Bounds on Random Infinite Urn Model

Abstract. Let N(n) be a Poisson random variable with parameter n. An in-finite urn model is defined as follows: N(n) balls are independently placed in an infinite set of urns and each ball has probability pk> 0 of being assigned to the k-th urn. We assume that pk ≥ pk+1 for all k and ∞X k=1 pk = 1. Let Un be the number of occupied urns after N(n) balls have been thrown. Dutko showed in 1989 that under the condition lim n→∞V ar(Un) = ∞ we have Un − E(Un)p V ar(Un) d− → N (0, 1) as n→ ∞ where N (0, 1) is the standard normal ran-dom variable. However, Dutko did not give a bound of his approximation. So in this paper, we give uniform and non-uniform bounds of the approximation