Bounds in a normal approximation of an infinite urn model

Abstract

Let N (n) be a Poisson random variable with parameter n. An infinite urn model is defined as follows: N (n) balls are indenpendently 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 is >= Pk+1 for all k and sigma [superscript infinity] [subscript k=1 Pk = 1. Let Zeta [subscript N (n)] be the number of occupied urns after N (n) balls have been thrown. Dutko showed in 1989 that under the condition lim [subscript n vector infinity Var (Zeta [subscript N (n))] = Infinity, We have lim [subscript n vector infinity] F [subscript n (x) = phi (x) Where F[subscript n] is the distribution function of Z [subscript (n) - E (Z [subscript N(n) / squairroot Var (Z [subscript N (n)] and phi is the standard normal distribution. However, Dutko did not give a bound of his approximation. In our work, We use the technique in Chen and Shao (2001) to give uniform and non-uniform bounds of the approximation.