Strong laws of large numbers for sub-linear expectations

TL;DR

This paper extends classical strong laws of large numbers to capacities under sub-linear expectations, providing a framework for understanding IID random variables in non-additive probability settings.

Contribution

It introduces three strong laws of large numbers for capacities with a new notion of IID under sub-linear expectations, generalizing classical results.

Findings

Established natural extensions of Kolmogorov's SLLN to non-additive probabilities

Provided a frequentist interpretation for capacities in the context of sub-linear expectations

Demonstrated the theoretical validity of these laws for IID random variables under the new framework

Abstract

We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.