\alpha]{t_{k}}\tilde{\zeta}_{1}\big{)}\big{]}.$

This implies that

$\mathbb{\tilde{E}}\big{[}\phi\big{(}\tilde{\zeta}_{t}\big{)}\big{]}=\mathbb{\tilde{E}}\big{[}\phi\big{(}\sqrt[\alpha]{t}\tilde{\zeta}_{1}\big{)}\big{]},\text{
for }\phi\in C_{b,Lip}(\mathbb{R}^{d}).$

Next, we shall verify that $\tilde{\zeta}_{\cdot}$ is symmetric. For any given
$\phi\in C_{b,Lip}(\mathbb{R}^{d})$, Theorem 4.7 implies that
$u(t,0)=\mathbb{\tilde{E}}[\phi(\tilde{\zeta}_{t})]$, where $u$ is the unique
viscosity solution of the PIDE (3.7) with initial condition $\phi$. Note that

$F_{\mu}(B)=F_{\mu}(-B)\text{, \ for }B\in\mathcal{B}(\mathbb{R}^{d}).$

For any $0\leq t\leq 1$, define $v(t,x):=u(t,-x)$. It follows from

$\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}u(t,-x)F_{\mu}(d\lambda)=\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}v(t,x)F_{\mu}(d\lambda),$

that $v$ is the unique viscosity solution of the PIDE (3.7) with initial
condition $\psi(x):=\phi(-x)$. From Theorem 4.7, we derive that
$v(t,0)=\mathbb{\tilde{E}}[\psi(\tilde{\zeta}_{t})]$. Therefore, for any
$\phi\in C_{b,Lip}(\mathbb{R}^{d})$,

$\mathbb{\tilde{E}}[\phi(\tilde{\zeta}_{t})]=u(t,0)=v(t,0)=\mathbb{\tilde{E}}[\psi(\tilde{\zeta}_{t})]=\mathbb{\tilde{E}}[\phi(-\tilde{\zeta}_{t})],$

and the proof is complete.

##  Appendix A Appendix

###  A.1 Tightness under sublinear expectation

The weak convergence approach plays an imperative role in establishing robust
limit theorems. We recall its main results for the reader’s convenience.
Further details can be found in section 2 of (2022).

###### Definition A.1

A sublinear expectation $\mathbb{\hat{E}}$ on
$(\mathbb{R}^{n},C_{b,Lip}(\mathbb{R}^{n}))$ is said to be tight if for each
$\varepsilon>0$, there exist an $N>0$ and $\varphi\in$
$C_{b,Lip}(\mathbb{R}^{n})$ with $\mathbbm{1}_{\\{|x|\geq N\\}}\leq\varphi$
such that $\mathbb{\hat{E}}[\varphi]<\varepsilon$.

###### Definition A.2

A family of sublinear expectations
$\\{\mathbb{\hat{E}}_{\alpha}\\}_{\alpha\in\mathcal{A}}$ on
$(\mathbb{R}^{n},C_{b,Lip}(\mathbb{R}^{n}))$ is said to be tight if there
exists a tight sublinear expectation $\mathbb{\hat{E}}$ on
$(\mathbb{R}^{n},C_{b,Lip}(\mathbb{R}^{n}))$ such that

$\mathbb{\hat{E}}_{\alpha}[\varphi]-\mathbb{\hat{E}}_{\alpha}[\varphi^{\prime}]\leq\mathbb{\hat{E}}[\varphi-\varphi^{\prime}],\text{
for each }\varphi,\varphi^{\prime}\in C_{b,Lip}(\mathbb{R}^{n}).$

###### Definition A.3

Let $\\{\mathbb{\hat{E}}_{n}\\}_{n=1}^{\infty}$ be a sequence of sublinear
expectations defined on $(\mathbb{R}^{n},C_{b,Lip}(\mathbb{R}^{n}))$. They are
said to be weakly convergent if, for each $\varphi\in
C_{b,Lip}(\mathbb{R}^{n})$,
$\\{\mathbb{\hat{E}}_{n}[\varphi]\\}_{n=1}^{\infty}$ is a Cauchy sequence. A
family of sublinear expectations
$\\{\mathbb{\hat{E}}_{\alpha}\\}_{\alpha\in\mathcal{A}}$ defined on
$(\mathbb{R}^{n},C_{b,Lip}(\mathbb{R}^{n}))$ is said to be weakly compact if
for each sequence $\\{\mathbb{\hat{E}}_{\alpha_{i}}\\}_{i=1}^{\infty}$ there
exists a weakly convergent subsequence.

