A Robust $\alpha$-Stable Central Limit Theorem Under Sublinear Expectation without Integrability Condition

Abstract

This article fills a gap in the literature by relaxing the integrability condition 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 law to ${\tilde{\zeta }}_{1}$, where $({\tilde{\zeta }}_{t})_{t\in [0,1]}$ is a multidimensional nonlinear symmetric $\alpha$-stable process with jump uncertainty set ${\mathcal {L}}$. The limiting $\alpha$-stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _{t}u(t,x)-\sup \limits _{F_{\mu }\in {\mathcal {L}}}\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }u(t,x)F_{\mu }(d\lambda )\right\} =0,\\ \displaystyle u(0,x)=\phi (x),\quad \forall (t,x)\in [0,1]\times {\mathbb {R}}^{d}, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \delta _{\lambda }^{\alpha }u(t,x):=\left\{ \begin{array}{ll} u(t,x+\lambda )-u(t,x)-\langle D_{x}u(t,x),\lambda \mathbbm {1}_{\{|\lambda |\le 1\}}\rangle , &{}\quad \alpha =1,\\ u(t,x+\lambda )-u(t,x), &{}\quad \alpha \in (0,1). \end{array} \right. \end{aligned}$$

The approach used in this study involves the utilization of several tools, including 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. In addition, the article presents a probabilistic method for proving the existence of a solution to the above fully nonlinear PIDE.

1 Introduction

Non-Gaussian $\alpha$-stable distributions allow for large fluctuations, making them well suited for modeling high variability and heavy tail events. These distributions serve as the only non-Gaussian limiting distributions, acting as attractors of normalized sums of i.i.d. random variables $\big \{n^{-\frac{1}{\alpha }}\sum _{i=1}^{n}Z_{i}\big \}_{n=1}^{\infty }$, which are commonly referred to as $\alpha$-stable central limit theorems. Recently, efforts have been made to generalize these central limit theorems to a robust case under model uncertainty (see [4, 14, 15]) for $\alpha \in (1,2)$. The case $\alpha \in (0,1]$ remains an open challenge because $Z_{i}$ is not integrable in such a situation. In this paper, we address this more challenging case by establishing a novel robust $\alpha$-stable central limit theorem without the requirement of an integrability condition.

The theory of robust probability and expectation has been developed by [26,27,28, 32], 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 [29, 31]. 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.,

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbb {{\hat{E}}}\bigg [\phi \bigg (\frac{1}{\sqrt{n} }\sum _{i=1}^{n}{X_{i}},\frac{1}{n}\sum _{i=1}^{n}Y_{i} \bigg )\bigg ]=\mathbb {{\tilde{E}}}[\phi (\xi ,\eta )], \end{aligned}$$

for any test function $\phi$. The G-distributed random variable $(\xi ,\eta )$ can describe both the volatility and the mean uncertainty of the model.. The corresponding convergence rate was established in [12, 34] using Stein’s method, in [21] using stochastic control method and in [18] using monotone approximation scheme method under different model assumptions. We refer the reader to [5, 8, 35] and the references therein for more research in this field.

A general G-Lévy process in the setting of sublinear expectation was introduced by [16, 17] to further describe Poisson jump uncertainty beyond volatility and mean uncertainty. They established a new type of Lévy–Khintchine representation for G-Lévy processes by connecting it to a class of fully nonlinear partial integro-differential equations (PIDEs). Furthermore, the case of infinite activity jumps has been studied in [11, 22, 24, 25]. An important class of nonlinear Lévy processes with infinite activity jumps is the integrable $\alpha$-stable process $(\zeta _{t})_{t\ge 0}$ 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 $\alpha$-stable central limit theorem under sublinear expectation was established in [4], 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.,

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbb {{\hat{E}}}\bigg [\phi \bigg (\frac{1}{\root \alpha \of {n}}\sum _{i=1}^{n}Z_{i}\bigg )\bigg ]=\mathbb {{\tilde{E}}} [\phi (\zeta )],\ \alpha \in (1,2), \end{aligned}$$

for any test function $\phi$. The corresponding convergence rate was established in [14] via a monotone approximation scheme.

Recently, in collaboration with Hu and Peng, the authors have established a universal robust limit theorem in [15], building upon the weak convergence approach introduced in [30]. This result covers Peng’s robust central limit theorem [29, 31] and Bayraktar–Munk’s robust stable limit theorem [4] as special cases. Under specific moment conditions of $(X_{i},Y_{i},Z_{i})$ and a consistency condition of $Z_{i}$, the authors have proved that, for $\alpha \in (1,2)$, the normalized sums of i.i.d. random variables $\left\{ \left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}X_{i},\frac{1}{n}\sum _{i=1}^{n}Y_{i},\frac{1}{\root \alpha \of {n}} \sum _{i=1}^{n}Z_{i}\right) \right\} _{n=1}^{\infty }$ converge in law 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 $\Theta$ consisting of Lévy triplets. They further proved that $({\tilde{L}}_{t})_{t\in [0,1]}$ can be characterized via a fully nonlinear and possibly degenerate PIDE:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _{t}u(t,x,y,z)-\sup \limits _{(F_{\mu },q,Q)\in \Theta }\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }u(t,x,y,z)F_{\mu } (d\lambda )\right. \\ \qquad \qquad \qquad \qquad \displaystyle \left. +\langle D_{y}u(t,x,y,z),q\rangle +\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right\} =0,\\ \displaystyle u(0,x,y,z)=\phi (x,y,z),\ \ \forall (t,x,y,z)\in [ 0,1]\times {\mathbb {R}}^{3d}, \end{array} \right. \end{aligned}$$

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 $\alpha \in (0,1]$ unexplored. This regime presents two main challenges. First, this case lies outside the scope of the integrability conditions of Lévy triplets in [25], and the PIDE characterization of the nonlinear $\alpha$-stable distribution remains unknown to date. Consequently, the existing method employed in [4] is rendered invalid, as [4] heavily relies on the regularity estimates of fully nonlinear PIDEs. Second, the lack of integrability for the normalized sums of i.i.d. random variables renders the sufficient condition for the weak convergence limit, as presented in [15], ineffective in this scenario.

We overcome the aforementioned difficulty by introducing a sublinear expectation space, where random variables are not required to have any moments, so the random variables do not need to belong to the space. Instead, it is sufficient for the composition of bounded Lipschitz continuous functions of the random variables to belong to the space. This new space enables us to define their distributions and study the corresponding robust central limit theorems. To ensure the tightness of the sequence and establish a weak convergence limit, we introduce a new $\delta$-moment condition on the normalized sums of i.i.d. random variables (see assumption (A.1) in Section 3). Additionally, by imposing a consistency condition (see assumption (A.2) in Sect. 3), we demonstrate that the limiting process is a nonlinear symmetric $\alpha$-stable process $({\tilde{\zeta }}_{t})_{t\in [0,1]}$. Specifically, we show that for any test function $\phi$:

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbb {{\hat{E}}}\bigg [\phi \bigg (\frac{1}{\root \alpha \of {n}}\sum _{i=1}^{n}Z_{i}\bigg )\bigg ]=\mathbb {{\tilde{E}}} [\phi ({\tilde{\zeta }}_{1})],\ \alpha \in (0,1], \end{aligned}$$

for any test function $\phi$. It is important to note that one remarkable feature of this process is that ${\tilde{\zeta }}_{t}$ is non-integrable, meaning that $\mathbb {{\tilde{E}}}[|{\tilde{\zeta }}_{1}|]=\infty$. Consequently, a direct generalization of the Lévy–Khintchine representation formula presented in [15] for $\alpha \in (0,1]$ is not possible. To overcome this challenge, we employ a truncation technique to estimate the $\alpha$-stable Lévy measure. This approach provides a novel type of Lévy–Khintchine representation for $({\tilde{\zeta }}_{t})_{t\in [0,1]}$ with $\alpha \in (0,1]$, circumventing the issue of non-integrability.

Together with the case $\alpha \in (1,2)$ studied in [15], we are able to provide a complete characterization of the nonlinear and non-Gaussian $\alpha$-stable process for $\alpha \in (0,2)$. This characterization is achieved through a fully nonlinear partial integro-differential equation (PIDE):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _{t}u(t,x)-\sup \limits _{F_{\mu }\in {\mathcal {L}}}\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }u(t,x)F_{\mu }(d\lambda )\right\} =0,\\ \displaystyle u(0,x)=\phi (x),\forall (t,x)\in [0,1]\times {\mathbb {R}}^{d}, \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \delta _{\lambda }^{\alpha }u(t,x):=\left\{ \begin{array}{ll} u(t,x+\lambda )-u(t,x)-\langle D_{x}u(t,x),\lambda \rangle , &{}\quad \alpha \in (1,2),\\ u(t,x+\lambda )-u(t,x)-\langle D_{x}u(t,x),\lambda \mathbbm {1}_{\{|\lambda |\le 1\}}\rangle , &{}\quad \alpha =1,\\ u(t,x+\lambda )-u(t,x), &{}\quad \alpha \in (0,1). \end{array} \right. \end{aligned}$$

Due to the connection with the fully nonlinear PIDE mentioned above, the existence of its viscosity solution can also be obtained as a byproduct. This means that the solution can be obtained through the weak convergence limit of the normalized sums of i.i.d. random variables. Furthermore, when we restrict the jump uncertainty set to a singleton, our work complements the classical $\alpha$-stable central limit theorems in the linear setting. Relevant references for the linear setting include [6, 7, 9, 13, 19, 20, 23] and the references cited therein.

The rest of the paper is organized as follows. In Sect. 2, we provide some necessary results for the $\alpha$-stable central limit theorem without integrability condition under the sublinear expectation framework. 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. Section 4 contains the proof of the main theorem along with some technical results provided in Appendix.

