Convergence to a self-normalized G-Brownian motion

TL;DR

This paper establishes a self-normalized functional central limit theorem for i.i.d. variables under sub-linear expectation, showing convergence to a G-Brownian motion normalized by its quadratic variation, extending classical results.

Contribution

It introduces a new self-normalized CLT under sub-linear expectation and proves a corresponding Donsker's invariance principle for G-Brownian motion.

Findings

Proves a self-normalized CLT for i.i.d. variables under sub-linear expectation.

Establishes a new Donsker's invariance principle for G-Brownian motion.

Shows convergence to a G-Brownian motion normalized by its quadratic variation.

Abstract

G-Brownian motion has a very rich and interesting new structure which nontrivially generalizes the classical one. 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-Browian motion self-normalized by its quadratic variation. To prove the self-normalized central limit theorem, we also establish a new Donsker's invariance principle.