Three series theorem for independent random variables under sub-linear expectations with applications

TL;DR

This paper extends the classical three series theorem to independent random variables under sub-linear expectations, providing new convergence results and applications like a Marcinkiewicz strong law of large numbers.

Contribution

It establishes a three series theorem under sub-linear expectations, a significant generalization from classical probability theory.

Findings

Proved a three series theorem for sub-linear expectations.

Derived a Marcinkiewicz strong law of large numbers in this setting.

Demonstrated the applicability of the theorem to convergence analysis.

Abstract

In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng, we give a theorem about the convergence of a random series and establish a three series theorem of independent random variables under the sub-linear expectations. As an application, we obtain the Marcinkiewicz's strong law of large numbers for independent and identically distributed random variables under the sub-linear expectations. The technical details are different from those for classical theorems because the sub-linear expectation and its related capacity are not additive.