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.
Categories: math.PR