Functional central limit theorems for random vectors under sub-linear expectations

TL;DR

This paper extends the functional central limit theorem to multi-dimensional martingale-like random vectors under sub-linear expectations, providing new convergence results and conditions for independent vectors.

Contribution

It introduces a functional CLT for multi-dimensional martingale-like vectors under sub-linear expectations, expanding previous one-dimensional results.

Findings

Established a functional CLT for multi-dimensional martingale-like vectors.

Derived a Lindeberg CLT for independent random vectors.

Identified necessary and sufficient conditions for CLT in i.i.d. vectors.

Abstract

The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes. The central limit theorem and functional central limit theorem are obtained for martingale like random variables under the sub-linear expectation by Zhang (2019). In this paper, we consider the multi-dimensional martingale like random vectors and establish a functional central limit theorem. As applications, the Lindeberg central limit theorem for independent random vectors is established, and the sufficient and necessary conditions of the central limit theorem for independent and identically distributed random vectors are obtained.