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

Abstract

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