On the Number of i.i.d. Samples Required to Observe All of the Balls in an Urn

Abstract

Suppose an urn contains m distinct balls, numbered 1,...,m, and let τ denote the number of i.i.d. samples required to observe all of the balls in the urn. We generalize the partial fraction expansion type arguments used by Pólya (Z Angew Math Mech 10:96–97, 1930) for approximating $\mathbb{E}(\tau)$ in the case of fixed sample sizes to obtain an approximation of $\mathbb{E}(\tau)$ when the sample sizes are i.i.d. random variables. The approximation agrees with that of Sellke (Ann Appl Probab 5(1):294–309, 1995), who made use of Wald’s equation and a Markov chain coupling argument. We also derive a new approximation of $\mathbb{V}(\tau)$, provide an (improved) bound on the error in these approximations, derive a recurrence for $\mathbb{E}(\tau)$, give a new large deviation type result for tail probabilities, and look at some special cases.