A Central Limit Theorem and Its Applications to Multicolor Randomly Reinforced Urns - Patrizia Berti; Irene Crimaldi; Luca Pratelli; Pietro Rigo (Università degli Studi di Pavia, Dipartimento di Economia Politica e Metodi Quantitativi (EPMQ), 2010)
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.
Keywords: 330; Bayesian statistics Central limit theorem Empirical distribution Poisson-Dirichlet process Predictive distribution Random probability measure Stable convergence Urn model