G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation. To prove the self-normalized central limit theorem, we also establish a new Donsker’s invariance principle with the limit process being a generalized G-Brownian motion.

中文翻译：

---

收敛到自归一化的G-布朗运动

G-布朗运动具有非常丰富和有趣的新结构，可以轻松地概括经典的布朗运动。它的二次变化过程也是具有独立和固定增量的连续过程。我们证明了在亚线性期望下针对独立且均匀分布的随机变量的自归一化函数中心极限定理，其中极限过程是通过二次方差自归化的G布朗运动。为了证明自归一化的中心极限定理，我们还建立了新的Donsker定律，其中极限过程是广义G-布朗运动。