Journal of Applied Probability

Asymptotics for randomly reinforced urns with random barriers

Patrizia Berti, Irene Crimaldi, Luca Pratelli, and Pietro Rigo

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

Article information

J. Appl. Probab., Volume 53, Number 4 (2016), 1206-1220.

First available in Project Euclid: 7 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures
Secondary: 60F05: Central limit and other weak theorems 60G57: Random measures 62F15: Bayesian inference

Bayesian nonparametric central limit theorem random probability measure stable convergence urn model


Berti, Patrizia; Crimaldi, Irene; Pratelli, Luca; Rigo, Pietro. Asymptotics for randomly reinforced urns with random barriers. J. Appl. Probab. 53 (2016), no. 4, 1206--1220.

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