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 $\{n^{-\frac{1}{\alpha }}{\sum }_{i=1}^n{Z}_i{\}}_{n=1}^{\infty }$ converge in distribution to ${\tilde{\zeta }}_1$, where $({\tilde{\zeta }}_t{)}_{t\in [0,1]}$ 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) \[ \left \{ \begin{array} [c]{l}\displaystyle \partial_{t}u(t,x)-\sup \limits_{F_μ\in \mathcal{L}}\left \{ \int_{\mathbb{R}^{d}}δ_λ^αu(t,x)F_μ(dλ)\right \} =0,\\ \displaystyle u(0,x)=ϕ(x),\ \ \ \forall(t,x)\in \lbrack0,1]\times \mathbb{R}^{d}, \end{array} \right. \] 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 $\alpha$-stable process and a truncation technique to estimate the corresponding $\alpha$-stable Lévy measures. As a byproduct, the article also provides a probabilistic approach to prove the existence of the above fully nonlinear PIDE.

26 pages. arXiv admin note: text overlap with arXiv:2205.00203