2 Preliminaries on Nonlinear $\alpha$-Stable Process for $\alpha \in (0,1]$

The sublinear expectation framework has been introduced in [27, 28, 30, 32]. 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 $(\Omega ,{\mathcal {H}},\mathbb {{\hat{E}})}$, called a sublinear expectation space, where $\Omega$ is a given set, ${\mathcal {H}}$ 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 ${\mathbb {R}}^{n}$, and $\mathbb {{\hat{E}}}:{\mathcal {H}}\rightarrow {\mathbb {R}}$ is said to a sublinear expectation satisfying

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

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

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

• (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

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

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 each $\varphi \in C_{b,Lip}({\mathbb {R}}^{m+n})$

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

which is denoted by $Y\perp \! \! \! \perp X$. Notably, Y is independent from X does not necessarily imply that X is independent from Y. If $Y\overset{d}{=}X$ and $Y\perp \! \! \! \perp X$, we call Y an independent copy of X.

Proposition 2.3

[15] 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.,

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

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

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

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

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 2.4

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

$$\begin{aligned} aX+bY\overset{d}{=}(a^{\alpha }+b^{\alpha })^{1/\alpha }X,\quad \text {for}\quad a,b\ge 0, \end{aligned}$$

where Y is an independent copy of X.

Remark 2.5

A special attention is deserved for the case $\alpha =1$. For any $k\ge 1$, if $|X|^{k}\in {{\mathcal {H}}}$ and $\mathbb {{\hat{E}}}[|X|^{k}]$ exist, in this situation, X is the maximum distribution as introduced in [29, 32]. 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, which is different from the maximum distribution.

We close this section by recalling nonlinear Lévy processes under sublinear expectations introduced in [17, 25] and, in particular, nonlinear symmetric $\alpha$-stable processes.

Definition 2.6

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

• (i) it starts at the origin, $X_{0}=0$;

• (ii) it has stationary increments, that is, for any $0\le s\le t$, $X_{t}-X_{s}$ and $X_{t-s}$ are identically distributed;

• (iii) it has independent increments, that is, for any choice of $n\in {\mathbb {N}}$ and $0\le t_{1}\le \cdots \le t_{n}\le s\le t$, $X_{t}-X_{s}$ is independent from $(X_{t_{1}},\ldots ,X_{t_{n}})$.

Furthermore, $(X_{t})_{t\ge 0}$ is called a nonlinear symmetric $\alpha$-stable process if for any $t\ge 0$, ${X}_{t}\overset{d}{=}t^{1/\alpha }X_{1}$ and $X_{t}\overset{d}{=}-X_{t}$.

3 Main Results

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

$$\begin{aligned} F_{\mu }(B)=\int _{S}\mu (dx)\int _{0}^{\infty }\mathbbm {1}_{B}(rx)\frac{dr}{r^{1+\alpha }},\quad \text {for}\quad B\in {\mathcal {B}}({\mathbb {R}}^{d}), \end{aligned}$$

(3.1)

where $\mu$ is a symmetric finite measure (cf. [33]) on S. Introduce a jump uncertainty set:

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

(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

  $$\begin{aligned} \frac{1}{s}\bigg \vert \mathbb {{\hat{E}}}\big [\varphi (x+s^{\frac{1}{\alpha } }Z_{1})-\varphi (x)\big ]-s\sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi (x)F_{\mu }(d\lambda )\bigg \vert \le l(s)\rightarrow 0 \end{aligned}$$

  uniformly on $x\in {\mathbb {R}}^{d}$ as $s\rightarrow 0$, where $l:[0,1]\rightarrow {\mathbb {R}}_{+}$ and

  $$\begin{aligned} \delta _{\lambda }^{\alpha }\varphi (x):=\left\{ \begin{array}{ll} \varphi (x+\lambda )-\varphi (x)-\langle D\varphi (x),\lambda \mathbbm {1}_{\{|\lambda |\le 1\}}\rangle , &{}\quad \alpha =1,\\ \varphi (x+\lambda )-\varphi (x), &{} \quad \alpha \in (0,1). \end{array} \right. \end{aligned}$$

3.1 Discussion on the $\delta$-Moment and Consistency Conditions (A1)–(A2)

The $\delta$-moment condition (A1) guarantees the weak convergence of the i.i.d. sequence $\{Z_{i}\}_{i=1}^{\infty }$, which is inspired by the existing moment conditions imposed in the literature. Indeed, for a centralized random variable $Z_i$, in the robust central limit theorem, the second-moment condition $\mathbb {{\hat{E}}}[|Z_{i}|^{2}]<\infty$ is equivalent to $M_{2}<\infty$ with $\alpha =2$. Furthermore, for the robust $\alpha$-stable central limit theorem in [15] 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. The assumption (A2) serves as a consistency condition for the distribution of $Z_{i}$, which has been widely used in the numerical analysis of monotone approximation schemes for nonlinear PDEs [1,2,3]. Specifically, the first part relates to the approximation schemes for $\alpha$-stable distribution based on $Z_{i}$, and the second part relates to the infinitesimal generator of $\alpha$-stable distribution.

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 [19] indicates that

$$\begin{aligned} \frac{1}{\root \alpha \of {n}}\sum _{i=1}^{n}Z_{i}\overset{{\mathcal {D}}}{\rightarrow }\zeta \text {, as}\ n\rightarrow \infty \end{aligned}$$

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

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

(3.3)

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

$$\begin{aligned} \lim _{x\rightarrow -\infty }\beta _{1}(x)=\lim _{x\rightarrow \infty }\beta _{2}(x)=0. \end{aligned}$$

Note that under the condition (3.3), it follows that

$$\begin{aligned} E[|Z_{1}|]\ge \int _{1}^{\infty }P(|Z_{1}|>z)dz=\frac{2k}{\alpha }\int _{1}^{\infty }z^{-\alpha }dz+\int _{1}^{\infty }\frac{\beta _{1}(-z)+\beta _{2} (z)}{z^{\alpha }}dz=\infty . \end{aligned}$$

Next, we verify that conditions (A1) and (A2) hold. To verify the $\delta$-moment condition (A1), for given $n>0$ and $0<\delta <\alpha$, we define the following recursive approximation scheme

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

We further assume that

$$\begin{aligned}{} & {} \text {the functions }\beta _{i},i\nonumber \\{} & {} =1,2,\text {are continuously differentiable on their respective half-lines,} \end{aligned}$$

(3.4)

and

$$\begin{aligned} \beta _{1}(x)=\beta _{2}(-x)\text {, for }x<0. \end{aligned}$$

(3.5)

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 1 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 $\gamma \ge 0$ such that

$$\begin{aligned} |\beta _{i}(x)|\le \frac{C}{|x|^{\gamma }}\text {, }|x|\ge 1. \end{aligned}$$

(3.6)

We remark that the above condition also appears in [7] under the classical linear case.

Proposition 3.1

Under the classical linear expectation, suppose that the conditions (3.3), (3.4), and (3.6) hold. Then, (A2) holds for $\alpha \in (0,1)$. If we further assume that (3.5) 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

$$\begin{aligned}&\frac{1}{s}\bigg \vert E\big [\varphi (x+s^{\frac{1}{\alpha }}Z_{1} )-\varphi (x)\big ]-s\int _{{\mathbb {R}}}\delta _{\lambda }^{\alpha }\varphi (x)F_{\mu }(d\lambda )\bigg \vert \\&=\bigg \vert \int _{\mathbb {-\infty }}^{0}\delta _{\lambda }^{\alpha } \varphi (x)\big [\alpha \beta _{1}(s^{-\frac{1}{\alpha }}\lambda )-\beta _{1} ^{\prime }(s^{-\frac{1}{\alpha }}\lambda )s^{-\frac{1}{\alpha }}\lambda \big ]|\lambda |^{-\alpha -1}d\lambda \ \\&\quad +\int _{0}^{\infty }\delta _{\lambda }^{\alpha }\varphi (x)\big [\alpha \beta _{2}(s^{-\frac{1}{\alpha }}\lambda )-\beta _{2}^{\prime }(s^{-\frac{1}{\alpha }}\lambda )s^{-\frac{1}{\alpha }}\lambda \big ]\lambda ^{-\alpha -1}d\lambda \bigg \vert \end{aligned}$$

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

$$\begin{aligned}&\bigg \vert \int _{0}^{\infty }\delta _{\lambda }^{\alpha }\varphi (x)[\alpha \beta _{2}(s^{-\frac{1}{\alpha }}\lambda )-\beta _{2}^{\prime }(s^{-\frac{1}{\alpha }}\lambda )s^{-\frac{1}{\alpha }}\lambda ]\lambda ^{-\alpha -1} d\lambda \bigg \vert \\&=\bigg \vert \int _{0}^{1}\delta _{\lambda }^{\alpha }\varphi (x)\big [\alpha \beta _{2}(s^{-\frac{1}{\alpha }}\lambda )-\beta _{2}^{\prime }(s^{-\frac{1}{\alpha }}\lambda )s^{-\frac{1}{\alpha }}\lambda \big ]\lambda ^{-\alpha -1} d\lambda \bigg \vert \\&\quad +\bigg \vert \int _{1}^{\infty }\delta _{\lambda }^{\alpha } \varphi (x)\big [\alpha \beta _{2}(s^{-\frac{1}{\alpha }}\lambda )-\beta _{2} ^{\prime }(s^{-\frac{1}{\alpha }}\lambda )s^{-\frac{1}{\alpha }}\lambda \big ]\lambda ^{-\alpha -1}d\lambda \bigg \vert :={\mathcal {I}}+\mathcal{I}\mathcal{I}\text {.} \end{aligned}$$

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

$$\begin{aligned} {\mathcal {I}}&=\bigg \vert -\delta _{1}^{\alpha }\varphi (x)\beta _{2} (s^{-\frac{1}{\alpha }})+\int _{0}^{1}D\varphi (x+\lambda )\beta _{2}(s^{-\frac{1}{\alpha }}\lambda )\lambda ^{-\alpha }d\lambda \bigg \vert \\&\le 2\left\| \varphi \right\| _{\infty }|\beta _{2}(s^{-\frac{1}{\alpha }})|+\left\| D\varphi \right\| _{\infty }\int _{0}^{1}|\beta _{2} (s^{-\frac{1}{\alpha }}\lambda )|\lambda ^{-\alpha }d\lambda \rightarrow 0,\,\text {as}\,s\rightarrow 0. \end{aligned}$$

