%0 Journal Article
%@ 0021-9002
%A Berti, Patrizia
%A Crimaldi, Irene
%A Pratelli, Luca
%A Rigo, Pietro
%D 2011
%F eprints:975
%I Applied Probability Trust
%J Journal of applied probability 
%K Bayesian statistics; central limit theorem; empirical distribution; Poisson-Dirichlet process; predictive distribution; random probability measure; stable convergence; urn model
%N 2
%P 527-546
%T A central limit theorem and its applications to multicolor randomly reinforced urns
%U http://eprints.imtlucca.it/975/
%V 48
%X Let Xn be a sequence of integrable real random variables, adapted to a filtration (Gn). Define Cn = √{(1 / n)∑k=1nXk - E(Xn+1 | Gn)} and Dn = √n{E(Xn+1 | Gn) - Z}, where Z is the almost-sure limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn, Dn) → 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} = Cn + Dn → 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.