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Bounds on Random Infinite Urn Model

Boonta, S., and Neammanee, K., (2007) Bounds on Random Infinite Urn Model. Bulletin of the Malaysian Mathematical Sciences Society, 30 (2). pp. 121-128. ISSN 0126-6705

Full text not available from this repository.

Official URL: http://math.usm.my/bulletin/pdf/v30n2/v30n2p4.pdf

Affiliations

Chulalongkorn University, Faculty of Science, Dept. of Mathematics

Abstract

Let $N(n)$ be a Poisson random variable with Parameter $n$. An infinite urn model is defined as follows: $N(n)$ balls are independently placed in an infinite set of urns and each ball has probability $p_k > 0$ of being assigned to the $k$-th urn. We assume that $p_k \geq p_{k+1}$ for all $k$ and $\sum_{k=1}^{\infty} p_k = 1$. Let $ U_n $, be the number of occupied urns after $N(n)$ balls have been thrown. Dutko showed in 1989 that under the condition $\lim_{n\rightarrow \infty} Var(U_{n}) = \infty$ we have $\frac{U_n - E(U_{n})}{\sqrt{Var(U_{n})}} \rightarrow^d \mathcal{N} (0,1)$ as $n \rightarrow \infty$ where $\mathcal{N} (0,1)$ is the standard random 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.


2000 Mathematics Subject Classification: 60F05, 60G50

Item Type:Journal
Keywords:Infinite urn model, Central limit theorem, Uniform and non-uniform bounds
Subjects:Q Science, Computer Science
ID Code:1454

[1] L.H.Y. Chen and Q.M. Shao, A non-uniform Berry-Esseen bound via Stein’s method, Probab. Theory Relat. Fields. 120(2001), 236—254.

[2] M. Dutko, Central limit theorems for infinite urn models, Ann. Probab. 17(3)(1989), 1255—1263.

[3] S. Karlin, Central limit theorems for certain infinite urn schemes, S. Math. Mech. 17(1967), 373—401.

[4] O. Milenkovic and K.J. Compton, Probabilistic transforms for combinatorial urn models, Combin. Probab. Comput. 13(2004), 645-675.

[5] C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proc. Sixth Berkeley Symp. Math. Stat. Prob. 2, 583-602, Univ. California Press, Berkeley, Calif., 1972.

[6] C. Stein, Approximation Computation of Expectations, Lecture Note 7, Inst. Math. Statist., Hayward, Calif., 1986.

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