For the part $\mathcal{I}\mathcal{I}$, when $\gamma \in (1-\alpha ,\infty )$, it follows from integration by parts and (3.6) that

$$\begin{aligned} \mathcal{I}\mathcal{I}&=\bigg \vert \delta _{1}^{\alpha }\varphi (x)\beta _{2} (s^{-\frac{1}{\alpha }})+\int _{1}^{\infty }D\varphi (x+\lambda )\beta _{2}(s^{-\frac{1}{\alpha }}\lambda )\lambda ^{-\alpha }d\lambda \bigg \vert \\&\le 2\left\| \varphi \right\| _{\infty }|\beta _{2}(s^{-\frac{1}{\alpha }})|+\left\| D\varphi \right\| _{\infty }\int _{1}^{\infty }|\beta _{2}(s^{-\frac{1}{\alpha }}\lambda )|\lambda ^{-\alpha }d\lambda \rightarrow 0\text {, as }s\rightarrow 0\text {.} \end{aligned}$$

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

$$\begin{aligned}&\bigg \vert \int _{N_{0}}^{\infty }\delta _{\lambda }^{\alpha }\varphi (x)\big [\alpha \beta _{2}(s^{-\frac{1}{\alpha }}\lambda )-\beta _{2}^{\prime }(s^{-\frac{1}{\alpha }}\lambda )s^{-\frac{1}{\alpha }}\lambda \big ]\lambda ^{-\alpha -1}d\lambda \bigg \vert \\&\quad \le 2\left\| \varphi \right\| _{\infty }\int _{N_{0}}^{\infty }\big \vert \alpha \beta _{2}(s^{-\frac{1}{\alpha }}\lambda )-\beta _{2}^{\prime }(s^{-\frac{1}{\alpha }}\lambda )s^{-\frac{1}{\alpha }}\lambda \big \vert \lambda ^{-\alpha -1}d\lambda \\&\quad =2\left\| \varphi \right\| _{\infty }|\beta _{2}(s^{-\frac{1}{\alpha } }N_{0})|N_{0}^{-\alpha }\le 2C\left\| \varphi \right\| _{\infty } N_{0}^{-\alpha }, \end{aligned}$$

and

$$\begin{aligned}&\bigg \vert \int _{1}^{N_{0}}\delta _{\lambda }^{\alpha }\varphi (x)\big [\alpha \beta _{2}(s^{-\frac{1}{\alpha }}\lambda )-\beta _{2}^{\prime }(s^{-\frac{1}{\alpha }}\lambda )s^{-\frac{1}{\alpha }}\lambda \big ]\lambda ^{-\alpha -1} d\lambda \bigg \vert \\&\quad =\bigg \vert \delta _{1}^{\alpha }\varphi (x)\beta _{2}(s^{-\frac{1}{\alpha } })-\delta _{N_{0}}^{\alpha }\varphi (x)\beta _{2}(s^{-\frac{1}{\alpha }}N_{0} )N_{0}^{-\alpha }\\&\qquad +\int _{1}^{N_{0}}D\varphi (x+\lambda )\beta _{2}(s^{-\frac{1}{\alpha }}\lambda )\lambda ^{-\alpha }d\lambda \bigg \vert \\&\quad \le 2\left\| \varphi \right\| _{\infty }|\beta _{2}(s^{-\frac{1}{\alpha }})|+2C\left\| \varphi \right\| _{\infty }N_{0}^{-\alpha }+\frac{1}{1-\alpha }\left\| D\varphi \right\| _{\infty }\sup _{|\lambda |\ge s^{-1/\alpha }}|\beta _{2}(\lambda )|N_{0}^{1-\alpha }. \end{aligned}$$

By letting $N_{0}=(\sup \limits _{|\lambda |\ge s^{-1/\alpha }}|\beta _{2} (\lambda )|)^{-1}$, we obtain that

$$\begin{aligned} \mathcal{I}\mathcal{I}\le & {} 2\left\| \varphi \right\| _{\infty }|\beta _{2} (s^{-\frac{1}{\alpha }})|\\{} & {} +\left[ 4C\left\| \varphi \right\| _{\infty }+\frac{1}{1-\alpha }\left\| D\varphi \right\| _{\infty }\right] \sup \limits _{|\lambda |\ge s^{-1/\alpha }}|\beta _{2}(\lambda )|^{\alpha }\rightarrow 0\text {, as }s\rightarrow 0\text {.} \end{aligned}$$

(ii) $\alpha =1$. We further impose (3.5), i.e., $\beta _{1}(x)=\beta _{2}(-x)$ for $x<0$. Then, for any $a>0$, we have $E[Z_{1}\mathbbm {1}_{\{|Z_{1}|\le a\}}]=0$, which implies that for $\varphi \in C_{b}^{3}({\mathbb {R}})$ and $s\in (0,1]$,

$$\begin{aligned}&\frac{1}{s}\bigg \vert E\big [\varphi (x+sZ_{1})-\varphi (x)\big ]-s\int _{{\mathbb {R}}}\delta _{\lambda }^{1}\varphi (x)F_{\mu }(d\lambda )\bigg \vert \\&\quad =\frac{1}{s}\bigg \vert E\big [\varphi (x+sZ_{1})-\varphi (x)-D\varphi (x)sZ_{1}\mathbbm {1}_{\{|sZ_{1}|\le 1\}}\big ]-s\int _{{\mathbb {R}}} \delta _{\lambda }^{1}\varphi (x)F_{\mu }(d\lambda )\bigg \vert \\&\quad =\bigg \vert \int _{\mathbb {-\infty }}^{0}\delta _{\lambda }^{1}\varphi (x)\big [\beta _{2}(-s^{-1}\lambda )+\beta _{2}^{\prime }(-s^{-1}\lambda )s^{-1}\lambda \big ]|\lambda |^{-\alpha -1}d\lambda \\&\qquad +\int _{0}^{\infty }\delta _{\lambda }^{1}\varphi (x)\big [\beta _{2}(s^{-1}\lambda )-\beta _{2}^{\prime }(s^{-1}\lambda )s^{-1}\lambda \big ]\lambda ^{-\alpha -1}d\lambda \bigg \vert , \end{aligned}$$

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

$$\begin{aligned}&\bigg \vert \int _{0}^{\infty }\delta _{\lambda }^{1}\varphi (x)\big [\beta _{2}(s^{-1}\lambda )-\beta _{2}^{\prime }(s^{-1}\lambda )s^{-1}\lambda \big ]\lambda ^{-2}d\lambda \bigg \vert \\&\quad =\bigg \vert \int _{0}^{1}\delta _{\lambda }^{1}\varphi (x)\big [\beta _{2} (s^{-1}\lambda )-\beta _{2}^{\prime }(s^{-1}\lambda )s^{-1}\lambda \big ]\lambda ^{-2}d\lambda \bigg \vert \\&\qquad +\bigg \vert \int _{1}^{\infty }\delta _{\lambda }^{1} \varphi (x)\big [\beta _{2}(s^{-1}\lambda )-\beta _{2}^{\prime }(s^{-1} \lambda )s^{-1}\lambda \big ]\lambda ^{-2}d\lambda \bigg \vert :={\mathcal {I}} +\mathcal{I}\mathcal{I}\text {.} \end{aligned}$$

Then, we can deduce that

$$\begin{aligned} {\mathcal {I}}&=\bigg \vert -\delta _{1}^{1}\varphi (x)\beta _{2}(s^{-1})+\int _{0}^{1}(D\varphi (x+\lambda )-D\varphi (x))\beta _{2}(s^{-1}\lambda )\lambda ^{-1}d\lambda \bigg \vert \\&\le (2\left\| \varphi \right\| _{\infty }+\left\| D\varphi \right\| _{\infty })|\beta _{2}(s^{-1})|+\left\| D^{2}\varphi \right\| _{\infty }\int _{0}^{1}|\beta _{2}(s^{-1}\lambda )|d\lambda \rightarrow 0,\\&\quad \text {as}\quad s\rightarrow 0, \end{aligned}$$

and

$$\begin{aligned} \mathcal{I}\mathcal{I}\le 2\left\| \varphi \right\| _{\infty }|\beta _{2} (s^{-1})|+\left\| D\varphi \right\| _{\infty }\int _{1}^{\infty }|\beta _{2}(s^{-1}\lambda )|\lambda ^{-1}d\lambda \rightarrow 0,\quad \text {as}\quad s\rightarrow 0. \end{aligned}$$

To sum up, the assumption (A2) holds. $\square$

3.2 Robust $\alpha$-Stable Central Limit Theorem

Now, we give the main theorem of this paper, which is called 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 $({\tilde{\zeta }}_{t})_{t\in [0,1]}$, connected with the jump uncertainty set ${\mathcal {L}}$ such that for any $\phi \in C_{b,Lip}({\mathbb {R}}^{d})$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbb {{\hat{E}}}\left[ \phi \left( \sum \limits _{i=1}^{n}\frac{Z_{i}}{\root \alpha \of {n}}\right) \right] =\mathbb {{\tilde{E}}}[\phi ({\tilde{\zeta }}_{1})]=u^{\phi }(1,0), \end{aligned}$$

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

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _{t}u(t,x)-\sup \limits _{F_{\mu }\in {\mathcal {L}}}\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }u(t,x)F_{\mu }(d\lambda )\right\} =0,\\ \displaystyle u(0,x)=\phi (x),\text {}\forall (t,x)\in [ 0,1]\times {\mathbb {R}}^{d}, \end{array} \right. \end{aligned}$$

(3.7)

where

$$\begin{aligned} \delta _{\lambda }^{\alpha }u(t,x):=\left\{ \begin{array}{ll} u(t,x+\lambda )-u(t,x)-\langle D_{x}u(t,x),\lambda \mathbbm {1}_{\{|\lambda |\le 1\}}\rangle , &{}\quad \alpha =1,\\ u(t,x+\lambda )-u(t,x), &{}\quad \alpha \in (0,1). \end{array} \right. \end{aligned}$$

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 $0\le t\le 1$.

