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, lim inf n EAn,1 > lim sup n 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. 1. The problem An urn contains aj > 0 balls of color j ∈ {1, . . . , d} where d ≥ 2. At each time n ≥ 1, a ball is drawn and then replaced together with a random number of balls of the same color. Say that An,j ≥ 0 balls of color j are added to the urn in case Xn,j = 1, where Xn,j is the indicator of {ball of color j at time n}. Let Nn,j = aj + n ∑