A Sub-Gaussian Berry-Esseen Theorem for the Hypergeometric Distribution
  In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and the Normal distribution functions are shown to decay at a sub-Gaussian rate. 
Soumendra N. Lahiri, A. Chatterjee, and T. Maiti