The moments of the maximum of normalized partial sums related to laws of the iterated logarithm under the sub-linear expectation

Abstract

Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(\Omega ,\mathcal{H},\hat{\mathbb{E}})$, $S_n=X_1+\dots +X_n$. We consider the moments of ${\max }_{n\ge 1}|S_n|/\sqrt{2n\log \log n}$. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.