Exponential inequalities under sub-linear expectations with applications to laws of the iterated logarithm

TL;DR

This paper develops exponential inequalities under sub-linear expectations for independent and negatively dependent variables, and applies these to establish laws of the iterated logarithm and a central limit theorem in this non-additive framework.

Contribution

It introduces Kolmogorov type exponential inequalities under sub-linear expectations and applies them to laws of the iterated logarithm and a CLT for dependent variables.

Findings

Established exponential inequalities for independent and negatively dependent variables.

Proved laws of the iterated logarithm under sub-linear expectations.

Derived a central limit theorem with finite variance under sub-linear expectations.

Abstract

Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments. For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sub-linear expectation for random variables with only finite variances.