Lindeberg's central limit theorems for martingale like sequences under sub-linear expectations

TL;DR

This paper extends the central limit theorem to martingale-like sequences under sub-linear expectations, providing new theoretical results and proofs relevant for stochastic processes and G-Brownian motion.

Contribution

It introduces general and functional CLTs for martingale-like variables under sub-linear expectations, including new proofs and inequalities.

Findings

Lindeberg CLT and functional CLT for independent, non-identically distributed variables

A new proof of Lévy's characterization of G-Brownian motion

Established Rosenthal's and exceptional inequalities for martingale-like variables

Abstract

The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. In this paper, general central limit theorems and functional central limit theorems are obtained for martingale like random variables under the sub-linear expectation. As applications, the Lindeberg central limit theorem and functional central limit theorem are obtained for independent but not necessarily identically distributed random variables, and a new proof of the L\'evy characterization of a G-Brownian motion without using stochastic calculus is given. For proving the results, we have also established Rosenthal's inequality and the exceptional inequality for the martingale like random variables.