Strassen's law of the iterated logarithm under sub-linear expectations

TL;DR

This paper extends Strassen's law of the iterated logarithm to the setting of sub-linear expectations, providing new theoretical results for i.i.d. variables with zero mean and finite variance under non-additive probability measures.

Contribution

It establishes the LIL under sub-linear expectations and capacities, broadening classical probabilistic laws to non-linear expectation frameworks.

Findings

Proves LIL for i.i.d. variables with zero mean under sub-linear expectations.

Shows LIL for upper capacity when certain variance conditions are met.

Extends classical probabilistic laws to non-linear expectation spaces.

Abstract

We establish the Strassen's law of the iterated logarithm for independent and identically distributed random variables with $\hat{\mathbb{E}}[X_1]=\hat{\mathcal{E}}[X_1]=0$ and $C_{\mathbb{V}}[X_1^2]<\infty$ under sub-linear expectation space with a countably sub-additive capacity $\mathbb{V}$. We also show the LIL for upper capacity with $\sigma=\bar{\sigma}$ under some certain conditions.