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

Berti, Patrizia and Crimaldi, Irene and Pratelli, Luca and Rigo, Pietro A central limit theorem and its applications to multicolor randomly reinforced urns. Journal of applied probability , 48 (2). pp. 527-546. ISSN 0021-9002 (2011)

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Abstract

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.

Item Type: Article 10.1239/jap/1308662642 Bayesian statistics; central limit theorem; empirical distribution; Poisson-Dirichlet process; predictive distribution; random probability measure; stable convergence; urn model H Social Sciences > HA StatisticsQ Science > QA Mathematics Economics and Institutional Change Irene Crimaldi 31 Oct 2011 11:23 17 Nov 2011 14:44 http://eprints.imtlucca.it/id/eprint/975