# The convergence of the sums of independent random variables under the sub-linear expectations

## TL;DR

This paper extends classical convergence results of sums of independent random variables to the setting of sub-linear expectations, establishing conditions and inequalities for convergence and the CLT.

## Contribution

It introduces new convergence criteria and maximal inequalities for independent variables under sub-linear expectations, generalizing classical probability results.

## Key findings

- Established Levy and Kolmogorov maximal inequalities under sub-linear expectations.
- Provided necessary and sufficient conditions for convergence of sums.
- Derived conditions for the central limit theorem in this framework.

## Abstract

Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables on a probability space $(\Omega, \mathcal{F}, P)$ and $S_n=\sum_{k=1}^n X_k$. It is well-known that the almost sure convergence, the convergence in probability and the convergence in distribution of $S_n$ are equivalent. In this paper, we prove similar results for the independent random variables under the sub-linear expectations, and give a group of sufficient and necessary conditions for these convergence. For proving the results, the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established. As an application of the maximal inequalities, the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.