3.3 An Example

We discuss a concrete example to illustrate 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 brevity, we will focus on the one-dimensional case. For any given ${\underline{\Lambda }},{\overline{\Lambda }}>0$, let $F_{\mu }$ be the $\alpha$-stable Lévy measure with the symmetric measure $\mu$ concentrated on the points $S_{0}=\{1,-1\}$ and

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

Denote $K=({\underline{\Lambda }},{\overline{\Lambda }})$ and $k=\mu \{1\}=\mu \{-1\}$. Let $0<\delta <\alpha$ and $\rho :{\mathbb {N}}\rightarrow \mathbb {R_{+}}$ be a function satisfying $\lim \limits _{n\rightarrow \infty }\rho (n)=0$. For each $k\in K$, let $W_{k}$ be a classical random variable such that

• (i) $W_{k}$ has a cumulative distribution function

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

  where $\beta _{k}:[0,\infty )\rightarrow {\mathbb {R}}$ is a continuously differentiable function such that there exist $\gamma \ge 0$ and $C>0$ satisfying

  $$\begin{aligned} |\beta _{k}(x)|\le \frac{C}{|x|^{\gamma }}\text {,}|x|\ge 1. \end{aligned}$$

• (ii) For any $n\ge 1$, the following items are bounded by $\rho (n)$

  $$\begin{aligned}{} & {} |\beta _{k}(n^{1/\alpha })|, \quad \int _{1}^{\infty }\frac{|\beta _{k}(n^{1/\alpha }x)|}{x^{1+\alpha -\delta }}dx, \quad \int _{0}^{1}\frac{|\beta _{k}(n^{1/\alpha }x)|}{x^{\alpha }}dx\ (if\ \alpha \in (0,1)),\\{} & {} \quad \int _{0}^{1}|\beta _{2,k}(nx)|dx\ (if\ \alpha =1). \end{aligned}$$

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

$$\begin{aligned} \mathbb {{\hat{E}}}[X]=\sup _{k\in K}\int _{{\mathbb {R}}}f(x)dF_{W_{k}}(x). \end{aligned}$$

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 [10, 32] for a detailed discussion.

Let $Z(z)=z$, $z\in {\mathbb {R}}$, be a random variable. 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 $|Z|^{\delta }\in {\mathcal {H}}$. Indeed, for any $Z^{\delta ,m}$, $Z^{\delta ,n}\in {\mathcal {H}}_0$, we have

$$\begin{aligned} \mathbb {{\hat{E}}}[|Z^{\delta ,m}-Z^{\delta ,n}|]&\le \mathbb {{\hat{E}}}\left[ \left| |Z|^{\delta }\wedge m-|Z|^{\delta }\right| \right] +\mathbb {{\hat{E}}}\left[ \left| |Z|^{\delta }\wedge n-|Z|^{\delta }\right| \right] \\&\le \sup _{k\in K}\int _{{\mathbb {R}}}|z|^{\delta }\mathbbm {1}_{\{|z|^{\delta }>m\}}dF_{W_{k}}(z)+\sup _{k\in K}\int _{{\mathbb {R}}}|z|^{\delta } \mathbbm {1}_{\{|z|^{\delta }>n\}}dF_{W_{k}}(z)\\&=2m^{\frac{\delta -\alpha }{\delta }}\sup _{k\in K}\left\{ \frac{k}{\alpha -\delta }+\delta \int _{1}^{\infty }\frac{\beta _{k}(m^{1/\delta } z)}{z^{1+\alpha -\delta }}dz+\beta _{k}(m^{1/\delta })\right\} \\&\quad +2n^{\frac{\delta -\alpha }{\delta }}\sup _{k\in K}\left\{ \frac{k}{\alpha -\delta }+\delta \int _{1}^{\infty }\frac{\beta _{k}(n^{1/\delta } z)}{z^{1+\alpha -\delta }}dz+\beta _{k}(n^{1/\delta })\right\} \end{aligned}$$

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

Next, we shall adapt the monotone scheme method to verify the assumption (A1). Let $\{Z_{i}\}_{i=1}^{\infty }$ be a sequence of i.i.d. ${\mathbb {R}}$-valued random variables. Specifically, we have $Z_{1}\overset{d}{=}Z$, $Z_{i+1} \overset{d}{=}Z_{i}$, and $Z_{i+1}\perp \! \! \! \perp (Z_{1},Z_{2},\ldots ,Z_{i})$ for every $i\in {\mathbb {N}}$. For each given $n>0$, we define a discrete scheme $u_{n}:[0,1]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ as follows

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

(3.8)

It is easy to check that $u_{n}(1,0)=n^{-\frac{\delta }{\alpha }}\mathbb {\hat{E}}[|S_{n}|^{\delta }]$ 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 Appendix 1 due to its technicality

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

where $\sup _{n\in {\mathbb {N}}}I_{n}<\infty$. It follows that

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

which implies (A1) holds.

For the assumption (A2), using a similar way as in Proposition 3.1, we can conclude that for $\varphi \in C_{b}^{3}({\mathbb {R}})$,

$$\begin{aligned} \frac{1}{s}\bigg \vert \mathbb {{\hat{E}}}\big [\varphi (z+s^{\frac{1}{\alpha } }Z_{1})-\varphi (z)\big ]-s\sup _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{2} }\delta _{\lambda }^{\alpha }\varphi (z)F_{\mu }(d\lambda )\bigg \vert \rightarrow 0, \end{aligned}$$

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

4 Proof of Theorem 3.2

4.1 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

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

Theorem 4.1

Assume that (A1)–(A2) hold. 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 $({\tilde{\Omega }},Lip({\tilde{\Omega }}), \mathbb {{\tilde{E}}})$.

In the proof of Theorem 4.1, we will need the following lemma.

Lemma 4.2

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

$$\begin{aligned} \mathbb {{\hat{F}}}[\phi ]:=\sup _{n}\mathbb {{\hat{E}}}\left[ \phi \left( \frac{S_{n}}{\root \alpha \of {n}}\right) \right] . \end{aligned}$$

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 Appendix.

Proof

It is easy to verify that $\mathbb {{\hat{F}}}$ is a sublinear expectation on $({\mathbb {R}}^{d},C_{b,Lip}({\mathbb {R}}^{d}))$, and we only prove its tightness. For any given $N>0$, we define

$$\begin{aligned} \varphi _{N}(x)=\left\{ \begin{array}{ll} 1, &{}\quad |x|>N,\\ |x|-N+1, &{}\quad N-1\le |x|\le N,\\ 0, &{} \quad |x|<N-1. \end{array} \right. \end{aligned}$$

Clearly, $\varphi _{N}\in C_{b,Lip}({\mathbb {R}})$ and

$$\begin{aligned} \mathbbm {1}_{\{|x|>N\}}\le \varphi _{N}(x)\le \mathbbm {1}_{\{|x|>N-1\}} \le \frac{|x|^{\delta }}{(N-1)^{\delta }}, \end{aligned}$$

for $0<\delta <\alpha$. Thus, from the assumption (A1), for each $\varepsilon >0$, choosing $N_{0}>\root \delta \of {M_{\delta }/\varepsilon }+1$ such that

$$\begin{aligned} \mathbb {{\hat{F}}}[\varphi _{N_{0}}(z)]=\sup _{n}\mathbb {{\hat{E}}}\left[ \varphi _{N_{0}}\left( \frac{S_{n}}{\root \alpha \of {n}}\right) \right] \le \frac{M_{\delta }}{(N_{0}-1)^{\delta }}<\varepsilon , \end{aligned}$$

we complete the proof. $\square$

Proof of Theorem 4.1

For simplicity, denote ${\bar{S}}_{n}=n^{-1/\alpha }S_{n}$. Since $\mathbb {{\hat{F}}}$ is tight and

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

it follows from Corollary A.4 that there exists a subsequence $\{ {\bar{S}}_{n_{i}}\}_{i=1}^{\infty }\subset \{ {\bar{S}}_{n} \}_{n=1}^{\infty }$ that converges in distribution to some $\zeta _{1}$ in $(\Omega ,{\mathcal {H}},\mathbb {{\hat{E}}}_{1})$. For any increasing integers $\{ {\tilde{n}}_{i}\}_{i=1}^{\infty }$ satisfying $|{\tilde{n}}_{i}-n_{i}|\le 1$, we note that both $\{ {\bar{S}}_{n_{i}}\}_{i=1}^{\infty }$ and $\{ {\bar{S}} _{{\tilde{n}}_{i}}\}_{i=1}^{\infty }$ converge in distribution to the same limit. In the following, without loss of generality, we assume that $n_{i}$, $i=1,2,\ldots$, are all even numbers. Then, we can decompose ${\bar{S}}_{n_{i}}$ into two parts

$$\begin{aligned} {\bar{S}}_{n_{i}}=\frac{1}{\root \alpha \of {2}}(n_{i}/2)^{-\frac{1}{\alpha } }S_{n_{i}/2}+\frac{1}{\root \alpha \of {2}}(n_{i}/2)^{-\frac{1}{\alpha }}(S_{n_{i} }-S_{n_{i}/2}):=\ {\bar{S}}_{n_{i}/2}^{1/2}+({\bar{S}}_{n_{i}}-{\bar{S}}_{n_{i} /2}^{1/2}), \end{aligned}$$

where ${\bar{S}}_{n}^{t}:=(t/n)^{1/\alpha }S_{n}$ for $t\in [0,1)$. For the first part, using the same argument again, we can find 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

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

where ${\bar{\zeta }}_{1/2}$ is an independent copy of $\zeta _{1/2}$. Since ${\bar{S}}_{n_{i}^{1}}\overset{{\mathcal {D}}}{\rightarrow }\zeta _{1}$, we can conclude that $\zeta _{1}\overset{d}{=}\zeta _{1/2}+{\bar{\zeta }}_{1/2}.$

Proceeding the procedure above for ${\bar{S}}_{n_{i}^{1}/2}^{1/2}$, a random variable $\zeta _{1/4}$ can be defined. This can be repeated to obtain the sequence $\{ \zeta _{1/2^{m}}\}_{m=0}^{\infty }$ in $(\Omega ,{\mathcal {H}},\mathbb {{\hat{E}}}_{1})$, such that each $\zeta _{1/2^{m}}$ has a convergent sequence $\{ {\bar{S}}_{n_{i}^{m}/2^{m}}^{1/2^{m}}\}_{i=1}^{\infty }$ which converges in distribution to it. Eventually, by using the random variables $\{ \zeta _{1/2^{m}}\}_{m=0}^{\infty }$, a sublinear expectation $\mathbb {\tilde{E}}$ can be constructed on $({\tilde{\Omega }},Lip({\tilde{\Omega }}))$ such that the canonical process $({\tilde{\zeta }}_{t})_{t\in [0,1]}$ is a nonlinear Lévy process.

