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

This article relaxes the integrability condition imposed in the literature for the robust $α$-stable central limit theorem under sublinear expectation. Specifically, for $α\in(0,1]$, we prove that the normalized sums of i.i.d. non-integrable random variables $\big \{n^{-\frac{1}α}\sum_{i=1}^{n}Z_{i}\big \}_{n=1}^{\infty}$ converge in distribution to $\tildeζ_{1}$, where $(\tildeζ_{t})_{t\in \lbrack0,1]}$ is a multidimensional nonlinear symmetric $α$-stable process with a jump uncertainty set $\mathcal{L}$. The limiting $α$-stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE) \[ \left \{ \begin{array} [c]{l}\displaystyle \partial_{t}u(t,x)-\sup \limits_{F_μ\in \mathcal{L}}\left \{ \int_{\mathbb{R}^{d}}δ_λ^αu(t,x)F_μ(dλ)\right \} =0,\\ \displaystyle u(0,x)=ϕ(x),\ \ \ \forall(t,x)\in \lbrack0,1]\times \mathbb{R}^{d}, \end{array} \right. \] where \[ δ_λ^α u(t,x):= \left \{ \begin{array} [c]{l} u(t,x+λ)-u(t,x)-\langle D_{x}u(t,x),λ\mathbb{1}_{\{|λ|\leq 1\}}\rangle,\ α=1,\\ u(t,x+λ)-u(t,x),\ α\in(0,1). \end{array} \right. \] The main tools are a weak convergence approach to obtain the limiting process, a Lévy-Khintchine representation of the nonlinear $α$-stable process and a truncation technique to estimate the corresponding $α$-stable Lévy measures. As a byproduct, the article also provides a probabilistic approach to prove the existence of the above fully nonlinear PIDE.

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