The following result is a generalization of the celebrated Prokhorov’s theorem
to the sublinear expectation case.

###### Theorem A.4

Let $\\{\mathbb{\hat{E}}_{\alpha}\\}_{\alpha\in\mathcal{A}}$ be a family of
tight sublinear expectations on $(\mathbb{R}^{n},C_{b,Lip}(\mathbb{R}^{n}))$.
Then $\\{\mathbb{\hat{E}}_{\alpha}\\}_{\alpha\in\mathcal{A}}$ is weakly
compact, namely, for each sequence
$\\{\mathbb{\hat{E}}_{\alpha_{n}}\\}_{n=1}^{\infty}$, there exists a
subsequence $\\{\mathbb{\hat{E}}_{\alpha_{n_{i}}}\\}_{i=1}^{\infty}$ such
that, for each $\varphi\in C_{b,Lip}(\mathbb{R}^{n})$,
$\\{\mathbb{\hat{E}}_{\alpha_{n_{i}}}[\varphi]\\}_{i=1}^{\infty}$ is a Cauchy
sequence.

An immediate corollary of Theorem A.4 and Definition 2.2 is the following
result.

###### Corollary A.5

Let $\\{X_{i}\\}_{i=1}^{\infty}$ be a sequence of $n$-dimensional random
variables on a sublinear expectation space
$(\Omega,\mathcal{H},\mathbb{\hat{E})}$. If
$\\{\mathbb{F}_{X_{i}}\\}_{i=1}^{\infty}$ is tight, then there exists a
subsequence $\\{X_{i_{j}}\\}_{j=1}^{\infty}\subset\\{X_{i}\\}_{i=1}^{\infty}$
which converges in distribution.

###  A.2 The regularity estimate for $u_{n}$

In this appendix, we shall state the regularity properties of $u_{n}$ defined
in (3.8). For any fixed $N>0$, define
$Z_{1}^{N}:=Z_{1}\mathbf{1}_{\\{|Z_{1}|\leq N\\}}$. It is easy to check that
$\mathbb{\hat{E}}[Z_{1}^{N}]=\mathbb{\hat{E}}[-Z_{1}^{N}]=0$. We introduce the
following truncated scheme $u_{n,N}:[0,1]\times\mathbb{R\rightarrow R}$
recursively by

$\begin{array}[c]{l}u_{n,N}(t,z)=|z|^{\delta},\text{ \ if\ }t\in[0,1/n),\\\
u_{n,N}(t,z)=\mathbb{\hat{E}}[u_{n,N}(t-1/n,z+n^{-1/\alpha}Z_{1}^{N})],\text{
\ if\ }t\in[1/n,1].\end{array}$ (A.1)

It is easy to obtain the following estimates.

###### Lemma A.6

For any given $N>0$, we have

$\begin{array}[c]{ll}\displaystyle\mathbb{\hat{E}}[|Z_{1}^{N}|^{2}]=N^{2-\alpha}I_{1,N},\text{
\
}&\displaystyle\mathbb{\hat{E}}[|Z_{1}-Z_{1}^{N}|^{\delta}]=N^{\delta-\alpha}I_{2,N},\end{array}$

where

$\displaystyle I_{1,N}$ $\displaystyle:=2\sup_{k\in
K}\left\\{\frac{k}{2-\alpha}+2\int_{0}^{1}\frac{\beta_{k}(zN)}{z^{\alpha-1}}dz-\beta_{k}(N)\right\\},$
$\displaystyle I_{2,N}$ $\displaystyle:=2\sup_{k\in
K}\left\\{\frac{k}{\alpha-\delta}+\delta\int_{1}^{\infty}\frac{\beta_{k}(zN)}{z^{1+\alpha-\delta}}dz+\beta_{k}(N)\right\\}.$

###### Lemma A.7

Given $N>0$. Then for any positive integer $k\leq n$ and $x\in\mathbb{R}$,

$\left|u_{n,N}(k/n,x)-u_{n,N}(0,x)\right|\leq(I_{1,N})^{\frac{\delta}{2}}N^{\frac{(2-\alpha)\delta}{2}}n^{\frac{(\alpha-2)\delta}{2\alpha}}(k/n)^{\frac{\delta}{2}},$

where $I_{1,N}$ is given in Lemma A.6.