Firstly, for each $m\ge 0$, we denote $\tau _{m}=2^{-m}$,

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

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 $n\in {\mathbb {N}}$. Given $m\ge 1$, $\phi \big ({\tilde{\zeta }}_{n\tau _{m}} -{\tilde{\zeta }}_{(n-1)\tau _{m}}\big )$ with $1\le n\le 2^{m}$, and $\phi \in C_{b,Lip}({\mathbb {R}}^{d})$, define

$$\begin{aligned} \mathbb {{\tilde{E}}}^{m}\big [\phi \big ({\tilde{\zeta }}_{n\tau _{m}}-\tilde{\zeta }_{(n-1)\tau _{m}}\big )\big ]=\mathbb {{\hat{E}}}_{1}\big [\phi (\zeta _{\tau _{m}} ^{n})\big ]. \end{aligned}$$

Then for any $\varphi \in C_{b,Lip}\big ({\mathbb {R}}^{d\times 2^{m}}\big )$, we can specify the distribution of $\varphi \big ({\tilde{\zeta }}_{\tau _{m}},\ldots ,{\tilde{\zeta }}_{2^{m}\tau _{m} }-{\tilde{\zeta }}_{(2^{m}-1)\tau _{m} }\big )\in {\mathcal {H}}^{m}$ as follows

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

where $\varphi _{0}$ is defined iteratively through

$$\begin{aligned} \varphi _{2^{m}-1}(x_{1},x_{2},\ldots ,x_{2^{m}-1})= & {} \mathbb {\tilde{E}}^{m}\big [\varphi \big (x_{1},x_{2},\ldots ,x_{2^{m}-1},{\tilde{\zeta }} _{2^{m}\tau _{m}}-{\tilde{\zeta }}_{(2^{m}-1)\tau _{m}}\big )\big ]\\ \varphi _{2^{m}-2}(x_{1},x_{2},\ldots ,x_{2^{m}-2})= & {} \mathbb {\tilde{E}}^{m}\big [\varphi _{2^{m}-1}\big (x_{1},x_{2},\ldots ,x_{2^{m}-2},\tilde{\zeta }_{(2^{m}-1)\tau _{m}}-{\tilde{\zeta }}_{(2^{m}-2)\tau _{m}}\big )\big ]\\{} & {} \vdots \\ \varphi _{1}(x_{1})= & {} \mathbb {{\tilde{E}}}^{m}\big [\varphi _{2}\big (x_{1},{\tilde{\zeta }}_{2\tau _{m}}-{\tilde{\zeta }}_{\tau _{m}}\big )\big ]\\ \varphi _{0}= & {} \mathbb {{\tilde{E}}}^{m}\big [\varphi _{1} \big ({\tilde{\zeta }}_{\tau _{m}}\big )\big ]. \end{aligned}$$

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

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

For each $m\ge 0$, denote ${\mathcal {D}}_{m}[0,1]:=\{l2^{-m}:0\le l\le 2^{m},l\in {\mathbb {N}}\}$. It is easy to verify that $\mathbb {{\tilde{E}}} ^{m}[\cdot ]$ is a consistent sublinear expectation on ${\mathcal {H}}^{m}$, 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 any $t_{i},s,t\in {\mathcal {D}}_{m}[0,1]$ with $t_{i}\le s\le t$.

Denote

$$\begin{aligned} {\mathcal {H}}^{\infty }=\bigcup _{m\ge 0}{\mathcal {H}}^{m}. \end{aligned}$$

Obviously, ${\mathcal {H}}^{\infty }\subset Lip({\tilde{\Omega }})$. Note that for any $\chi \in {\mathcal {H}}^{\infty }$, there is an $m_{0}\in {\mathbb {N}}$ such that $\chi \in {\mathcal {H}}^{m_{0}}$. Then, we can define a sublinear expectation $\mathbb {{\tilde{E}}}:{\mathcal {H}}^{\infty }\rightarrow {\mathbb {R}}$ by

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

Denote ${\mathcal {D}}_{\infty }[0,1]:=\cup _{m\ge 0}{\mathcal {D}}_{m}[0,1]$. 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 $t_{k}\in [0,1]$, $1\le k\le n$, choosing 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 $i\rightarrow \infty$, we can generalize $\mathbb {{\tilde{E}}}$ on $Lip({\tilde{\Omega }})$ as follows

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

We clarify that the limit is independent of the choice of $\{t_{k}^{i} \}_{i=1}^{\infty }$. In fact, for the two descending sequences $\{t_{k} ^{i}\}_{i=1}^{\infty }$ and $\{t_{k}^{i^{\prime }}\}_{i^{\prime }=1}^{\infty }$ that converge to 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 $1\le l_{k}\le 2^{m_{k}}$. In view of the construction process above and Proposition 2.3, we know that there exists a convergent sequence $\{ {\bar{S}}_{n_{j}^{*}}^{\tau _{m_{k}}}\}_{j=1}^{\infty }$ such that for $\varphi \in C_{b,Lip}({\mathbb {R}}^{d\times n})$

$$\begin{aligned}{} & {} \left| \mathbb {{\tilde{E}}}\big [\varphi \big (\tilde{\zeta }_{t_{1}^{i}},\ldots ,{\tilde{\zeta }}_{t_{n}^{i}}-{\tilde{\zeta }}_{t_{n-1}^{i} }\big )\big ]-\mathbb {{\tilde{E}}}\big [\varphi \big ({\tilde{\zeta }}_{t_{1} ^{i^{\prime }}},\ldots ,{\tilde{\zeta }}_{t_{n}^{i^{\prime }}}-\tilde{\zeta }_{t_{n-1}^{i^{\prime }}}\big )\big ]\right| \\{} & {} \quad \le L_{\varphi }\mathbb {{\tilde{E}}}\big [\big (\sum _{k=1} ^{n}|{\tilde{\zeta }}_{t_{k}^{i}}-{\tilde{\zeta }}_{t_{k-1}^{i}}-\tilde{\zeta }_{t_{k}^{i^{\prime }}}+{\tilde{\zeta }}_{t_{k-1}^{i^{\prime }}}|\big )\wedge N_{\varphi }\big ]\\{} & {} \quad \le 2L_{\varphi }\sum _{k=1}^{n}\mathbb {{\tilde{E}}}\big [|\tilde{\zeta }_{t_{k}^{i}-t_{k}^{i^{\prime }}}|\wedge N_{\varphi }\big ]\\{} & {} \quad =2L_{\varphi }\sum \limits _{k=1}^{n}\mathbb {{\hat{E}}}_{1} \big [|\zeta _{\tau _{m_{k}}}^{1}+\cdots +\zeta _{\tau _{m_{k}}}^{l_{k}}|\wedge N_{\varphi }\big ]\\{} & {} \quad =2L_{\varphi }\sum _{k=1}^{n}\lim \limits _{j\rightarrow \infty }\mathbb {{\hat{E}}}\big [|{\bar{S}}_{l_{k}n_{j}^{*}}^{t_{k}^{i}-t_{k} ^{i^{\prime }}}|\wedge N_{\varphi }\big ]\\{} & {} \quad =2L_{\varphi }\sum _{k=1}^{n}\lim \limits _{j\rightarrow \infty }\mathbb {{\hat{E}}}\big [\big |(t_{k}^{i}-t_{k}^{i^{\prime }})^{1/\alpha } (l_{k}n_{j}^{*})^{-1/\alpha }S_{l_{k}n_{j}^{*}}\big |\wedge N_{\varphi }\big ]\\{} & {} \quad \le 2L_{\varphi }\sum _{k=1}^{n}|t_{k}^{i}-t_{k}^{i^{\prime } }|^{\delta /\alpha }\lim \limits _{j\rightarrow \infty }\mathbb {{\hat{E}}} \big [|(l_{k}n_{j}^{*})^{-1/\alpha }S_{l_{k}n_{j}^{*}}|^{\delta }\big ]N_{\varphi }^{1-\delta }\\{} & {} \quad \le 2L_{\varphi }M_{\delta }N_{\varphi }^{1-\delta }\sum _{k=1} ^{n}|t_{k}^{i}-t_{k}^{i^{\prime }}|^{\delta /\alpha }\\{} & {} \quad \rightarrow 0,\quad \text {as}\quad i,i^{\prime }\rightarrow \infty , \end{aligned}$$

where $L_{\varphi }>0$ is the Lipschitz constant of $\varphi$, $K_{\varphi }:=\left\| \varphi \right\| _{\infty }$ and $N_{\varphi }:=\frac{2K_{\varphi }}{L_{\varphi }}$. In addition, 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 $\varphi ,\varphi ^{\prime }\in C_{b,Lip}({\mathbb {R}}^{d\times n})$, we have

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

This implies that $\mathbb {{\tilde{E}}}:Lip({\tilde{\Omega }})\rightarrow {\mathbb {R}}$ is a well-defined sublinear expectation, on which $({\tilde{\zeta }}_{t})_{t\in [0,1]}$ is a nonlinear Lévy process.

We point out that while the above process indicates that Theorem 4.1 holds for a subsequence under the assumption (A1), it is sufficient for the following results. Furthermore, note that the distribution of ${\tilde{\zeta }}_{1}$ is uniquely determined by the viscosity solution of the fully nonlinear PIDE (4.3) (see Theorem 4.7). We finally complete the proof by claiming that

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

Suppose not. We can find a sequence $\{ {\bar{S}}_{{\tilde{n}}}\}_{{\tilde{n}} =1}^{\infty }\subset \{ {\bar{S}}_{n}\}_{n=1}^{\infty }$ such that 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 $\{ {\bar{S}}_{\tilde{n}_{i}}\}_{i=1}^{\infty } \subset \{ {\bar{S}}_{{\tilde{n}}}\}_{{\tilde{n}} =1}^{\infty }$ converging in distribution to ${\tilde{\zeta }}_{1}$, which induces a contradiction. The proof is completed. $\square$

4.2 Lévy–Khintchine Representation of Nonlinear $\alpha$-Stable Process

For $\alpha \in (1,2)$, the Lévy–Khintchine representation for nonlinear $\alpha$-stable processes has been established in [15]. However, for nonlinear $\alpha$-stable processes with $\alpha \in (0,1]$, the corresponding Lévy–Khintchine representation is still lacking, which is crucial for the proof of the robust limit theorem.

