Limit theorems with rate of convergence under sublinear expectations

TL;DR

This paper develops new limit theorems under sublinear expectations, providing convergence rates, special cases, and representations of the G-normal distribution, advancing the theoretical foundation of probability under uncertainty.

Contribution

It introduces new laws of large numbers and central limit theorems with convergence rates under sublinear expectations, extending Peng's G-normal distribution framework.

Findings

New law of large numbers under sublinear expectations

Central limit theorem with convergence rate under sublinear expectations

Representation of the G-normal distribution in a probability space

Abstract

Under the sublinear expectation $\mathbb{E}[\cdot]:=\sup_{\theta\in \Theta} E_\theta[\cdot]$ for a given set of linear expectations $\{E_\theta: \theta\in \Theta\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the $G$-normal distribution, which was used as the limit in Peng (2007)'s central limit theorem, in a probability space.