Proof. Applying induction to (A.1), one can get that for any positive integer
$k\leq n$ and $x,y\in\mathbb{R}$

$\left|u_{n,N}(k/n,x)-u_{n,N}(k/n,y)\right|\leq|x-y|^{\delta}.$

From Young’s inequality we know that for any $a,b>0$,
$ab\leq\frac{\delta}{2}a^{\frac{2}{\delta}}+(1-\frac{\delta}{2})b^{\frac{2}{2-\delta}}$.
For any $\varepsilon>0$, let $x=|x-y|^{\delta}$ and $y=\frac{1}{\varepsilon}$,
then it follows from (i) that

$u_{n,N}(k/n,x)\leq u_{n,N}(k/n,y)+A|x-y|^{2}+B,$

where $A=\frac{\delta}{2}\varepsilon$ and
$B=(1-\frac{\delta}{2})\varepsilon^{\frac{-\delta}{2-\delta}}$.

We claim that, for any positive integer $k\leq n$ and $x,y\in\mathbb{R}$, it
holds that

$u_{n,N}(k/n,x)\leq
u_{n,N}(0,y)+A|x-y|^{2}+AM_{N}^{2}kn^{-\frac{2}{\alpha}}+B,$ (A.2)

where $M_{N}^{2}:=\mathbb{\hat{E}}[|Z_{1}^{N}|^{2}]$. Indeed, (A.2) obviously
holds for $k=0$. Assume that for some integer $1\leq k<n$ the assertion (A.2)
holds. Notice that

$\displaystyle u_{n,N}((k+1)/n,x)$
$\displaystyle=\mathbb{\hat{E}}[u_{n,N}(k/n,x+n^{-\frac{1}{\alpha}}Z_{1}^{N})]$
(A.3) $\displaystyle\leq
u_{n,N}(0,y)+A\mathbb{\hat{E}}[|x-y+n^{-\frac{1}{\alpha}}Z_{1}^{N}|^{2}]+AM_{N}^{2}kn^{-\frac{2}{\alpha}}+B.$

Seeing that, $\mathbb{\hat{E}}[Z_{1}^{N}]=\mathbb{\hat{E}}[-Z_{1}^{N}]=0$, we
have

$\mathbb{\hat{E}}[|x-y+n^{-\frac{1}{\alpha}}Z_{1}^{N}|^{2}]=|x-y|^{2}+M_{N}^{2}n^{-\frac{2}{\alpha}}.$
(A.4)

Combining (A.3)-(A.4), we obtain that

$u_{n,N}((k+1)/n,x)\leq
u_{n,N}(0,y)+A|x-y|^{2}+AM_{N}^{2}(k+1)n^{-\frac{2}{\alpha}}+B,$

which shows that (A.2) also holds for $k+1$. By the principle of induction our
claim is true for all positive integer $k\leq n$ and $x,y\in\mathbb{R}$. By
taking $y=x$ in (A.2), we have for any $\varepsilon>0$,

$u_{n,N}(k/n,x)\leq
u_{n,N}(0,x)+\frac{\delta}{2}M_{N}^{2}kn^{-\frac{2}{\alpha}}\varepsilon+(1-\frac{\delta}{2})\varepsilon^{\frac{-\delta}{2-\delta}}.$

By minimizing of the right-hand side with respect to $\varepsilon$, we obtain
that

$u_{n,N}(k/n,x)\leq
u_{n,N}(0,x)+(M_{N}^{2})^{\frac{\delta}{2}}n^{\frac{\delta(\alpha-2)}{2\alpha}}(k/n)^{\frac{\delta}{2}}.$

Similarly, we can also obtain the other side. From Lemma A.6, we obtain our
desired result.

Analogous to Lemma 4.5 in (2021), we have the following error estimate of
$u_{n,N}$ for $u_{n}$.