Before presenting the Lévy–Khintchine representation for nonlinear $\alpha$-stable processes with $\alpha \in (0,1]$, we first give the following useful results. To this end, we denote

$$\begin{aligned} {\mathcal {K}}_{\alpha }:=\left\{ \begin{array}{ll} \sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}|\lambda |^{2} \wedge 1F_{\mu }(d\lambda ), &{}\quad \alpha =1,\\ \sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}|\lambda |\wedge 1F_{\mu }(d\lambda ), &{}\quad \alpha \in (0,1). \end{array} \right. \end{aligned}$$

(4.1)

It follows from the Sato-type result (see [33, Remark 14.4]) that ${\mathcal {K}}_{\alpha }<\infty$.

Lemma 4.3

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

$$\begin{aligned} \sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\left| \delta _{\lambda }^{\alpha }\varphi (x^{\prime })-\delta _{\lambda }^{\alpha } \varphi (x)\right| F_{\mu }(d\lambda )\le C_{\alpha }|x^{\prime }-x|^{\delta }, \end{aligned}$$

where $0<\delta <\alpha$ and

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

Proof

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

$$\begin{aligned} \delta _{\lambda }^{\alpha }\varphi (x)=\varphi \left( x+\lambda \right) -\varphi (x)=\int _{0}^{1}\langle D\varphi (x+\theta \lambda ),\lambda \rangle d\theta . \end{aligned}$$

Note that for $f\in C_{b}^{1}({\mathbb {R}}^{d})$ and $x,x^{\prime }\in {\mathbb {R}}^{d}$,

$$\begin{aligned} \left| f(x)-f(x^{\prime })\right|&\le \left\| Df\right\| _{\infty }\left( |x^{\prime }-x|\wedge \frac{2\left\| f\right\| _{\infty } }{\left\| Df\right\| _{\infty }}\right) \\&\le \left\| Df\right\| _{\infty }|x^{\prime }-x|^{\delta } \big (2\left\| f\right\| _{\infty }\left\| Df\right\| _{\infty } ^{-1}\big )^{1-\delta }\\&=\left\| Df\right\| _{\infty }^{\delta }(2\left\| f\right\| _{\infty })^{1-\delta }|x^{\prime }-x|^{\delta }, \end{aligned}$$

with $0<\delta <\alpha$. This implies that for $x,x^{\prime }\in {\mathbb {R}}^{d}$,

$$\begin{aligned}&\int _{|\lambda |\le 1}\left| \delta _{\lambda }^{\alpha }\varphi (x^{\prime })-\delta _{\lambda }^{\alpha }\varphi (x)\right| F_{\mu }(d\lambda )\\&\quad =\int _{|\lambda |\le 1}\left| \int _{0}^{1}\langle D\varphi (x^{\prime }+\theta \lambda )-D\varphi (x+\theta \lambda ),\lambda \rangle d\theta \right| F_{\mu }(d\lambda )\\&\quad \le \int _{|\lambda |\le 1}|\lambda |F_{\mu }(d\lambda )\left\| D^{2} \varphi \right\| _{\infty }^{\delta }(2\left\| D\varphi \right\| _{\infty })^{1-\delta }|x^{\prime }-x|^{\delta }, \end{aligned}$$

and

$$\begin{aligned}&\int _{|\lambda |>1}\left| \delta _{\lambda }^{\alpha }\varphi (x^{\prime })-\delta _{\lambda }^{\alpha }\varphi (x)\right| F_{\mu }(d\lambda )\\&\quad =\int _{|\lambda |>1}\left| \varphi (x^{\prime }+\lambda )-\varphi (x+\lambda )-(\varphi (x^{\prime })-\varphi (x))\right| F_{\mu }(d\lambda )\\&\quad \le 2\int _{|\lambda |>1}F_{\mu }(d\lambda )\left\| D\varphi \right\| _{\infty }^{\delta }(2\left\| \varphi \right\| _{\infty })^{1-\delta }|x^{\prime }-x|^{\delta }. \end{aligned}$$

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

$$\begin{aligned} \delta _{\lambda }^{1}\varphi (x)=\int _{0}^{1}\int _{0}^{1}\langle D^{2} \varphi (x+\tau \theta \lambda )\lambda ,\lambda \rangle \theta d\tau d\theta \text {, for }|\lambda |\le 1, \end{aligned}$$

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

$$\begin{aligned}&\int _{|\lambda |\le 1}\left| \delta _{\lambda }^{1}\varphi (x^{\prime })-\delta _{\lambda }^{1}\varphi (x)\right| F_{\mu }(d\lambda )\\&\quad =\int _{|\lambda |\le 1}\left| \int _{0}^{1}\int _{0}^{1}\langle (D^{2}\varphi (x^{\prime }+\tau \theta \lambda )-D^{2}\varphi (x+\tau \theta \lambda ))\lambda ,\lambda \rangle \theta d\tau d\theta \right| F_{\mu }(d\lambda )\\&\quad \le \int _{|\lambda |\le 1}|\lambda |^{2}F_{\mu }(d\lambda )\left\| D^{3}\varphi \right\| _{\infty }^{\delta }(2\left\| D^{2}\varphi \right\| _{\infty })^{1-\delta }|x^{\prime }-x|^{\delta }, \end{aligned}$$

and

$$\begin{aligned} \int _{|\lambda |>1}\left| \delta _{\lambda }^{1}\varphi (x^{\prime } )-\delta _{\lambda }^{1}\varphi (x)\right| F_{\mu }(d\lambda )\le 2\int _{|\lambda |>1}F_{\mu }(d\lambda )\left\| D\varphi \right\| _{\infty }^{\delta }(2\left\| \varphi \right\| _{\infty })^{1-\delta }|x^{\prime }-x|^{\delta }. \end{aligned}$$

The proof is completed. $\square$

Theorem 4.4

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

$$\begin{aligned} \lim _{n\rightarrow \infty }\bigg \vert \mathbb {{\hat{E}}}\bigg [\varphi \left( x+(s/n)^{\frac{1}{\alpha }}S_{n}\right) \bigg ]-\varphi (x)-s\sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi (x)F_{\mu }(d\lambda )\bigg \vert =o(s), \end{aligned}$$

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

Proof

Notice that

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

where

$$\begin{aligned} \omega _{n-1}:=\sum \limits _{k=1}^{n-1}z_{k}\quad \text {and}\quad \epsilon (x):=\sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi (x)F_{\mu }(d\lambda ). \end{aligned}$$

In view of the assumptions (A1)–(A2) and Lemma 4.3, we can obtain that

$$\begin{aligned}&\mathbb {{\hat{E}}}\bigg [\varphi \left( x+(s/n)^{\frac{1}{\alpha }} (\omega _{n-1}+Z_{n})\right) \bigg ]\\&\quad =\mathbb {{\hat{E}}}\bigg [\varphi \left( x+(s/n)^{\frac{1}{\alpha }} (\omega _{n-1}+Z_{n})\right) -\varphi \left( x+(s/n)^{\frac{1}{\alpha }} \omega _{n-1}\right) \\&\qquad -\frac{s}{n}\sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi \left( x+(s/n)^{\frac{1}{\alpha }}\omega _{n-1}\right) F_{\mu }(d\lambda )\bigg ]\\&\qquad +\frac{s}{n}\sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi \left( x+(s/n)^{\frac{1}{\alpha }}\omega _{n-1}\right) F_{\mu }(d\lambda )\\&\qquad -\frac{s}{n}\sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi \left( x\right) F_{\mu }(d\lambda )+\varphi \left( x+(s/n)^{\frac{1}{\alpha }}\omega _{n-1}\right) +\frac{s}{n}\epsilon (x)\\&\quad \le \frac{s}{n}l\left( \frac{s}{n}\right) + C\left( \frac{s}{n}\right) ^{1+\frac{\delta }{\alpha }}\left| \omega _{n-1}\right| ^{\delta }+ \varphi \left( x+(s/n)^{\frac{1}{\alpha }}\omega _{n-1}\right) +\frac{s}{n}\epsilon (x), \end{aligned}$$

which indicates that

$$\begin{aligned}&\mathbb {{\hat{E}}}\left[ \varphi \left( x+(s/n)^{\frac{1}{\alpha }} S_{n}\right) \right] \\&\quad \le \mathbb {{\hat{E}}}\left[ \varphi \left( x+(s/n)^{\frac{1}{\alpha } }S_{n-1}\right) \right] +\frac{s}{n}\epsilon (x)+\frac{s}{n}l\left( \frac{s}{n}\right) +CM_{\delta }s^{1+\frac{\delta }{\alpha }}\frac{1}{n}\left( \frac{n-1}{n}\right) ^{\frac{\delta }{\alpha }}. \end{aligned}$$

Proceeding the above estimate recursively, one get

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

Analogously, we have

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

Thus,

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

where we have used the fact that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\sum \limits _{k=1}^{n-1}\left( \frac{k}{n}\right) ^{\frac{\delta }{\alpha }}=\frac{\alpha }{\delta +\alpha }. \end{aligned}$$

This implies the desired result. $\square$

Denote

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

In the following, we will provide the characterization for $\lim \limits _{s\rightarrow 0}\mathbb {{\tilde{E}}}[\varphi ({\tilde{\zeta }}_{s})]s^{-1}$ for $\varphi \in {\mathfrak {F}}$, which can be considered as a new type of Lévy–Khintchine representation for the nonlinear pure jump Lévy process $({\tilde{\zeta }}_{t})_{t\in [0,1]}$.

