Note on Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of I.I.D.: Random Variables under Sublinear Expectations

Abstract

Let $X,X_n,n\ge 1$ be a sequence of independent, identically distributed random variables under sublinear expectations with $C_{\mathbb{V}}\left(X^2\right)<\infty$ , ${\text{lim}}_{c⟶\infty }\mathbb{E}\left[{\left(X^2-c\right)}^{+}\right]=0$ , and $\breve{\mathbb{E}}\left(X\right)=\breve{\mathbb{E}}\left(-X\right)=0$ . Write $S_0=0$ , $S_n={\displaystyle {\sum }_{k=1}^n{X}_n}$ , and $M_n={\max }_{0\le k\le n}\left|S_k\right|$ , $n\ge 1$ . For $d>0$ and $a_n=o\left({\left(\log \log n\right)}^{-d}\right)$ , we obtain the exact rates in the law of iterated logarithm of a kind of weighted infinite series of $C_{\mathbb{V}}\left\{{\left[M_n-\left(\epsilon +a_n\right)\overline{\sigma }\sqrt{n}{\left(\log \log n\right)}^d\right]}^{+}\right\}$ as $\epsilon \downarrow 0$ .