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A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns

Part of: Probability theory on algebraic and topological structures Stochastic processes Limit theorems Parametric inference

Published online by Cambridge University Press:  14 July 2016

Patrizia Berti ,
Irene Crimaldi ,
Luca Pratelli  and
Pietro Rigo
Patrizia Berti*
Affiliation:
Università di Modena e Reggio-Emilia
Irene Crimaldi*
Affiliation:
Università di Bologna
Luca Pratelli*
Affiliation:
Accademia Navale
Pietro Rigo*
Affiliation:
Università di Pavia
*
∗Postal address: Dipartimento di Matematica Pura ed Applicata ‘G. Vitali’, Università di Modena e Reggio-Emilia, via Campi 213/B, 41100 Modena, Italy. Email address: patrizia.berti@unimore.it
∗∗Postal address: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy. Email address: crimaldi@dm.unibo.it
∗∗∗Postal address: Accademia Navale, viale Italia 72, 57100 Livorno, Italy. Email address: pratel@mail.dm.unipi.it
∗∗∗∗Postal address: Dipartimento di Economia Politica e Metodi Quantitativi, Università di Pavia, via S. Felice 5, 27100 Pavia, Italy. Email address: prigo@eco.unipv.it
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Abstract

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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.

Keywords

Bayesian statisticscentral limit theoremempirical distributionPoisson-Dirichlet processpredictive distributionrandom probability measurestable convergenceurn model

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G57: Random measures 60B10: Convergence of probability measures 62F15: Bayesian inference
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 527 - 546
Copyright
Copyright © Applied Probability Trust 2011 

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