Theorem 4.5

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

$$\begin{aligned} \lim \limits _{s\rightarrow 0}\mathbb {{\tilde{E}}}[\varphi ({\tilde{\zeta }} _{s})]s^{-1}=\sup _{F_{\mu }\in {\mathcal {L}}}\left\{ \int _{{\mathbb {R}}^{d}} \delta _{\lambda }^{\alpha }\varphi (0)F_{\mu }(d\lambda )\right\} . \end{aligned}$$

Proof

For each $\varphi \in {\mathfrak {F}}$ and $s\in [0,1]$, Theorem 4.1 states 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 $\big \{(s_{k}/n_{i}^{*})^{\frac{1}{\alpha }}S_{n_{i}^{*}}\big \}_{i=1} ^{\infty }$ for each $s_{k}$, such that

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

Together with the assumption (A2) and Theorem 4.4, we have

$$\begin{aligned}&\bigg \vert \mathbb {{\tilde{E}}}[\varphi ({\tilde{\zeta }}_{s})]-\varphi (0)-s\sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi (0)F_{\mu }(d\lambda )\bigg \vert \nonumber \\&\quad \le \lim _{k\rightarrow \infty }\bigg \vert \lim _{i\rightarrow \infty }\mathbb {{\hat{E}}}\big [\varphi ((s_{k}/n_{i}^{*})^{\frac{1}{\alpha }} S_{n_{i}^{*}})\big ]-\varphi (0)-s_{k}\sup \limits _{F_{\mu }\in {\mathcal {L}}} \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi (0)F_{\mu }(d\lambda )\bigg \vert \nonumber \\&\qquad =o(s), \end{aligned}$$

(4.2)

which implies that

$$\begin{aligned} \lim \limits _{s\rightarrow 0}\mathbb {{\tilde{E}}}[\varphi ({\tilde{\zeta }} _{s})]s^{-1}&=\lim _{s\rightarrow 0}\bigg (\mathbb {{\tilde{E}}}[\varphi ({\tilde{\zeta }}_{s})]-\varphi (0)-s\sup \limits _{F_{\mu }\in {\mathcal {L}}} \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi (0)F_{\mu }(d\lambda )\bigg )s^{-1}\\&\quad +\sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}} \delta _{\lambda }^{\alpha }\varphi (0)F_{\mu }(d\lambda ) \le \sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\varphi (0)F_{\mu }(d\lambda ), \end{aligned}$$

and similarly,

$$\begin{aligned} \lim \limits _{s\rightarrow 0}\mathbb {{\tilde{E}}}[\varphi ({\tilde{\zeta }} _{s})]s^{-1}\ge \sup \limits _{F_{\mu }\in {\mathcal {L}}}\int _{{\mathbb {R}}^{d}} \delta _{\lambda }^{\alpha }\varphi (0)F_{\mu }(d\lambda ). \end{aligned}$$

Therefore, by the squeeze theorem, we complete the proof. $\square$

4.3 Connection to PIDE

In this section, we establish a connection between the nonlinear pure jump Lévy process $({\tilde{\zeta }}_{t})_{t\in [0,1]}$ and the fully nonlinear PIDE (3.7). We denote $C_{b}^{2,3}([0,1]\times {\mathbb {R}} ^{d})$ as the set of functions on $[0,1]\times {\mathbb {R}}^{d}$ with bounded, continuous partial derivatives up to the second order in t and third order in x. To proceed, we introduce the definition of a 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 $u(0,\cdot )\le \phi (\cdot )$ (resp. $\ge \phi (\cdot ))$ and for each $(t,x)\in (0,1]\times {\mathbb {R}}^{d}$,

$$\begin{aligned} \partial _{t}\psi (t,x)-\sup \limits _{F_{\mu }\in {\mathcal {L}}}\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\psi (t,x)F_{\mu }(d\lambda )\right\} \le 0\text { }(\text {resp. }\ge 0) \end{aligned}$$

whenever $\psi \in C_{b}^{2,3}((0,1]\times {\mathbb {R}}^{d})$ is such that $\psi \ge u$ (resp. $\psi \le u$) 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.

Theorem 4.7

Suppose that the assumptions (A1)–(A2) hold. Then, for each $\phi \in C_{b,Lip}({\mathbb {R}}^{d})$ and $(t,x)\in [0,1]\times {\mathbb {R}}^{d}$, the value function $u(t,x):=\mathbb {\tilde{E}}[\phi (x+{\tilde{\zeta }}_{t})]$ is the unique viscosity solution of the following fully nonlinear PIDE

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _{t}u(t,x)-\sup \limits _{F_{\mu }\in {\mathcal {L}}}\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }u(t,x)F_{\mu }(d\lambda )\right\} =0,\\ \displaystyle u(0,x)=\phi (x),\text {}\forall (t,x)\in [ 0,1]\times {\mathbb {R}}^{d}, \end{array} \right. \end{aligned}$$

(4.3)

where

$$\begin{aligned} \delta _{\lambda }^{\alpha }u(t,x)=\left\{ \begin{array}{ll} u(t,x+\lambda )-u(t,x)-\langle D_{x}u(t,x),\lambda \mathbbm {1}_{\{|\lambda |\le 1\}}\rangle , &{}\quad \alpha =1,\\ u(t,x+\lambda )-u(t,x), &{}\quad \alpha \in (0,1). \end{array} \right. \end{aligned}$$

Proof

To begin with, we demonstrate the continuity of u. It is apparent that $u(t,\cdot )$ is uniformly Lipschitz continuous with the Lipschitz constant $L_{\phi }>0$. Note that for each $t,s\in [0,1]$ such that $t+s\le 1$, we have the following dynamic programming principle

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

(4.4)

Then, from Theorem 4.1, we know 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 $\big \{(s_{k}/n_{i}^{*})^{\frac{1}{\alpha }}S_{n_{i}^{*} }\big \}_{i=1}^{\infty }$ for each $s_{k}$, such that

$$\begin{aligned} \left| \mathbb {{\tilde{E}}}\big [u(t,x+{\tilde{\zeta }}_{s} )\big ]-u(t,x)\right|&\le L_{\phi }\mathbb {{\tilde{E}}}\big [|\tilde{\zeta }_{s}|\wedge N_{\phi }\big ]\nonumber \\&=L_{\phi }\lim \limits _{k\rightarrow \infty }\lim \limits _{i\rightarrow \infty }\mathbb {{\hat{E}}}\big [|(s_{k}/n_{i}^{*})^{\frac{1}{\alpha }}S_{n_{i}^{*} }|\wedge N_{\phi }\big ]\nonumber \\&\le L_{\phi }s^{\frac{\delta }{\alpha }}\sup _{i}\mathbb {{\hat{E}} }\big [|(1/n_{i}^{*})^{\frac{1}{\alpha }}S_{n_{i}^{*}}|^{\delta }\big ]N_{\phi }^{1-\delta }, \end{aligned}$$

(4.5)

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

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

We will now demonstrate that u is the unique viscosity solution of (4.3). The uniqueness is outlined in Corollary 55 of [17]. To establish this, we only need to prove that u is a viscosity subsolution, as the same method can be applied for the supersolution scenario. Assume that $\psi$ is a smooth test function on $(0,1]\times {\mathbb {R}}^{d}$ satisfying $\psi \ge u$ and $\psi ({\bar{t}},{\bar{x}})=u({\bar{t}},{\bar{x}})$ for some point $({\bar{t}},{\bar{x}})\in (0,1]\times {\mathbb {R}}^{d}$. From the dynamic programming principle (4.4), it can be shown that for each $s\in (0,{\bar{t}})$

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

(4.6)

Using Taylor’s expansion, we obtain that

$$\begin{aligned}&\psi ({\bar{t}}-s,{\bar{x}}+{\tilde{\zeta }}_{s})-\psi ({\bar{t}},{\bar{x}})\nonumber \\&\quad =\psi ({\bar{t}}-s,{\bar{x}}+{\tilde{\zeta }}_{s})-\psi ({\bar{t}},{\bar{x}} +{\tilde{\zeta }}_{s})+\psi ({\bar{t}},{\bar{x}}+{\tilde{\zeta }}_{s})-\psi ({\bar{t}} ,{\bar{x}})\nonumber \\&\quad =-\partial _{t}\psi ({\bar{t}},{\bar{x}})s+\psi ({\bar{t}},{\bar{x}}+{\tilde{\zeta }} _{s})-\psi ({\bar{t}},{\bar{x}})+\epsilon . \end{aligned}$$

(4.7)

where

$$\begin{aligned} \epsilon =s\int _{0}^{1}[-\partial _{t}\psi ({\bar{t}}-\theta s,{\bar{x}}+\tilde{\zeta }_{s})+\partial _{t}\psi ({\bar{t}},{\bar{x}})]d\theta . \end{aligned}$$

Since $\psi$ is the smooth function, similar to (4.5), one easily gets $\mathbb {{\tilde{E}}}[|\epsilon |]\le C_{\psi }s^{1+\frac{\delta }{\alpha }}$. Then, it follows from (4.6)–(4.7) that

$$\begin{aligned} 0\le & {} \lim _{s\rightarrow 0}\mathbb {{\tilde{E}}}[\psi ({\bar{t}}-s,{\bar{x}} +{\tilde{\zeta }}_{s})-\psi ({\bar{t}},{\bar{x}})]s^{-1}=-\partial _{t}\psi (\bar{t},{\bar{x}})\nonumber \\{} & {} +\lim _{s\rightarrow 0}\mathbb {{\tilde{E}}}[\psi ({\bar{t}},\bar{x}+{\tilde{\zeta }}_{s})-\psi ({\bar{t}},{\bar{x}})]s^{-1}. \end{aligned}$$

(4.8)

Moreover, Theorem 4.5 implies that

$$\begin{aligned} \lim _{s\rightarrow 0}\mathbb {{\tilde{E}}}\big [\psi ({\bar{t}},{\bar{x}}+\tilde{\zeta }_{s})-\psi ({\bar{t}},{\bar{x}})\big ]s^{-1}=\sup \limits _{F_{\mu }\in {\mathcal {L}} }\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\psi ({\bar{t}},\bar{x})F_{\mu }(d\lambda )\right\} . \end{aligned}$$

(4.9)

Combining (4.8) with (4.9), we conclude that

$$\begin{aligned} \partial _{t}\psi ({\bar{t}},{\bar{x}})-\sup \limits _{F_{\mu }\in {\mathcal {L}}}\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }\psi ({\bar{t}},{\bar{x}})F_{\mu }(d\lambda )\right\} \le 0. \end{aligned}$$

The proof is completed. $\square$

4.4 The Scaling and Symmetric Properties

Theorem 4.8

Assume that (A1)–(A2) hold. Then, $({\tilde{\zeta }}_{t})_{t\in [0,1]}$ is a nonlinear symmetric $\alpha$-stable process, that is, ${\tilde{\zeta }}_{t}\overset{d}{=}t^{1/\alpha } {\tilde{\zeta }}_{1}$ and ${\tilde{\zeta }}_{t}\overset{d}{=}-{\tilde{\zeta }}_{t}$.

