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

Abstract

Let X_n be a sequence of integrable real random variables, adapted to a filtration (G_n). Define C_n = √{(1 / n)∑_k=1ⁿX_k - E(X_n+1 | G_n)} and D_n = √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 √n{(1 / n)∑_k=1ⁿX_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.