# Central limit theorems for multicolor urns with dominated colors

Berti, Patrizia and Crimaldi, Irene and Pratelli, Luca and Rigo, Pietro Central limit theorems for multicolor urns with dominated colors. Stochastic processes and their applications , 120 (8). pp. 1473-1491. ISSN 0304-4149 (2010)

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## Abstract

An urn contains balls of d≥2 colors. At each time n≥1, a ball is drawn and then replaced together with a random number of balls of the same color. Let A n = diag (An,1,…,An,d) be the n-th reinforce matrix. Assuming that EAn,j=EAn,1 for all n and j, a few central limit theorems (CLTs) are available for such urns. In real problems, however, it is more reasonable to assume that EA n,j = EA n,1 whenever  n ≥ 1  and  1 ≤ j ≤ d0 , liminfn EAn,1 > limsupn EAn,j whenever  j > d0 for some integer 1≤d0≤d. Under this condition, the usual weak limit theorems may fail, but it is still possible to prove the CLTs for some slightly different random quantities. These random quantities are obtained by neglecting dominated colors, i.e., colors from d0+1 to d, and they allow the same inference on the urn structure. The sequence (An : n ≥ 1) is independent but need not be identically distributed. Some statistical applications are given as well.

Item Type: Article 10.1016/j.spa.2010.04.005 Central limit theorem; Clinical trials; Random probability measure; Stable convergence; Urn model H Social Sciences > HA StatisticsQ Science > QA Mathematics Economics and Institutional Change Irene Crimaldi 31 Oct 2011 12:04 03 Nov 2011 13:19 http://eprints.imtlucca.it/id/eprint/978

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