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

Abstract

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.