A general and precise Berry –Esseen bound is proved for the Studentized mean based on $N$ random observations drawn without replacement from a finite population. The bound yields the optimal rate $O(N^ -1 /2)$ under minimal conditions. If the Erdős–Rényi condition holds this bound implies the asymptotic normality of Student’s statistic and the self-normalized sum.
"A Berry-Esseen Bound for Finite Population Student’s Statistic." Ann. Probab. 27 (4) 2089 - 2108, October 1999. https://doi.org/10.1214/aop/1022874830