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Asymptotics for randomly reinforced urns with random barriers

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

Published online by Cambridge University Press:  09 December 2016

Patrizia Berti ,
Irene Crimaldi ,
Luca Pratelli  and
Pietro Rigo
Patrizia Berti*
Affiliation:
Università di Modena e Reggio-Emilia
Irene Crimaldi*
Affiliation:
IMT School for Advanced Studies Lucca
Luca Pratelli*
Affiliation:
Accademia Navale Livorno
Pietro Rigo*
Affiliation:
Università di Pavia
*
* Postal address: Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio-Emilia, via Campi 213/B, 41100 Modena, Italy.
** Postal address: IMT School for Advanced Studies, Piazza San Ponziano 6, 55100 Lucca, Italy.
*** Postal address: Accademia Navale Livorno, viale Italia 72, 57100 Livorno, Italy.
**** Postal address: Dipartimento di Matematica `F. Casorati', Università di Pavia, via Ferrata 1, 27100 Pavia, Italy. Email address: pietro.rigo@unipv.it
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Abstract

An urn contains black and red balls. Let Z n be the proportion of black balls at time n and 0≤L<U≤1 random barriers. At each time n, a ball b n is drawn. If b n is black and Z n-1<U, then b n is replaced together with a random number B n of black balls. If b n is red and Z n-1>L, then b n is replaced together with a random number R n of red balls. Otherwise, no additional balls are added, and b n alone is replaced. In this paper we assume that R n =B n . Then, under mild conditions, it is shown that Z n →a.s. Z for some random variable Z, and D n ≔√n(Z n -Z)→𝒩(0,σ2) conditionally almost surely (a.s.), where σ2 is a certain random variance. Almost sure conditional convergence means that ℙ(D n ∈⋅|𝒢n )→w 𝒩(0,σ2) a.s., where ℙ(D n ∈⋅|𝒢n ) is a regular version of the conditional distribution of D n given the past 𝒢n . Thus, in particular, one obtains D n →𝒩(0,σ2) stably. It is also shown that L<Z<U a.s. and Z has nonatomic distribution.

Keywords

Bayesian nonparametriccentral limit theoremrandom probability measurestable convergenceurn model

MSC classification

Primary: 60B10: Convergence of probability measures
Secondary: 60F05: Central limit and other weak theorems 60G57: Random measures 62F15: Bayesian inference
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1206 - 1220
Copyright
Copyright © Applied Probability Trust 2016 

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