Convergence rates in the strong law of large numbers for negatively orthant dependent random variables with general moment conditions

Let $\{a_n,n\ge 1\}$ be a sequence of real numbers with $0<a_n/n^{1/p}\uparrow$ for some $1\leq p<2$, and let $\{X,X_n,n\ge 1\}$ be a sequence of identically distributed negatively orthant dependent random variables. In this paper, it is shown that $\sum^\infty_{n=1}n^{r-1}\times P\{|X|>a_n\}<\infty$ is equivalent to $\sum^\infty_{n=1}n^{r-2}P\{\max_{1\leq m\leq n}|S_m-m EXI(|X|\leq a_n)|>\varepsilon a_n\}<\infty,\,\forall\ \varepsilon>0$, where $r\ge 1$ and $S_n=\sum^n_{k=1}X_k$.