Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications

TL;DR

This paper extends classical inequalities to negatively dependent variables under sub-linear expectations, providing new tools for analyzing laws of large numbers in non-linear probability frameworks.

Contribution

It introduces negative dependence under sub-linear expectations and establishes Kolmogorov's and Rosenthal's inequalities in this setting, with applications to strong laws of large numbers.

Findings

Kolmogorov's and Rosenthal's inequalities are valid for negatively dependent variables under sub-linear expectations.

Strong law of large numbers holds if the Choquet integral is finite under a continuous sub-linear expectation.

The results generalize classical inequalities to a non-linear expectation context.

Abstract

Classical Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of random variables are basic tools for studying the strong laws of large numbers. In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng (2006, 2008b), we introduce the concept of negative dependence of random variables and establish Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of negatively dependent random variables under the sub-linear expectations. As an application, we show that Kolmogorov's strong law of larger numbers holds for independent and identically distributed under a continuous sub-linear expectation if and only if the corresponding Choquet integral is finite.