A note on the cluster set of the law of the iterated logarithm under sub-linear expectations

TL;DR

This paper proves a version of the law of the iterated logarithm under sub-linear expectations, extending classical results to a more general setting with upper capacity, and introduces a self-normalized version for i.i.d. variables.

Contribution

It establishes a compact law of the iterated logarithm under sub-linear expectations and develops a self-normalized law of the iterated logarithm for i.i.d. variables.

Findings

Law of the iterated logarithm under upper capacity established

Self-normalized law of the iterated logarithm proved

Extension of classical results to sub-linear expectation framework

Abstract

In this note, we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space. For showing the result, a self-normalized law of the iterated logarithm is established.