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This article relaxes the integrability condition imposed in the literaturefor the robust $\alpha$-stable central limit theorem under sublinearexpectation. Specifically, for $\alpha \in(0,1]$, we prove that the normalizedsums of i.i.d. non-integrable random variables $\big\{n^{-\frac{1}{\alpha}}\sum_{i=1}^{n}Z_{i}\big \}_{n=1}^{\infty}$ converge indistribution to $\tilde{\zeta}_{1}$, where $(\tilde{\zeta}_{t})_{t\in\lbrack0,1]}$ is a multidimensional nonlinear symmetric $\alpha$-stable processwith a jump uncertainty set $\mathcal{L}$. The limiting $\alpha $-stableprocess is further characterized by a fully nonlinear partialintegro-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 \[ \delta_{\lambda}^{\alpha}u(t,x):= \left \{ \begin{array} [c]{l} u(t,x+\lambda)-u(t,x)-\langleD_{x}u(t,x),\lambda \mathbb{1}_{\{|\lambda |\leq 1\}}\rangle,\ \alpha=1,\\u(t,x+\lambda)-u(t,x),\ \alpha \in(0,1). \end{array} \right. \] The main toolsare a weak convergence approach to obtain the limiting process, aL\'evy-Khintchine representation of the nonlinear $\alpha$-stable process and atruncation technique to estimate the corresponding $\alpha$-stable L\'{e}vymeasures. As a byproduct, the article also provides a probabilistic approach toprove the existence of the above fully nonlinear PIDE.

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