A Central Limit Theorem and Its Applications to Multicolor Randomly Reinforced Urns

Let (Xn) be a sequence of integrable real random variables, adapted to a filtration (Gn). Define: Cn = n^(1/2) {1/n SUM(k=1:n) Xk - E(Xn+1 | Gn) } and Dn = n^(1/2){ E(Xn+1 | Gn)-Z } where Z is the a.s. limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn,Dn) --> N(0,U) × N(0,V) stably are given, where U, V are certain random variables. In particular, under such conditions, one obtains n^(1/2) { 1/n SUM(k=1:n) Xk - Z } = Cn + Dn --> N(0,U+V) stably. This CLT has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced generalized Polya urns.