A central limit theorem and its applications to multicolor randomly reinforced urns

Let X_n be a sequence of integrable real random variables, adapted to a filtration (G_n). Define C_n = \sqrt{n}{(1/n)\sum_{k=1^n}X_k - E(X_n+1|G_n)} and D_n = \sqrt{n}{E(X_n+1 | G_n) - Z}, where Z is the almost-sure limit of E(X_n+1 | G_n) (assumed to exist). Conditions for (C_n, D_n) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain \sqrt{n}{(1/n)\sum_{k=1^n}X_k - Z} = C_n + D_n → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.