The coverage of a multinomial random sample is the sum of the probabilities of the observed classes. A normal limit law is rigorously proved for Good's (1953) coverage estimator. The result is valid under very general conditions and all terms except the coverage itself are observable. Nevertheless the implied confidence intervals are not much wider than those developed under restrictive assumptions such as in the classical occupancy problem. The asymptotic variance is somewhat unexpected. The proof utilizes a method of Holst (1979).
"A Normal Limit Law for a Nonparametric Estimator of the Coverage of a Random Sample." Ann. Statist. 11 (3) 905 - 912, September, 1983. https://doi.org/10.1214/aos/1176346256