A central limit theorem and remainder term estimates are given for the distribution of the sum of scores based on the occupancy numbers resulting from the random allocation of $N$ balls to $n$ boxes. The proof involves bivariate characteristic functions, exploiting the equivalence of multinomial and conditioned Poisson variables. The results are shown to include the statistics for the empty cell test, the chi-squared test and the likelihood ratio test.
"Normal Approximations to Sums of Scores Based on Occupancy Numbers." Ann. Probab. 12 (3) 794 - 804, August, 1984. https://doi.org/10.1214/aop/1176993228