###### Lemma A.8

Given $N>0$. Then for any positive integer $k\leq n$ and $x\in\mathbb{R}$,

$\left|u_{n}(k/n,x)-u_{n,N}(k/n,x)\right|\leq
I_{2,N}N^{\delta-\alpha}n^{\frac{\alpha-\delta}{\alpha}}(k/n),$

where $I_{2,N}$ is given in Lemma A.6.

Finally, we are in a position to obtain the regularity estimate of $u_{n}$.

###### Theorem A.9

For any $t,s\in[0,1]$ and $x\in\mathbb{R}$, we have

$\left|u_{n}(t,x)-u_{n}(s,x)\right|\leq
I_{n}(|t-s|^{\delta/2}+n^{-\delta/2}),$

where $I_{n}:=(I_{1,n^{1/\alpha}})^{\frac{\delta}{2}}+I_{2,n^{1/\alpha}}$ with
$I_{n}<\infty$ as $n\rightarrow\infty$.

Proof. Noting that $u_{n,N}(0,x)=u_{n}(0,x)=\phi(x)$, we have for any positive
integer $k\leq n$

$\left|u_{n}(k/n,x)-u_{n}(0,x)\right|\leq\left|u_{n}(k/n,x)-u_{n,N}(k/n,x)\right|+\left|u_{n,N}(k/n,x)-u_{n,N}(0,x)\right|.$

In view of Lemmas A.7 and A.8, by choosing $N=n^{\frac{1}{\alpha}}$, we obtain

$\begin{split}\left|u_{n}(k/n,x)-u_{n}(0,x)\right|&\leq(I_{1,N})^{\frac{\delta}{2}}N^{\frac{(2-\alpha)\delta}{2}}n^{\frac{(\alpha-2)\delta}{2\alpha}}(k/n)^{\frac{\delta}{2}}+I_{2,N}N^{\delta-\alpha}n^{\frac{\alpha-\delta}{\alpha}}(k/n)\\\
&\leq\big{(}(I_{1,n^{1/\alpha}})^{\frac{\delta}{2}}+I_{2,n^{1/\alpha}}\big{)}(k/n)^{\frac{\delta}{2}}.\end{split}$
(A.5)

Moreover, it is easy to verify that $I_{1,n^{1/\alpha}}$ and
$I_{2,n^{1/\alpha}}$ are finite as $n\rightarrow\infty$. Without loss of
generality, we assume $k\geq l$. By using induction (3.8) and the estimate
(A.5), we obtain that for any $k\geq l$ and $x\in\mathbb{R}$,

$\begin{split}&\left|u_{n}(k/n,x)-u_{n}(l/n,x)\right|\\\
&=\big{|}\mathbb{\hat{E}}\big{[}u_{n}\big{(}(k-l)/n,x+n^{-\frac{1}{\alpha}}\sum_{i=1}^{l}Z_{i}\big{)}\big{]}-\mathbb{\hat{E}}\big{[}u_{n}\big{(}0,x+n^{-\frac{1}{\alpha}}\sum_{i=1}^{l}Z_{i}\big{)}\big{]}\big{|}\\\
&\leq\big{(}(I_{1,n^{1/\alpha}})^{\frac{\delta}{2}}+I_{2,n^{1/\alpha}}\big{)}((k-l)/n)^{\frac{\delta}{2}}.\end{split}$
(A.6)

In general, for $s,t\in[0,1]$, let $\beta_{s},\beta_{t}\in[0,1/n)$ such that
$s-\beta_{s}$ and $t-\beta_{t}$ are in the grid points
$\\{k/n:k\in\mathbb{N}\\}$. Then, from (A.6), we have

$\displaystyle u_{n}(t,x)=u_{n}(t-\beta_{t},x)$ $\displaystyle\leq
u_{n}(s-\beta_{s},x)+\big{(}(I_{1,n^{1/\alpha}})^{\frac{\delta}{2}}+I_{2,n^{1/\alpha}}\big{)}|t-s-\beta_{t}+\beta_{s}|^{\frac{\delta}{2}}$
$\displaystyle\leq
u_{n}(s,x)+\big{(}(I_{1,n^{1/\alpha}})^{\frac{\delta}{2}}+I_{2,n^{1/\alpha}}\big{)}(|t-s|^{\frac{\delta}{2}}+n^{-\frac{\delta}{2}}).$

Similarly, we can prove the other side. The proof is complete.

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