A CENTRAL LIMIT THEOREM AND ITS APPLICATIONS TO MULTICOLOR RANDOMLY REINFORCED URNS

作者：Berti Patrizia^*; Crimaldi Irene; Pratelli Luca; Rigo Pietro

来源：Journal of Applied Probability, 2011, 48(2): 527-546.

DOI：10.1017/s0021900200008020

• Bayesian statistics
• central limit theorem
• empirical distribution
• Poisson-Dirichlet process
• predictive distribution
• random probability measure
• stable convergence
• urn model

Let (X(n)) be a sequence of integrable real random variables, adapted to a filtration (g(n)). Define C(n) = root n{(1/n) Sigma(n)(k=1) X(k) - E(X(n+1) | g(n))}, and D(n) = root n{E(X(n+1) | g(n) - Z}, where Z is the almost-sure limit of E(Xn+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 root n{(1/n) Sigma(n)(k=1) 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.

• 出版日期2011-6