Central limit theorems for multicolor urns with dominated colors

An urn contains balls of d≥2d≥2 colors. At each time n≥1n≥1, a ball is drawn and then replaced together with a random number of balls of the same color. Let An= diag (An,1,…,An,d)(An,1,…,An,d) be the nn-th reinforce matrix. Assuming that EAn,j=EAn,1EAn,j=EAn,1 for all nn and jj, a few central limit theorems (CLTs) are available for such urns. In real problems, however, it is more reasonable to assume that EAn,j=EAn,1whenever n≥1 and 1≤j≤d0,lim infnEAn,1>lim supnEAn,jwhenever j>d0, for some integer 1≤d0≤d1≤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+1d0+1 to dd, 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.