Let X_n be a sequence of integrable real random variables, adapted to a filtration (G_n). Define C_n = \sqrt{n}{(1/n)\sum_{k=1^n}X_k - E(X_n+1|G_n)} and D_n = \sqrt{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 \sqrt{n}{(1/n)\sum_{k=1^n}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.

### 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 = \sqrt{n}{(1/n)\sum_{k=1^n}X_k - E(X_n+1|G_n)} and D_n = \sqrt{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 \sqrt{n}{(1/n)\sum_{k=1^n}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.
##### Scheda breve Scheda completa Scheda completa (DC)
2011
urns models; central limit theorem; stable convergence; Poisson-Dirichlet sequences; Bayesian statistics; empirical and predictive distributions; random probability measures
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11771/4107