  This article establishes a universal robust limit theorem under a sublinear
expectation framework. Under moment and consistency conditions, we show that,
for $\alpha \in(1,2)$, the i.i.d. sequence \[ \left \{ \left(
\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum
_{i=1}^{n}Y_{i},\frac{1}{\sqrt[\alpha]{n}}\sum_{i=1}^{n}Z_{i}\right) \right \}
_{n=1}^{\infty} \] converges in distribution to $\tilde{L}_{1}$, where
$\tilde{L}_{t}=(\tilde {\xi}_{t},\tilde{\eta}_{t},\tilde{\zeta}_{t})$, $t\in
[0,1]$, is a multidimensional nonlinear L\'{e}vy process with an uncertainty
set $\Theta$ as a set of L\'{e}vy triplets. This nonlinear L\'{e}vy process is
characterized by a fully nonlinear and possibly degenerate partial
integro-differential equation (PIDE) \[ \left \{ \begin{array} [c]{l}
\displaystyle \partial_{t}u(t,x,y,z)-\sup \limits_{(F_{\mu},q,Q)\in \Theta
}\left \{
\int_{\mathbb{R}^{d}}\delta_{\lambda}u(t,x,y,z)F_{\mu}(d\lambda)\right. \\
\displaystyle \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. +\langle
D_{y}u(t,x,y,z),q\rangle+\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right \} =0,\\
\displaystyle u(0,x,y,z)=\phi(x,y,z),\ \ \forall(t,x,y,z)\in \lbrack 0,1]\times
\mathbb{R}^{3d}, \end{array} \right. \] with
$\delta_{\lambda}u(t,x,y,z):=u(t,x,y,z+\lambda)-u(t,x,y,z)-\langle
D_{z}u(t,x,y,z),\lambda \rangle$. To construct the limit process
$(\tilde{L}_{t})_{t\in \lbrack0,1]}$, we develop a novel weak convergence
approach based on the notions of tightness and weak compactness on a sublinear
expectation space. We further prove a new type of L\'{e}vy-Khintchine
representation formula to characterize $(\tilde{L}_{t})_{t\in [0,1]}$. As a
byproduct, we also provide a probabilistic approach to prove the existence of
the above fully nonlinear degenerate PIDE.
