Limit Theorems for Some Occupancy and Sequential Occupancy Problems

Consider a situation in which balls are falling into N cells with arbitrary probabilities. A limiting distribution for the number of occupied cells after n falls is obtained, when n and N → ∞, so that n²/N → ∞ and n/N → 0. This result completes some theorems given by Chistyakov (1964), (1967). Limiting distributions of the number of falls to achieve aN + 1 occupied cells are obtained when lim aN/N < 1. These theorems generalize theorems given by Baum and Billingsley (1965), and David and Barton (1962), when the balls fall into cells with the same probability for every cell.