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December 2016 Asymptotics for randomly reinforced urns with random barriers
Patrizia Berti, Irene Crimaldi, Luca Pratelli, Pietro Rigo
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J. Appl. Probab. 53(4): 1206-1220 (December 2016).

Abstract

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.

Citation

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Patrizia Berti. Irene Crimaldi. Luca Pratelli. Pietro Rigo. "Asymptotics for randomly reinforced urns with random barriers." J. Appl. Probab. 53 (4) 1206 - 1220, December 2016.

Information

Published: December 2016
First available in Project Euclid: 7 December 2016

zbMATH: 1358.60005
MathSciNet: MR3581252

Subjects:
Primary: 60B10
Secondary: 60F05, 60G57, 62F15

Rights: Copyright © 2016 Applied Probability Trust

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Vol.53 • No. 4 • December 2016
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