Asymptotic Distributions for Occupancy and Waiting Time Problems with Positive Probability of Falling Through the Cells

Consider N cells into which balls are being dropped independently, in such a way that the cells are equiprobable, and each ball has probability pN > 0 of staying in the cell. Let WN(pN, kN) denote the waiting time until kN + 1 cells are occupied, and let SN(pN, jN) denote the number of distinct cells occupied after jN balls have been dropped. The full characterization of the limiting distributions of these two random variables is obtained, depending upon the joint behaviour of pN, kN and pN, jN respectively, as N → ∞. The limit distributions obtained are the negative binomial, binomial, Poisson, chi-square and normal distributions.