| An urn model is defined as follows: n balls are thrown at an infinite array of urns, where each ball has probability p(,k) > 0 of independently hitting the k('th) urn, and.;We assume that p(,k) (GREATERTHEQ) p(,k+1) for all k.;Let N(t) be a Poisson process with mean t. A random variable z(,n) is defined to be the number of occupied urns after n balls (fixed sample size) have been thrown. Correspondingly, a random variable Z(,N(t)) is defined to be the number of occupied urns after N(t) balls have been thrown (randomized Poisson sample size). Let (mu)(,n) and (mu)(t) denote the means of Z(,n) and of Z(,N(t)), respectively, and let (sigma)(,n) and (sigma)(t) denote the standard deviations of Z(,n) and Z(,N(t)), respectively. Sometimes we will use the discrete parameter n instead of t in the Poisson process. We let (alpha)(x) = max k(VBAR)p(,k) (GREATERTHEQ) 1/x .;The following results are established: (1) For all sequences p(,k) such that.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;Z(,n) - (mu)(,n) /(sigma)(n) converges in distribution to N(0,1) (standard normal) as n (--->) (INFIN). (2) The limiting behavior of Z(,N(t)) - (mu)(t) is described for t (--->) (INFIN) by studying its characteristic function. The discussion is in two parts. First, it is shown that p(,k) must satisfy the necessary condition (A) (alpha)(x) (TURN) a log x, x (--->) (INFIN), a > 0, if Z(,N(t)) - (mu)(t) is to converge in distribution. Second, the case p(,k) = 1/2('k) is examined (which satisfies (A)), and it is proved that the characteristic function of Z(,N(t)) - (mu)(t) does not have a pointwise limit as t (--->) (INFIN). Condition (A) eliminates the need to consider p(,k) outside this class. (3) For the case p(,k) = 1/2('k), the class of limiting distributions of Z(,N(t)) - (mu)(t) is identified along all convergent subsequences. This class consists of a one-parameter family of characteristic functions, each characteristic function being a two-tailed infinite product.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;(4) The limiting behavior of the quantity.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;is described as t (--->) (INFIN), for the case p(,k) = (1-(theta))(theta)('k-1), 0 < (theta) < 1. It is proved that the limit of this integral exists if and only if (theta) = 2('-1/q), q a positive integer.;The result in (1) extends a previous result of Karlin to a wider class of sequences p(,k) . The results in (2), (3), and (4) settle the question of the limiting behavior of Z(,N(t)) - (mu)(t) raised by Karlin. |
