On the law of the iterated logarithm under the sub-linear expectations

TL;DR

This paper extends the law of the iterated logarithm to independent, non-identically distributed random variables within sub-linear expectation spaces, providing new inequalities and conditions.

Contribution

It introduces general forms of the law of the iterated logarithm under sub-linear expectations, including exponential inequalities and necessary conditions.

Findings

Established exponential inequalities for maximum sums

Derived necessary and sufficient conditions for i.i.d. variables

Extended LIL to non-identically distributed variables

Abstract

In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov's converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.