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

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.