Asymptotic normality associated with generalized occupancy problems

We consider a scheme of independent allocation of particles of ss types into an infinite number of cells. The object of interest here are the statistics that are the sum of functions of the frequencies of the cells. The asymptotic normality of these statistics is proved under weaker conditions than those known. This result is then applied in the construction of a goodness of fit test of sequential type to verify a hypothesis about discrete distributions.