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Asymptotics for randomly reinforced urns with random barriers

Part of: Stochastic processes Probability theory on algebraic and topological structures 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: [email protected]
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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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References

[1] Aletti, G.,Ghiglietti, A. and Paganoni, A. M. (2013).Randomly reinforced urn designs with prespecified allocations.J. Appl. Prob. 50,486–498.Google Scholar
[2] Aletti, G.,May, C. and Secchi, P. (2007).On the distribution of the limit proportion for a two-color, randomly reinforced urn with equal reinforcement distributions.Adv. Appl. Prob. 39,690–707.Google Scholar
[3] Aletti, G.,May, C. and Secchi, P. (2009).A central limit theorem, and related results, for a two-color randomly reinforced urn.Adv. Appl. Prob. 41,829–844.CrossRefGoogle Scholar
[4] Bai, Z. D. and Hu, F. (2005).Asymptotics in randomized URN models.Ann. Appl. Prob. 15,914–940.Google Scholar
[5] Berti, P.,Crimaldi, I.,Pratelli, L. and Rigo, P. (2010).Central limit theorems for multicolor urns with dominated colors.Stoch. Process. Appl. 120,1473–1491.Google Scholar
[6] Berti, P.,Crimaldi, I.,Pratelli, L. and Rigo, P. (2011).A central limit theorem and its applications to multicolor randomly reinforced urns.J. Appl. Prob. 48,527–546.Google Scholar
[7] Blackwell, D. and Dubins, L. (1962).Merging of opinions with increasing information.Ann. Math. Statist. 33,882–886.Google Scholar
[8] Chauvin, B.,Pouyanne, N. and Sahnoun, R. (2011).Limit distributions for large Pólya urns.Ann. Appl. Prob. 21,1–32.Google Scholar
[9] Crimaldi, I.,Letta, G. and Pratelli, L. (2007).A strong form of stable convergence.In Séminaire de Probabilités XL (Lecture Notes Math. 1899),Springer,Berlin,pp. 203–225.Google Scholar
[10] Crimaldi, I. (2009).An almost sure conditional convergence result and an application to a generalized Pólya urn.Internat. Math. Forum 4,1139–1156.Google Scholar
[11] Laruelle, S. and Pagès, G. (2013).Randomized urn models revisited using stochastic approximation.Ann. Appl. Prob. 23,1409–1436.CrossRefGoogle Scholar
[12] Mahmoud, H. M. (2009).Pólya Urn Models.CRC,Boca Raton, FL.Google Scholar
[13] May, C. and Flournoy, N. (2009).Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn.Ann. Statist. 37,1058–1078.Google Scholar
[14] Zhang, L.-X. (2014).A Gaussian process approximation for two-color randomly reinforced urns.Electron. J. Prob. 19,86.Google Scholar
[15] Zhang, L.-X.,Hu, F.,Cheung, S. H. and Chan, W. S. (2014).Asymptotic properties of multicolor randomly reinforced Pólya urns.Adv. Appl. Prob. 46,585–602.Google Scholar