Central Limit Theorems For Multicolor Urns With Dominated Colors

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 An =diag (An,1, . . . ,An,d) be the n-th reinforce matrix. Assuming EAn,j = EAn,1 for all n and j, a few CLT s are available for such urns. In real problems, however, it is more reasonable to assume EAn,j = EAn,1 whenever n >= 1 and 1 <= j <= d0, liminf EAn,1 > limsup 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 CLT s for some slightly different random quantities. These random quantities are obtained neglecting dominated colors, i.e., colors from d0 + 1 to d, and 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.