Proof

We start with the scaling property. In view of Theorem 4.1, for each $0\le t\le 1$, 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 $0\le l_{k}\le 2^{m_{k}}$. From the construction of ${\tilde{\zeta }}_{t_{k}}$ and Proposition 2.3, there exists a convergent sequence $\{ {\bar{S}}_{n_{i}^{*}}^{\tau _{m_{k}} }\}_{i=1}^{\infty }$ such that

$$\begin{aligned} \mathbb {{\tilde{E}}}\big [\phi \big ({\tilde{\zeta }}_{t_{k}}\big )\big ]=\mathbb {\tilde{E}}^{m_{k}}\big [\phi \big ({\tilde{\zeta }}_{l_{k}\tau _{m_{k}}} \big )\big ]=\mathbb {{\hat{E}}}_{1}\big [\phi \big (\zeta _{\tau _{m_{k}}}^{1} +\cdots +\zeta _{\tau _{m_{k}}}^{l_{k}}\big )\big ]=\lim _{i\rightarrow \infty }\mathbb {{\hat{E}}}\big [\phi \big ({\bar{S}}_{l_{k}n_{i}^{*}}^{t_{k}}\big )\big ], \end{aligned}$$

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

$$\begin{aligned} {\bar{S}}_{l_{k}n_{i}^{*}}^{t_{k}}= & {} (\tau _{m_{k}}/n_{i}^{*})^{1/\alpha }(S_{l_{k}n_{i}^{*}}-S_{(l_{k}-1)n_{i}^{*}})+\cdots +(\tau _{m_{k}} /n_{i}^{*})^{1/\alpha }(S_{2n_{i}^{*}}-S_{n_{i}^{*}})\\{} & {} +(\tau _{m_{k} }/n_{i}^{*})^{1/\alpha }S_{n_{i}^{*}}. \end{aligned}$$

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

$$\begin{aligned} \lim _{i\rightarrow \infty }\mathbb {{\hat{E}}}\big [\phi \big ({\bar{S}}_{l_{k} n_{i}^{*}}^{t_{k}}\big )\big ]=\lim _{i\rightarrow \infty }\mathbb {{\hat{E}} }\big [\phi \big (\root \alpha \of {t_{k}}{\bar{S}}_{l_{k}n_{i}^{*}} \big )\big ]=\mathbb {{\tilde{E}}}\big [\phi \big (\root \alpha \of {t_{k}}\tilde{\zeta }_{1}\big )\big ]. \end{aligned}$$

This implies that

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

Next, we shall verify that ${\tilde{\zeta }}_{\cdot }$ is symmetric. For any given $\phi \in C_{b,Lip}({\mathbb {R}}^{d})$, Theorem 4.7 shows 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

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

For any $0\le t\le 1$, define $v(t,x):=u(t,-x)$. It can be verified that

$$\begin{aligned} \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }u(t,-x)F_{\mu }(d\lambda )=\int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }v(t,x)F_{\mu }(d\lambda ), \end{aligned}$$

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

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

and thus, we complete the proof. $\square$

Proof of Theorem 3.2

The existence and uniqueness of weak convergence limit for the sequences $\big \{n^{-\frac{1}{\alpha }}\sum _{i=1}^{n}Z_{i}\big \}_{n=1}^{\infty }$ can be immediately deduced from Theorem 4.1 and Theorem 4.7, respectively. We further know from Theorem 4.8 that the limit process $(\tilde{\zeta }_{t})_{t\in [0,1]}$ is a nonlinear symmetric $\alpha$-stable process, which yields the desired result. $\square$

A Appendix

1.1 A.1 Tightness Under Sublinear Expectation

The weak convergence approach provides a powerful tool in establishing robust limit theorems. We recall its main results for the reader’s convenience. Further details can be found in Sect. 2 of [15].

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 $\mathbbm {1}_{\{|x|\ge N\}}\le \varphi$ 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

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

The result presented below is an extension of Prokhorov’s theorem to the sublinear expectation case.

Theorem A.3

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, i.e., 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.

The following result is an immediate corollary of Theorem A.3 and Definition 2.2.

Corollary A.4

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.

1.2 A.2 The Regularity Estimate for $u_{n}$

In this appendix, we shall generalize the truncation method developed in [14] to obtain the regularity properties of $u_{n}$ defined in (3.8). For any fixed $N>0$, define $Z_{1}^{N}:=Z_{1}{\textbf{1}} _{\{|Z_{1}|\le N\}}$. 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

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

(A.1)

The following estimates can be easily obtained.

Lemma A.5

For any given $N>0$, we have

$$\begin{aligned} \mathbb {{\hat{E}}}[|Z_{1}^{N}|^{2}]=N^{2-\alpha }I_{1,N},\quad \mathbb {{\hat{E}}}[|Z_{1}-Z_{1}^{N}|^{\delta }]=N^{\delta -\alpha }I_{2,N}, \end{aligned}$$

where

$$\begin{aligned} I_{1,N}&:=2\sup _{k\in K}\left\{ \frac{k}{2-\alpha }+2\int _{0}^{1} \frac{\beta _{k}(zN)}{z^{\alpha -1}}dz-\beta _{k}(N)\right\} ,\\ I_{2,N}&:=2\sup _{k\in K}\left\{ \frac{k}{\alpha -\delta }+\delta \int _{1}^{\infty }\frac{\beta _{k}(zN)}{z^{1+\alpha -\delta }}dz+\beta _{k}(N)\right\} . \end{aligned}$$

Lemma A.6

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

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

where $I_{1,N}$ is given in Lemma A.5.

Proof

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

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

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

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

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

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

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

(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 $1\le k<n$ the assertion (A.2) holds. Notice that

$$\begin{aligned} u_{n,N}((k+1)/n,x)&=\mathbb {{\hat{E}}}[u_{n,N}(k/n,x+n^{-\frac{1}{\alpha } }Z_{1}^{N})]\nonumber \\&\le u_{n,N}(0,y)+A\mathbb {{\hat{E}}}[|x-y+n^{-\frac{1}{\alpha }}Z_{1} ^{N}|^{2}]+AM_{N}^{2}kn^{-\frac{2}{\alpha }}+B. \end{aligned}$$

(A.3)

Since $\mathbb {{\hat{E}}}[Z_{1}^{N}]=\mathbb {{\hat{E}}}[-Z_{1}^{N}]=0$, it follows that

$$\begin{aligned} \mathbb {{\hat{E}}}[|x-y+n^{-\frac{1}{\alpha }}Z_{1}^{N}|^{2}]=|x-y|^{2}+M_{N} ^{2}n^{-\frac{2}{\alpha }}. \end{aligned}$$

(A.4)

Combining (A.3)–(A.4), we obtain that

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

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

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

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

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

Similarly, we can also obtain the other side. From Lemma A.5, we obtain our desired result. $\square$

Analogous to Lemma 4.5 in [14], we have the following error estimate of $u_{n,N}$ for $u_{n}$.

Lemma A.7

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

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

where $I_{2,N}$ is given in Lemma A.5.

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

Theorem A.8

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

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

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 $k\le n$

$$\begin{aligned}{} & {} \left| u_{n}(k/n,x)-u_{n}(0,x)\right| \\{} & {} \quad \le \left| u_{n} (k/n,x)-u_{n,N}(k/n,x)\right| +\left| u_{n,N}(k/n,x)-u_{n,N} (0,x)\right| . \end{aligned}$$

In view of Lemmas A.6 and A.7, by choosing $N=n^{\frac{1}{\alpha }}$, we obtain

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

(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 $k\ge l$. By using induction (3.8) and the estimate (A.5), we obtain that for any $k\ge l$ and $x\in {\mathbb {R}}$,

$$\begin{aligned} \begin{aligned}&\left| u_{n}(k/n,x)-u_{n}(l/n,x)\right| \\&\quad =\Bigg \vert \mathbb {{\hat{E}}}\Bigg [u_{n}\Bigg ((k-l)/n,x+n^{-\frac{1}{\alpha }}\sum _{i=1}^{l}Z_{i}\Bigg )\Bigg ]-\mathbb {{\hat{E}}}\Bigg [u_{n} \Bigg (0,x+n^{-\frac{1}{\alpha }}\sum _{i=1}^{l}Z_{i}\Bigg )\Bigg ]\Bigg \vert \\&\quad \le \left( (I_{1,n^{1/\alpha }})^{\frac{\delta }{2}}+I_{2,n^{1/\alpha } }\right) ((k-l)/n)^{\frac{\delta }{2}}. \end{aligned}\nonumber \\ \end{aligned}$$

(A.6)

Generally, 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

$$\begin{aligned} u_{n}(t,x)=u_{n}(t-\beta _{t},x)&\le u_{n}(s-\beta _{s} ,x)+\big ((I_{1,n^{1/\alpha }})^{\frac{\delta }{2}}+I_{2,n^{1/\alpha } }\big )|t-s-\beta _{t}+\beta _{s}|^{\frac{\delta }{2}}\\&\le u_{n}(s,x)+\big ((I_{1,n^{1/\alpha }})^{\frac{\delta }{2}} +I_{2,n^{1/\alpha }}\big )(|t-s|^{\frac{\delta }{2}}+n^{-\frac{\delta }{2}}). \end{aligned}$$

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