  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 $(\tilde{\zeta}_{t})_{t\in
\lbrack0,1]}$ 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) \[ \left \{ \begin{array}
[c]{l}\displaystyle \partial_{t}u(t,x)-\sup \limits_{F_{\mu}\in
\mathcal{L}}\left \{
\int_{\mathbb{R}^{d}}\delta_{\lambda}^{\alpha}u(t,x)F_{\mu}(d\lambda)\right \}
=0,\\ \displaystyle u(0,x)=\phi(x),\ \ \ \forall(t,x)\in \lbrack0,1]\times
\mathbb{R}^{d}, \end{array} \right. \] where \[ \delta_{\lambda}^{\alpha}
u(t,x):= \left \{ \begin{array} [c]{l} u(t,x+\lambda)-u(t,x)-\langle
D_{x}u(t,x),\lambda \mathbb{1}_{\{|\lambda |\leq 1\}}\rangle,\ \alpha=1,\\
u(t,x+\lambda)-u(t,x),\ \alpha \in(0,1). \end{array} \right. \] The main tools
are a weak convergence approach to obtain the limiting process, a
L\'evy-Khintchine representation of the nonlinear $\alpha$-stable process and a
truncation technique to estimate the corresponding $\alpha$-stable L\'{e}vy
measures. As a byproduct, the article also provides a probabilistic approach to
prove the existence of the above fully nonlinear PIDE.
